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

Optimal. Leaf size=16 \[ -\frac{1}{6 b \left (a+b x^3\right )^2} \]

[Out]

-1/(6*b*(a + b*x^3)^2)

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Rubi [A]  time = 0.00903664, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{1}{6 b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x^3)^3,x]

[Out]

-1/(6*b*(a + b*x^3)^2)

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Rubi in Sympy [A]  time = 2.19093, size = 14, normalized size = 0.88 \[ - \frac{1}{6 b \left (a + b x^{3}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(6*b*(a + b*x**3)**2)

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Mathematica [A]  time = 0.00865362, size = 16, normalized size = 1. \[ -\frac{1}{6 b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b*x^3)^3,x]

[Out]

-1/(6*b*(a + b*x^3)^2)

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Maple [A]  time = 0.002, size = 15, normalized size = 0.9 \[ -{\frac{1}{6\,b \left ( b{x}^{3}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^3+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.42751, size = 19, normalized size = 1.19 \[ -\frac{1}{6 \,{\left (b x^{3} + a\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6/((b*x^3 + a)^2*b)

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Fricas [A]  time = 0.242473, size = 35, normalized size = 2.19 \[ -\frac{1}{6 \,{\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6/(b^3*x^6 + 2*a*b^2*x^3 + a^2*b)

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Sympy [A]  time = 2.11547, size = 27, normalized size = 1.69 \[ - \frac{1}{6 a^{2} b + 12 a b^{2} x^{3} + 6 b^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x**3+a)**3,x)

[Out]

-1/(6*a**2*b + 12*a*b**2*x**3 + 6*b**3*x**6)

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GIAC/XCAS [A]  time = 0.219005, size = 19, normalized size = 1.19 \[ -\frac{1}{6 \,{\left (b x^{3} + a\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6/((b*x^3 + a)^2*b)