Integral de x^5sqrt(x^2+4) dx

TrigSubstitutionRule(theta=_theta, func=2*tan(_theta), rewritten=128*sin(_theta)**5/cos(_theta)**8, substep=ConstantTimesRule(constant=128, other=sin(_theta)**5/cos(_theta)**8, substep=RewriteRule(rewritten=(1 - cos(_theta)**2)**2*sin(_theta)/cos(_theta)**8, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=(_u**4 - 2*_u**2 + 1)/_u**8, substep=RewriteRule(rewritten=_u**(-4) - 2/_u**6 + _u**(-8), substep=AddRule(substeps=[PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), ConstantTimesRule(constant=-2, other=_u**(-6), substep=PowerRule(base=_u, exp=-6, context=_u**(-6), symbol=_u), context=-2/_u**6, symbol=_u), PowerRule(base=_u, exp=-8, context=_u**(-8), symbol=_u)], context=_u**(-4) - 2/_u**6 + _u**(-8), symbol=_u), context=(_u**4 - 2*_u**2 + 1)/_u**8, symbol=_u), context=(_u**4 - 2*_u**2 + 1)/_u**8, symbol=_u), context=(1 - cos(_theta)**2)**2*sin(_theta)/cos(_theta)**8, symbol=_theta), RewriteRule(rewritten=(sin(_theta)*cos(_theta)**4 - 2*sin(_theta)*cos(_theta)**2 + sin(_theta))/cos(_theta)**8, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=(_u**4 - 2*_u**2 + 1)/_u**8, substep=RewriteRule(rewritten=_u**(-4) - 2/_u**6 + _u**(-8), substep=AddRule(substeps=[PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), ConstantTimesRule(constant=-2, other=_u**(-6), substep=PowerRule(base=_u, exp=-6, context=_u**(-6), symbol=_u), context=-2/_u**6, symbol=_u), PowerRule(base=_u, exp=-8, context=_u**(-8), symbol=_u)], context=_u**(-4) - 2/_u**6 + _u**(-8), symbol=_u), context=(_u**4 - 2*_u**2 + 1)/_u**8, symbol=_u), context=(_u**4 - 2*_u**2 + 1)/_u**8, symbol=_u), context=(sin(_theta)*cos(_theta)**4 - 2*sin(_theta)*cos(_theta)**2 + sin(_theta))/cos(_theta)**8, symbol=_theta), RewriteRule(rewritten=sin(_theta)/cos(_theta)**4 - 2*sin(_theta)/cos(_theta)**6 + sin(_theta)/cos(_theta)**8, substep=AddRule(substeps=[URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**(-4), substep=PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), context=_u**(-4), symbol=_u), context=sin(_theta)/cos(_theta)**4, symbol=_theta), ConstantTimesRule(constant=-2, other=sin(_theta)/cos(_theta)**6, substep=URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**(-6), substep=PowerRule(base=_u, exp=-6, context=_u**(-6), symbol=_u), context=_u**(-6), symbol=_u), context=sin(_theta)/cos(_theta)**6, symbol=_theta), context=-2*sin(_theta)/cos(_theta)**6, symbol=_theta), URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**(-8), substep=PowerRule(base=_u, exp=-8, context=_u**(-8), symbol=_u), context=_u**(-8), symbol=_u), context=sin(_theta)/cos(_theta)**8, symbol=_theta)], context=sin(_theta)/cos(_theta)**4 - 2*sin(_theta)/cos(_theta)**6 + sin(_theta)/cos(_theta)**8, symbol=_theta), context=(sin(_theta)*cos(_theta)**4 - 2*sin(_theta)*cos(_theta)**2 + sin(_theta))/cos(_theta)**8, symbol=_theta)], context=(sin(_theta)*cos(_theta)**4 - 2*sin(_theta)*cos(_theta)**2 + sin(_theta))/cos(_theta)**8, symbol=_theta), context=(1 - cos(_theta)**2)**2*sin(_theta)/cos(_theta)**8, symbol=_theta), RewriteRule(rewritten=sin(_theta)/cos(_theta)**4 - 2*sin(_theta)/cos(_theta)**6 + sin(_theta)/cos(_theta)**8, substep=AddRule(substeps=[URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**(-4), substep=PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), context=_u**(-4), symbol=_u), context=sin(_theta)/cos(_theta)**4, symbol=_theta), ConstantTimesRule(constant=-2, other=sin(_theta)/cos(_theta)**6, substep=URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**(-6), substep=PowerRule(base=_u, exp=-6, context=_u**(-6), symbol=_u), context=_u**(-6), symbol=_u), context=sin(_theta)/cos(_theta)**6, symbol=_theta), context=-2*sin(_theta)/cos(_theta)**6, symbol=_theta), URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**(-8), substep=PowerRule(base=_u, exp=-8, context=_u**(-8), symbol=_u), context=_u**(-8), symbol=_u), context=sin(_theta)/cos(_theta)**8, symbol=_theta)], context=sin(_theta)/cos(_theta)**4 - 2*sin(_theta)/cos(_theta)**6 + sin(_theta)/cos(_theta)**8, symbol=_theta), context=(1 - cos(_theta)**2)**2*sin(_theta)/cos(_theta)**8, symbol=_theta)], context=(1 - cos(_theta)**2)**2*sin(_theta)/cos(_theta)**8, symbol=_theta), context=sin(_theta)**5/cos(_theta)**8, symbol=_theta), context=128*sin(_theta)**5/cos(_theta)**8, symbol=_theta), restriction=True, context=x**5*sqrt(x**2 + 4), symbol=x)

1. Ahora simplificar:

   $\frac{\left(x^{2} + 4\right)^{\frac{3}{2}} \left(- 168 x^{2} + 15 \left(x^{2} + 4\right)^{2} - 112\right)}{105}$

2. Añadimos la constante de integración:

   $\frac{\left(x^{2} + 4\right)^{\frac{3}{2}} \left(- 168 x^{2} + 15 \left(x^{2} + 4\right)^{2} - 112\right)}{105}+ \mathrm{constant}$

Respuesta:

$\frac{\left(x^{2} + 4\right)^{\frac{3}{2}} \left(- 168 x^{2} + 15 \left(x^{2} + 4\right)^{2} - 112\right)}{105}+ \mathrm{constant}$