3.1966 \(\int \frac{x^2}{a+\frac{b}{x^3}} \, dx\)

Optimal. Leaf size=27 \[ \frac{x^3}{3 a}-\frac{b \log \left (a x^3+b\right )}{3 a^2} \]

[Out]

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

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Rubi [A]  time = 0.062126, antiderivative size = 27, 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 \[ \frac{x^3}{3 a}-\frac{b \log \left (a x^3+b\right )}{3 a^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b/x^3),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/a, (x, x**3))/3 - b*log(a*x**3 + b)/(3*a**2)

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Mathematica [A]  time = 0.00769591, size = 27, normalized size = 1. \[ \frac{x^3}{3 a}-\frac{b \log \left (a x^3+b\right )}{3 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b/x^3),x]

[Out]

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

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Maple [A]  time = 0.003, size = 24, normalized size = 0.9 \[{\frac{{x}^{3}}{3\,a}}-{\frac{b\ln \left ( a{x}^{3}+b \right ) }{3\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b/x^3),x)

[Out]

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

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Maxima [A]  time = 1.41605, size = 31, normalized size = 1.15 \[ \frac{x^{3}}{3 \, a} - \frac{b \log \left (a x^{3} + b\right )}{3 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^3/a - 1/3*b*log(a*x^3 + b)/a^2

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Fricas [A]  time = 0.233483, size = 30, normalized size = 1.11 \[ \frac{a x^{3} - b \log \left (a x^{3} + b\right )}{3 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(a*x^3 - b*log(a*x^3 + b))/a^2

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Sympy [A]  time = 1.27087, size = 20, normalized size = 0.74 \[ \frac{x^{3}}{3 a} - \frac{b \log{\left (a x^{3} + b \right )}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(a+b/x**3),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.227272, size = 32, normalized size = 1.19 \[ \frac{x^{3}}{3 \, a} - \frac{b{\rm ln}\left ({\left | a x^{3} + b \right |}\right )}{3 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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