Optimal. Leaf size=184 \[ \frac {e^5 (a e+c d x)^4}{4 c^6 d^6}-\frac {\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac {5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6}+\frac {5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac {5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac {10 e^2 x \left (c d^2-a e^2\right )^3}{c^5 d^5} \]
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Rubi [A] time = 0.20, antiderivative size = 184, 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} \[ \frac {e^5 (a e+c d x)^4}{4 c^6 d^6}+\frac {5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac {5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac {10 e^2 x \left (c d^2-a e^2\right )^3}{c^5 d^5}-\frac {\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac {5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6} \]
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 )^2} \, dx &=\int \frac {(d+e x)^5}{(a e+c d x)^2} \, dx\\ &=\int \left (\frac {10 e^2 \left (c d^2-a e^2\right )^3}{c^5 d^5}+\frac {\left (c d^2-a e^2\right )^5}{c^5 d^5 (a e+c d x)^2}+\frac {5 e \left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac {10 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)}{c^5 d^5}+\frac {5 \left (c d^2 e^4-a e^6\right ) (a e+c d x)^2}{c^5 d^5}+\frac {e^5 (a e+c d x)^3}{c^5 d^5}\right ) \, dx\\ &=\frac {10 e^2 \left (c d^2-a e^2\right )^3 x}{c^5 d^5}-\frac {\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac {5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac {5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac {e^5 (a e+c d x)^4}{4 c^6 d^6}+\frac {5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 263, normalized size = 1.43 \[ \frac {12 a^5 e^{10}-12 a^4 c d e^8 (5 d+4 e x)+30 a^3 c^2 d^2 e^6 \left (4 d^2+6 d e x-e^2 x^2\right )-10 a^2 c^3 d^3 e^4 \left (12 d^3+24 d^2 e x-12 d e^2 x^2-e^3 x^3\right )+5 a c^4 d^4 e^2 \left (12 d^4+24 d^3 e x-36 d^2 e^2 x^2-8 d e^3 x^3-e^4 x^4\right )+60 e \left (c d^2-a e^2\right )^4 (a e+c d x) \log (a e+c d x)+c^5 d^5 \left (-12 d^5+120 d^3 e^2 x^2+60 d^2 e^3 x^3+20 d e^4 x^4+3 e^5 x^5\right )}{12 c^6 d^6 (a e+c d x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.38, size = 417, normalized size = 2.27 \[ \frac {3 \, c^{5} d^{5} e^{5} x^{5} - 12 \, c^{5} d^{10} + 60 \, a c^{4} d^{8} e^{2} - 120 \, a^{2} c^{3} d^{6} e^{4} + 120 \, a^{3} c^{2} d^{4} e^{6} - 60 \, a^{4} c d^{2} e^{8} + 12 \, a^{5} e^{10} + 5 \, {\left (4 \, c^{5} d^{6} e^{4} - a c^{4} d^{4} e^{6}\right )} x^{4} + 10 \, {\left (6 \, c^{5} d^{7} e^{3} - 4 \, a c^{4} d^{5} e^{5} + a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 30 \, {\left (4 \, c^{5} d^{8} e^{2} - 6 \, a c^{4} d^{6} e^{4} + 4 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 12 \, {\left (10 \, a c^{4} d^{7} e^{3} - 20 \, a^{2} c^{3} d^{5} e^{5} + 15 \, a^{3} c^{2} d^{3} e^{7} - 4 \, a^{4} c d e^{9}\right )} x + 60 \, {\left (a c^{4} d^{8} e^{2} - 4 \, a^{2} c^{3} d^{6} e^{4} + 6 \, a^{3} c^{2} d^{4} e^{6} - 4 \, a^{4} c d^{2} e^{8} + a^{5} e^{10} + {\left (c^{5} d^{9} e - 4 \, a c^{4} d^{7} e^{3} + 6 \, a^{2} c^{3} d^{5} e^{5} - 4 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x\right )} \log \left (c d x + a e\right )}{12 \, {\left (c^{7} d^{7} x + a c^{6} d^{6} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 673, normalized size = 3.66 \[ \frac {5 \, {\left (c^{7} d^{14} e - 7 \, a c^{6} d^{12} e^{3} + 21 \, a^{2} c^{5} d^{10} e^{5} - 35 \, a^{3} c^{4} d^{8} e^{7} + 35 \, a^{4} c^{3} d^{6} e^{9} - 21 \, a^{5} c^{2} d^{4} e^{11} + 7 \, a^{6} c d^{2} e^{13} - a^{7} e^{15}\right )} \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{8} d^{10} - 2 \, a c^{7} d^{8} e^{2} + a^{2} c^{6} d^{6} e^{4}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac {5 \, {\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{6} d^{6}} - \frac {c^{7} d^{15} - 7 \, a c^{6} d^{13} e^{2} + 21 \, a^{2} c^{5} d^{11} e^{4} - 35 \, a^{3} c^{4} d^{9} e^{6} + 35 \, a^{4} c^{3} d^{7} e^{8} - 21 \, a^{5} c^{2} d^{5} e^{10} + 7 \, a^{6} c d^{3} e^{12} - a^{7} d e^{14} + {\left (c^{7} d^{14} e - 7 \, a c^{6} d^{12} e^{3} + 21 \, a^{2} c^{5} d^{10} e^{5} - 35 \, a^{3} c^{4} d^{8} e^{7} + 35 \, a^{4} c^{3} d^{6} e^{9} - 21 \, a^{5} c^{2} d^{4} e^{11} + 7 \, a^{6} c d^{2} e^{13} - a^{7} e^{15}\right )} x}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )} c^{6} d^{6}} + \frac {{\left (3 \, c^{6} d^{6} x^{4} e^{13} + 20 \, c^{6} d^{7} x^{3} e^{12} + 60 \, c^{6} d^{8} x^{2} e^{11} + 120 \, c^{6} d^{9} x e^{10} - 8 \, a c^{5} d^{5} x^{3} e^{14} - 60 \, a c^{5} d^{6} x^{2} e^{13} - 240 \, a c^{5} d^{7} x e^{12} + 18 \, a^{2} c^{4} d^{4} x^{2} e^{15} + 180 \, a^{2} c^{4} d^{5} x e^{14} - 48 \, a^{3} c^{3} d^{3} x e^{16}\right )} e^{\left (-8\right )}}{12 \, c^{8} d^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 378, normalized size = 2.05 \[ \frac {e^{5} x^{4}}{4 c^{2} d^{2}}-\frac {2 a \,e^{6} x^{3}}{3 c^{3} d^{3}}+\frac {5 e^{4} x^{3}}{3 c^{2} d}+\frac {a^{5} e^{10}}{\left (c d x +a e \right ) c^{6} d^{6}}-\frac {5 a^{4} e^{8}}{\left (c d x +a e \right ) c^{5} d^{4}}+\frac {10 a^{3} e^{6}}{\left (c d x +a e \right ) c^{4} d^{2}}-\frac {10 a^{2} e^{4}}{\left (c d x +a e \right ) c^{3}}+\frac {3 a^{2} e^{7} x^{2}}{2 c^{4} d^{4}}+\frac {5 a \,d^{2} e^{2}}{\left (c d x +a e \right ) c^{2}}-\frac {5 a \,e^{5} x^{2}}{c^{3} d^{2}}-\frac {d^{4}}{\left (c d x +a e \right ) c}+\frac {5 e^{3} x^{2}}{c^{2}}+\frac {5 a^{4} e^{9} \ln \left (c d x +a e \right )}{c^{6} d^{6}}-\frac {20 a^{3} e^{7} \ln \left (c d x +a e \right )}{c^{5} d^{4}}-\frac {4 a^{3} e^{8} x}{c^{5} d^{5}}+\frac {30 a^{2} e^{5} \ln \left (c d x +a e \right )}{c^{4} d^{2}}+\frac {15 a^{2} e^{6} x}{c^{4} d^{3}}-\frac {20 a \,e^{4} x}{c^{3} d}-\frac {20 a \,e^{3} \ln \left (c d x +a e \right )}{c^{3}}+\frac {5 d^{2} e \ln \left (c d x +a e \right )}{c^{2}}+\frac {10 d \,e^{2} x}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 300, normalized size = 1.63 \[ -\frac {c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}}{c^{7} d^{7} x + a c^{6} d^{6} e} + \frac {3 \, c^{3} d^{3} e^{5} x^{4} + 4 \, {\left (5 \, c^{3} d^{4} e^{4} - 2 \, a c^{2} d^{2} e^{6}\right )} x^{3} + 6 \, {\left (10 \, c^{3} d^{5} e^{3} - 10 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{2} + 12 \, {\left (10 \, c^{3} d^{6} e^{2} - 20 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} - 4 \, a^{3} e^{8}\right )} x}{12 \, c^{5} d^{5}} + \frac {5 \, {\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \log \left (c d x + a e\right )}{c^{6} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.60, size = 387, normalized size = 2.10 \[ x\,\left (\frac {10\,d\,e^2}{c^2}+\frac {2\,a\,e\,\left (\frac {a^2\,e^7}{c^4\,d^4}-\frac {10\,e^3}{c^2}+\frac {2\,a\,e\,\left (\frac {5\,e^4}{c^2\,d}-\frac {2\,a\,e^6}{c^3\,d^3}\right )}{c\,d}\right )}{c\,d}-\frac {a^2\,e^2\,\left (\frac {5\,e^4}{c^2\,d}-\frac {2\,a\,e^6}{c^3\,d^3}\right )}{c^2\,d^2}\right )+x^3\,\left (\frac {5\,e^4}{3\,c^2\,d}-\frac {2\,a\,e^6}{3\,c^3\,d^3}\right )-x^2\,\left (\frac {a^2\,e^7}{2\,c^4\,d^4}-\frac {5\,e^3}{c^2}+\frac {a\,e\,\left (\frac {5\,e^4}{c^2\,d}-\frac {2\,a\,e^6}{c^3\,d^3}\right )}{c\,d}\right )+\frac {e^5\,x^4}{4\,c^2\,d^2}+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (5\,a^4\,e^9-20\,a^3\,c\,d^2\,e^7+30\,a^2\,c^2\,d^4\,e^5-20\,a\,c^3\,d^6\,e^3+5\,c^4\,d^8\,e\right )}{c^6\,d^6}+\frac {a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}}{c\,d\,\left (x\,c^6\,d^6+a\,e\,c^5\,d^5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.19, size = 264, normalized size = 1.43 \[ x^{3} \left (- \frac {2 a e^{6}}{3 c^{3} d^{3}} + \frac {5 e^{4}}{3 c^{2} d}\right ) + x^{2} \left (\frac {3 a^{2} e^{7}}{2 c^{4} d^{4}} - \frac {5 a e^{5}}{c^{3} d^{2}} + \frac {5 e^{3}}{c^{2}}\right ) + x \left (- \frac {4 a^{3} e^{8}}{c^{5} d^{5}} + \frac {15 a^{2} e^{6}}{c^{4} d^{3}} - \frac {20 a e^{4}}{c^{3} d} + \frac {10 d e^{2}}{c^{2}}\right ) + \frac {a^{5} e^{10} - 5 a^{4} c d^{2} e^{8} + 10 a^{3} c^{2} d^{4} e^{6} - 10 a^{2} c^{3} d^{6} e^{4} + 5 a c^{4} d^{8} e^{2} - c^{5} d^{10}}{a c^{6} d^{6} e + c^{7} d^{7} x} + \frac {e^{5} x^{4}}{4 c^{2} d^{2}} + \frac {5 e \left (a e^{2} - c d^{2}\right )^{4} \log {\left (a e + c d x \right )}}{c^{6} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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