Optimal. Leaf size=38 \[ \frac{\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2}+\frac{e x}{c d} \]
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Rubi [A] time = 0.0323342, antiderivative size = 38, 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, 43} \[ \frac{\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2}+\frac{e x}{c d} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\int \frac{d+e x}{a e+c d x} \, dx\\ &=\int \left (\frac{e}{c d}+\frac{c d^2-a e^2}{c d (a e+c d x)}\right ) \, dx\\ &=\frac{e x}{c d}+\frac{\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2}\\ \end{align*}
Mathematica [A] time = 0.0089051, size = 35, normalized size = 0.92 \[ \frac{\left (c d^2-a e^2\right ) \log (a e+c d x)+c d e x}{c^2 d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 45, normalized size = 1.2 \begin{align*}{\frac{ex}{cd}}-{\frac{\ln \left ( cdx+ae \right ) a{e}^{2}}{{c}^{2}{d}^{2}}}+{\frac{\ln \left ( cdx+ae \right ) }{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0063, size = 51, normalized size = 1.34 \begin{align*} \frac{e x}{c d} + \frac{{\left (c d^{2} - a e^{2}\right )} \log \left (c d x + a e\right )}{c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55843, size = 76, normalized size = 2. \begin{align*} \frac{c d e x +{\left (c d^{2} - a e^{2}\right )} \log \left (c d x + a e\right )}{c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.376055, size = 32, normalized size = 0.84 \begin{align*} \frac{e x}{c d} - \frac{\left (a e^{2} - c d^{2}\right ) \log{\left (a e + c d x \right )}}{c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25951, size = 215, normalized size = 5.66 \begin{align*} \frac{x e}{c d} + \frac{{\left (c d^{2} - a e^{2}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{2} d^{2}} + \frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\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 )}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}} c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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