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

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

[Out]

-1/(3*b*(a + b*x^3))

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Rubi [A]  time = 0.00967853, 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 \[ -\frac{1}{3 b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x^3)^2,x]

[Out]

-1/(3*b*(a + b*x^3))

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Rubi in Sympy [A]  time = 2.34826, size = 12, normalized size = 0.75 \[ - \frac{1}{3 b \left (a + b x^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(3*b*(a + b*x**3))

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Mathematica [A]  time = 0.00877585, size = 16, normalized size = 1. \[ -\frac{1}{3 b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b*x^3)^2,x]

[Out]

-1/(3*b*(a + b*x^3))

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Maple [A]  time = 0.001, size = 15, normalized size = 0.9 \[ -{\frac{1}{3\,b \left ( b{x}^{3}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^3+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.43843, size = 19, normalized size = 1.19 \[ -\frac{1}{3 \,{\left (b x^{3} + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/((b*x^3 + a)*b)

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Fricas [A]  time = 0.205677, size = 20, normalized size = 1.25 \[ -\frac{1}{3 \,{\left (b^{2} x^{3} + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/(b^2*x^3 + a*b)

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Sympy [A]  time = 1.37364, size = 15, normalized size = 0.94 \[ - \frac{1}{3 a b + 3 b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x**3+a)**2,x)

[Out]

-1/(3*a*b + 3*b**2*x**3)

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GIAC/XCAS [A]  time = 0.219017, size = 19, normalized size = 1.19 \[ -\frac{1}{3 \,{\left (b x^{3} + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/((b*x^3 + a)*b)