Optimal. Leaf size=111 \[ -\frac {3 e \left (c d^2-a e^2\right )^2}{c^4 d^4 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^3}{2 c^4 d^4 (a e+c d x)^2}+\frac {3 e^2 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^4 d^4}+\frac {e^3 x}{c^3 d^3} \]
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Rubi [A] time = 0.09, antiderivative size = 111, normalized size of antiderivative = 1.00, 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} \[ -\frac {3 e \left (c d^2-a e^2\right )^2}{c^4 d^4 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^3}{2 c^4 d^4 (a e+c d x)^2}+\frac {3 e^2 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^4 d^4}+\frac {e^3 x}{c^3 d^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {(d+e x)^6}{\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)^3}{(a e+c d x)^3} \, dx\\ &=\int \left (\frac {e^3}{c^3 d^3}+\frac {\left (c d^2-a e^2\right )^3}{c^3 d^3 (a e+c d x)^3}+\frac {3 e \left (c d^2-a e^2\right )^2}{c^3 d^3 (a e+c d x)^2}+\frac {3 \left (c d^2 e^2-a e^4\right )}{c^3 d^3 (a e+c d x)}\right ) \, dx\\ &=\frac {e^3 x}{c^3 d^3}-\frac {\left (c d^2-a e^2\right )^3}{2 c^4 d^4 (a e+c d x)^2}-\frac {3 e \left (c d^2-a e^2\right )^2}{c^4 d^4 (a e+c d x)}+\frac {3 e^2 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^4 d^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 139, normalized size = 1.25 \[ \frac {-5 a^3 e^6+a^2 c d e^4 (9 d-4 e x)+a c^2 d^2 e^2 \left (-3 d^2+12 d e x+4 e^2 x^2\right )-6 e^2 \left (a e^2-c d^2\right ) (a e+c d x)^2 \log (a e+c d x)-c^3 \left (d^6+6 d^5 e x-2 d^3 e^3 x^3\right )}{2 c^4 d^4 (a e+c d x)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.22, size = 227, normalized size = 2.05 \[ \frac {2 \, c^{3} d^{3} e^{3} x^{3} + 4 \, a c^{2} d^{2} e^{4} x^{2} - c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6} - 2 \, {\left (3 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 2 \, a^{2} c d e^{5}\right )} x + 6 \, {\left (a^{2} c d^{2} e^{4} - a^{3} e^{6} + {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )} \log \left (c d x + a e\right )}{2 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 699, normalized size = 6.30 \[ \frac {3 \, {\left (c^{6} d^{12} e^{2} - 6 \, a c^{5} d^{10} e^{4} + 15 \, a^{2} c^{4} d^{8} e^{6} - 20 \, a^{3} c^{3} d^{6} e^{8} + 15 \, a^{4} c^{2} d^{4} e^{10} - 6 \, a^{5} c d^{2} e^{12} + a^{6} e^{14}\right )} \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{8} d^{12} - 4 \, a c^{7} d^{10} e^{2} + 6 \, a^{2} c^{6} d^{8} e^{4} - 4 \, a^{3} c^{5} d^{6} e^{6} + a^{4} c^{4} d^{4} e^{8}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac {x e^{3}}{c^{3} d^{3}} + \frac {3 \, {\left (c d^{2} e^{2} - a e^{4}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{4} d^{4}} - \frac {c^{7} d^{16} - a c^{6} d^{14} e^{2} - 15 \, a^{2} c^{5} d^{12} e^{4} + 55 \, a^{3} c^{4} d^{10} e^{6} - 85 \, a^{4} c^{3} d^{8} e^{8} + 