Optimal. Leaf size=217 \[ \frac{e^4 x \left (10 a^2 e^4-24 a c d^2 e^2+15 c^2 d^4\right )}{c^6 d^6}+\frac{e^5 x^2 \left (3 c d^2-2 a e^2\right )}{c^5 d^5}-\frac{15 e^2 \left (c d^2-a e^2\right )^4}{c^7 d^7 (a e+c d x)}-\frac{3 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^6}{3 c^7 d^7 (a e+c d x)^3}+\frac{20 e^3 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^7 d^7}+\frac{e^6 x^3}{3 c^4 d^4} \]
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Rubi [A] time = 0.271207, antiderivative size = 217, 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} \[ \frac{e^4 x \left (10 a^2 e^4-24 a c d^2 e^2+15 c^2 d^4\right )}{c^6 d^6}+\frac{e^5 x^2 \left (3 c d^2-2 a e^2\right )}{c^5 d^5}-\frac{15 e^2 \left (c d^2-a e^2\right )^4}{c^7 d^7 (a e+c d x)}-\frac{3 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^6}{3 c^7 d^7 (a e+c d x)^3}+\frac{20 e^3 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^7 d^7}+\frac{e^6 x^3}{3 c^4 d^4} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^{10}}{\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)^6}{(a e+c d x)^4} \, dx\\ &=\int \left (\frac{15 c^2 d^4 e^4-24 a c d^2 e^6+10 a^2 e^8}{c^6 d^6}+\frac{2 e^5 \left (3 c d^2-2 a e^2\right ) x}{c^5 d^5}+\frac{e^6 x^2}{c^4 d^4}+\frac{\left (c d^2-a e^2\right )^6}{c^6 d^6 (a e+c d x)^4}+\frac{6 e \left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)^3}+\frac{15 e^2 \left (c d^2-a e^2\right )^4}{c^6 d^6 (a e+c d x)^2}+\frac{20 \left (c d^2 e-a e^3\right )^3}{c^6 d^6 (a e+c d x)}\right ) \, dx\\ &=\frac{e^4 \left (15 c^2 d^4-24 a c d^2 e^2+10 a^2 e^4\right ) x}{c^6 d^6}+\frac{e^5 \left (3 c d^2-2 a e^2\right ) x^2}{c^5 d^5}+\frac{e^6 x^3}{3 c^4 d^4}-\frac{\left (c d^2-a e^2\right )^6}{3 c^7 d^7 (a e+c d x)^3}-\frac{3 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)^2}-\frac{15 e^2 \left (c d^2-a e^2\right )^4}{c^7 d^7 (a e+c d x)}+\frac{20 e^3 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^7 d^7}\\ \end{align*}
Mathematica [A] time = 0.141924, size = 335, normalized size = 1.54 \[ \frac{3 a^4 c^2 d^2 e^8 \left (-65 d^2+81 d e x+13 e^2 x^2\right )+a^3 c^3 d^3 e^6 \left (-405 d^2 e x+110 d^3-27 d e^2 x^2+73 e^3 x^3\right )-3 a^2 c^4 d^4 e^4 \left (45 d^2 e^2 x^2-90 d^3 e x+5 d^4+63 d e^3 x^3-5 e^4 x^4\right )+3 a^5 c d e^{10} (47 d-17 e x)-37 a^6 e^{12}-3 a c^5 d^5 e^2 \left (-60 d^3 e^2 x^2-45 d^2 e^3 x^3+15 d^4 e x+d^5+15 d e^4 x^4+e^5 x^5\right )-60 e^3 \left (a e^2-c d^2\right )^3 (a e+c d x)^3 \log (a e+c d x)+c^6 d^6 \left (-45 d^4 e^2 x^2+45 d^2 e^4 x^4-9 d^5 e x-d^6+9 d e^5 x^5+e^6 x^6\right )}{3 c^7 d^7 (a e+c d x)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.054, size = 578, normalized size = 2.7 \begin{align*} -5\,{\frac{{a}^{4}{e}^{8}}{{c}^{5}{d}^{3} \left ( cdx+ae \right ) ^{3}}}+{\frac{20\,{a}^{3}{e}^{6}}{3\,{c}^{4}d \left ( cdx+ae \right ) ^{3}}}-5\,{\frac{{a}^{2}d{e}^{4}}{{c}^{3} \left ( cdx+ae \right ) ^{3}}}+2\,{\frac{{d}^{3}a{e}^{2}}{{c}^{2} \left ( cdx+ae \right ) ^{3}}}+3\,{\frac{{a}^{5}{e}^{11}}{{c}^{7}{d}^{7} \left ( cdx+ae \right ) ^{2}}}-15\,{\frac{{e}^{9}{a}^{4}}{{d}^{5}{c}^{6} \left ( cdx+ae \right ) ^{2}}}+30\,{\frac{{e}^{7}{a}^{3}}{{c}^{5}{d}^{3} \left ( cdx+ae \right ) ^{2}}}-30\,{\frac{{a}^{2}{e}^{5}}{{c}^{4}d \left ( cdx+ae \right ) ^{2}}}+15\,{\frac{ad{e}^{3}}{{c}^{3} \left ( cdx+ae \right ) ^{2}}}-15\,{\frac{{e}^{10}{a}^{4}}{{c}^{7}{d}^{7} \left ( cdx+ae \right ) }}+60\,{\frac{{e}^{8}{a}^{3}}{{d}^{5}{c}^{6} \left ( cdx+ae \right ) }}-90\,{\frac{{e}^{6}{a}^{2}}{{c}^{5}{d}^{3} \left ( cdx+ae \right ) }}+60\,{\frac{{e}^{4}a}{{c}^{4}d \left ( cdx+ae \right ) }}-20\,{\frac{{e}^{9}\ln \left ( cdx+ae \right ){a}^{3}}{{c}^{7}{d}^{7}}}+60\,{\frac{{e}^{7}\ln \left ( cdx+ae \right ){a}^{2}}{{d}^{5}{c}^{6}}}-60\,{\frac{{e}^{5}\ln \left ( cdx+ae \right ) a}{{c}^{5}{d}^{3}}}+10\,{\frac{{a}^{2}{e}^{8}x}{{c}^{6}{d}^{6}}}-2\,{\frac{{e}^{7}{x}^{2}a}{{c}^{5}{d}^{5}}}-24\,{\frac{a{e}^{6}x}{{c}^{5}{d}^{4}}}+3\,{\frac{{e}^{5}{x}^{2}}{{c}^{4}{d}^{3}}}+15\,{\frac{{e}^{4}x}{{c}^{4}{d}^{2}}}+20\,{\frac{{e}^{3}\ln \left ( cdx+ae \right ) }{{c}^{4}d}}-3\,{\frac{{d}^{3}e}{{c}^{2} \left ( cdx+ae \right ) ^{2}}}-15\,{\frac{d{e}^{2}}{{c}^{3} \left ( cdx+ae \right ) }}-{\frac{{d}^{5}}{3\,c \left ( cdx+ae \right ) ^{3}}}-{\frac{{a}^{6}{e}^{12}}{3\,{c}^{7}{d}^{7} \left ( cdx+ae \right ) ^{3}}}+2\,{\frac{{a}^{5}{e}^{10}}{{d}^{5}{c}^{6} \left ( cdx+ae \right ) ^{3}}}+{\frac{{e}^{6}{x}^{3}}{3\,{c}^{4}{d}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0903, size = 572, normalized size = 2.64 \begin{align*} -\frac{c^{6} d^{12} + 3 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 110 \, a^{3} c^{3} d^{6} e^{6} + 195 \, a^{4} c^{2} d^{4} e^{8} - 141 \, a^{5} c d^{2} e^{10} + 37 \, a^{6} e^{12} + 45 \,{\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 9 \,{\left (c^{6} d^{11} e + 5 \, a c^{5} d^{9} e^{3} - 30 \, a^{2} c^{4} d^{7} e^{5} + 50 \, a^{3} c^{3} d^{5} e^{7} - 35 \, a^{4} c^{2} d^{3} e^{9} + 9 \, a^{5} c d e^{11}\right )} x}{3 \,{\left (c^{10} d^{10} x^{3} + 3 \, a c^{9} d^{9} e x^{2} + 3 \, a^{2} c^{8} d^{8} e^{2} x + a^{3} c^{7} d^{7} e^{3}\right )}} + \frac{c^{2} d^{2} e^{6} x^{3} + 3 \,{\left (3 \, c^{2} d^{3} e^{5} - 2 \, a c d e^{7}\right )} x^{2} + 3 \,{\left (15 \, c^{2} d^{4} e^{4} - 24 \, a c d^{2} e^{6} + 10 \, a^{2} e^{8}\right )} x}{3 \, c^{6} d^{6}} + \frac{20 \,{\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} \log \left (c d x + a e\right )}{c^{7} d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.95182, size = 1292, normalized