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Simplificación de expresiones/ Descomposición en fracciones simples/ tan(5*x+7)^3*(20+20*tan(5*x+7)^2)/(2*x^2+9)-4*x*tan(5*x+7)^4/(2*x^2+9)^2

¿Cómo vas a descomponer esta tan(5*x+7)^3*(20+20*tan(5*x+7)^2)/(2*x^2+9)-4*x*tan(5*x+7)^4/(2*x^2+9)^2 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   3          /           2         \          4         
tan (5*x + 7)*\20 + 20*tan (5*x + 7)/   4*x*tan (5*x + 7)
------------------------------------- - -----------------
                  2                                  2   
               2*x  + 9                    /   2    \    
                                           \2*x  + 9/
−4xtan⁡4(5x+7)(2x2+9)2+(20tan⁡2(5x+7)+20)tan⁡3(5x+7)2x2+9- \frac{4 x \tan^{4}{\left(5 x + 7 \right)}}{\left(2 x^{2} + 9\right)^{2}} + \frac{\left(20 \tan^{2}{\left(5 x + 7 \right)} + 20\right) \tan^{3}{\left(5 x + 7 \right)}}{2 x^{2} + 9} 
(tan(5*x + 7)^3*(20 + 20*tan(5*x + 7)^2))/(2*x^2 + 9) - (4*x)*tan(5*x + 7)^4/(2*x^2 + 9)^2
Simplificación general [src] 
     3          //       2         \ /         2\                 \
4*tan (7 + 5*x)*\\1 + tan (7 + 5*x)/*\45 + 10*x / - x*tan(7 + 5*x)/
-------------------------------------------------------------------
                                      2                            
                            /       2\                             
                            \9 + 2*x /
4(−xtan⁡(5x+7)+(10x2+45)(tan⁡2(5x+7)+1))tan⁡3(5x+7)(2x2+9)2\frac{4 \left(- x \tan{\left(5 x + 7 \right)} + \left(10 x^{2} + 45\right) \left(\tan^{2}{\left(5 x + 7 \right)} + 1\right)\right) \tan^{3}{\left(5 x + 7 \right)}}{\left(2 x^{2} + 9\right)^{2}} 
4*tan(7 + 5*x)^3*((1 + tan(7 + 5*x)^2)*(45 + 10*x^2) - x*tan(7 + 5*x))/(9 + 2*x^2)^2
Combinatoria [src] 
     3          /         2         2                                 2    2         \
4*tan (7 + 5*x)*\45 + 10*x  + 45*tan (7 + 5*x) - x*tan(7 + 5*x) + 10*x *tan (7 + 5*x)/
--------------------------------------------------------------------------------------
                                               2                                      
                                     /       2\                                       
                                     \9 + 2*x /
4(10x2tan⁡2(5x+7)+10x2−xtan⁡(5x+7)+45tan⁡2(5x+7)+45)tan⁡3(5x+7)(2x2+9)2\frac{4 \left(10 x^{2} \tan^{2}{\left(5 x + 7 \right)} + 10 x^{2} - x \tan{\left(5 x + 7 \right)} + 45 \tan^{2}{\left(5 x + 7 \right)} + 45\right) \tan^{3}{\left(5 x + 7 \right)}}{\left(2 x^{2} + 9\right)^{2}} 
4*tan(7 + 5*x)^3*(45 + 10*x^2 + 45*tan(7 + 5*x)^2 - x*tan(7 + 5*x) + 10*x^2*tan(7 + 5*x)^2)/(9 + 2*x^2)^2
Unión de expresiones racionales [src] 
     3          /                    /       2         \ /       2\\
4*tan (7 + 5*x)*\-x*tan(7 + 5*x) + 5*\1 + tan (7 + 5*x)/*\9 + 2*x //
--------------------------------------------------------------------
                                      2                             
                            /       2\                              
                            \9 + 2*x /
4(−xtan⁡(5x+7)+5(2x2+9)(tan⁡2(5x+7)+1))tan⁡3(5x+7)(2x2+9)2\frac{4 \left(- x \tan{\left(5 x + 7 \right)} + 5 \left(2 x^{2} + 9\right) \left(\tan^{2}{\left(5 x + 7 \right)} + 1\right)\right) \tan^{3}{\left(5 x + 7 \right)}}{\left(2 x^{2} + 9\right)^{2}} 
4*tan(7 + 5*x)^3*(-x*tan(7 + 5*x) + 5*(1 + tan(7 + 5*x)^2)*(9 + 2*x^2))/(9 + 2*x^2)^2
Potencias [src] 
                                                                                    /                                        2\
                                                                                  3 |        /   I*(7 + 5*x)    I*(-7 - 5*x)\ |
                                                  /   I*(7 + 5*x)    I*(-7 - 5*x)\  |     20*\- e            + e            / |
                                                I*\- e            + e            / *|20 - ------------------------------------|
                                         4                                          |                                     2   |
