3.354 \(\int \frac{x^2}{a-b x^3} \, dx\)

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

[Out]

-Log[a - b*x^3]/(3*b)

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Rubi [A]  time = 0.00930703, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{\log \left (a-b x^3\right )}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a - b*x^3),x]

[Out]

-Log[a - b*x^3]/(3*b)

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Rubi in Sympy [A]  time = 2.43986, size = 12, normalized size = 0.75 \[ - \frac{\log{\left (a - b x^{3} \right )}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(a - b*x**3)/(3*b)

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Mathematica [A]  time = 0.00482246, size = 16, normalized size = 1. \[ -\frac{\log \left (a-b x^3\right )}{3 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a - b*x^3),x]

[Out]

-Log[a - b*x^3]/(3*b)

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Maple [A]  time = 0.002, size = 16, normalized size = 1. \[ -{\frac{\ln \left ( b{x}^{3}-a \right ) }{3\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 1.4387, size = 20, normalized size = 1.25 \[ -\frac{\log \left (b x^{3} - a\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*log(b*x^3 - a)/b

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Fricas [A]  time = 0.221694, size = 20, normalized size = 1.25 \[ -\frac{\log \left (b x^{3} - a\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*log(b*x^3 - a)/b

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Sympy [A]  time = 0.326634, size = 12, normalized size = 0.75 \[ - \frac{\log{\left (- a + b x^{3} \right )}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(-b*x**3+a),x)

[Out]

-log(-a + b*x**3)/(3*b)

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GIAC/XCAS [A]  time = 0.240908, size = 22, normalized size = 1.38 \[ -\frac{{\rm ln}\left ({\left | b x^{3} - a \right |}\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*ln(abs(b*x^3 - a))/b