Optimal. Leaf size=73 \[ \frac{1}{(d+e x) \left (c d^2-a e^2\right )}+\frac{c d \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}-\frac{c d \log (d+e x)}{\left (c d^2-a e^2\right )^2} \]
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Rubi [A] time = 0.0505175, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 44} \[ \frac{1}{(d+e x) \left (c d^2-a e^2\right )}+\frac{c d \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}-\frac{c d \log (d+e x)}{\left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx &=\int \frac{1}{(a e+c d x) (d+e x)^2} \, dx\\ &=\int \left (\frac{c^2 d^2}{\left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{e}{\left (c d^2-a e^2\right ) (d+e x)^2}-\frac{c d e}{\left (c d^2-a e^2\right )^2 (d+e x)}\right ) \, dx\\ &=\frac{1}{\left (c d^2-a e^2\right ) (d+e x)}+\frac{c d \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}-\frac{c d \log (d+e x)}{\left (c d^2-a e^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0292618, size = 66, normalized size = 0.9 \[ \frac{c d (d+e x) \log (a e+c d x)-a e^2+c d^2-c d (d+e x) \log (d+e x)}{(d+e x) \left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 75, normalized size = 1. \begin{align*} -{\frac{1}{ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( ex+d \right ) }}-{\frac{cd\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}}+{\frac{cd\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 = 0.998684, size = 144, normalized size = 1.97 \begin{align*} \frac{c d \log \left (c d x + a e\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac{c d \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac{1}{c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91676, size = 225, normalized size = 3.08 \begin{align*} \frac{c d^{2} - a e^{2} +{\left (c d e x + c d^{2}\right )} \log \left (c d x + a e\right ) -{\left (c d e x + c d^{2}\right )} \log \left (e x + d\right )}{c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} +{\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.00351, size = 301, normalized size = 4.12 \begin{align*} - \frac{c d \log{\left (x + \frac{- \frac{a^{3} c d e^{6}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{3 a^{2} c^{2} d^{3} e^{4}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{3 a c^{3} d^{5} e^{2}}{\left (a e^{2} - c d^{2}\right )^{2}} + a c d e^{2} + \frac{c^{4} d^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + c^{2} d^{3}}{2 c^{2} d^{2} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{c d \log{\left (x + \frac{\frac{a^{3} c d e^{6}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{3 a^{2} c^{2} d^{3} e^{4}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{3 a c^{3} d^{5} e^{2}}{\left (a e^{2} - c d^{2}\right )^{2}} + a c d e^{2} - \frac{c^{4} d^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + c^{2} d^{3}}{2 c^{2} d^{2} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{1}{a d e^{2} - c d^{3} + x \left (a e^{3} - c d^{2} e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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