Optimal. Leaf size=219 \[ \frac{e^6 (a e+c d x)^5}{5 c^7 d^7}+\frac{3 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^4}{2 c^7 d^7}+\frac{5 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^7 d^7}+\frac{10 e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^2}{c^7 d^7}+\frac{15 e^2 x \left (c d^2-a e^2\right )^4}{c^6 d^6}-\frac{\left (c d^2-a e^2\right )^6}{c^7 d^7 (a e+c d x)}+\frac{6 e \left (c d^2-a e^2\right )^5 \log (a e+c d x)}{c^7 d^7} \]
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Rubi [A] time = 0.27523, antiderivative size = 219, 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} \[ \frac{e^6 (a e+c d x)^5}{5 c^7 d^7}+\frac{3 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^4}{2 c^7 d^7}+\frac{5 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^7 d^7}+\frac{10 e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^2}{c^7 d^7}+\frac{15 e^2 x \left (c d^2-a e^2\right )^4}{c^6 d^6}-\frac{\left (c d^2-a e^2\right )^6}{c^7 d^7 (a e+c d x)}+\frac{6 e \left (c d^2-a e^2\right )^5 \log (a e+c d x)}{c^7 d^7} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^8}{\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)^6}{(a e+c d x)^2} \, dx\\ &=\int \left (\frac{15 e^2 \left (c d^2-a e^2\right )^4}{c^6 d^6}+\frac{\left (c d^2-a e^2\right )^6}{c^6 d^6 (a e+c d x)^2}+\frac{6 e \left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac{20 \left (c d^2 e-a e^3\right )^3 (a e+c d x)}{c^6 d^6}+\frac{15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac{6 \left (c d^2 e^5-a e^7\right ) (a e+c d x)^3}{c^6 d^6}+\frac{e^6 (a e+c d x)^4}{c^6 d^6}\right ) \, dx\\ &=\frac{15 e^2 \left (c d^2-a e^2\right )^4 x}{c^6 d^6}-\frac{\left (c d^2-a e^2\right )^6}{c^7 d^7 (a e+c d x)}+\frac{10 e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^2}{c^7 d^7}+\frac{5 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^7 d^7}+\frac{3 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^4}{2 c^7 d^7}+\frac{e^6 (a e+c d x)^5}{5 c^7 d^7}+\frac{6 e \left (c d^2-a e^2\right )^5 \log (a e+c d x)}{c^7 d^7}\\ \end{align*}
Mathematica [A] time = 0.124329, size = 339, normalized size = 1.55 \[ \frac{-30 a^4 c^2 d^2 e^8 \left (5 d^2+8 d e x-e^2 x^2\right )+10 a^3 c^3 d^3 e^6 \left (45 d^2 e x+20 d^3-15 d e^2 x^2-e^3 x^3\right )-5 a^2 c^4 d^4 e^4 \left (-60 d^2 e^2 x^2+80 d^3 e x+30 d^4-10 d e^3 x^3-e^4 x^4\right )+10 a^5 c d e^{10} (6 d+5 e x)-10 a^6 e^{12}+a c^5 d^5 e^2 \left (-300 d^3 e^2 x^2-100 d^2 e^3 x^3+150 d^4 e x+60 d^5-25 d e^4 x^4-3 e^5 x^5\right )-60 e \left (a e^2-c d^2\right )^5 (a e+c d x) \log (a e+c d x)+c^6 d^6 \left (150 d^4 e^2 x^2+100 d^3 e^3 x^3+50 d^2 e^4 x^4-10 d^6+15 d e^5 x^5+2 e^6 x^6\right )}{10 c^7 d^7 (a e+c d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 502, normalized size = 2.3 \begin{align*} -{\frac{{a}^{6}{e}^{12}}{{c}^{7}{d}^{7} \left ( cdx+ae \right ) }}+6\,{\frac{{a}^{5}{e}^{10}}{{c}^{6}{d}^{5} \left ( cdx+ae \right ) }}-4\,{\frac{{e}^{6}{x}^{3}a}{{c}^{3}{d}^{2}}}-15\,{\frac{{a}^{4}{e}^{8}}{{c}^{5}{d}^{3} \left ( cdx+ae \right ) }}-2\,{\frac{{e}^{9}{x}^{2}{a}^{3}}{{c}^{5}{d}^{5}}}-40\,{\frac{a{e}^{4}x}{{c}^{3}}}+{\frac{{e}^{6}{x}^{5}}{5\,{c}^{2}{d}^{2}}}+{\frac{3\,{e}^{5}{x}^{4}}{2\,{c}^{2}d}}+10\,{\frac{{e}^{3}d{x}^{2}}{{c}^{2}}}+{\frac{{e}^{8}{x}^{3}{a}^{2}}{{c}^{4}{d}^{4}}}-{\frac{{e}^{7}{x}^{4}a}{2\,{c}^{3}{d}^{3}}}+15\,{\frac{{d}^{2}{e}^{2}x}{{c}^{2}}}+6\,{\frac{{d}^{3}e\ln \left ( cdx+ae \right ) }{{c}^{2}}}+9\,{\frac{{e}^{7}{x}^{2}{a}^{2}}{{c}^{4}{d}^{3}}}-15\,{\frac{{e}^{5}{x}^{2}a}{{c}^{3}d}}+5\,{\frac{{a}^{4}{e}^{10}x}{{c}^{6}{d}^{6}}}-24\,{\frac{{a}^{3}{e}^{8}x}{{c}^{5}{d}^{4}}}+20\,{\frac{{a}^{3}{e}^{6}}{{c}^{4}d \left ( cdx+ae \right ) }}-15\,{\frac{{a}^{2}d{e}^{4}}{{c}^{3} \left ( cdx+ae \right ) }}-60\,{\frac{{e}^{7}\ln \left ( cdx+ae \right ){a}^{3}}{{c}^{5}{d}^{3}}}+60\,{\frac{{e}^{5}\ln \left ( cdx+ae \right ){a}^{2}}{{c}^{4}d}}-30\,{\frac{{e}^{3}d\ln \left ( cdx+ae \right ) a}{{c}^{3}}}+5\,{\frac{{e}^{4}{x}^{3}}{{c}^{2}}}-{\frac{{d}^{5}}{c \left ( cdx+ae \right ) }}+45\,{\frac{{a}^{2}{e}^{6}x}{{c}^{4}{d}^{2}}}+6\,{\frac{{d}^{3}{e}^{2}a}{{c}^{2} \left ( cdx+ae \right ) }}-6\,{\frac{{e}^{11}\ln \left ( cdx+ae \right ){a}^{5}}{{c}^{7}{d}^{7}}}+30\,{\frac{{e}^{9}\ln \left ( cdx+ae \right ){a}^{4}}{{c}^{6}{d}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0221, size = 537, normalized size = 2.45 \begin{align*} -\frac{c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{c^{8} d^{8} x + a c^{7} d^{7} e} + \frac{2 \, c^{4} d^{4} e^{6} x^{5} + 5 \,{\left (3 \, c^{4} d^{5} e^{5} - a c^{3} d^{3} e^{7}\right )} x^{4} + 10 \,{\left (5 \, c^{4} d^{6} e^{4} - 4 \, a c^{3} d^{4} e^{6} + a^{2} c^{2} d^{2} e^{8}\right )} x^{3} + 10 \,{\left (10 \, c^{4} d^{7} e^{3} - 15 \, a c^{3} d^{5} e^{5} + 9 \, a^{2} c^{2} d^{3} e^{7} - 2 \, a^{3} c d e^{9}\right )} x^{2} + 10 \,{\left (15 \, c^{4} d^{8} e^{2} - 40 \, a c^{3} d^{6} e^{4} + 45 \, a^{2} c^{2} d^{4} e^{6} - 24 \, a^{3} c d^{2} e^{8} + 5 \, a^{4} e^{10}\right )} x}{10 \, c^{6} d^{6}} + \frac{6 \,{\left (c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}\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 = 1.80114, size = 1112, normalized size = 5.08 \begin{align*} \frac{2 \, c^{6} d^{6} e^{6} x^{6} - 10 \, c^{6} d^{12} + 60 \, a c^{5} d^{10} e^{2} - 150 \, a^{2} c^{4} d^{8} e^{4} + 200 \, a^{3} c^{3} d^{6} e^{6} - 150 \, a^{4} c^{2} d^{4} e^{8} + 60 \, a^{5} c d^{2} e^{10} - 10 \, a^{6} e^{12} + 3 \,{\left (5 \, c^{6} d^{7} e^{5} - a c^{5} d^{5} e^{7}\right )} x^{5} + 5 \,{\left (10 \, c^{6} d^{8} e^{4} - 5 \, a c^{5} d^{6} e^{6} + a^{2} c^{4} d^{4} e^{8}\right )} x^{4} + 10 \,{\left (10 \, c^{6} d^{9} e^{3} - 10 \, a c^{5} d^{7} e^{5} + 5 \, a^{2} c^{4} d^{5} e^{7} - a^{3} c^{3} d^{3} e^{9}\right )} x^{3} + 30 \,{\left (5 \, c^{6} d^{10} e^{2} - 10 \, a c^{5} d^{8} e^{4} + 10 \, a^{2} c^{4} d^{6} e^{6} - 5 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 10 \,{\left (15 \, a c^{5} d^{9} e^{3} - 40 \, a^{2} c^{4} d^{7} e^{5} + 45 \, a^{3} c^{3} d^{5} e^{7} - 24 \, a^{4} c^{2} d^{3} e^{9} + 5 \, a^{5} c d e^{11}\right )} x + 60 \,{\left (a c^{5} d^{10} e^{2} - 5 \, a^{2} c^{4} d^{8} e^{4} + 10 \, a^{3} c^{3} d^{6} e^{6} - 10 \, a^{4} c^{2} d^{4} e^{8} + 5 \, a^{5} c d^{2} e^{10} - a^{6} e^{12} +{\left (c^{6} d^{11} e - 5 \, a c^{5} d^{9} e^{3} + 10 \, a^{2} c^{4} d^{7} e^{5} - 10 \, a^{3} c^{3} d^{5} e^{7} + 5 \, 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 )}{10 \,{\left (c^{8} d^{8} x + a c^{7} d^{7} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.33021, size = 348, normalized size = 1.59 \begin{align*} - \frac{a^{6} e^{12} - 6 a^{5} c d^{2} e^{10} + 15 a^{4} c^{2} d^{4} e^{8} - 20 a^{3} c^{3} d^{6} e^{6} + 15 a^{2} c^{4} d^{8} e^{4} - 6 a c^{5} d^{10} e^{2} + c^{6} d^{12}}{a c^{7} d^{7} e + c^{8} d^{8} x} + \frac{e^{6} x^{5}}{5 c^{2} d^{2}} - \frac{x^{4} \left (a e^{7} - 3 c d^{2} e^{5}\right )}{2 c^{3} d^{3}} + \frac{x^{3} \left (a^{2} e^{8} - 4 a c d^{2} e^{6} + 5 c^{2} d^{4} e^{4}\right )}{c^{4} d^{4}} - \frac{x^{2} \left (2 a^{3} e^{9} - 9 a^{2} c d^{2} e^{7} + 15 a c^{2} d^{4} e^{5} - 10 c^{3} d^{6} e^{3}\right )}{c^{5} d^{5}} + \frac{x \left (5 a^{4} e^{10} - 24 a^{3} c d^{2} e^{8} + 45 a^{2} c^{2} d^{4} e^{6} - 40 a c^{3} d^{6} e^{4} + 15 c^{4} d^{8} e^{2}\right )}{c^{6} d^{6}} - \frac{6 e \left (a e^{2} - c d^{2}\right )^{5} \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 [B] time = 1.54775, size = 1075, normalized size = 4.91 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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