Optimal. Leaf size=35 \[ -\frac {b \log \left (a-b x^2\right )}{2 a^2}+\frac {b \log (x)}{a^2}-\frac {1}{2 a x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 35, 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} \[ -\frac {b \log \left (a-b x^2\right )}{2 a^2}+\frac {b \log (x)}{a^2}-\frac {1}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a-b x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (a-b x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a x^2}+\frac {b}{a^2 x}+\frac {b^2}{a^2 (a-b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a x^2}+\frac {b \log (x)}{a^2}-\frac {b \log \left (a-b x^2\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 1.00 \[ -\frac {b \log \left (a-b x^2\right )}{2 a^2}+\frac {b \log (x)}{a^2}-\frac {1}{2 a x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 33, normalized size = 0.94 \[ -\frac {b x^{2} \log \left (b x^{2} - a\right ) - 2 \, b x^{2} \log \relax (x) + a}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 43, normalized size = 1.23 \[ \frac {b \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {b \log \left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{2}} - \frac {b x^{2} + a}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 33, normalized size = 0.94 \[ \frac {b \ln \relax (x )}{a^{2}}-\frac {b \ln \left (b \,x^{2}-a \right )}{2 a^{2}}-\frac {1}{2 a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 35, normalized size = 1.00 \[ -\frac {b \log \left (b x^{2} - a\right )}{2 \, a^{2}} + \frac {b \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {1}{2 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 31, normalized size = 0.89 \[ \frac {b\,\ln \relax (x)}{a^2}-\frac {b\,\ln \left (a-b\,x^2\right )}{2\,a^2}-\frac {1}{2\,a\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 31, normalized size = 0.89 \[ - \frac {1}{2 a x^{2}} + \frac {b \log {\relax (x )}}{a^{2}} - \frac {b \log {\left (- \frac {a}{b} + x^{2} \right )}}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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