Optimal. Leaf size=262 \[ -\frac {c d^2+a e^2+2 c d e x}{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}+\frac {5 c d e \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}-\frac {10 c^2 d^2 e^2 \left (c d^2+a e^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^6 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}-\frac {20 c^3 d^3 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^7}+\frac {20 c^3 d^3 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^7} \]
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Rubi [A]
time = 0.07, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {628, 630, 31}
\begin {gather*} -\frac {20 c^3 d^3 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^7}+\frac {20 c^3 d^3 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^7}-\frac {10 c^2 d^2 e^2 \left (a e^2+c d^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^6 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )}+\frac {5 c d e \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2}-\frac {a e^2+c d^2+2 c d e x}{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 628
Rule 630
Rubi steps
\begin {align*} \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=-\frac {c d^2+a e^2+2 c d e x}{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}-\frac {(10 c d e) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx}{3 \left (c d^2-a e^2\right )^2}\\ &=-\frac {c d^2+a e^2+2 c d e x}{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}+\frac {5 c d e \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}+\frac {\left (10 c^2 d^2 e^2\right ) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx}{\left (c d^2-a e^2\right )^4}\\ &=-\frac {c d^2+a e^2+2 c d e x}{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}+\frac {5 c d e \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}-\frac {10 c^2 d^2 e^2 \left (c d^2+a e^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^6 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}-\frac {\left (20 c^3 d^3 e^3\right ) \int \frac {1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{\left (c d^2-a e^2\right )^6}\\ &=-\frac {c d^2+a e^2+2 c d e x}{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}+\frac {5 c d e \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}-\frac {10 c^2 d^2 e^2 \left (c d^2+a e^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^6 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}+\frac {\left (20 c^4 d^4 e^4\right ) \int \frac {1}{c d^2+c d e x} \, dx}{\left (c d^2-a e^2\right )^7}-\frac {\left (20 c^4 d^4 e^4\right ) \int \frac {1}{a e^2+c d e x} \, dx}{\left (c d^2-a e^2\right )^7}\\ &=-\frac {c d^2+a e^2+2 c d e x}{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}+\frac {5 c d e \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}-\frac {10 c^2 d^2 e^2 \left (c d^2+a e^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^6 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}-\frac {20 c^3 d^3 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^7}+\frac {20 c^3 d^3 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^7}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 234, normalized size = 0.89 \begin {gather*} \frac {\frac {c^3 d^3 \left (c d^2-a e^2\right )^3}{(a e+c d x)^3}-\frac {6 c^3 d^3 e \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {30 c^3 d^3 e^2 \left (c d^2-a e^2\right )}{a e+c d x}+\frac {\left (c d^2 e-a e^3\right )^3}{(d+e x)^3}+\frac {6 c d e^3 \left (c d^2-a e^2\right )^2}{(d+e x)^2}+\frac {30 c^2 d^2 e^3 \left (c d^2-a e^2\right )}{d+e x}+60 c^3 d^3 e^3 \log (a e+c d x)-60 c^3 d^3 e^3 \log (d+e x)}{3 \left (-c d^2+a e^2\right )^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.78, size = 253, normalized size = 0.97 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1193 vs.
\(2 (256) = 512\).
