Optimal. Leaf size=49 \[ \frac {a \log \left (a x^2+b\right )}{b^3}-\frac {2 a \log (x)}{b^3}-\frac {a}{2 b^2 \left (a x^2+b\right )}-\frac {1}{2 b^2 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 49, 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} \[ -\frac {a}{2 b^2 \left (a x^2+b\right )}+\frac {a \log \left (a x^2+b\right )}{b^3}-\frac {2 a \log (x)}{b^3}-\frac {1}{2 b^2 x^2} \]
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 )^2 x^7} \, dx &=\int \frac {1}{x^3 \left (b+a x^2\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (b+a x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{b^2 x^2}-\frac {2 a}{b^3 x}+\frac {a^2}{b^2 (b+a x)^2}+\frac {2 a^2}{b^3 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 b^2 x^2}-\frac {a}{2 b^2 \left (b+a x^2\right )}-\frac {2 a \log (x)}{b^3}+\frac {a \log \left (b+a x^2\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 41, normalized size = 0.84 \[ -\frac {b \left (\frac {a}{a x^2+b}+\frac {1}{x^2}\right )-2 a \log \left (a x^2+b\right )+4 a \log (x)}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 73, normalized size = 1.49 \[ -\frac {2 \, a b x^{2} + b^{2} - 2 \, {\left (a^{2} x^{4} + a b x^{2}\right )} \log \left (a x^{2} + b\right ) + 4 \, {\left (a^{2} x^{4} + a b x^{2}\right )} \log \relax (x)}{2 \, {\left (a b^{3} x^{4} + b^{4} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 51, normalized size = 1.04 \[ -\frac {a \log \left (x^{2}\right )}{b^{3}} + \frac {a \log \left ({\left | a x^{2} + b \right |}\right )}{b^{3}} - \frac {2 \, a x^{2} + b}{2 \, {\left (a x^{4} + b x^{2}\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 46, normalized size = 0.94 \[ -\frac {a}{2 \left (a \,x^{2}+b \right ) b^{2}}-\frac {2 a \ln \relax (x )}{b^{3}}+\frac {a \ln \left (a \,x^{2}+b \right )}{b^{3}}-\frac {1}{2 b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 52, normalized size = 1.06 \[ -\frac {2 \, a x^{2} + b}{2 \, {\left (a b^{2} x^{4} + b^{3} x^{2}\right )}} + \frac {a \log \left (a x^{2} + b\right )}{b^{3}} - \frac {a \log \left (x^{2}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 51, normalized size = 1.04 \[ \frac {a\,\ln \left (a\,x^2+b\right )}{b^3}-\frac {\frac {1}{2\,b}+\frac {a\,x^2}{b^2}}{a\,x^4+b\,x^2}-\frac {2\,a\,\ln \relax (x)}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 51, normalized size = 1.04 \[ - \frac {2 a \log {\relax (x )}}{b^{3}} + \frac {a \log {\left (x^{2} + \frac {b}{a} \right )}}{b^{3}} + \frac {- 2 a x^{2} - b}{2 a b^{2} x^{4} + 2 b^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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