Optimal. Leaf size=40 \[ \frac {a^3 x^2}{2}+3 a^2 b \log (x)-\frac {3 a b^2}{2 x^2}-\frac {b^3}{4 x^4} \]
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Rubi [A] time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {263, 266, 43} \[ 3 a^2 b \log (x)+\frac {a^3 x^2}{2}-\frac {3 a b^2}{2 x^2}-\frac {b^3}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right )^3 x \, dx &=\int \frac {\left (b+a x^2\right )^3}{x^5} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(b+a x)^3}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a^3+\frac {b^3}{x^3}+\frac {3 a b^2}{x^2}+\frac {3 a^2 b}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {b^3}{4 x^4}-\frac {3 a b^2}{2 x^2}+\frac {a^3 x^2}{2}+3 a^2 b \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.00 \[ \frac {a^3 x^2}{2}+3 a^2 b \log (x)-\frac {3 a b^2}{2 x^2}-\frac {b^3}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 39, normalized size = 0.98 \[ \frac {2 \, a^{3} x^{6} + 12 \, a^{2} b x^{4} \log \relax (x) - 6 \, a b^{2} x^{2} - b^{3}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 36, normalized size = 0.90 \[ \frac {1}{2} \, a^{3} x^{2} + 3 \, a^{2} b \log \left ({\left | x \right |}\right ) - \frac {6 \, a b^{2} x^{2} + b^{3}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 0.88 \[ \frac {a^{3} x^{2}}{2}+3 a^{2} b \ln \relax (x )-\frac {3 a \,b^{2}}{2 x^{2}}-\frac {b^{3}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 37, normalized size = 0.92 \[ \frac {1}{2} \, a^{3} x^{2} + \frac {3}{2} \, a^{2} b \log \left (x^{2}\right ) - \frac {6 \, a b^{2} x^{2} + b^{3}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 37, normalized size = 0.92 \[ \frac {a^3\,x^2}{2}-\frac {\frac {b^3}{4}+\frac {3\,a\,b^2\,x^2}{2}}{x^4}+3\,a^2\,b\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 37, normalized size = 0.92 \[ \frac {a^{3} x^{2}}{2} + 3 a^{2} b \log {\relax (x )} + \frac {- 6 a b^{2} x^{2} - b^{3}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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