Optimal. Leaf size=54 \[ -\frac {\log \left (a x^2+b\right )}{2 b^3}+\frac {1}{2 b^2 \left (a x^2+b\right )}+\frac {1}{4 b \left (a x^2+b\right )^2}+\frac {\log (x)}{b^3} \]
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Rubi [A] time = 0.04, antiderivative size = 54, 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, 44} \[ \frac {1}{2 b^2 \left (a x^2+b\right )}-\frac {\log \left (a x^2+b\right )}{2 b^3}+\frac {1}{4 b \left (a x^2+b\right )^2}+\frac {\log (x)}{b^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^7} \, dx &=\int \frac {1}{x \left (b+a x^2\right )^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (b+a x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{b^3 x}-\frac {a}{b (b+a x)^3}-\frac {a}{b^2 (b+a x)^2}-\frac {a}{b^3 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{4 b \left (b+a x^2\right )^2}+\frac {1}{2 b^2 \left (b+a x^2\right )}+\frac {\log (x)}{b^3}-\frac {\log \left (b+a x^2\right )}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 43, normalized size = 0.80 \[ \frac {\frac {b \left (2 a x^2+3 b\right )}{\left (a x^2+b\right )^2}-2 \log \left (a x^2+b\right )+4 \log (x)}{4 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 90, normalized size = 1.67 \[ \frac {2 \, a b x^{2} + 3 \, b^{2} - 2 \, {\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \log \left (a x^{2} + b\right ) + 4 \, {\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \log \relax (x)}{4 \, {\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 59, normalized size = 1.09 \[ \frac {\log \left (x^{2}\right )}{2 \, b^{3}} - \frac {\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, b^{3}} + \frac {3 \, a^{2} x^{4} + 8 \, a b x^{2} + 6 \, b^{2}}{4 \, {\left (a x^{2} + b\right )}^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 49, normalized size = 0.91 \[ \frac {1}{4 \left (a \,x^{2}+b \right )^{2} b}+\frac {1}{2 \left (a \,x^{2}+b \right ) b^{2}}+\frac {\ln \relax (x )}{b^{3}}-\frac {\ln \left (a \,x^{2}+b \right )}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 60, normalized size = 1.11 \[ \frac {2 \, a x^{2} + 3 \, b}{4 \, {\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )}} - \frac {\log \left (a x^{2} + b\right )}{2 \, b^{3}} + \frac {\log \left (x^{2}\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 56, normalized size = 1.04 \[ \frac {\ln \relax (x)}{b^3}+\frac {\frac {3}{4\,b}+\frac {a\,x^2}{2\,b^2}}{a^2\,x^4+2\,a\,b\,x^2+b^2}-\frac {\ln \left (a\,x^2+b\right )}{2\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 56, normalized size = 1.04 \[ \frac {2 a x^{2} + 3 b}{4 a^{2} b^{2} x^{4} + 8 a b^{3} x^{2} + 4 b^{4}} + \frac {\log {\relax (x )}}{b^{3}} - \frac {\log {\left (x^{2} + \frac {b}{a} \right )}}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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