Optimal. Leaf size=22 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^2\right )}{2 a} \]
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Rubi [A] time = 0.0120126, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {1593, 266, 36, 29, 31} \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^2\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a x+b x^3} \, dx &=\int \frac{1}{x \left (a+b x^2\right )} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (a+b x)} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 a}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,x^2\right )}{2 a}\\ &=\frac{\log (x)}{a}-\frac{\log \left (a+b x^2\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0042559, size = 22, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^2\right )}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 21, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06132, size = 27, normalized size = 1.23 \begin{align*} -\frac{\log \left (b x^{2} + a\right )}{2 \, a} + \frac{\log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40941, size = 49, normalized size = 2.23 \begin{align*} -\frac{\log \left (b x^{2} + a\right ) - 2 \, \log \left (x\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.178339, size = 15, normalized size = 0.68 \begin{align*} \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x^{2} \right )}}{2 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18213, size = 32, normalized size = 1.45 \begin{align*} \frac{\log \left (x^{2}\right )}{2 \, a} - \frac{\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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