Optimal. Leaf size=22 \[ \frac{\log (x)}{b}-\frac{\log \left (b+d x^2\right )}{2 b} \]
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Rubi [A] time = 0.0115629, 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} \[ \frac{\log (x)}{b}-\frac{\log \left (b+d x^2\right )}{2 b} \]
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}{b x+d x^3} \, dx &=\int \frac{1}{x \left (b+d x^2\right )} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (b+d x)} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 b}-\frac{d \operatorname{Subst}\left (\int \frac{1}{b+d x} \, dx,x,x^2\right )}{2 b}\\ &=\frac{\log (x)}{b}-\frac{\log \left (b+d x^2\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0045165, size = 22, normalized size = 1. \[ \frac{\log (x)}{b}-\frac{\log \left (b+d x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 21, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( d{x}^{2}+b \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0347, size = 27, normalized size = 1.23 \begin{align*} -\frac{\log \left (d x^{2} + b\right )}{2 \, b} + \frac{\log \left (x\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28758, size = 49, normalized size = 2.23 \begin{align*} -\frac{\log \left (d x^{2} + b\right ) - 2 \, \log \left (x\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.188181, size = 15, normalized size = 0.68 \begin{align*} \frac{\log{\left (x \right )}}{b} - \frac{\log{\left (\frac{b}{d} + x^{2} \right )}}{2 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18816, size = 32, normalized size = 1.45 \begin{align*} \frac{\log \left (x^{2}\right )}{2 \, b} - \frac{\log \left ({\left | d x^{2} + b \right |}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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