Integral de x^3sqrt(1+3x^2) dx

TrigSubstitutionRule(theta=_theta, func=sqrt(3)*tan(_theta)/3, rewritten=tan(_theta)**3/(9*cos(_theta)**3), substep=ConstantTimesRule(constant=1/9, other=tan(_theta)**3/cos(_theta)**3, substep=RewriteRule(rewritten=(sec(_theta)**2 - 1)*tan(_theta)*sec(_theta)**3, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sec(_theta), constant=1, substep=AddRule(substeps=[PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), ConstantTimesRule(constant=-1, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=-_u**2, symbol=_u)], context=_u**4 - _u**2, symbol=_u), context=(sec(_theta)**2 - 1)*tan(_theta)*sec(_theta)**3, symbol=_theta), RewriteRule(rewritten=tan(_theta)*sec(_theta)**5 - tan(_theta)*sec(_theta)**3, substep=AddRule(substeps=[URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=tan(_theta)*sec(_theta)**5, symbol=_theta), ConstantTimesRule(constant=-1, other=tan(_theta)*sec(_theta)**3, substep=URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=tan(_theta)*sec(_theta)**3, symbol=_theta), context=-tan(_theta)*sec(_theta)**3, symbol=_theta)], context=tan(_theta)*sec(_theta)**5 - tan(_theta)*sec(_theta)**3, symbol=_theta), context=(sec(_theta)**2 - 1)*tan(_theta)*sec(_theta)**3, symbol=_theta), RewriteRule(rewritten=tan(_theta)*sec(_theta)**5 - tan(_theta)*sec(_theta)**3, substep=AddRule(substeps=[URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=tan(_theta)*sec(_theta)**5, symbol=_theta), ConstantTimesRule(constant=-1, other=tan(_theta)*sec(_theta)**3, substep=URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=tan(_theta)*sec(_theta)**3, symbol=_theta), context=-tan(_theta)*sec(_theta)**3, symbol=_theta)], context=tan(_theta)*sec(_theta)**5 - tan(_theta)*sec(_theta)**3, symbol=_theta), context=(sec(_theta)**2 - 1)*tan(_theta)*sec(_theta)**3, symbol=_theta)], context=(sec(_theta)**2 - 1)*tan(_theta)*sec(_theta)**3, symbol=_theta), context=tan(_theta)**3*sec(_theta)**3, symbol=_theta), context=tan(_theta)**3/(9*cos(_theta)**3), symbol=_theta), restriction=True, context=x**3*sqrt(3*x**2 + 1), symbol=x)

1. Ahora simplificar:

   $\frac{\left(3 x^{2} + 1\right)^{\frac{3}{2}} \left(9 x^{2} - 2\right)}{135}$

2. Añadimos la constante de integración:

   $\frac{\left(3 x^{2} + 1\right)^{\frac{3}{2}} \left(9 x^{2} - 2\right)}{135}+ \mathrm{constant}$

Respuesta:

$\frac{\left(3 x^{2} + 1\right)^{\frac{3}{2}} \left(9 x^{2} - 2\right)}{135}+ \mathrm{constant}$