         /   I*(7 + 5*x)    I*(-7 - 5*x)\                                           |       / I*(-7 - 5*x)    I*(7 + 5*x)\    |
     4*x*\- e            + e            /                                           \       \e             + e           /    /
- ------------------------------------------- - -------------------------------------------------------------------------------
            2                               4                                                               3                  
  /       2\  / I*(-7 - 5*x)    I*(7 + 5*x)\                       /       2\ / I*(-7 - 5*x)    I*(7 + 5*x)\                   
  \9 + 2*x / *\e             + e           /                       \9 + 2*x /*\e             + e           /
−4x(ei(−5x−7)−ei(5x+7))4(2x2+9)2(ei(−5x−7)+ei(5x+7))4−i(−20(ei(−5x−7)−ei(5x+7))2(ei(−5x−7)+ei(5x+7))2+20)(ei(−5x−7)−ei(5x+7))3(2x2+9)(ei(−5x−7)+ei(5x+7))3- \frac{4 x \left(e^{i \left(- 5 x - 7\right)} - e^{i \left(5 x + 7\right)}\right)^{4}}{\left(2 x^{2} + 9\right)^{2} \left(e^{i \left(- 5 x - 7\right)} + e^{i \left(5 x + 7\right)}\right)^{4}} - \frac{i \left(- \frac{20 \left(e^{i \left(- 5 x - 7\right)} - e^{i \left(5 x + 7\right)}\right)^{2}}{\left(e^{i \left(- 5 x - 7\right)} + e^{i \left(5 x + 7\right)}\right)^{2}} + 20\right) \left(e^{i \left(- 5 x - 7\right)} - e^{i \left(5 x + 7\right)}\right)^{3}}{\left(2 x^{2} + 9\right) \left(e^{i \left(- 5 x - 7\right)} + e^{i \left(5 x + 7\right)}\right)^{3}} 
-4*x*(-exp(i*(7 + 5*x)) + exp(i*(-7 - 5*x)))^4/((9 + 2*x^2)^2*(exp(i*(-7 - 5*x)) + exp(i*(7 + 5*x)))^4) - i*(-exp(i*(7 + 5*x)) + exp(i*(-7 - 5*x)))^3*(20 - 20*(-exp(i*(7 + 5*x)) + exp(i*(-7 - 5*x)))^2/(exp(i*(-7 - 5*x)) + exp(i*(7 + 5*x)))^2)/((9 + 2*x^2)*(exp(i*(-7 - 5*x)) + exp(i*(7 + 5*x)))^3)
Denominador racional [src] 
                                                         2                              2              
          4               3    4               /       2\     3               /       2\     5         
- 36*x*tan (7 + 5*x) - 8*x *tan (7 + 5*x) + 20*\9 + 2*x / *tan (7 + 5*x) + 20*\9 + 2*x / *tan (7 + 5*x)
-------------------------------------------------------------------------------------------------------
                                                        3                                              
                                              /       2\                                               
                                              \9 + 2*x /
−8x3tan⁡4(5x+7)−36xtan⁡4(5x+7)+20(2x2+9)2tan⁡5(5x+7)+20(2x2+9)2tan⁡3(5x+7)(2x2+9)3\frac{- 8 x^{3} \tan^{4}{\left(5 x + 7 \right)} - 36 x \tan^{4}{\left(5 x + 7 \right)} + 20 \left(2 x^{2} + 9\right)^{2} \tan^{5}{\left(5 x + 7 \right)} + 20 \left(2 x^{2} + 9\right)^{2} \tan^{3}{\left(5 x + 7 \right)}}{\left(2 x^{2} + 9\right)^{3}} 
(-36*x*tan(7 + 5*x)^4 - 8*x^3*tan(7 + 5*x)^4 + 20*(9 + 2*x^2)^2*tan(7 + 5*x)^3 + 20*(9 + 2*x^2)^2*tan(7 + 5*x)^5)/(9 + 2*x^2)^3
Respuesta numérica [src] 
tan(5*x + 7)^3*(20.0 + 20.0*tan(5*x + 7)^2)/(9.0 + 2.0*x^2) - 0.0493827160493827*x*tan(5*x + 7)^4/(1 + 0.222222222222222*x^2)^2
tan(5*x + 7)^3*(20.0 + 20.0*tan(5*x + 7)^2)/(9.0 + 2.0*x^2) - 0.0493827160493827*x*tan(5*x + 7)^4/(1 + 0.222222222222222*x^2)^2
Denominador común [src] 
       3                   5                   4                2    3                2    5         
180*tan (7 + 5*x) + 180*tan (7 + 5*x) - 4*x*tan (7 + 5*x) + 40*x *tan (7 + 5*x) + 40*x *tan (7 + 5*x)
-----------------------------------------------------------------------------------------------------
                                                  4       2                                          
                                          81 + 4*x  + 36*x
40x2tan⁡5(5x+7)+40x2tan⁡3(5x+7)−4xtan⁡4(5x+7)+180tan⁡5(5x+7)+180tan⁡3(5x+7)4x4+36x2+81\frac{40 x^{2} \tan^{5}{\left(5 x + 7 \right)} + 40 x^{2} \tan^{3}{\left(5 x + 7 \right)} - 4 x \tan^{4}{\left(5 x + 7 \right)} + 180 \tan^{5}{\left(5 x + 7 \right)} + 180 \tan^{3}{\left(5 x + 7 \right)}}{4 x^{4} + 36 x^{2} + 81} 
(180*tan(7 + 5*x)^3 + 180*tan(7 + 5*x)^5 - 4*x*tan(7 + 5*x)^4 + 40*x^2*tan(7 + 5*x)^3 + 40*x^2*tan(7 + 5*x)^5)/(81 + 4*x^4 + 36*x^2)
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