Optimal. Leaf size=85 \[ -\frac{2 e \left (c d^2-a e^2\right )}{c^3 d^3 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^2}{2 c^3 d^3 (a e+c d x)^2}+\frac{e^2 \log (a e+c d x)}{c^3 d^3} \]
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Rubi [A] time = 0.0614312, antiderivative size = 85, 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} \[ -\frac{2 e \left (c d^2-a e^2\right )}{c^3 d^3 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^2}{2 c^3 d^3 (a e+c d x)^2}+\frac{e^2 \log (a e+c d x)}{c^3 d^3} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac{(d+e x)^2}{(a e+c d x)^3} \, dx\\ &=\int \left (\frac{\left (c d^2-a e^2\right )^2}{c^2 d^2 (a e+c d x)^3}+\frac{2 \left (c d^2 e-a e^3\right )}{c^2 d^2 (a e+c d x)^2}+\frac{e^2}{c^2 d^2 (a e+c d x)}\right ) \, dx\\ &=-\frac{\left (c d^2-a e^2\right )^2}{2 c^3 d^3 (a e+c d x)^2}-\frac{2 e \left (c d^2-a e^2\right )}{c^3 d^3 (a e+c d x)}+\frac{e^2 \log (a e+c d x)}{c^3 d^3}\\ \end{align*}
Mathematica [A] time = 0.0336488, size = 65, normalized size = 0.76 \[ \frac{2 e^2 \log (a e+c d x)-\frac{\left (c d^2-a e^2\right ) \left (3 a e^2+c d (d+4 e x)\right )}{(a e+c d x)^2}}{2 c^3 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 123, normalized size = 1.5 \begin{align*} -{\frac{{a}^{2}{e}^{4}}{2\,{c}^{3}{d}^{3} \left ( cdx+ae \right ) ^{2}}}+{\frac{a{e}^{2}}{{c}^{2}d \left ( cdx+ae \right ) ^{2}}}-{\frac{d}{2\,c \left ( cdx+ae \right ) ^{2}}}+2\,{\frac{a{e}^{3}}{{c}^{3}{d}^{3} \left ( cdx+ae \right ) }}-2\,{\frac{e}{{c}^{2}d \left ( cdx+ae \right ) }}+{\frac{{e}^{2}\ln \left ( cdx+ae \right ) }{{c}^{3}{d}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07021, size = 142, normalized size = 1.67 \begin{align*} -\frac{c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 4 \,{\left (c^{2} d^{3} e - a c d e^{3}\right )} x}{2 \,{\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}} + \frac{e^{2} \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 = 1.98946, size = 255, normalized size = 3. \begin{align*} -\frac{c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 4 \,{\left (c^{2} d^{3} e - a c d e^{3}\right )} x - 2 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \log \left (c d x + a e\right )}{2 \,{\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.852945, size = 109, normalized size = 1.28 \begin{align*} \frac{3 a^{2} e^{4} - 2 a c d^{2} e^{2} - c^{2} d^{4} + x \left (4 a c d e^{3} - 4 c^{2} d^{3} e\right )}{2 a^{2} c^{3} d^{3} e^{2} + 4 a c^{4} d^{4} e x + 2 c^{5} d^{5} x^{2}} + \frac{e^{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 = 1.33168, size = 811, normalized size = 9.54 \begin{align*} \frac{{\left (c^{5} d^{10} e^{2} - 5 \, a c^{4} d^{8} e^{4} + 10 \, a^{2} c^{3} d^{6} e^{6} - 10 \, a^{3} c^{2} d^{4} e^{8} + 5 \, a^{4} c d^{2} e^{10} - a^{5} e^{12}\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^{7} d^{11} - 4 \, a c^{6} d^{9} e^{2} + 6 \, a^{2} c^{5} d^{7} e^{4} - 4 \, a^{3} c^{4} d^{5} e^{6} + a^{4} c^{3} d^{3} e^{8}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac{e^{2} \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{c^{6} d^{14} - 2 \, a c^{5} d^{12} e^{2} - 5 \, a^{2} c^{4} d^{10} e^{4} + 20 \, a^{3} c^{3} d^{8} e^{6} - 25 \, a^{4} c^{2} d^{6} e^{8} + 14 \, a^{5} c d^{4} e^{10} - 3 \, a^{6} d^{2} e^{12} + 4 \,{\left (c^{6} d^{11} e^{3} - 5 \, a c^{5} d^{9} e^{5} + 10 \, a^{2} c^{4} d^{7} e^{7} - 10 \, a^{3} c^{3} d^{5} e^{9} + 5 \, a^{4} c^{2} d^{3} e^{11} - a^{5} c d e^{13}\right )} x^{3} + 3 \,{\left (3 \, c^{6} d^{12} e^{2} - 14 \, a c^{5} d^{10} e^{4} + 25 \, a^{2} c^{4} d^{8} e^{6} - 20 \, a^{3} c^{3} d^{6} e^{8} + 5 \, a^{4} c^{2} d^{4} e^{10} + 2 \, a^{5} c d^{2} e^{12} - a^{6} e^{14}\right )} x^{2} + 6 \,{\left (c^{6} d^{13} e - 4 \, a c^{5} d^{11} e^{3} + 5 \, a^{2} c^{4} d^{9} e^{5} - 5 \, a^{4} c^{2} d^{5} e^{9} + 4 \, a^{5} c d^{3} e^{11} - a^{6} d e^{13}\right )} x}{2 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}^{2}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{2} c^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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