Optimal. Leaf size=49 \[ -\frac {b^2 \log \left (a+b x^2\right )}{2 a^3}+\frac {b^2 \log (x)}{a^3}+\frac {b}{2 a^2 x^2}-\frac {1}{4 a x^4} \]
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Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 44} \[ -\frac {b^2 \log \left (a+b x^2\right )}{2 a^3}+\frac {b^2 \log (x)}{a^3}+\frac {b}{2 a^2 x^2}-\frac {1}{4 a x^4} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a+b x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^3 (a+b x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 a x^4}+\frac {b}{2 a^2 x^2}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log \left (a+b x^2\right )}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 1.00 \[ -\frac {b^2 \log \left (a+b x^2\right )}{2 a^3}+\frac {b^2 \log (x)}{a^3}+\frac {b}{2 a^2 x^2}-\frac {1}{4 a x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 45, normalized size = 0.92 \[ -\frac {2 \, b^{2} x^{4} \log \left (b x^{2} + a\right ) - 4 \, b^{2} x^{4} \log \relax (x) - 2 \, a b x^{2} + a^{2}}{4 \, a^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 57, normalized size = 1.16 \[ \frac {b^{2} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac {b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3}} - \frac {3 \, b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}{4 \, a^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 44, normalized size = 0.90 \[ \frac {b^{2} \ln \relax (x )}{a^{3}}-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{3}}+\frac {b}{2 a^{2} x^{2}}-\frac {1}{4 a \,x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 47, normalized size = 0.96 \[ -\frac {b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac {b^{2} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac {2 \, b x^{2} - a}{4 \, a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 46, normalized size = 0.94 \[ \frac {b^2\,\ln \relax (x)}{a^3}-\frac {b^2\,\ln \left (b\,x^2+a\right )}{2\,a^3}-\frac {\frac {1}{4\,a}-\frac {b\,x^2}{2\,a^2}}{x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 42, normalized size = 0.86 \[ \frac {- a + 2 b x^{2}}{4 a^{2} x^{4}} + \frac {b^{2} \log {\relax (x )}}{a^{3}} - \frac {b^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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