3.1979 \(\int \frac{x^2}{\left (a+\frac{b}{x^3}\right )^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{b^2}{3 a^3 \left (a x^3+b\right )}-\frac{2 b \log \left (a x^3+b\right )}{3 a^3}+\frac{x^3}{3 a^2} \]

[Out]

x^3/(3*a^2) - b^2/(3*a^3*(b + a*x^3)) - (2*b*Log[b + a*x^3])/(3*a^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0934779, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{b^2}{3 a^3 \left (a x^3+b\right )}-\frac{2 b \log \left (a x^3+b\right )}{3 a^3}+\frac{x^3}{3 a^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b/x^3)^2,x]

[Out]

x^3/(3*a^2) - b^2/(3*a^3*(b + a*x^3)) - (2*b*Log[b + a*x^3])/(3*a^3)

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{x^{3}} \frac{1}{a^{2}}\, dx}{3} - \frac{b^{2}}{3 a^{3} \left (a x^{3} + b\right )} - \frac{2 b \log{\left (a x^{3} + b \right )}}{3 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(a+b/x**3)**2,x)

[Out]

Integral(a**(-2), (x, x**3))/3 - b**2/(3*a**3*(a*x**3 + b)) - 2*b*log(a*x**3 + b
)/(3*a**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0281812, size = 38, normalized size = 0.83 \[ \frac{-\frac{b^2}{a x^3+b}-2 b \log \left (a x^3+b\right )+a x^3}{3 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b/x^3)^2,x]

[Out]

(a*x^3 - b^2/(b + a*x^3) - 2*b*Log[b + a*x^3])/(3*a^3)

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 41, normalized size = 0.9 \[{\frac{{x}^{3}}{3\,{a}^{2}}}-{\frac{{b}^{2}}{3\,{a}^{3} \left ( a{x}^{3}+b \right ) }}-{\frac{2\,b\ln \left ( a{x}^{3}+b \right ) }{3\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b/x^3)^2,x)

[Out]

1/3*x^3/a^2-1/3*b^2/a^3/(a*x^3+b)-2/3*b*ln(a*x^3+b)/a^3

_______________________________________________________________________________________

Maxima [A]  time = 1.4272, size = 58, normalized size = 1.26 \[ -\frac{b^{2}}{3 \,{\left (a^{4} x^{3} + a^{3} b\right )}} + \frac{x^{3}}{3 \, a^{2}} - \frac{2 \, b \log \left (a x^{3} + b\right )}{3 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(a + b/x^3)^2,x, algorithm="maxima")

[Out]

-1/3*b^2/(a^4*x^3 + a^3*b) + 1/3*x^3/a^2 - 2/3*b*log(a*x^3 + b)/a^3

_______________________________________________________________________________________

Fricas [A]  time = 0.232567, size = 76, normalized size = 1.65 \[ \frac{a^{2} x^{6} + a b x^{3} - b^{2} - 2 \,{\left (a b x^{3} + b^{2}\right )} \log \left (a x^{3} + b\right )}{3 \,{\left (a^{4} x^{3} + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(a + b/x^3)^2,x, algorithm="fricas")

[Out]

1/3*(a^2*x^6 + a*b*x^3 - b^2 - 2*(a*b*x^3 + b^2)*log(a*x^3 + b))/(a^4*x^3 + a^3*
b)

_______________________________________________________________________________________

Sympy [A]  time = 1.74545, size = 42, normalized size = 0.91 \[ - \frac{b^{2}}{3 a^{4} x^{3} + 3 a^{3} b} + \frac{x^{3}}{3 a^{2}} - \frac{2 b \log{\left (a x^{3} + b \right )}}{3 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(a+b/x**3)**2,x)

[Out]

-b**2/(3*a**4*x**3 + 3*a**3*b) + x**3/(3*a**2) - 2*b*log(a*x**3 + b)/(3*a**3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.22986, size = 55, normalized size = 1.2 \[ \frac{x^{3}}{3 \, a^{2}} - \frac{2 \, b{\rm ln}\left ({\left | a x^{3} + b \right |}\right )}{3 \, a^{3}} - \frac{b^{2}}{3 \,{\left (a x^{3} + b\right )} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(a + b/x^3)^2,x, algorithm="giac")

[Out]

1/3*x^3/a^2 - 2/3*b*ln(abs(a*x^3 + b))/a^3 - 1/3*b^2/((a*x^3 + b)*a^3)