time = 0.37, size = 1193, normalized size = 4.55 \begin {gather*} -\frac {20 \, c^{3} d^{3} e^{3} \log \left (c d x + a e\right )}{c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}} + \frac {20 \, c^{3} d^{3} e^{3} \log \left (x e + d\right )}{c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}} - \frac {60 \, c^{5} d^{5} x^{5} e^{5} + c^{5} d^{10} - 8 \, a c^{4} d^{8} e^{2} + 37 \, a^{2} c^{3} d^{6} e^{4} + 37 \, a^{3} c^{2} d^{4} e^{6} - 8 \, a^{4} c d^{2} e^{8} + a^{5} e^{10} + 150 \, {\left (c^{5} d^{6} e^{4} + a c^{4} d^{4} e^{6}\right )} x^{4} + 10 \, {\left (11 \, c^{5} d^{7} e^{3} + 38 \, a c^{4} d^{5} e^{5} + 11 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 15 \, {\left (c^{5} d^{8} e^{2} + 19 \, a c^{4} d^{6} e^{4} + 19 \, a^{2} c^{3} d^{4} e^{6} + a^{3} c^{2} d^{2} e^{8}\right )} x^{2} - 3 \, {\left (c^{5} d^{9} e - 14 \, a c^{4} d^{7} e^{3} - 74 \, a^{2} c^{3} d^{5} e^{5} - 14 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x}{3 \, {\left (a^{3} c^{6} d^{15} e^{3} - 6 \, a^{4} c^{5} d^{13} e^{5} + 15 \, a^{5} c^{4} d^{11} e^{7} - 20 \, a^{6} c^{3} d^{9} e^{9} + 15 \, a^{7} c^{2} d^{7} e^{11} - 6 \, a^{8} c d^{5} e^{13} + a^{9} d^{3} e^{15} + {\left (c^{9} d^{15} e^{3} - 6 \, a c^{8} d^{13} e^{5} + 15 \, a^{2} c^{7} d^{11} e^{7} - 20 \, a^{3} c^{6} d^{9} e^{9} + 15 \, a^{4} c^{5} d^{7} e^{11} - 6 \, a^{5} c^{4} d^{5} e^{13} + a^{6} c^{3} d^{3} e^{15}\right )} x^{6} + 3 \, {\left (c^{9} d^{16} e^{2} - 5 \, a c^{8} d^{14} e^{4} + 9 \, a^{2} c^{7} d^{12} e^{6} - 5 \, a^{3} c^{6} d^{10} e^{8} - 5 \, a^{4} c^{5} d^{8} e^{10} + 9 \, a^{5} c^{4} d^{6} e^{12} - 5 \, a^{6} c^{3} d^{4} e^{14} + a^{7} c^{2} d^{2} e^{16}\right )} x^{5} + 3 \, {\left (c^{9} d^{17} e - 3 \, a c^{8} d^{15} e^{3} - 2 \, a^{2} c^{7} d^{13} e^{5} + 19 \, a^{3} c^{6} d^{11} e^{7} - 30 \, a^{4} c^{5} d^{9} e^{9} + 19 \, a^{5} c^{4} d^{7} e^{11} - 2 \, a^{6} c^{3} d^{5} e^{13} - 3 \, a^{7} c^{2} d^{3} e^{15} + a^{8} c d e^{17}\right )} x^{4} + {\left (c^{9} d^{18} + 3 \, a c^{8} d^{16} e^{2} - 30 \, a^{2} c^{7} d^{14} e^{4} + 62 \, a^{3} c^{6} d^{12} e^{6} - 36 \, a^{4} c^{5} d^{10} e^{8} - 36 \, a^{5} c^{4} d^{8} e^{10} + 62 \, a^{6} c^{3} d^{6} e^{12} - 30 \, a^{7} c^{2} d^{4} e^{14} + 3 \, a^{8} c d^{2} e^{16} + a^{9} e^{18}\right )} x^{3} + 3 \, {\left (a c^{8} d^{17} e - 3 \, a^{2} c^{7} d^{15} e^{3} - 2 \, a^{3} c^{6} d^{13} e^{5} + 19 \, a^{4} c^{5} d^{11} e^{7} - 30 \, a^{5} c^{4} d^{9} e^{9} + 19 \, a^{6} c^{3} d^{7} e^{11} - 2 \, a^{7} c^{2} d^{5} e^{13} - 3 \, a^{8} c d^{3} e^{15} + a^{9} d e^{17}\right )} x^{2} + 3 \, {\left (a^{2} c^{7} d^{16} e^{2} - 5 \, a^{3} c^{6} d^{14} e^{4} + 9 \, a^{4} c^{5} d^{12} e^{6} - 5 \, a^{5} c^{4} d^{10} e^{8} - 5 \, a^{6} c^{3} d^{8} e^{10} + 9 \, a^{7} c^{2} d^{6} e^{12} - 5 \, a^{8} c d^{4} e^{14} + a^{9} d^{2} e^{16}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1619 vs.
\(2 (256) = 512\).
