Optimal. Leaf size=66 \[ -\frac {3 b^2 \log \left (a+b x^2\right )}{2 a^4}+\frac {3 b^2 \log (x)}{a^4}+\frac {b^2}{2 a^3 \left (a+b x^2\right )}+\frac {b}{a^3 x^2}-\frac {1}{4 a^2 x^4} \]
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Rubi [A] time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 266, 44} \[ \frac {b^2}{2 a^3 \left (a+b x^2\right )}-\frac {3 b^2 \log \left (a+b x^2\right )}{2 a^4}+\frac {3 b^2 \log (x)}{a^4}+\frac {b}{a^3 x^2}-\frac {1}{4 a^2 x^4} \]
Antiderivative was successfully verified.
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Rule 28
Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx &=b^2 \int \frac {1}{x^5 \left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac {1}{2} b^2 \operatorname {Subst}\left (\int \frac {1}{x^3 \left (a b+b^2 x\right )^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^2 \operatorname {Subst}\left (\int \left (\frac {1}{a^2 b^2 x^3}-\frac {2}{a^3 b x^2}+\frac {3}{a^4 x}-\frac {b}{a^3 (a+b x)^2}-\frac {3 b}{a^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 a^2 x^4}+\frac {b}{a^3 x^2}+\frac {b^2}{2 a^3 \left (a+b x^2\right )}+\frac {3 b^2 \log (x)}{a^4}-\frac {3 b^2 \log \left (a+b x^2\right )}{2 a^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 57, normalized size = 0.86 \[ \frac {-6 b^2 \log \left (a+b x^2\right )+a \left (\frac {2 b^2}{a+b x^2}-\frac {a}{x^4}+\frac {4 b}{x^2}\right )+12 b^2 \log (x)}{4 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 90, normalized size = 1.36 \[ \frac {6 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - a^{3} - 6 \, {\left (b^{3} x^{6} + a b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 12 \, {\left (b^{3} x^{6} + a b^{2} x^{4}\right )} \log \relax (x)}{4 \, {\left (a^{4} b x^{6} + a^{5} x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 86, normalized size = 1.30 \[ \frac {3 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac {3 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4}} + \frac {3 \, b^{3} x^{2} + 4 \, a b^{2}}{2 \, {\left (b x^{2} + a\right )} a^{4}} - \frac {9 \, b^{2} x^{4} - 4 \, a b x^{2} + a^{2}}{4 \, a^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 0.92 \[ \frac {b^{2}}{2 \left (b \,x^{2}+a \right ) a^{3}}+\frac {3 b^{2} \ln \relax (x )}{a^{4}}-\frac {3 b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{4}}+\frac {b}{a^{3} x^{2}}-\frac {1}{4 a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 70, normalized size = 1.06 \[ \frac {6 \, b^{2} x^{4} + 3 \, a b x^{2} - a^{2}}{4 \, {\left (a^{3} b x^{6} + a^{4} x^{4}\right )}} - \frac {3 \, b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac {3 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 67, normalized size = 1.02 \[ \frac {\frac {3\,b\,x^2}{4\,a^2}-\frac {1}{4\,a}+\frac {3\,b^2\,x^4}{2\,a^3}}{b\,x^6+a\,x^4}-\frac {3\,b^2\,\ln \left (b\,x^2+a\right )}{2\,a^4}+\frac {3\,b^2\,\ln \relax (x)}{a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 68, normalized size = 1.03 \[ \frac {- a^{2} + 3 a b x^{2} + 6 b^{2} x^{4}}{4 a^{4} x^{4} + 4 a^{3} b x^{6}} + \frac {3 b^{2} \log {\relax (x )}}{a^{4}} - \frac {3 b^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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