Optimal. Leaf size=26 \[ \left (a-\frac{c d^2}{e^2}\right ) \log (d+e x)+\frac{c d x}{e} \]
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Rubi [A] time = 0.0281776, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {24, 43} \[ \left (a-\frac{c d^2}{e^2}\right ) \log (d+e x)+\frac{c d x}{e} \]
Antiderivative was successfully verified.
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Rule 24
Rule 43
Rubi steps
\begin{align*} \int \frac{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^2} \, dx &=\frac{\int \frac{a e^3+c d e^2 x}{d+e x} \, dx}{e^2}\\ &=\frac{\int \left (c d e+\frac{-c d^2 e+a e^3}{d+e x}\right ) \, dx}{e^2}\\ &=\frac{c d x}{e}+\left (a-\frac{c d^2}{e^2}\right ) \log (d+e x)\\ \end{align*}
Mathematica [A] time = 0.0091383, size = 30, normalized size = 1.15 \[ \frac{\left (a e^2-c d^2\right ) \log (d+e x)}{e^2}+\frac{c d x}{e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 32, normalized size = 1.2 \begin{align*}{\frac{cdx}{e}}+\ln \left ( ex+d \right ) a-{\frac{\ln \left ( ex+d \right ) c{d}^{2}}{{e}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01912, size = 42, normalized size = 1.62 \begin{align*} \frac{c d x}{e} - \frac{{\left (c d^{2} - a e^{2}\right )} \log \left (e x + d\right )}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56012, size = 62, normalized size = 2.38 \begin{align*} \frac{c d e x -{\left (c d^{2} - a e^{2}\right )} \log \left (e x + d\right )}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.392545, size = 26, normalized size = 1. \begin{align*} \frac{c d x}{e} + \frac{\left (a e^{2} - c d^{2}\right ) \log{\left (d + e x \right )}}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24215, size = 158, normalized size = 6.08 \begin{align*}{\left (2 \, d e^{\left (-3\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (x e + d\right )} e^{\left (-3\right )} - \frac{d^{2} e^{\left (-3\right )}}{x e + d}\right )} c d e -{\left (c d^{2} + a e^{2}\right )}{\left (e^{\left (-1\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac{d e^{\left (-1\right )}}{x e + d}\right )} e^{\left (-1\right )} - \frac{a d}{x e + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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