Optimal. Leaf size=142 \[ \frac{e^3 x \left (4 c d^2-3 a e^2\right )}{c^4 d^4}-\frac{4 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^4}{2 c^5 d^5 (a e+c d x)^2}+\frac{6 e^2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^5 d^5}+\frac{e^4 x^2}{2 c^3 d^3} \]
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Rubi [A] time = 0.135345, antiderivative size = 142, 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} \[ \frac{e^3 x \left (4 c d^2-3 a e^2\right )}{c^4 d^4}-\frac{4 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^4}{2 c^5 d^5 (a e+c d x)^2}+\frac{6 e^2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^5 d^5}+\frac{e^4 x^2}{2 c^3 d^3} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^7}{\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)^4}{(a e+c d x)^3} \, dx\\ &=\int \left (\frac{4 c d^2 e^3-3 a e^5}{c^4 d^4}+\frac{e^4 x}{c^3 d^3}+\frac{\left (c d^2-a e^2\right )^4}{c^4 d^4 (a e+c d x)^3}+\frac{4 e \left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)^2}+\frac{6 \left (c d^2 e-a e^3\right )^2}{c^4 d^4 (a e+c d x)}\right ) \, dx\\ &=\frac{e^3 \left (4 c d^2-3 a e^2\right ) x}{c^4 d^4}+\frac{e^4 x^2}{2 c^3 d^3}-\frac{\left (c d^2-a e^2\right )^4}{2 c^5 d^5 (a e+c d x)^2}-\frac{4 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}+\frac{6 e^2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^5 d^5}\\ \end{align*}
Mathematica [A] time = 0.0702236, size = 191, normalized size = 1.35 \[ \frac{a^2 c^2 d^2 e^4 \left (18 d^2-16 d e x-11 e^2 x^2\right )+2 a^3 c d e^6 (e x-10 d)+7 a^4 e^8-4 a c^3 d^3 e^2 \left (-6 d^2 e x+d^3-4 d e^2 x^2+e^3 x^3\right )+12 e^2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2 \log (a e+c d x)+c^4 d^4 \left (-8 d^3 e x-d^4+8 d e^3 x^3+e^4 x^4\right )}{2 c^5 d^5 (a e+c d x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 302, normalized size = 2.1 \begin{align*}{\frac{{e}^{4}{x}^{2}}{2\,{c}^{3}{d}^{3}}}-3\,{\frac{a{e}^{5}x}{{c}^{4}{d}^{4}}}+4\,{\frac{{e}^{3}x}{{c}^{3}{d}^{2}}}-{\frac{{a}^{4}{e}^{8}}{2\,{c}^{5}{d}^{5} \left ( cdx+ae \right ) ^{2}}}+2\,{\frac{{a}^{3}{e}^{6}}{{c}^{4}{d}^{3} \left ( cdx+ae \right ) ^{2}}}-3\,{\frac{{a}^{2}{e}^{4}}{{c}^{3}d \left ( cdx+ae \right ) ^{2}}}+2\,{\frac{ad{e}^{2}}{{c}^{2} \left ( cdx+ae \right ) ^{2}}}-{\frac{{d}^{3}}{2\,c \left ( cdx+ae \right ) ^{2}}}+4\,{\frac{{a}^{3}{e}^{7}}{{c}^{5}{d}^{5} \left ( cdx+ae \right ) }}-12\,{\frac{{a}^{2}{e}^{5}}{{c}^{4}{d}^{3} \left ( cdx+ae \right ) }}+12\,{\frac{a{e}^{3}}{{c}^{3}d \left ( cdx+ae \right ) }}-4\,{\frac{de}{{c}^{2} \left ( cdx+ae \right ) }}+6\,{\frac{{e}^{6}\ln \left ( cdx+ae \right ){a}^{2}}{{c}^{5}{d}^{5}}}-12\,{\frac{{e}^{4}\ln \left ( cdx+ae \right ) a}{{c}^{4}{d}^{3}}}+6\,{\frac{{e}^{2}\ln \left ( cdx+ae \right ) }{{c}^{3}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07075, size = 302, normalized size = 2.13 \begin{align*} -\frac{c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 7 \, a^{4} e^{8} + 8 \,{\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x}{2 \,{\left (c^{7} d^{7} x^{2} + 2 \, a c^{6} d^{6} e x + a^{2} c^{5} d^{5} e^{2}\right )}} + \frac{c d e^{4} x^{2} + 2 \,{\left (4 \, c d^{2} e^{3} - 3 \, a e^{5}\right )} x}{2 \, c^{4} d^{4}} + \frac{6 \,{\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.89035, size = 664, normalized size = 4.68 \begin{align*} \frac{c^{4} d^{4} e^{4} x^{4} - c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 18 \, a^{2} c^{2} d^{4} e^{4} - 20 \, a^{3} c d^{2} e^{6} + 7 \, a^{4} e^{8} + 4 \,{\left (2 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} +{\left (16 \, a c^{3} d^{4} e^{4} - 11 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 2 \,{\left (4 \, c^{4} d^{7} e - 12 \, a c^{3} d^{5} e^{3} + 8 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x + 12 \,{\left (a^{2} c^{2} d^{4} e^{4} - 2 \, a^{3} c d^{2} e^{6} + a^{4} e^{8} +{\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (a c^{3} d^{5} e^{3} - 2 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \log \left (c d x + a e\right )}{2 \,{\left (c^{7} d^{7} x^{2} + 2 \, a c^{6} d^{6} e x + a^{2} c^{5} d^{5} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.65236, size = 224, normalized size = 1.58 \begin{align*} \frac{7 a^{4} e^{8} - 20 a^{3} c d^{2} e^{6} + 18 a^{2} c^{2} d^{4} e^{4} - 4 a c^{3} d^{6} e^{2} - c^{4} d^{8} + x \left (8 a^{3} c d e^{7} - 24 a^{2} c^{2} d^{3} e^{5} + 24 a c^{3} d^{5} e^{3} - 8 c^{4} d^{7} e\right )}{2 a^{2} c^{5} d^{5} e^{2} + 4 a c^{6} d^{6} e x + 2 c^{7} d^{7} x^{2}} + \frac{e^{4} x^{2}}{2 c^{3} d^{3}} - \frac{x \left (3 a e^{5} - 4 c d^{2} e^{3}\right )}{c^{4} d^{4}} + \frac{6 e^{2} \left (a e^{2} - c d^{2}\right )^{2} \log{\left (a e + c d x \right )}}{c^{5} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 14.7952, size = 1084, normalized size = 7.63 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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