Integral de 1/(xxxx(9+xx)^(1/2)) dx

TrigSubstitutionRule(theta=_theta, func=3*tan(_theta), rewritten=cos(_theta)**3/(81*sin(_theta)**4), substep=ConstantTimesRule(constant=1/81, other=cos(_theta)**3/sin(_theta)**4, substep=RewriteRule(rewritten=(1 - sin(_theta)**2)*cos(_theta)/sin(_theta)**4, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=(_u**2 - 1)/_u**4, substep=RewriteRule(rewritten=_u**(-2) - 1/_u**4, substep=AddRule(substeps=[PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), ConstantTimesRule(constant=-1, other=_u**(-4), substep=PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), context=-1/_u**4, symbol=_u)], context=_u**(-2) - 1/_u**4, symbol=_u), context=(_u**2 - 1)/_u**4, symbol=_u), context=(_u**2 - 1)/_u**4, symbol=_u), context=(1 - sin(_theta)**2)*cos(_theta)/sin(_theta)**4, symbol=_theta), RewriteRule(rewritten=-(sin(_theta)**2*cos(_theta) - cos(_theta))/sin(_theta)**4, substep=ConstantTimesRule(constant=-1, other=(sin(_theta)**2*cos(_theta) - cos(_theta))/sin(_theta)**4, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(_theta), constant=1, substep=RewriteRule(rewritten=_u**(-2) - 1/_u**4, substep=AddRule(substeps=[PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), ConstantTimesRule(constant=-1, other=_u**(-4), substep=PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), context=-1/_u**4, symbol=_u)], context=_u**(-2) - 1/_u**4, symbol=_u), context=(_u**2 - 1)/_u**4, symbol=_u), context=(sin(_theta)**2*cos(_theta) - cos(_theta))/sin(_theta)**4, symbol=_theta), RewriteRule(rewritten=cos(_theta)/sin(_theta)**2 - cos(_theta)/sin(_theta)**4, substep=AddRule(substeps=[URule(u_var=_u, u_func=sin(_theta), constant=1, substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=cos(_theta)/sin(_theta)**2, symbol=_theta), ConstantTimesRule(constant=-1, other=cos(_theta)/sin(_theta)**4, substep=URule(u_var=_u, u_func=sin(_theta), constant=1, substep=PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), context=cos(_theta)/sin(_theta)**4, symbol=_theta), context=-cos(_theta)/sin(_theta)**4, symbol=_theta)], context=cos(_theta)/sin(_theta)**2 - cos(_theta)/sin(_theta)**4, symbol=_theta), context=(sin(_theta)**2*cos(_theta) - cos(_theta))/sin(_theta)**4, symbol=_theta)], context=(sin(_theta)**2*cos(_theta) - cos(_theta))/sin(_theta)**4, symbol=_theta), context=-(sin(_theta)**2*cos(_theta) - cos(_theta))/sin(_theta)**4, symbol=_theta), context=(1 - sin(_theta)**2)*cos(_theta)/sin(_theta)**4, symbol=_theta), RewriteRule(rewritten=-cos(_theta)/sin(_theta)**2 + cos(_theta)/sin(_theta)**4, substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=cos(_theta)/sin(_theta)**2, substep=URule(u_var=_u, u_func=sin(_theta), constant=1, substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=cos(_theta)/sin(_theta)**2, symbol=_theta), context=-cos(_theta)/sin(_theta)**2, symbol=_theta), URule(u_var=_u, u_func=sin(_theta), constant=1, substep=PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), context=cos(_theta)/sin(_theta)**4, symbol=_theta)], context=-cos(_theta)/sin(_theta)**2 + cos(_theta)/sin(_theta)**4, symbol=_theta), context=(1 - sin(_theta)**2)*cos(_theta)/sin(_theta)**4, symbol=_theta)], context=(1 - sin(_theta)**2)*cos(_theta)/sin(_theta)**4, symbol=_theta), context=cos(_theta)**3/sin(_theta)**4, symbol=_theta), context=cos(_theta)**3/(81*sin(_theta)**4), symbol=_theta), restriction=True, context=1/((x*(x*(x*x)))*sqrt(x*x + 9)), symbol=x)

1. Ahora simplificar:

   $\frac{\sqrt{x^{2} + 9} \left(2 x^{2} - 9\right)}{243 x^{3}}$

2. Añadimos la constante de integración:

   $\frac{\sqrt{x^{2} + 9} \left(2 x^{2} - 9\right)}{243 x^{3}}+ \mathrm{constant}$

Respuesta:

$\frac{\sqrt{x^{2} + 9} \left(2 x^{2} - 9\right)}{243 x^{3}}+ \mathrm{constant}$