Optimal. Leaf size=27 \[ -\frac {b^2}{2 x^2}+\frac {a^2 x^2}{2}+2 a b \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 27, 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 = {269, 272, 45}
\begin {gather*} \frac {a^2 x^2}{2}+2 a b \log (x)-\frac {b^2}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rule 272
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right )^2 x \, dx &=\int \frac {\left (b+a x^2\right )^2}{x^3} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(b+a x)^2}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {b^2}{2 x^2}+\frac {a^2 x^2}{2}+2 a b \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} -\frac {b^2}{2 x^2}+\frac {a^2 x^2}{2}+2 a b \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 24, normalized size = 0.89
method | result | size |
default | \(-\frac {b^{2}}{2 x^{2}}+\frac {a^{2} x^{2}}{2}+2 a b \ln \left (x \right )\) | \(24\) |
risch | \(-\frac {b^{2}}{2 x^{2}}+\frac {a^{2} x^{2}}{2}+2 a b \ln \left (x \right )\) | \(24\) |
norman | \(\frac {\frac {1}{2} a^{2} x^{5}-\frac {1}{2} b^{2} x}{x^{3}}+2 a b \ln \left (x \right )\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 24, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, a^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac {b^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 27, normalized size = 1.00 \begin {gather*} \frac {a^{2} x^{4} + 4 \, a b x^{2} \log \left (x\right ) - b^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 24, normalized size = 0.89 \begin {gather*} \frac {a^{2} x^{2}}{2} + 2 a b \log {\left (x \right )} - \frac {b^{2}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.86, size = 24, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, a^{2} x^{2} + 2 \, a b \log \left ({\left | x \right |}\right ) - \frac {b^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.06, size = 23, normalized size = 0.85 \begin {gather*} \frac {a^2\,x^2}{2}-\frac {b^2}{2\,x^2}+2\,a\,b\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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