time = 2.51, size = 1619, normalized size = 6.18 \begin {gather*} \frac {3 \, c^{6} d^{11} x e - c^{6} d^{12} - 3 \, a^{5} c d x e^{11} + a^{6} e^{12} + 3 \, {\left (5 \, a^{4} c^{2} d^{2} x^{2} - 3 \, a^{5} c d^{2}\right )} e^{10} + 5 \, {\left (22 \, a^{3} c^{3} d^{3} x^{3} + 9 \, a^{4} c^{2} d^{3} x\right )} e^{9} + 15 \, {\left (10 \, a^{2} c^{4} d^{4} x^{4} + 18 \, a^{3} c^{3} d^{4} x^{2} + 3 \, a^{4} c^{2} d^{4}\right )} e^{8} + 30 \, {\left (2 \, a c^{5} d^{5} x^{5} + 9 \, a^{2} c^{4} d^{5} x^{3} + 6 \, a^{3} c^{3} d^{5} x\right )} e^{7} - 30 \, {\left (2 \, c^{6} d^{7} x^{5} + 9 \, a c^{5} d^{7} x^{3} + 6 \, a^{2} c^{4} d^{7} x\right )} e^{5} - 15 \, {\left (10 \, c^{6} d^{8} x^{4} + 18 \, a c^{5} d^{8} x^{2} + 3 \, a^{2} c^{4} d^{8}\right )} e^{4} - 5 \, {\left (22 \, c^{6} d^{9} x^{3} + 9 \, a c^{5} d^{9} x\right )} e^{3} - 3 \, {\left (5 \, c^{6} d^{10} x^{2} - 3 \, a c^{5} d^{10}\right )} e^{2} - 60 \, {\left (c^{6} d^{9} x^{3} e^{3} + a^{3} c^{3} d^{3} x^{3} e^{9} + 3 \, {\left (a^{2} c^{4} d^{4} x^{4} + a^{3} c^{3} d^{4} x^{2}\right )} e^{8} + 3 \, {\left (a c^{5} d^{5} x^{5} + 3 \, a^{2} c^{4} d^{5} x^{3} + a^{3} c^{3} d^{5} x\right )} e^{7} + {\left (c^{6} d^{6} x^{6} + 9 \, a c^{5} d^{6} x^{4} + 9 \, a^{2} c^{4} d^{6} x^{2} + a^{3} c^{3} d^{6}\right )} e^{6} + 3 \, {\left (c^{6} d^{7} x^{5} + 3 \, a c^{5} d^{7} x^{3} + a^{2} c^{4} d^{7} x\right )} e^{5} + 3 \, {\left (c^{6} d^{8} x^{4} + a c^{5} d^{8} x^{2}\right )} e^{4}\right )} \log \left (c d x + a e\right ) + 60 \, {\left (c^{6} d^{9} x^{3} e^{3} + a^{3} c^{3} d^{3} x^{3} e^{9} + 3 \, {\left (a^{2} c^{4} d^{4} x^{4} + a^{3} c^{3} d^{4} x^{2}\right )} e^{8} + 3 \, {\left (a c^{5} d^{5} x^{5} + 3 \, a^{2} c^{4} d^{5} x^{3} + a^{3} c^{3} d^{5} x\right )} e^{7} + {\left (c^{6} d^{6} x^{6} + 9 \, a c^{5} d^{6} x^{4} + 9 \, a^{2} c^{4} d^{6} x^{2} + a^{3} c^{3} d^{6}\right )} e^{6} + 3 \, {\left (c^{6} d^{7} x^{5} + 3 \, a c^{5} d^{7} x^{3} + a^{2} c^{4} d^{7} x\right )} e^{5} + 3 \, {\left (c^{6} d^{8} x^{4} + a c^{5} d^{8} x^{2}\right )} e^{4}\right )} \log \left (x e + d\right )}{3 \, {\left (c^{10} d^{20} x^{3} - a^{10} x^{3} e^{20} - 3 \, {\left (a^{9} c d x^{4} + a^{10} d x^{2}\right )} e^{19} - {\left (3 \, a^{8} c^{2} d^{2} x^{5} + 2 \, a^{9} c d^{2} x^{3} + 3 \, a^{10} d^{2} x\right )} e^{18} - {\left (a^{7} c^{3} d^{3} x^{6} - 12 \, a^{8} c^{2} d^{3} x^{4} - 12 \, a^{9} c d^{3} x^{2} + a^{10} d^{3}\right )} e^{17} + 3 \, {\left (6 \, a^{7} c^{3} d^{4} x^{5} + 11 \, a^{8} c^{2} d^{4} x^{3} + 6 \, a^{9} c d^{4} x\right )} e^{16} + {\left (7 \, a^{6} c^{4} d^{5} x^{6} - 3 \, a^{7} c^{3} d^{5} x^{4} - 3 \, a^{8} c^{2} d^{5} x^{2} + 7 \, a^{9} c d^{5}\right )} e^{15} - 2 \, {\left (21 \, a^{6} c^{4} d^{6} x^{5} + 46 \, a^{7} c^{3} d^{6} x^{3} + 21 \, a^{8} c^{2} d^{6} x\right )} e^{14} - 21 \, {\left (a^{5} c^{5} d^{7} x^{6} + 3 \, a^{6} c^{4} d^{7} x^{4} + 3 \, a^{7} c^{3} d^{7} x^{2} + a^{8} c^{2} d^{7}\right )} e^{13} + 14 \, {\left (3 \, a^{5} c^{5} d^{8} x^{5} + 7 \, a^{6} c^{4} d^{8} x^{3} + 3 \, a^{7} c^{3} d^{8} x\right )} e^{12} + 7 \, {\left (5 \, a^{4} c^{6} d^{9} x^{6} + 21 \, a^{5} c^{5} d^{9} x^{4} + 21 \, a^{6} c^{4} d^{9} x^{2} + 5 \, a^{7} c^{3} d^{9}\right )} e^{11} - 7 \, {\left (5 \, a^{3} c^{7} d^{11} x^{6} + 21 \, a^{4} c^{6} d^{11} x^{4} + 21 \, a^{5} c^{5} d^{11} x^{2} + 5 \, a^{6} c^{4} d^{11}\right )} e^{9} - 14 \, {\left (3 \, a^{3} c^{7} d^{12} x^{5} + 7 \, a^{4} c^{6} d^{12} x^{3} + 3 \, a^{5} c^{5} d^{12} x\right )} e^{8} + 21 \, {\left (a^{2} c^{8} d^{13} x^{6} + 3 \, a^{3} c^{7} d^{13} x^{4} + 3 \, a^{4} c^{6} d^{13} x^{2} + a^{5} c^{5} d^{13}\right )} e^{7} + 2 \, {\left (21 \, a^{2} c^{8} d^{14} x^{5} + 46 \, a^{3} c^{7} d^{14} x^{3} + 21 \, a^{4} c^{6} d^{14} x\right )} e^{6} - {\left (7 \, a c^{9} d^{15} x^{6} - 3 \, a^{2} c^{8} d^{15} x^{4} - 3 \, a^{3} c^{7} d^{15} x^{2} + 7 \, a^{4} c^{6} d^{15}\right )} e^{5} - 3 \, {\left (6 \, a c^{9} d^{16} x^{5} + 11 \, a^{2} c^{8} d^{16} x^{3} + 6 \, a^{3} c^{7} d^{16} x\right )} e^{4} + {\left (c^{10} d^{17} x^{6} - 12 \, a c^{9} d^{17} x^{4} - 12 \, a^{2} c^{8} d^{17} x^{2} + a^{3} c^{7} d^{17}\right )} e^{3} + {\left (3 \, c^{10} d^{18} x^{5} + 2 \, a c^{9} d^{18} x^{3} + 3 \, a^{2} c^{8} d^{18} x\right )} e^{2} + 3 \, {\left (c^{10} d^{19} x^{4} + a c^{9} d^{19} x^{2}\right )} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1748 vs.
\(2 (262) = 524\).
time = 28.25, size = 1748, normalized size = 6.67 \begin {gather*} - \frac {20 c^{3} d^{3} e^{3} \log {\left (x + \frac {- \frac {20 a^{8} c^{3} d^{3} e^{19}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {160 a^{7} c^{4} d^{5} e^{17}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac {560 a^{6} c^{5} d^{7} e^{15}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {1120 a^{5} c^{6} d^{9} e^{13}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac {1400 a^{4} c^{7} d^{11} e^{11}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {1120 a^{3} c^{8} d^{13} e^{9}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac {560 a^{2} c^{9} d^{15} e^{7}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {160 a c^{10} d^{17} e^{5}}{\left (a e^{2} - c d^{2}\right )^{7}} + 20 a c^{3} d^{3} e^{5} - \frac {20 c^{11} d^{19} e^{3}}{\left (a e^{2} - c d^{2}\right )^{7}} + 20 c^{4} d^{5} e^{3}}{40 c^{4} d^{4} e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {20 c^{3} d^{3} e^{3} \log {\left (x + \frac {\frac {20 a^{8} c^{3} d^{3} e^{19}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac {160 a^{7} c^{4} d^{5} e^{17}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {560 a^{6} c^{5} d^{7} e^{15}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac {1120 a^{5} c^{6} d^{9} e^{13}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {1400 