Optimal. Leaf size=220 \[ -\frac {b^2}{5 (b d-a e)^3 (a+b x)^5}+\frac {3 b^2 e}{4 (b d-a e)^4 (a+b x)^4}-\frac {2 b^2 e^2}{(b d-a e)^5 (a+b x)^3}+\frac {5 b^2 e^3}{(b d-a e)^6 (a+b x)^2}-\frac {15 b^2 e^4}{(b d-a e)^7 (a+b x)}-\frac {e^5}{2 (b d-a e)^6 (d+e x)^2}-\frac {6 b e^5}{(b d-a e)^7 (d+e x)}-\frac {21 b^2 e^5 \log (a+b x)}{(b d-a e)^8}+\frac {21 b^2 e^5 \log (d+e x)}{(b d-a e)^8} \]
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Rubi [A]
time = 0.18, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 46}
\begin {gather*} -\frac {21 b^2 e^5 \log (a+b x)}{(b d-a e)^8}+\frac {21 b^2 e^5 \log (d+e x)}{(b d-a e)^8}-\frac {15 b^2 e^4}{(a+b x) (b d-a e)^7}+\frac {5 b^2 e^3}{(a+b x)^2 (b d-a e)^6}-\frac {2 b^2 e^2}{(a+b x)^3 (b d-a e)^5}+\frac {3 b^2 e}{4 (a+b x)^4 (b d-a e)^4}-\frac {b^2}{5 (a+b x)^5 (b d-a e)^3}-\frac {6 b e^5}{(d+e x) (b d-a e)^7}-\frac {e^5}{2 (d+e x)^2 (b d-a e)^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 46
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^6 (d+e x)^3} \, dx\\ &=\int \left (\frac {b^3}{(b d-a e)^3 (a+b x)^6}-\frac {3 b^3 e}{(b d-a e)^4 (a+b x)^5}+\frac {6 b^3 e^2}{(b d-a e)^5 (a+b x)^4}-\frac {10 b^3 e^3}{(b d-a e)^6 (a+b x)^3}+\frac {15 b^3 e^4}{(b d-a e)^7 (a+b x)^2}-\frac {21 b^3 e^5}{(b d-a e)^8 (a+b x)}+\frac {e^6}{(b d-a e)^6 (d+e x)^3}+\frac {6 b e^6}{(b d-a e)^7 (d+e x)^2}+\frac {21 b^2 e^6}{(b d-a e)^8 (d+e x)}\right ) \, dx\\ &=-\frac {b^2}{5 (b d-a e)^3 (a+b x)^5}+\frac {3 b^2 e}{4 (b d-a e)^4 (a+b x)^4}-\frac {2 b^2 e^2}{(b d-a e)^5 (a+b x)^3}+\frac {5 b^2 e^3}{(b d-a e)^6 (a+b x)^2}-\frac {15 b^2 e^4}{(b d-a e)^7 (a+b x)}-\frac {e^5}{2 (b d-a e)^6 (d+e x)^2}-\frac {6 b e^5}{(b d-a e)^7 (d+e x)}-\frac {21 b^2 e^5 \log (a+b x)}{(b d-a e)^8}+\frac {21 b^2 e^5 \log (d+e x)}{(b d-a e)^8}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 204, normalized size = 0.93 \begin {gather*} -\frac {\frac {4 b^2 (b d-a e)^5}{(a+b x)^5}-\frac {15 b^2 e (b d-a e)^4}{(a+b x)^4}+\frac {40 b^2 e^2 (b d-a e)^3}{(a+b x)^3}-\frac {100 b^2 e^3 (b d-a e)^2}{(a+b x)^2}+\frac {300 b^2 e^4 (b d-a e)}{a+b x}+\frac {10 e^5 (b d-a e)^2}{(d+e x)^2}+\frac {120 b e^5 (b d-a e)}{d+e x}+420 b^2 e^5 \log (a+b x)-420 b^2 e^5 \log (d+e x)}{20 (b d-a e)^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 215, normalized size = 0.98 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. 1468 vs.
