Optimal. Leaf size=74 \[ -\frac{\left (c d^2-a e^2\right )^2}{c^3 d^3 (a e+c d x)}+\frac{2 e \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^3 d^3}+\frac{e^2 x}{c^2 d^2} \]
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Rubi [A] time = 0.0591841, antiderivative size = 74, 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} \[ -\frac{\left (c d^2-a e^2\right )^2}{c^3 d^3 (a e+c d x)}+\frac{2 e \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^3 d^3}+\frac{e^2 x}{c^2 d^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac{(d+e x)^2}{(a e+c d x)^2} \, dx\\ &=\int \left (\frac{e^2}{c^2 d^2}+\frac{\left (c d^2-a e^2\right )^2}{c^2 d^2 (a e+c d x)^2}+\frac{2 \left (c d^2 e-a e^3\right )}{c^2 d^2 (a e+c d x)}\right ) \, dx\\ &=\frac{e^2 x}{c^2 d^2}-\frac{\left (c d^2-a e^2\right )^2}{c^3 d^3 (a e+c d x)}+\frac{2 e \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^3 d^3}\\ \end{align*}
Mathematica [A] time = 0.0461451, size = 65, normalized size = 0.88 \[ \frac{-\frac{\left (c d^2-a e^2\right )^2}{a e+c d x}+2 \left (c d^2 e-a e^3\right ) \log (a e+c d x)+c d e^2 x}{c^3 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 114, normalized size = 1.5 \begin{align*}{\frac{{e}^{2}x}{{c}^{2}{d}^{2}}}-{\frac{{a}^{2}{e}^{4}}{{c}^{3}{d}^{3} \left ( cdx+ae \right ) }}+2\,{\frac{a{e}^{2}}{{c}^{2}d \left ( cdx+ae \right ) }}-{\frac{d}{c \left ( cdx+ae \right ) }}-2\,{\frac{{e}^{3}\ln \left ( cdx+ae \right ) a}{{c}^{3}{d}^{3}}}+2\,{\frac{e\ln \left ( cdx+ae \right ) }{{c}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02031, size = 120, normalized size = 1.62 \begin{align*} -\frac{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{c^{4} d^{4} x + a c^{3} d^{3} e} + \frac{e^{2} x}{c^{2} d^{2}} + \frac{2 \,{\left (c d^{2} e - a e^{3}\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 = 1.92026, size = 227, normalized size = 3.07 \begin{align*} \frac{c^{2} d^{2} e^{2} x^{2} + a c d e^{3} x - c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4} + 2 \,{\left (a c d^{2} e^{2} - a^{2} e^{4} +{\left (c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \log \left (c d x + a e\right )}{c^{4} d^{4} x + a c^{3} d^{3} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.68862, size = 85, normalized size = 1.15 \begin{align*} - \frac{a^{2} e^{4} - 2 a c d^{2} e^{2} + c^{2} d^{4}}{a c^{3} d^{3} e + c^{4} d^{4} x} + \frac{e^{2} x}{c^{2} d^{2}} - \frac{2 e \left (a e^{2} - c d^{2}\right ) \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 = 1.29919, size = 525, normalized size = 7.09 \begin{align*} \frac{2 \,{\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\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^{5} d^{7} - 2 \, a c^{4} d^{5} e^{2} + a^{2} c^{3} d^{3} e^{4}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac{x e^{2}}{c^{2} d^{2}} + \frac{{\left (c d^{2} e - a e^{3}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{c^{3} d^{3}} - \frac{\frac{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{c} + \frac{{\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} x}{c d}}{{\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 )} c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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