69 \, a^{5} c^{2} d^{6} e^{10} - 29 \, a^{6} c d^{4} e^{12} + 5 \, a^{7} d^{2} e^{14} + 6 \, {\left (c^{7} d^{13} e^{3} - 6 \, a c^{6} d^{11} e^{5} + 15 \, a^{2} c^{5} d^{9} e^{7} - 20 \, a^{3} c^{4} d^{7} e^{9} + 15 \, a^{4} c^{3} d^{5} e^{11} - 6 \, a^{5} c^{2} d^{3} e^{13} + a^{6} c d e^{15}\right )} x^{3} + {\left (13 \, c^{7} d^{14} e^{2} - 73 \, a c^{6} d^{12} e^{4} + 165 \, a^{2} c^{5} d^{10} e^{6} - 185 \, a^{3} c^{4} d^{8} e^{8} + 95 \, a^{4} c^{3} d^{6} e^{10} - 3 \, a^{5} c^{2} d^{4} e^{12} - 17 \, a^{6} c d^{2} e^{14} + 5 \, a^{7} e^{16}\right )} x^{2} + 2 \, {\left (4 \, c^{7} d^{15} e - 19 \, a c^{6} d^{13} e^{3} + 30 \, a^{2} c^{5} d^{11} e^{5} - 5 \, a^{3} c^{4} d^{9} e^{7} - 40 \, a^{4} c^{3} d^{7} e^{9} + 51 \, a^{5} c^{2} d^{5} e^{11} - 26 \, a^{6} c d^{3} e^{13} + 5 \, a^{7} d e^{15}\right )} x}{2 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}^{2} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{2} c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 201, normalized size = 1.81 \[ \frac {a^{3} e^{6}}{2 \left (c d x +a e \right )^{2} c^{4} d^{4}}-\frac {3 a^{2} e^{4}}{2 \left (c d x +a e \right )^{2} c^{3} d^{2}}+\frac {3 a \,e^{2}}{2 \left (c d x +a e \right )^{2} c^{2}}-\frac {d^{2}}{2 \left (c d x +a e \right )^{2} c}-\frac {3 a^{2} e^{5}}{\left (c d x +a e \right ) c^{4} d^{4}}+\frac {6 a \,e^{3}}{\left (c d x +a e \right ) c^{3} d^{2}}-\frac {3 e}{\left (c d x +a e \right ) c^{2}}-\frac {3 a \,e^{4} \ln \left (c d x +a e \right )}{c^{4} d^{4}}+\frac {3 e^{2} \ln \left (c d x +a e \right )}{c^{3} d^{2}}+\frac {e^{3} x}{c^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 156, normalized size = 1.41 \[ -\frac {c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} - 9 \, a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}} + \frac {e^{3} x}{c^{3} d^{3}} + \frac {3 \, {\left (c d^{2} e^{2} - a e^{4}\right )} \log \left (c d x + a e\right )}{c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 163, normalized size = 1.47 \[ \frac {e^3\,x}{c^3\,d^3}-\frac {x\,\left (3\,a^2\,e^5-6\,a\,c\,d^2\,e^3+3\,c^2\,d^4\,e\right )+\frac {5\,a^3\,e^6-9\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6}{2\,c\,d}}{a^2\,c^3\,d^3\,e^2+2\,a\,c^4\,d^4\,e\,x+c^5\,d^5\,x^2}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (3\,a\,e^4-3\,c\,d^2\,e^2\right )}{c^4\,d^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.04, size = 163, normalized size = 1.47 \[ \frac {- 5 a^{3} e^{6} + 9 a^{2} c d^{2} e^{4} - 3 a c^{2} d^{4} e^{2} - c^{3} d^{6} + x \left (- 6 a^{2} c d e^{5} + 12 a c^{2} d^{3} e^{3} - 6 c^{3} d^{5} e\right )}{2 a^{2} c^{4} d^{4} e^{2} + 4 a c^{5} d^{5} e x + 2 c^{6} d^{6} x^{2}} + \frac {e^{3} x}{c^{3} d^{3}} - \frac {3 e^{2} \left (a e^{2} - c d^{2}\right ) \log {\left (a e + c d x \right )}}{c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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