\(2 (220) = 440\).
time = 0.47, size = 1468, normalized size = 6.67 \begin {gather*} -\frac {21 \, b^{2} e^{5} \log \left (b x + a\right )}{b^{8} d^{8} - 8 \, a b^{7} d^{7} e + 28 \, a^{2} b^{6} d^{6} e^{2} - 56 \, a^{3} b^{5} d^{5} e^{3} + 70 \, a^{4} b^{4} d^{4} e^{4} - 56 \, a^{5} b^{3} d^{3} e^{5} + 28 \, a^{6} b^{2} d^{2} e^{6} - 8 \, a^{7} b d e^{7} + a^{8} e^{8}} + \frac {21 \, b^{2} e^{5} \log \left (x e + d\right )}{b^{8} d^{8} - 8 \, a b^{7} d^{7} e + 28 \, a^{2} b^{6} d^{6} e^{2} - 56 \, a^{3} b^{5} d^{5} e^{3} + 70 \, a^{4} b^{4} d^{4} e^{4} - 56 \, a^{5} b^{3} d^{3} e^{5} + 28 \, a^{6} b^{2} d^{2} e^{6} - 8 \, a^{7} b d e^{7} + a^{8} e^{8}} - \frac {420 \, b^{6} x^{6} e^{6} + 4 \, b^{6} d^{6} - 31 \, a b^{5} d^{5} e + 109 \, a^{2} b^{4} d^{4} e^{2} - 241 \, a^{3} b^{3} d^{3} e^{3} + 459 \, a^{4} b^{2} d^{2} e^{4} + 130 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} + 630 \, {\left (b^{6} d e^{5} + 3 \, a b^{5} e^{6}\right )} x^{5} + 70 \, {\left (2 \, b^{6} d^{2} e^{4} + 41 \, a b^{5} d e^{5} + 47 \, a^{2} b^{4} e^{6}\right )} x^{4} - 35 \, {\left (b^{6} d^{3} e^{3} - 19 \, a b^{5} d^{2} e^{4} - 145 \, a^{2} b^{4} d e^{5} - 77 \, a^{3} b^{3} e^{6}\right )} x^{3} + 7 \, {\left (2 \, b^{6} d^{4} e^{2} - 23 \, a b^{5} d^{3} e^{3} + 177 \, a^{2} b^{4} d^{2} e^{4} + 607 \, a^{3} b^{3} d e^{5} + 137 \, a^{4} b^{2} e^{6}\right )} x^{2} - 7 \, {\left (b^{6} d^{5} e - 9 \, a b^{5} d^{4} e^{2} + 41 \, a^{2} b^{4} d^{3} e^{3} - 159 \, a^{3} b^{3} d^{2} e^{4} - 224 \, a^{4} b^{2} d e^{5} - 10 \, a^{5} b e^{6}\right )} x}{20 \, {\left (a^{5} b^{7} d^{9} - 7 \, a^{6} b^{6} d^{8} e + 21 \, a^{7} b^{5} d^{7} e^{2} - 35 \, a^{8} b^{4} d^{6} e^{3} + 35 \, a^{9} b^{3} d^{5} e^{4} - 21 \, a^{10} b^{2} d^{4} e^{5} + 7 \, a^{11} b d^{3} e^{6} - a^{12} d^{2} e^{7} + {\left (b^{12} d^{7} e^{2} - 7 \, a b^{11} d^{6} e^{3} + 21 \, a^{2} b^{10} d^{5} e^{4} - 35 \, a^{3} b^{9} d^{4} e^{5} + 35 \, a^{4} b^{8} d^{3} e^{6} - 21 \, a^{5} b^{7} d^{2} e^{7} + 7 \, a^{6} b^{6} d e^{8} - a^{7} b^{5} e^{9}\right )} x^{7} + {\left (2 \, b^{12} d^{8} e - 9 \, a b^{11} d^{7} e^{2} + 7 \, a^{2} b^{10} d^{6} e^{3} + 35 \, a^{3} b^{9} d^{5} e^{4} - 105 \, a^{4} b^{8} d^{4} e^{5} + 133 \, a^{5} b^{7} d^{3} e^{6} - 91 \, a^{6} b^{6} d^{2} e^{7} + 33 \, a^{7} b^{5} d e^{8} - 5 \, a^{8} b^{4} e^{9}\right )} x^{6} + {\left (b^{12} d^{9} + 3 \, a b^{11} d^{8} e - 39 \, a^{2} b^{10} d^{7} e^{2} + 105 \, a^{3} b^{9} d^{6} e^{3} - 105 \, a^{4} b^{8} d^{5} e^{4} - 