Optimal. Leaf size=24 \[ a^2 \log (x)-\frac {a b}{x^2}-\frac {b^2}{4 x^4} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ a^2 \log (x)-\frac {a b}{x^2}-\frac {b^2}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x} \, dx &=\int \frac {\left (b+a x^2\right )^2}{x^5} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(b+a x)^2}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {b^2}{x^3}+\frac {2 a b}{x^2}+\frac {a^2}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {b^2}{4 x^4}-\frac {a b}{x^2}+a^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 24, normalized size = 1.00 \[ a^2 \log (x)-\frac {a b}{x^2}-\frac {b^2}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 28, normalized size = 1.17 \[ \frac {4 \, a^{2} x^{4} \log \relax (x) - 4 \, a b x^{2} - b^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 34, normalized size = 1.42 \[ \frac {1}{2} \, a^{2} \log \left (x^{2}\right ) - \frac {3 \, a^{2} x^{4} + 4 \, a b x^{2} + b^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 23, normalized size = 0.96 \[ a^{2} \ln \relax (x )-\frac {a b}{x^{2}}-\frac {b^{2}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 26, normalized size = 1.08 \[ \frac {1}{2} \, a^{2} \log \left (x^{2}\right ) - \frac {4 \, a b x^{2} + b^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 24, normalized size = 1.00 \[ a^2\,\ln \relax (x)-\frac {\frac {b^2}{4}+a\,b\,x^2}{x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 24, normalized size = 1.00 \[ a^{2} \log {\relax (x )} + \frac {- 4 a b x^{2} - b^{2}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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