Optimal. Leaf size=44 \[ -\frac {b^2}{2 a^3 \left (a x^2+b\right )}-\frac {b \log \left (a x^2+b\right )}{a^3}+\frac {x^2}{2 a^2} \]
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Rubi [A] time = 0.03, antiderivative size = 44, 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} \[ -\frac {b^2}{2 a^3 \left (a x^2+b\right )}-\frac {b \log \left (a x^2+b\right )}{a^3}+\frac {x^2}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {x}{\left (a+\frac {b}{x^2}\right )^2} \, dx &=\int \frac {x^5}{\left (b+a x^2\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{(b+a x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a^2}+\frac {b^2}{a^2 (b+a x)^2}-\frac {2 b}{a^2 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 a^2}-\frac {b^2}{2 a^3 \left (b+a x^2\right )}-\frac {b \log \left (b+a x^2\right )}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 0.86 \[ \frac {-\frac {b^2}{a x^2+b}-2 b \log \left (a x^2+b\right )+a x^2}{2 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 56, normalized size = 1.27 \[ \frac {a^{2} x^{4} + a b x^{2} - b^{2} - 2 \, {\left (a b x^{2} + b^{2}\right )} \log \left (a x^{2} + b\right )}{2 \, {\left (a^{4} x^{2} + a^{3} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 41, normalized size = 0.93 \[ \frac {x^{2}}{2 \, a^{2}} - \frac {b \log \left ({\left | a x^{2} + b \right |}\right )}{a^{3}} - \frac {b^{2}}{2 \, {\left (a x^{2} + b\right )} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 41, normalized size = 0.93 \[ \frac {x^{2}}{2 a^{2}}-\frac {b^{2}}{2 \left (a \,x^{2}+b \right ) a^{3}}-\frac {b \ln \left (a \,x^{2}+b \right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 43, normalized size = 0.98 \[ -\frac {b^{2}}{2 \, {\left (a^{4} x^{2} + a^{3} b\right )}} + \frac {x^{2}}{2 \, a^{2}} - \frac {b \log \left (a x^{2} + b\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 45, normalized size = 1.02 \[ \frac {x^2}{2\,a^2}-\frac {b^2}{2\,\left (a^4\,x^2+b\,a^3\right )}-\frac {b\,\ln \left (a\,x^2+b\right )}{a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 39, normalized size = 0.89 \[ - \frac {b^{2}}{2 a^{4} x^{2} + 2 a^{3} b} + \frac {x^{2}}{2 a^{2}} - \frac {b \log {\left (a x^{2} + b \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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