21 \, a^{5} b^{7} d^{4} e^{5} + 147 \, a^{6} b^{6} d^{3} e^{6} - 141 \, a^{7} b^{5} d^{2} e^{7} + 60 \, a^{8} b^{4} d e^{8} - 10 \, a^{9} b^{3} e^{9}\right )} x^{5} + 5 \, {\left (a b^{11} d^{9} - 3 \, a^{2} b^{10} d^{8} e - 5 \, a^{3} b^{9} d^{7} e^{2} + 35 \, a^{4} b^{8} d^{6} e^{3} - 63 \, a^{5} b^{7} d^{5} e^{4} + 49 \, a^{6} b^{6} d^{4} e^{5} - 7 \, a^{7} b^{5} d^{3} e^{6} - 15 \, a^{8} b^{4} d^{2} e^{7} + 10 \, a^{9} b^{3} d e^{8} - 2 \, a^{10} b^{2} e^{9}\right )} x^{4} + 5 \, {\left (2 \, a^{2} b^{10} d^{9} - 10 \, a^{3} b^{9} d^{8} e + 15 \, a^{4} b^{8} d^{7} e^{2} + 7 \, a^{5} b^{7} d^{6} e^{3} - 49 \, a^{6} b^{6} d^{5} e^{4} + 63 \, a^{7} b^{5} d^{4} e^{5} - 35 \, a^{8} b^{4} d^{3} e^{6} + 5 \, a^{9} b^{3} d^{2} e^{7} + 3 \, a^{10} b^{2} d e^{8} - a^{11} b e^{9}\right )} x^{3} + {\left (10 \, a^{3} b^{9} d^{9} - 60 \, a^{4} b^{8} d^{8} e + 141 \, a^{5} b^{7} d^{7} e^{2} - 147 \, a^{6} b^{6} d^{6} e^{3} + 21 \, a^{7} b^{5} d^{5} e^{4} + 105 \, a^{8} b^{4} d^{4} e^{5} - 105 \, a^{9} b^{3} d^{3} e^{6} + 39 \, a^{10} b^{2} d^{2} e^{7} - 3 \, a^{11} b d e^{8} - a^{12} e^{9}\right )} x^{2} + {\left (5 \, a^{4} b^{8} d^{9} - 33 \, a^{5} b^{7} d^{8} e + 91 \, a^{6} b^{6} d^{7} e^{2} - 133 \, a^{7} b^{5} d^{6} e^{3} + 105 \, a^{8} b^{4} d^{5} e^{4} - 35 \, a^{9} b^{3} d^{4} e^{5} - 7 \, a^{10} b^{2} d^{3} e^{6} + 9 \, a^{11} b d^{2} e^{7} - 2 \, a^{12} d e^{8}\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. 1917 vs.
\(2 (220) = 440\).
time = 2.79, size = 1917, normalized size = 8.71 \begin {gather*} -\frac {4 \, b^{7} d^{7} - {\left (420 \, a b^{6} x^{6} + 1890 \, a^{2} b^{5} x^{5} + 3290 \, a^{3} b^{4} x^{4} + 2695 \, a^{4} b^{3} x^{3} + 959 \, a^{5} b^{2} x^{2} + 70 \, a^{6} b x - 10 \, a^{7}\right )} e^{7} + 14 \, {\left (30 \, b^{7} d x^{6} + 90 \, a b^{6} d x^{5} + 30 \, a^{2} b^{5} d x^{4} - 170 \, a^{3} b^{4} d x^{3} - 235 \, a^{4} b^{3} d x^{2} - 107 \, a^{5} b^{2} d x - 10 \, a^{6} b d\right )} e^{6} + 7 \, {\left (90 \, b^{7} d^{2} x^{5} + 390 \, a b^{6} d^{2} x^{4} + 630 \, a^{2} b^{5} d^{2} x^{3} + 430 \, a^{3} b^{4} d^{2} x^{2} + 65 \, a^{4} b^{3} d^{2} x - 47 \, a^{5} b^{2} d^{2}\right )} e^{5} + 140 \, {\left (b^{7} d^{3} x^{4} + 5 \, a b^{6} d^{3} x^{3} + 10 \, a^{2} b^{5} d^{3} x^{2} + 10 \, a^{3} b^{4} d^{3} x + 5 \, a^{4} b^{3} d^{3}\right )} e^{4} - 35 \, {\left (b^{7} d^{4} x^{3} + 5 \, a b^{6} d^{4} x^{2} + 10 \, a^{2} b^{5} d^{4} x + 10 \, a^{3} b^{4} d^{4}\right )} e^{3} + 14 \, {\left (b^{7} d^{5} x^{2} + 