Optimal. Leaf size=122 \[ \frac {e^3 \log (a e+c d x)}{c^4 d^4}-\frac {3 e^2 \left (c d^2-a e^2\right )}{c^4 d^4 (a e+c d x)}-\frac {3 e \left (c d^2-a e^2\right )^2}{2 c^4 d^4 (a e+c d x)^2}-\frac {\left (c d^2-a e^2\right )^3}{3 c^4 d^4 (a e+c d x)^3} \]
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Rubi [A] time = 0.09, antiderivative size = 122, 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} \[ -\frac {3 e^2 \left (c d^2-a e^2\right )}{c^4 d^4 (a e+c d x)}-\frac {3 e \left (c d^2-a e^2\right )^2}{2 c^4 d^4 (a e+c d x)^2}-\frac {\left (c d^2-a e^2\right )^3}{3 c^4 d^4 (a e+c d x)^3}+\frac {e^3 \log (a e+c d x)}{c^4 d^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 626
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*}
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Mathematica [A] time = 0.05, 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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fricas [A] time = 1.13, size = 217, normalized size = 1.78 \[ -\frac {2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 18 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 9 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x - 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{4} x^{2} + 3 \, a^{2} c d e^{5} x + a^{3} e^{6}\right )} \log \left (c d x + a e\right )}{6 \, {\left (c^{7} d^{7} x^{3} + 3 \, a c^{6} d^{6} e x^{2} + 3 \, a^{2} c^{5} d^{5} e^{2} x + a^{3} c^{4} d^{4} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 210, normalized size = 1.72 \[ \frac {a^{3} e^{6}}{3 \left (c d x +a e \right )^{3} c^{4} d^{4}}-\frac {a^{2} e^{4}}{\left (c d x +a e \right )^{3} c^{3} d^{2}}+\frac {a \,e^{2}}{\left (c d x +a e \right )^{3} c^{2}}-\frac {d^{2}}{3 \left (c d x +a e \right )^{3} c}-\frac {3 a^{2} e^{5}}{2 \left (c d x +a e \right )^{2} c^{4} d^{4}}+\frac {3 a \,e^{3}}{\left (c d x +a e \right )^{2} c^{3} d^{2}}-\frac {3 e}{2 \left (c d x +a e \right )^{2} c^{2}}+\frac {3 a \,e^{4}}{\left (c d x +a e \right ) c^{4} d^{4}}-\frac {3 e^{2}}{\left (c d x +a e \right ) c^{3} d^{2}}+\frac {e^{3} \ln \left (c d x +a e \right )}{c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 179, normalized size = 1.47 \[ -\frac {2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 18 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 9 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x}{6 \, {\left (c^{7} d^{7} x^{3} + 3 \, a c^{6} d^{6} e x^{2} + 3 \, a^{2} c^{5} d^{5} e^{2} x + a^{3} c^{4} d^{4} e^{3}\right )}} + \frac {e^{3} \log \left (c d x + a e\right )}{c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 178, normalized size = 1.46 \[ \frac {e^3\,\ln \left (a\,e+c\,d\,x\right )}{c^4\,d^4}-\frac {\frac {-11\,a^3\,e^6+6\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+2\,c^3\,d^6}{6\,c^4\,d^4}+\frac {3\,x\,\left (-3\,a^2\,e^5+2\,a\,c\,d^2\,e^3+c^2\,d^4\,e\right )}{2\,c^3\,d^3}-\frac {3\,e^2\,x^2\,\left (a\,e^2-c\,d^2\right )}{c^2\,d^2}}{a^3\,e^3+3\,a^2\,c\,d\,e^2\,x+3\,a\,c^2\,d^2\,e\,x^2+c^3\,d^3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.63, size = 189, normalized size = 1.55 \[ \frac {11 a^{3} e^{6} - 6 a^{2} c d^{2} e^{4} - 3 a c^{2} d^{4} e^{2} - 2 c^{3} d^{6} + x^{2} \left (18 a c^{2} d^{2} e^{4} - 18 c^{3} d^{4} e^{2}\right ) + x \left (27 a^{2} c d e^{5} - 18 a c^{2} d^{3} e^{3} - 9 c^{3} d^{5} e\right )}{6 a^{3} c^{4} d^{4} e^{3} + 18 a^{2} c^{5} d^{5} e^{2} x + 18 a c^{6} d^{6} e x^{2} + 6 c^{7} d^{7} x^{3}} + \frac {e^{3} \log {\left (a e + c d x \right )}}{c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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