3.1835 \(\int \frac{\left (a+\frac{b}{x^2}\right )^3}{x} \, dx\)

Optimal. Leaf size=39 \[ a^3 \log (x)-\frac{3 a^2 b}{2 x^2}-\frac{3 a b^2}{4 x^4}-\frac{b^3}{6 x^6} \]

[Out]

-b^3/(6*x^6) - (3*a*b^2)/(4*x^4) - (3*a^2*b)/(2*x^2) + a^3*Log[x]

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Rubi [A]  time = 0.0608169, antiderivative size = 39, 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 \[ a^3 \log (x)-\frac{3 a^2 b}{2 x^2}-\frac{3 a b^2}{4 x^4}-\frac{b^3}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^3/x,x]

[Out]

-b^3/(6*x^6) - (3*a*b^2)/(4*x^4) - (3*a^2*b)/(2*x^2) + a^3*Log[x]

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Rubi in Sympy [A]  time = 10.1853, size = 41, normalized size = 1.05 \[ \frac{a^{3} \log{\left (x^{2} \right )}}{2} - \frac{3 a^{2} b}{2 x^{2}} - \frac{3 a b^{2}}{4 x^{4}} - \frac{b^{3}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*log(x**2)/2 - 3*a**2*b/(2*x**2) - 3*a*b**2/(4*x**4) - b**3/(6*x**6)

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Mathematica [A]  time = 0.00717146, size = 39, normalized size = 1. \[ a^3 \log (x)-\frac{3 a^2 b}{2 x^2}-\frac{3 a b^2}{4 x^4}-\frac{b^3}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^3/x,x]

[Out]

-b^3/(6*x^6) - (3*a*b^2)/(4*x^4) - (3*a^2*b)/(2*x^2) + a^3*Log[x]

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Maple [A]  time = 0.008, size = 34, normalized size = 0.9 \[ -{\frac{{b}^{3}}{6\,{x}^{6}}}-{\frac{3\,a{b}^{2}}{4\,{x}^{4}}}-{\frac{3\,{a}^{2}b}{2\,{x}^{2}}}+{a}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^3/x,x)

[Out]

-1/6*b^3/x^6-3/4*a*b^2/x^4-3/2*a^2*b/x^2+a^3*ln(x)

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Maxima [A]  time = 1.48768, size = 53, normalized size = 1.36 \[ \frac{1}{2} \, a^{3} \log \left (x^{2}\right ) - \frac{18 \, a^{2} b x^{4} + 9 \, a b^{2} x^{2} + 2 \, b^{3}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a^3*log(x^2) - 1/12*(18*a^2*b*x^4 + 9*a*b^2*x^2 + 2*b^3)/x^6

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Fricas [A]  time = 0.218572, size = 53, normalized size = 1.36 \[ \frac{12 \, a^{3} x^{6} \log \left (x\right ) - 18 \, a^{2} b x^{4} - 9 \, a b^{2} x^{2} - 2 \, b^{3}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(12*a^3*x^6*log(x) - 18*a^2*b*x^4 - 9*a*b^2*x^2 - 2*b^3)/x^6

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Sympy [A]  time = 1.58206, size = 36, normalized size = 0.92 \[ a^{3} \log{\left (x \right )} - \frac{18 a^{2} b x^{4} + 9 a b^{2} x^{2} + 2 b^{3}}{12 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**3/x,x)

[Out]

a**3*log(x) - (18*a**2*b*x**4 + 9*a*b**2*x**2 + 2*b**3)/(12*x**6)

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GIAC/XCAS [A]  time = 0.22929, size = 63, normalized size = 1.62 \[ \frac{1}{2} \, a^{3}{\rm ln}\left (x^{2}\right ) - \frac{11 \, a^{3} x^{6} + 18 \, a^{2} b x^{4} + 9 \, a b^{2} x^{2} + 2 \, b^{3}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a^3*ln(x^2) - 1/12*(11*a^3*x^6 + 18*a^2*b*x^4 + 9*a*b^2*x^2 + 2*b^3)/x^6