5 \, a b^{6} d^{5} x + 10 \, a^{2} b^{5} d^{5}\right )} e^{2} - 7 \, {\left (b^{7} d^{6} x + 5 \, a b^{6} d^{6}\right )} e + 420 \, {\left ({\left (b^{7} x^{7} + 5 \, a b^{6} x^{6} + 10 \, a^{2} b^{5} x^{5} + 10 \, a^{3} b^{4} x^{4} + 5 \, a^{4} b^{3} x^{3} + a^{5} b^{2} x^{2}\right )} e^{7} + 2 \, {\left (b^{7} d x^{6} + 5 \, a b^{6} d x^{5} + 10 \, a^{2} b^{5} d x^{4} + 10 \, a^{3} b^{4} d x^{3} + 5 \, a^{4} b^{3} d x^{2} + a^{5} b^{2} d x\right )} e^{6} + {\left (b^{7} d^{2} x^{5} + 5 \, a b^{6} d^{2} x^{4} + 10 \, a^{2} b^{5} d^{2} x^{3} + 10 \, a^{3} b^{4} d^{2} x^{2} + 5 \, a^{4} b^{3} d^{2} x + a^{5} b^{2} d^{2}\right )} e^{5}\right )} \log \left (b x + a\right ) - 420 \, {\left ({\left (b^{7} x^{7} + 5 \, a b^{6} x^{6} + 10 \, a^{2} b^{5} x^{5} + 10 \, a^{3} b^{4} x^{4} + 5 \, a^{4} b^{3} x^{3} + a^{5} b^{2} x^{2}\right )} e^{7} + 2 \, {\left (b^{7} d x^{6} + 5 \, a b^{6} d x^{5} + 10 \, a^{2} b^{5} d x^{4} + 10 \, a^{3} b^{4} d x^{3} + 5 \, a^{4} b^{3} d x^{2} + a^{5} b^{2} d x\right )} e^{6} + {\left (b^{7} d^{2} x^{5} + 5 \, a b^{6} d^{2} x^{4} + 10 \, a^{2} b^{5} d^{2} x^{3} + 10 \, a^{3} b^{4} d^{2} x^{2} + 5 \, a^{4} b^{3} d^{2} x + a^{5} b^{2} d^{2}\right )} e^{5}\right )} \log \left (x e + d\right )}{20 \, {\left (b^{13} d^{10} x^{5} + 5 \, a b^{12} d^{10} x^{4} + 10 \, a^{2} b^{11} d^{10} x^{3} + 10 \, a^{3} b^{10} d^{10} x^{2} + 5 \, a^{4} b^{9} d^{10} x + a^{5} b^{8} d^{10} + {\left (a^{8} b^{5} x^{7} + 5 \, a^{9} b^{4} x^{6} + 10 \, a^{10} b^{3} x^{5} + 10 \, a^{11} b^{2} x^{4} + 5 \, a^{12} b x^{3} + a^{13} x^{2}\right )} e^{10} - 2 \, {\left (4 \, a^{7} b^{6} d x^{7} + 19 \, a^{8} b^{5} d x^{6} + 35 \, a^{9} b^{4} d x^{5} + 30 \, a^{10} b^{3} d x^{4} + 10 \, a^{11} b^{2} d x^{3} - a^{12} b d x^{2} - a^{13} d x\right )} e^{9} + {\left (28 \, a^{6} b^{7} d^{2} x^{7} + 124 \, a^{7} b^{6} d^{2} x^{6} + 201 \, a^{8} b^{5} d^{2} x^{5} + 125 \, a^{9} b^{4} d^{2} x^{4} - 10 \, a^{10} b^{3} d^{2} x^{3} - 42 \, a^{11} b^{2} d^{2} x^{2} - 11 \, a^{12} b d^{2} x + a^{13} d^{2}\right )} e^{8} - 8 \, {\left (7 \, a^{5} b^{8} d^{3} x^{7} + 28 \, a^{6} b^{7} d^{3} x^{6} + 36 \, a^{7} b^{6} d^{3} x^{5} + 5 \, a^{8} b^{5} d^{3} x^{4} - 25 \, a^{9} b^{4} d^{3} x^{3} - 18 \, a^{10} b^{3} d^{3} x^{2} - 2 \, a^{11} b^{2} d^{3} x + a^{12} b d^{3}\right )} e^{7} + 14 \, {\left (5 \, a^{4} b^{9} d^{4} x^{7} + 17 \, a^{5} b^{8} d^{4} x^{6} + 12 \, a^{6} b^{7} d^{4} x^{5} - 20 \, a^{7} b^{6} d^{4} x^{4} - 35 \, a^{8} b^{5} d^{4} x^{3} - 