3.1844 \(\int \frac{x^3}{a+\frac{b}{x^2}} \, dx\)

Optimal. Leaf size=40 \[ \frac{b^2 \log \left (a x^2+b\right )}{2 a^3}-\frac{b x^2}{2 a^2}+\frac{x^4}{4 a} \]

[Out]

-(b*x^2)/(2*a^2) + x^4/(4*a) + (b^2*Log[b + a*x^2])/(2*a^3)

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Rubi [A]  time = 0.0779655, antiderivative size = 40, 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 \[ \frac{b^2 \log \left (a x^2+b\right )}{2 a^3}-\frac{b x^2}{2 a^2}+\frac{x^4}{4 a} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b/x^2),x]

[Out]

-(b*x^2)/(2*a^2) + x^4/(4*a) + (b^2*Log[b + a*x^2])/(2*a^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00863538, size = 40, normalized size = 1. \[ \frac{b^2 \log \left (a x^2+b\right )}{2 a^3}-\frac{b x^2}{2 a^2}+\frac{x^4}{4 a} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b/x^2),x]

[Out]

-(b*x^2)/(2*a^2) + x^4/(4*a) + (b^2*Log[b + a*x^2])/(2*a^3)

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Maple [A]  time = 0.005, size = 35, normalized size = 0.9 \[ -{\frac{b{x}^{2}}{2\,{a}^{2}}}+{\frac{{x}^{4}}{4\,a}}+{\frac{{b}^{2}\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x^2),x)

[Out]

-1/2*b*x^2/a^2+1/4*x^4/a+1/2*b^2*ln(a*x^2+b)/a^3

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Maxima [A]  time = 1.42446, size = 46, normalized size = 1.15 \[ \frac{b^{2} \log \left (a x^{2} + b\right )}{2 \, a^{3}} + \frac{a x^{4} - 2 \, b x^{2}}{4 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^2*log(a*x^2 + b)/a^3 + 1/4*(a*x^4 - 2*b*x^2)/a^2

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Fricas [A]  time = 0.223489, size = 45, normalized size = 1.12 \[ \frac{a^{2} x^{4} - 2 \, a b x^{2} + 2 \, b^{2} \log \left (a x^{2} + b\right )}{4 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(a^2*x^4 - 2*a*b*x^2 + 2*b^2*log(a*x^2 + b))/a^3

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Sympy [A]  time = 1.21156, size = 32, normalized size = 0.8 \[ \frac{x^{4}}{4 a} - \frac{b x^{2}}{2 a^{2}} + \frac{b^{2} \log{\left (a x^{2} + b \right )}}{2 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a+b/x**2),x)

[Out]

x**4/(4*a) - b*x**2/(2*a**2) + b**2*log(a*x**2 + b)/(2*a**3)

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GIAC/XCAS [A]  time = 0.22876, size = 47, normalized size = 1.18 \[ \frac{b^{2}{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{3}} + \frac{a x^{4} - 2 \, b x^{2}}{4 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^2*ln(abs(a*x^2 + b))/a^3 + 1/4*(a*x^4 - 2*b*x^2)/a^2