Optimal. Leaf size=65 \[ \frac {b^3}{4 a^4 \left (a x^2+b\right )^2}-\frac {3 b^2}{2 a^4 \left (a x^2+b\right )}-\frac {3 b \log \left (a x^2+b\right )}{2 a^4}+\frac {x^2}{2 a^3} \]
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Rubi [A] time = 0.05, antiderivative size = 65, 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} \[ \frac {b^3}{4 a^4 \left (a x^2+b\right )^2}-\frac {3 b^2}{2 a^4 \left (a x^2+b\right )}-\frac {3 b \log \left (a x^2+b\right )}{2 a^4}+\frac {x^2}{2 a^3} \]
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 )^3} \, dx &=\int \frac {x^7}{\left (b+a x^2\right )^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{(b+a x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a^3}-\frac {b^3}{a^3 (b+a x)^3}+\frac {3 b^2}{a^3 (b+a x)^2}-\frac {3 b}{a^3 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 a^3}+\frac {b^3}{4 a^4 \left (b+a x^2\right )^2}-\frac {3 b^2}{2 a^4 \left (b+a x^2\right )}-\frac {3 b \log \left (b+a x^2\right )}{2 a^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 48, normalized size = 0.74 \[ -\frac {\frac {b^2 \left (6 a x^2+5 b\right )}{\left (a x^2+b\right )^2}+6 b \log \left (a x^2+b\right )-2 a x^2}{4 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 91, normalized size = 1.40 \[ \frac {2 \, a^{3} x^{6} + 4 \, a^{2} b x^{4} - 4 \, a b^{2} x^{2} - 5 \, b^{3} - 6 \, {\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )} \log \left (a x^{2} + b\right )}{4 \, {\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 53, normalized size = 0.82 \[ \frac {x^{2}}{2 \, a^{3}} - \frac {3 \, b \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{4}} - \frac {6 \, a b^{2} x^{2} + 5 \, b^{3}}{4 \, {\left (a x^{2} + b\right )}^{2} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.89 \[ \frac {x^{2}}{2 a^{3}}+\frac {b^{3}}{4 \left (a \,x^{2}+b \right )^{2} a^{4}}-\frac {3 b^{2}}{2 \left (a \,x^{2}+b \right ) a^{4}}-\frac {3 b \ln \left (a \,x^{2}+b \right )}{2 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 66, normalized size = 1.02 \[ -\frac {6 \, a b^{2} x^{2} + 5 \, b^{3}}{4 \, {\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}} + \frac {x^{2}}{2 \, a^{3}} - \frac {3 \, b \log \left (a x^{2} + b\right )}{2 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 68, normalized size = 1.05 \[ \frac {x^2}{2\,a^3}-\frac {\frac {5\,b^3}{4\,a}+\frac {3\,b^2\,x^2}{2}}{a^5\,x^4+2\,a^4\,b\,x^2+a^3\,b^2}-\frac {3\,b\,\ln \left (a\,x^2+b\right )}{2\,a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 68, normalized size = 1.05 \[ \frac {- 6 a b^{2} x^{2} - 5 b^{3}}{4 a^{6} x^{4} + 8 a^{5} b x^{2} + 4 a^{4} b^{2}} + \frac {x^{2}}{2 a^{3}} - \frac {3 b \log {\left (a x^{2} + b \right )}}{2 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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