size = 5.95 \begin{align*} \frac{c^{6} d^{6} e^{6} x^{6} - c^{6} d^{12} - 3 \, a c^{5} d^{10} e^{2} - 15 \, a^{2} c^{4} d^{8} e^{4} + 110 \, a^{3} c^{3} d^{6} e^{6} - 195 \, a^{4} c^{2} d^{4} e^{8} + 141 \, a^{5} c d^{2} e^{10} - 37 \, a^{6} e^{12} + 3 \,{\left (3 \, c^{6} d^{7} e^{5} - a c^{5} d^{5} e^{7}\right )} x^{5} + 15 \,{\left (3 \, c^{6} d^{8} e^{4} - 3 \, a c^{5} d^{6} e^{6} + a^{2} c^{4} d^{4} e^{8}\right )} x^{4} +{\left (135 \, a c^{5} d^{7} e^{5} - 189 \, a^{2} c^{4} d^{5} e^{7} + 73 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{3} - 3 \,{\left (15 \, c^{6} d^{10} e^{2} - 60 \, a c^{5} d^{8} e^{4} + 45 \, a^{2} c^{4} d^{6} e^{6} + 9 \, a^{3} c^{3} d^{4} e^{8} - 13 \, a^{4} c^{2} d^{2} e^{10}\right )} x^{2} - 3 \,{\left (3 \, c^{6} d^{11} e + 15 \, a c^{5} d^{9} e^{3} - 90 \, a^{2} c^{4} d^{7} e^{5} + 135 \, a^{3} c^{3} d^{5} e^{7} - 81 \, a^{4} c^{2} d^{3} e^{9} + 17 \, a^{5} c d e^{11}\right )} x + 60 \,{\left (a^{3} c^{3} d^{6} e^{6} - 3 \, a^{4} c^{2} d^{4} e^{8} + 3 \, a^{5} c d^{2} e^{10} - a^{6} e^{12} +{\left (c^{6} d^{9} e^{3} - 3 \, a c^{5} d^{7} e^{5} + 3 \, a^{2} c^{4} d^{5} e^{7} - a^{3} c^{3} d^{3} e^{9}\right )} x^{3} + 3 \,{\left (a c^{5} d^{8} e^{4} - 3 \, a^{2} c^{4} d^{6} e^{6} + 3 \, a^{3} c^{3} d^{4} e^{8} - a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 3 \,{\left (a^{2} c^{4} d^{7} e^{5} - 3 \, a^{3} c^{3} d^{5} e^{7} + 3 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} x\right )} \log \left (c d x + a e\right )}{3 \,{\left (c^{10} d^{10} x^{3} + 3 \, a c^{9} d^{9} e x^{2} + 3 \, a^{2} c^{8} d^{8} e^{2} x + a^{3} c^{7} d^{7} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 134.566, size = 422, normalized size = 1.94 \begin{align*} - \frac{37 a^{6} e^{12} - 141 a^{5} c d^{2} e^{10} + 195 a^{4} c^{2} d^{4} e^{8} - 110 a^{3} c^{3} d^{6} e^{6} + 15 a^{2} c^{4} d^{8} e^{4} + 3 a c^{5} d^{10} e^{2} + c^{6} d^{12} + x^{2} \left (45 a^{4} c^{2} d^{2} e^{10} - 180 a^{3} c^{3} d^{4} e^{8} + 270 a^{2} c^{4} d^{6} e^{6} - 180 a c^{5} d^{8} e^{4} + 45 c^{6} d^{10} e^{2}\right ) + x \left (81 a^{5} c d e^{11} - 315 a^{4} c^{2} d^{3} e^{9} + 450 a^{3} c^{3} d^{5} e^{7} - 270 a^{2} c^{4} d^{7} e^{5} + 45 a c^{5} d^{9} e^{3} + 9 c^{6} d^{11} e\right )}{3 a^{3} c^{7} d^{7} e^{3} + 9 a^{2} c^{8} d^{8} e^{2} x + 9 a c^{9} d^{9} e x^{2} + 3 c^{10} d^{10} x^{3}} + \frac{e^{6} x^{3}}{3 c^{4} d^{4}} - \frac{x^{2} \left (2 a e^{7} - 3 c d^{2} e^{5}\right )}{c^{5} d^{5}} + \frac{x \left (10 a^{2} e^{8} - 24 a c d^{2} e^{6} + 15 c^{2} d^{4} e^{4}\right )}{c^{6} d^{6}} - \frac{20 e^{3} \left (a e^{2} - c d^{2}\right )^{3} \log{\left (a e + c d x \right )}}{c^{7} d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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