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

Optimal. Leaf size=28 \[ -\frac{a \log \left (a-b x^2\right )}{2 b^2}-\frac{x^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0586737, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{a \log \left (a-b x^2\right )}{2 b^2}-\frac{x^2}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a - b*x^2),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00873298, size = 28, normalized size = 1. \[ -\frac{a \log \left (a-b x^2\right )}{2 b^2}-\frac{x^2}{2 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a - b*x^2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 26, normalized size = 0.9 \[ -{\frac{{x}^{2}}{2\,b}}-{\frac{a\ln \left ( b{x}^{2}-a \right ) }{2\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(-b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.33983, size = 34, normalized size = 1.21 \[ -\frac{x^{2}}{2 \, b} - \frac{a \log \left (b x^{2} - a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x^2/b - 1/2*a*log(b*x^2 - a)/b^2

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Fricas [A]  time = 0.226322, size = 31, normalized size = 1.11 \[ -\frac{b x^{2} + a \log \left (b x^{2} - a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(b*x^2 + a*log(b*x^2 - a))/b^2

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Sympy [A]  time = 1.2523, size = 22, normalized size = 0.79 \[ - \frac{a \log{\left (- a + b x^{2} \right )}}{2 b^{2}} - \frac{x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(-b*x**2+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.208559, size = 35, normalized size = 1.25 \[ -\frac{x^{2}}{2 \, b} - \frac{a{\rm ln}\left ({\left | b x^{2} - a \right |}\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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