Optimal. Leaf size=57 \[ -\frac {\log \left (a-b x^2\right )}{2 a^3}+\frac {\log (x)}{a^3}+\frac {1}{2 a^2 \left (a-b x^2\right )}+\frac {1}{4 a \left (a-b x^2\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {266, 44} \[ \frac {1}{2 a^2 \left (a-b x^2\right )}-\frac {\log \left (a-b x^2\right )}{2 a^3}+\frac {\log (x)}{a^3}+\frac {1}{4 a \left (a-b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \left (a-b x^2\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (a-b x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a^3 x}+\frac {b}{a (a-b x)^3}+\frac {b}{a^2 (a-b x)^2}+\frac {b}{a^3 (a-b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{4 a \left (a-b x^2\right )^2}+\frac {1}{2 a^2 \left (a-b x^2\right )}+\frac {\log (x)}{a^3}-\frac {\log \left (a-b x^2\right )}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 45, normalized size = 0.79 \[ \frac {\frac {a \left (3 a-2 b x^2\right )}{\left (a-b x^2\right )^2}-2 \log \left (a-b x^2\right )+4 \log (x)}{4 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 92, normalized size = 1.61 \[ -\frac {2 \, a b x^{2} - 3 \, a^{2} + 2 \, {\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )} \log \left (b x^{2} - a\right ) - 4 \, {\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )} \log \relax (x)}{4 \, {\left (a^{3} b^{2} x^{4} - 2 \, a^{4} b x^{2} + a^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 63, normalized size = 1.11 \[ \frac {\log \left (x^{2}\right )}{2 \, a^{3}} - \frac {\log \left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{3}} + \frac {3 \, b^{2} x^{4} - 8 \, a b x^{2} + 6 \, a^{2}}{4 \, {\left (b x^{2} - a\right )}^{2} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 55, normalized size = 0.96 \[ \frac {1}{4 \left (b \,x^{2}-a \right )^{2} a}-\frac {1}{2 \left (b \,x^{2}-a \right ) a^{2}}+\frac {\ln \relax (x )}{a^{3}}-\frac {\ln \left (b \,x^{2}-a \right )}{2 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 62, normalized size = 1.09 \[ -\frac {2 \, b x^{2} - 3 \, a}{4 \, {\left (a^{2} b^{2} x^{4} - 2 \, a^{3} b x^{2} + a^{4}\right )}} - \frac {\log \left (b x^{2} - a\right )}{2 \, a^{3}} + \frac {\log \left (x^{2}\right )}{2 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 57, normalized size = 1.00 \[ \frac {\ln \relax (x)}{a^3}+\frac {\frac {3}{4\,a}-\frac {b\,x^2}{2\,a^2}}{a^2-2\,a\,b\,x^2+b^2\,x^4}-\frac {\ln \left (a-b\,x^2\right )}{2\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 56, normalized size = 0.98 \[ - \frac {- 3 a + 2 b x^{2}}{4 a^{4} - 8 a^{3} b x^{2} + 4 a^{2} b^{2} x^{4}} + \frac {\log {\relax (x )}}{a^{3}} - \frac {\log {\left (- \frac {a}{b} + x^{2} \right )}}{2 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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