Optimal. Leaf size=69 \[ \frac{e x \left (c d^2-a e^2\right )}{c^2 d^2}+\frac{\left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^3 d^3}+\frac{(d+e x)^2}{2 c d} \]
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Rubi [A] time = 0.0331473, antiderivative size = 69, 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} \[ \frac{e x \left (c d^2-a e^2\right )}{c^2 d^2}+\frac{\left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^3 d^3}+\frac{(d+e x)^2}{2 c d} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\int \frac{(d+e x)^2}{a e+c d x} \, dx\\ &=\int \left (\frac{e \left (c d^2-a e^2\right )}{c^2 d^2}+\frac{\left (c d^2-a e^2\right )^2}{c^2 d^2 (a e+c d x)}+\frac{e (d+e x)}{c d}\right ) \, dx\\ &=\frac{e \left (c d^2-a e^2\right ) x}{c^2 d^2}+\frac{(d+e x)^2}{2 c d}+\frac{\left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^3 d^3}\\ \end{align*}
Mathematica [A] time = 0.020929, size = 58, normalized size = 0.84 \[ \frac{2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)+c d e x \left (c d (4 d+e x)-2 a e^2\right )}{2 c^3 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 93, normalized size = 1.4 \begin{align*}{\frac{{e}^{2}{x}^{2}}{2\,cd}}-{\frac{a{e}^{3}x}{{c}^{2}{d}^{2}}}+2\,{\frac{ex}{c}}+{\frac{\ln \left ( cdx+ae \right ){a}^{2}{e}^{4}}{{c}^{3}{d}^{3}}}-2\,{\frac{\ln \left ( cdx+ae \right ) a{e}^{2}}{{c}^{2}d}}+{\frac{d\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 = 1.05332, size = 104, normalized size = 1.51 \begin{align*} \frac{c d e^{2} x^{2} + 2 \,{\left (2 \, c d^{2} e - a e^{3}\right )} x}{2 \, c^{2} d^{2}} + \frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (c d x + a e\right )}{c^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57952, size = 165, normalized size = 2.39 \begin{align*} \frac{c^{2} d^{2} e^{2} x^{2} + 2 \,{\left (2 \, c^{2} d^{3} e - a c d e^{3}\right )} x + 2 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (c d x + a e\right )}{2 \, c^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.439244, size = 61, normalized size = 0.88 \begin{align*} \frac{e^{2} x^{2}}{2 c d} - \frac{x \left (a e^{3} - 2 c d^{2} e\right )}{c^{2} d^{2}} + \frac{\left (a e^{2} - c d^{2}\right )^{2} \log{\left (a e + c d x \right )}}{c^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27223, size = 282, normalized size = 4.09 \begin{align*} \frac{{\left (c d x^{2} e^{4} + 4 \, c d^{2} x e^{3} - 2 \, a x e^{5}\right )} e^{\left (-2\right )}}{2 \, c^{2} d^{2}} + \frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{3} d^{3}} + \frac{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\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^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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