Optimal. Leaf size=47 \[ \frac{\log (a e+c d x)}{c d^2-a e^2}-\frac{\log (d+e x)}{c d^2-a e^2} \]
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Rubi [A] time = 0.0154168, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {616, 31} \[ \frac{\log (a e+c d x)}{c d^2-a e^2}-\frac{\log (d+e x)}{c d^2-a e^2} \]
Antiderivative was successfully verified.
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Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=-\frac{(c d e) \int \frac{1}{c d^2+c d e x} \, dx}{c d^2-a e^2}+\frac{(c d e) \int \frac{1}{a e^2+c d e x} \, dx}{c d^2-a e^2}\\ &=\frac{\log (a e+c d x)}{c d^2-a e^2}-\frac{\log (d+e x)}{c d^2-a e^2}\\ \end{align*}
Mathematica [A] time = 0.0137618, size = 33, normalized size = 0.7 \[ \frac{\log (a e+c d x)-\log (d+e x)}{c d^2-a e^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 48, normalized size = 1. \begin{align*}{\frac{\ln \left ( ex+d \right ) }{a{e}^{2}-c{d}^{2}}}-{\frac{\ln \left ( cdx+ae \right ) }{a{e}^{2}-c{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06451, size = 63, normalized size = 1.34 \begin{align*} \frac{\log \left (c d x + a e\right )}{c d^{2} - a e^{2}} - \frac{\log \left (e x + d\right )}{c d^{2} - a e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54686, size = 69, normalized size = 1.47 \begin{align*} \frac{\log \left (c d x + a e\right ) - \log \left (e x + d\right )}{c d^{2} - a e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.361001, size = 172, normalized size = 3.66 \begin{align*} \frac{\log{\left (x + \frac{- \frac{a^{2} e^{4}}{a e^{2} - c d^{2}} + \frac{2 a c d^{2} e^{2}}{a e^{2} - c d^{2}} + a e^{2} - \frac{c^{2} d^{4}}{a e^{2} - c d^{2}} + c d^{2}}{2 c d e} \right )}}{a e^{2} - c d^{2}} - \frac{\log{\left (x + \frac{\frac{a^{2} e^{4}}{a e^{2} - c d^{2}} - \frac{2 a c d^{2} e^{2}}{a e^{2} - c d^{2}} + a e^{2} + \frac{c^{2} d^{4}}{a e^{2} - c d^{2}} + c d^{2}}{2 c d e} \right )}}{a e^{2} - c d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28223, size = 101, normalized size = 2.15 \begin{align*} \frac{2 \, \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 )}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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