a^{4} c^{7} d^{11} e^{11}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac {1120 a^{3} c^{8} d^{13} e^{9}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {560 a^{2} c^{9} d^{15} e^{7}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac {160 a c^{10} d^{17} e^{5}}{\left (a e^{2} - c d^{2}\right )^{7}} + 20 a c^{3} d^{3} e^{5} + \frac {20 c^{11} d^{19} e^{3}}{\left (a e^{2} - c d^{2}\right )^{7}} + 20 c^{4} d^{5} e^{3}}{40 c^{4} d^{4} e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {- a^{5} e^{10} + 8 a^{4} c d^{2} e^{8} - 37 a^{3} c^{2} d^{4} e^{6} - 37 a^{2} c^{3} d^{6} e^{4} + 8 a c^{4} d^{8} e^{2} - c^{5} d^{10} - 60 c^{5} d^{5} e^{5} x^{5} + x^{4} \left (- 150 a c^{4} d^{4} e^{6} - 150 c^{5} d^{6} e^{4}\right ) + x^{3} \left (- 110 a^{2} c^{3} d^{3} e^{7} - 380 a c^{4} d^{5} e^{5} - 110 c^{5} d^{7} e^{3}\right ) + x^{2} \left (- 15 a^{3} c^{2} d^{2} e^{8} - 285 a^{2} c^{3} d^{4} e^{6} - 285 a c^{4} d^{6} e^{4} - 15 c^{5} d^{8} e^{2}\right ) + x \left (3 a^{4} c d e^{9} - 42 a^{3} c^{2} d^{3} e^{7} - 222 a^{2} c^{3} d^{5} e^{5} - 42 a c^{4} d^{7} e^{3} + 3 c^{5} d^{9} e\right )}{3 a^{9} d^{3} e^{15} - 18 a^{8} c d^{5} e^{13} + 45 a^{7} c^{2} d^{7} e^{11} - 60 a^{6} c^{3} d^{9} e^{9} + 45 a^{5} c^{4} d^{11} e^{7} - 18 a^{4} c^{5} d^{13} e^{5} + 3 a^{3} c^{6} d^{15} e^{3} + x^{6} \cdot \left (3 a^{6} c^{3} d^{3} e^{15} - 18 a^{5} c^{4} d^{5} e^{13} + 45 a^{4} c^{5} d^{7} e^{11} - 60 a^{3} c^{6} d^{9} e^{9} + 45 a^{2} c^{7} d^{11} e^{7} - 18 a c^{8} d^{13} e^{5} + 3 c^{9} d^{15} e^{3}\right ) + x^{5} \cdot \left (9 a^{7} c^{2} d^{2} e^{16} - 45 a^{6} c^{3} d^{4} e^{14} + 81 a^{5} c^{4} d^{6} e^{12} - 45 a^{4} c^{5} d^{8} e^{10} - 45 a^{3} c^{6} d^{10} e^{8} + 81 a^{2} c^{7} d^{12} e^{6} - 45 a c^{8} d^{14} e^{4} + 9 c^{9} d^{16} e^{2}\right ) + x^{4} \cdot \left (9 a^{8} c d e^{17} - 27 a^{7} c^{2} d^{3} e^{15} - 18 a^{6} c^{3} d^{5} e^{13} + 171 a^{5} c^{4} d^{7} e^{11} - 270 a^{4} c^{5} d^{9} e^{9} + 171 a^{3} c^{6} d^{11} e^{7} - 18 a^{2} c^{7} d^{13} e^{5} - 27 a c^{8} d^{15} e^{3} + 9 c^{9} d^{17} e\right ) + x^{3} \cdot \left (3 a^{9} e^{18} + 9 a^{8} c d^{2} e^{16} - 90 a^{7} c^{2} d^{4} e^{14} + 186 a^{6} c^{3} d^{6} e^{12} - 108 a^{5} c^{4} d^{8} e^{10} - 108 a^{4} c^{5} d^{10} e^{8} + 186 a^{3} c^{6} d^{12} e^{6} - 90 a^{2} c^{7} d^{14} e^{4} + 9 a c^{8} d^{16} e^{2} + 3 c^{9} d^{18}\right ) + x^{2} \cdot \left (9 a^{9} d e^{17} - 27 a^{8} c d^{3} e^{15} - 18 a^{7} c^{2} d^{5} e^{13} + 171 a^{6} c^{3} d^{7} e^{11} - 270 a^{5} c^{4} d^{9} e^{9} + 171 a^{4} c^{5} d^{11} e^{7} - 18 a^{3} c^{6} d^{13} e^{5} - 27 a^{2} c^{7} d^{15} e^{3} + 9 a c^{8} d^{17} e\right ) + x \left (9 a^{9} d^{2} e^{16} - 45 a^{8} c d^{4} e^{14} + 81 a^{7} c^{2} d^{6} e^{12} - 45 a^{6} c^{3} d^{8} e^{10} - 45 a^{5} c^{4} d^{10} e^{8} + 81 a^{4} c^{5} d^{12} e^{6} - 45 a^{3} c^{6} d^{14} e^{4} + 9 a^{2} c^{7} d^{16} e^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 599 vs.
