Optimal. Leaf size=33 \[ \frac{c d \log (d+e x)}{e^2}-\frac{a-\frac{c d^2}{e^2}}{d+e x} \]
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Rubi [A] time = 0.0282534, antiderivative size = 33, 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} \[ \frac{c d \log (d+e x)}{e^2}-\frac{a-\frac{c d^2}{e^2}}{d+e x} \]
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)^3} \, dx &=\frac{\int \frac{a e^3+c d e^2 x}{(d+e x)^2} \, dx}{e^2}\\ &=\frac{\int \left (\frac{-c d^2 e+a e^3}{(d+e x)^2}+\frac{c d e}{d+e x}\right ) \, dx}{e^2}\\ &=-\frac{a-\frac{c d^2}{e^2}}{d+e x}+\frac{c d \log (d+e x)}{e^2}\\ \end{align*}
Mathematica [A] time = 0.0127439, size = 36, normalized size = 1.09 \[ \frac{c d^2-a e^2}{e^2 (d+e x)}+\frac{c d \log (d+e x)}{e^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 39, normalized size = 1.2 \begin{align*}{\frac{cd\ln \left ( ex+d \right ) }{{e}^{2}}}-{\frac{a}{ex+d}}+{\frac{c{d}^{2}}{{e}^{2} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1247, size = 53, normalized size = 1.61 \begin{align*} \frac{c d \log \left (e x + d\right )}{e^{2}} + \frac{c d^{2} - a e^{2}}{e^{3} x + d e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52604, size = 89, normalized size = 2.7 \begin{align*} \frac{c d^{2} - a e^{2} +{\left (c d e x + c d^{2}\right )} \log \left (e x + d\right )}{e^{3} x + d e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.538249, size = 32, normalized size = 0.97 \begin{align*} \frac{c d \log{\left (d + e x \right )}}{e^{2}} - \frac{a e^{2} - c d^{2}}{d e^{2} + e^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2081, size = 70, normalized size = 2.12 \begin{align*} c d e^{\left (-2\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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