Optimal. Leaf size=28 \[ -\frac{d \log (x)}{c^2}+\frac{d \log (c+d x)}{c^2}-\frac{1}{c x} \]
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Rubi [A] time = 0.0151371, antiderivative size = 28, normalized size of antiderivative = 1., 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 = {1593, 44} \[ -\frac{d \log (x)}{c^2}+\frac{d \log (c+d x)}{c^2}-\frac{1}{c x} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{c x^2+d x^3} \, dx &=\int \frac{1}{x^2 (c+d x)} \, dx\\ &=\int \left (\frac{1}{c x^2}-\frac{d}{c^2 x}+\frac{d^2}{c^2 (c+d x)}\right ) \, dx\\ &=-\frac{1}{c x}-\frac{d \log (x)}{c^2}+\frac{d \log (c+d x)}{c^2}\\ \end{align*}
Mathematica [A] time = 0.004111, size = 28, normalized size = 1. \[ -\frac{d \log (x)}{c^2}+\frac{d \log (c+d x)}{c^2}-\frac{1}{c x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 29, normalized size = 1. \begin{align*} -{\frac{1}{cx}}-{\frac{d\ln \left ( x \right ) }{{c}^{2}}}+{\frac{d\ln \left ( dx+c \right ) }{{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14695, size = 38, normalized size = 1.36 \begin{align*} \frac{d \log \left (d x + c\right )}{c^{2}} - \frac{d \log \left (x\right )}{c^{2}} - \frac{1}{c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22864, size = 61, normalized size = 2.18 \begin{align*} \frac{d x \log \left (d x + c\right ) - d x \log \left (x\right ) - c}{c^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.435983, size = 19, normalized size = 0.68 \begin{align*} - \frac{1}{c x} + \frac{d \left (- \log{\left (x \right )} + \log{\left (\frac{c}{d} + x \right )}\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26542, size = 41, normalized size = 1.46 \begin{align*} \frac{d \log \left ({\left | d x + c \right |}\right )}{c^{2}} - \frac{d \log \left ({\left | x \right |}\right )}{c^{2}} - \frac{1}{c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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