Optimal. Leaf size=75 \[ -\frac{1}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac{e \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}+\frac{e \log (d+e x)}{\left (c d^2-a e^2\right )^2} \]
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Rubi [A] time = 0.0446562, antiderivative size = 75, 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 = {626, 44} \[ -\frac{1}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac{e \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}+\frac{e \log (d+e x)}{\left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 44
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac{1}{(a e+c d x)^2 (d+e x)} \, dx\\ &=\int \left (\frac{c d}{\left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac{c d e}{\left (c d^2-a e^2\right )^2 (a e+c d x)}+\frac{e^2}{\left (c d^2-a e^2\right )^2 (d+e x)}\right ) \, dx\\ &=-\frac{1}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac{e \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}+\frac{e \log (d+e x)}{\left (c d^2-a e^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0297919, size = 74, normalized size = 0.99 \[ \frac{1}{\left (a e^2-c d^2\right ) (a e+c d x)}-\frac{e \log (a e+c d x)}{\left (a e^2-c d^2\right )^2}+\frac{e \log (d+e x)}{\left (a e^2-c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 75, normalized size = 1. \begin{align*}{\frac{e\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}}+{\frac{1}{ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( cdx+ae \right ) }}-{\frac{e\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02519, size = 153, normalized size = 2.04 \begin{align*} -\frac{e \log \left (c d x + a e\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac{e \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac{1}{a c d^{2} e - a^{2} e^{3} +{\left (c^{2} d^{3} - a c d e^{2}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90353, size = 238, normalized size = 3.17 \begin{align*} -\frac{c d^{2} - a e^{2} +{\left (c d e x + a e^{2}\right )} \log \left (c d x + a e\right ) -{\left (c d e x + a e^{2}\right )} \log \left (e x + d\right )}{a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} +{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.05649, size = 287, normalized size = 3.83 \begin{align*} \frac{e \log{\left (x + \frac{- \frac{a^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{3 a^{2} c d^{2} e^{5}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{3 a c^{2} d^{4} e^{3}}{\left (a e^{2} - c d^{2}\right )^{2}} + a e^{3} + \frac{c^{3} d^{6} e}{\left (a e^{2} - c d^{2}\right )^{2}} + c d^{2} e}{2 c d e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{e \log{\left (x + \frac{\frac{a^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{3 a^{2} c d^{2} e^{5}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{3 a c^{2} d^{4} e^{3}}{\left (a e^{2} - c d^{2}\right )^{2}} + a e^{3} - \frac{c^{3} d^{6} e}{\left (a e^{2} - c d^{2}\right )^{2}} + c d^{2} e}{2 c d e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{1}{a^{2} e^{3} - a c d^{2} e + x \left (a c d e^{2} - c^{2} d^{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17913, size = 262, normalized size = 3.49 \begin{align*} -\frac{2 \,{\left (c d^{2} e - a e^{3}\right )} \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} - \frac{c d^{2} x e + c d^{3} - a x e^{3} - a d e^{2}}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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