Optimal. Leaf size=122 \[ -\frac{3 e^2 \left (c d^2-a e^2\right )}{c^4 d^4 (a e+c d x)}-\frac{3 e \left (c d^2-a e^2\right )^2}{2 c^4 d^4 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^3}{3 c^4 d^4 (a e+c d x)^3}+\frac{e^3 \log (a e+c d x)}{c^4 d^4} \]
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Rubi [A] time = 0.0882848, antiderivative size = 122, 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} \[ -\frac{3 e^2 \left (c d^2-a e^2\right )}{c^4 d^4 (a e+c d x)}-\frac{3 e \left (c d^2-a e^2\right )^2}{2 c^4 d^4 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^3}{3 c^4 d^4 (a e+c d x)^3}+\frac{e^3 \log (a e+c d x)}{c^4 d^4} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=\int \frac{(d+e x)^3}{(a e+c d x)^4} \, dx\\ &=\int \left (\frac{\left (c d^2-a e^2\right )^3}{c^3 d^3 (a e+c d x)^4}+\frac{3 e \left (c d^2-a e^2\right )^2}{c^3 d^3 (a e+c d x)^3}+\frac{3 \left (c d^2 e^2-a e^4\right )}{c^3 d^3 (a e+c d x)^2}+\frac{e^3}{c^3 d^3 (a e+c d x)}\right ) \, dx\\ &=-\frac{\left (c d^2-a e^2\right )^3}{3 c^4 d^4 (a e+c d x)^3}-\frac{3 e \left (c d^2-a e^2\right )^2}{2 c^4 d^4 (a e+c d x)^2}-\frac{3 e^2 \left (c d^2-a e^2\right )}{c^4 d^4 (a e+c d x)}+\frac{e^3 \log (a e+c d x)}{c^4 d^4}\\ \end{align*}
Mathematica [A] time = 0.0516119, size = 99, normalized size = 0.81 \[ \frac{6 e^3 \log (a e+c d x)-\frac{\left (c d^2-a e^2\right ) \left (11 a^2 e^4+a c d e^2 (5 d+27 e x)+c^2 d^2 \left (2 d^2+9 d e x+18 e^2 x^2\right )\right )}{(a e+c d x)^3}}{6 c^4 d^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 210, normalized size = 1.7 \begin{align*}{\frac{{a}^{3}{e}^{6}}{3\,{c}^{4}{d}^{4} \left ( cdx+ae \right ) ^{3}}}-{\frac{{a}^{2}{e}^{4}}{{c}^{3}{d}^{2} \left ( cdx+ae \right ) ^{3}}}+{\frac{a{e}^{2}}{{c}^{2} \left ( cdx+ae \right ) ^{3}}}-{\frac{{d}^{2}}{3\,c \left ( cdx+ae \right ) ^{3}}}-{\frac{3\,{a}^{2}{e}^{5}}{2\,{c}^{4}{d}^{4} \left ( cdx+ae \right ) ^{2}}}+3\,{\frac{a{e}^{3}}{{c}^{3}{d}^{2} \left ( cdx+ae \right ) ^{2}}}-{\frac{3\,e}{2\,{c}^{2} \left ( cdx+ae \right ) ^{2}}}+3\,{\frac{{e}^{4}a}{{c}^{4}{d}^{4} \left ( cdx+ae \right ) }}-3\,{\frac{{e}^{2}}{{c}^{3}{d}^{2} \left ( cdx+ae \right ) }}+{\frac{{e}^{3}\ln \left ( cdx+ae \right ) }{{c}^{4}{d}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17454, size = 242, normalized size = 1.98 \begin{align*} -\frac{2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 18 \,{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 9 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x}{6 \,{\left (c^{7} d^{7} x^{3} + 3 \, a c^{6} d^{6} e x^{2} + 3 \, a^{2} c^{5} d^{5} e^{2} x + a^{3} c^{4} d^{4} e^{3}\right )}} + \frac{e^{3} \log \left (c d x + a e\right )}{c^{4} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80898, size = 433, normalized size = 3.55 \begin{align*} -\frac{2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 18 \,{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 9 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x - 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{4} x^{2} + 3 \, a^{2} c d e^{5} x + a^{3} e^{6}\right )} \log \left (c d x + a e\right )}{6 \,{\left (c^{7} d^{7} x^{3} + 3 \, a c^{6} d^{6} e x^{2} + 3 \, a^{2} c^{5} d^{5} e^{2} x + a^{3} c^{4} d^{4} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.17158, size = 189, normalized size = 1.55 \begin{align*} \frac{11 a^{3} e^{6} - 6 a^{2} c d^{2} e^{4} - 3 a c^{2} d^{4} e^{2} - 2 c^{3} d^{6} + x^{2} \left (18 a c^{2} d^{2} e^{4} - 18 c^{3} d^{4} e^{2}\right ) + x \left (27 a^{2} c d e^{5} - 18 a c^{2} d^{3} e^{3} - 9 c^{3} d^{5} e\right )}{6 a^{3} c^{4} d^{4} e^{3} + 18 a^{2} c^{5} d^{5} e^{2} x + 18 a c^{6} d^{6} e x^{2} + 6 c^{7} d^{7} x^{3}} + \frac{e^{3} \log{\left (a e + c d x \right )}}{c^{4} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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