\(2 (256) = 512\).
time = 0.66, size = 599, normalized size = 2.29 \begin {gather*} -\frac {20 \, c^{4} d^{4} e^{3} \log \left ({\left | c d x + a e \right |}\right )}{c^{8} d^{15} - 7 \, a c^{7} d^{13} e^{2} + 21 \, a^{2} c^{6} d^{11} e^{4} - 35 \, a^{3} c^{5} d^{9} e^{6} + 35 \, a^{4} c^{4} d^{7} e^{8} - 21 \, a^{5} c^{3} d^{5} e^{10} + 7 \, a^{6} c^{2} d^{3} e^{12} - a^{7} c d e^{14}} + \frac {20 \, c^{3} d^{3} e^{4} \log \left ({\left | x e + d \right |}\right )}{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}} - \frac {60 \, c^{5} d^{5} x^{5} e^{5} + 150 \, c^{5} d^{6} x^{4} e^{4} + 110 \, c^{5} d^{7} x^{3} e^{3} + 15 \, c^{5} d^{8} x^{2} e^{2} - 3 \, c^{5} d^{9} x e + c^{5} d^{10} + 150 \, a c^{4} d^{4} x^{4} e^{6} + 380 \, a c^{4} d^{5} x^{3} e^{5} + 285 \, a c^{4} d^{6} x^{2} e^{4} + 42 \, a c^{4} d^{7} x e^{3} - 8 \, a c^{4} d^{8} e^{2} + 110 \, a^{2} c^{3} d^{3} x^{3} e^{7} + 285 \, a^{2} c^{3} d^{4} x^{2} e^{6} + 222 \, a^{2} c^{3} d^{5} x e^{5} + 37 \, a^{2} c^{3} d^{6} e^{4} + 15 \, a^{3} c^{2} d^{2} x^{2} e^{8} + 42 \, a^{3} c^{2} d^{3} x e^{7} + 37 \, a^{3} c^{2} d^{4} e^{6} - 3 \, a^{4} c d x e^{9} - 8 \, a^{4} c d^{2} e^{8} + a^{5} e^{10}}{3 \, {\left (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}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \left \{\begin {array}{cl} -\frac {20\,c^3\,d^3\,e^3\,\ln \left (\frac {\frac {a\,e^2}{2}+\frac {c\,d^2}{2}-\sqrt {\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}-a\,c\,d^2\,e^2}+c\,d\,e\,x}{\frac {a\,e^2}{2}+\frac {c\,d^2}{2}+\sqrt {\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}-a\,c\,d^2\,e^2}+c\,d\,e\,x}\right )}{{\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )}^{7/2}}-\frac {20\,\left (\frac {c\,d^2}{2}+c\,x\,d\,e+\frac {a\,e^2}{2}\right )\,\left (\frac {c\,d\,e}{30\,\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^3}-\frac {c^2\,d^2\,e^2}{6\,{\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^2}+\frac {c^3\,d^3\,e^3}{{\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )}^3\,\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}\right )}{c\,d\,e} & \text {\ if\ \ }0<{\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\\ -\frac {20\,\left (\frac {c\,d^2}{2}+c\,x\,d\,e+\frac {a\,e^2}{2}\right )\,\left (\frac {c\,d\,e}{30\,\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^3}-\frac {c^2\,d^2\,e^2}{6\,{\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^2}+\frac {c^3\,d^3\,e^3}{{\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )}^3\,\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}\right )}{c\,d\,e}-\frac {20\,c^3\,d^3\,e^3\,\mathrm {atan}\left (\frac {\frac {c\,d^2}{2}+c\,x\,d\,e+\frac {a\,e^2}{2}}{\sqrt {a\,c\,d^2\,e^2-\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}}}\right )}{\sqrt {a\,c\,d^2\,e^2-\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}}\,{\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )}^3} & \text {\ if\ \ }{\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2<0\\ \int \frac {1}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^4} \,d x & \text {\ if\ \ }{\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\notin \mathbb {R}\vee {\left (c\,d^2+a\,e^2\right )}^2=4\,a\,c\,d^2\,e^2 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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