Optimal. Leaf size=26 \[ \frac {1}{2} b \log \left (b x^2+1\right )-b \log (x)-\frac {1}{2 x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 26, 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} \[ \frac {1}{2} b \log \left (b x^2+1\right )-b \log (x)-\frac {1}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (1+b x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (1+b x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{x^2}-\frac {b}{x}+\frac {b^2}{1+b x}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}-b \log (x)+\frac {1}{2} b \log \left (1+b x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.00 \[ \frac {1}{2} b \log \left (b x^2+1\right )-b \log (x)-\frac {1}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 28, normalized size = 1.08 \[ \frac {b x^{2} \log \left (b x^{2} + 1\right ) - 2 \, b x^{2} \log \relax (x) - 1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 32, normalized size = 1.23 \[ -\frac {1}{2} \, b \log \left (x^{2}\right ) + \frac {1}{2} \, b \log \left ({\left | b x^{2} + 1 \right |}\right ) + \frac {b x^{2} - 1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 23, normalized size = 0.88 \[ -b \ln \relax (x )+\frac {b \ln \left (b \,x^{2}+1\right )}{2}-\frac {1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 24, normalized size = 0.92 \[ \frac {1}{2} \, b \log \left (b x^{2} + 1\right ) - \frac {1}{2} \, b \log \left (x^{2}\right ) - \frac {1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 22, normalized size = 0.85 \[ \frac {b\,\ln \left (b\,x^2+1\right )}{2}-b\,\ln \relax (x)-\frac {1}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 22, normalized size = 0.85 \[ - b \log {\relax (x )} + \frac {b \log {\left (x^{2} + \frac {1}{b} \right )}}{2} - \frac {1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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