15 \, a^{9} b^{4} d^{4} x^{2} + 2 \, a^{10} b^{3} d^{4} x + 2 \, a^{11} b^{2} d^{4}\right )} e^{6} - 28 \, {\left (2 \, a^{3} b^{10} d^{5} x^{7} + 5 \, a^{4} b^{9} d^{5} x^{6} - 3 \, a^{5} b^{8} d^{5} x^{5} - 20 \, a^{6} b^{7} d^{5} x^{4} - 20 \, a^{7} b^{6} d^{5} x^{3} - 3 \, a^{8} b^{5} d^{5} x^{2} + 5 \, a^{9} b^{4} d^{5} x + 2 \, a^{10} b^{3} d^{5}\right )} e^{5} + 14 \, {\left (2 \, a^{2} b^{11} d^{6} x^{7} + 2 \, a^{3} b^{10} d^{6} x^{6} - 15 \, a^{4} b^{9} d^{6} x^{5} - 35 \, a^{5} b^{8} d^{6} x^{4} - 20 \, a^{6} b^{7} d^{6} x^{3} + 12 \, a^{7} b^{6} d^{6} x^{2} + 17 \, a^{8} b^{5} d^{6} x + 5 \, a^{9} b^{4} d^{6}\right )} e^{4} - 8 \, {\left (a b^{12} d^{7} x^{7} - 2 \, a^{2} b^{11} d^{7} x^{6} - 18 \, a^{3} b^{10} d^{7} x^{5} - 25 \, a^{4} b^{9} d^{7} x^{4} + 5 \, a^{5} b^{8} d^{7} x^{3} + 36 \, a^{6} b^{7} d^{7} x^{2} + 28 \, a^{7} b^{6} d^{7} x + 7 \, a^{8} b^{5} d^{7}\right )} e^{3} + {\left (b^{13} d^{8} x^{7} - 11 \, a b^{12} d^{8} x^{6} - 42 \, a^{2} b^{11} d^{8} x^{5} - 10 \, a^{3} b^{10} d^{8} x^{4} + 125 \, a^{4} b^{9} d^{8} x^{3} + 201 \, a^{5} b^{8} d^{8} x^{2} + 124 \, a^{6} b^{7} d^{8} x + 28 \, a^{7} b^{6} d^{8}\right )} e^{2} + 2 \, {\left (b^{13} d^{9} x^{6} + a b^{12} d^{9} x^{5} - 10 \, a^{2} b^{11} d^{9} x^{4} - 30 \, a^{3} b^{10} d^{9} x^{3} - 35 \, a^{4} b^{9} d^{9} x^{2} - 19 \, a^{5} b^{8} d^{9} x - 4 \, a^{6} b^{7} d^{9}\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. 1974 vs.
\(2 (201) = 402\).
time = 40.62, size = 1974, normalized size = 8.97 \begin {gather*} \frac {21 b^{2} e^{5} \log {\left (x + \frac {- \frac {21 a^{9} b^{2} e^{14}}{\left (a e - b d\right )^{8}} + \frac {189 a^{8} b^{3} d e^{13}}{\left (a e - b d\right )^{8}} - \frac {756 a^{7} b^{4} d^{2} e^{12}}{\left (a e - b d\right )^{8}} + \frac {1764 a^{6} b^{5} d^{3} e^{11}}{\left (a e - b d\right )^{8}} - \frac {2646 a^{5} b^{6} d^{4} e^{10}}{\left (a e - b d\right )^{8}} + \frac {2646 a^{4} b^{7} d^{5} e^{9}}{\left (a e - b d\right )^{8}} - \frac {1764 a^{3} b^{8} d^{6} e^{8}}{\left (a e - b d\right )^{8}} + \frac {756 a^{2} b^{9} d^{7} e^{7}}{\left (a e - b d\right )^{8}} - \frac {189 a b^{10} d^{8} e^{6}}{\left (a e - b d\right )^{8}} + 21 a b^{2} e^{6} + \frac {21 b^{11} d^{9} e^{5}}{\left (a e - b d\right )^{8}} + 21 b^{3} d e^{5}}{42 b^{3} e^{6}} \right )}}{\left (a e - b d\right )^{8}} - \frac {21 b^{2} e^{5} \log {\left (x + \frac {\frac {21 a^{9} b^{2} e^{14}}{\left (a e - b d\right )^{8}} - \frac {189 a^{8} b^{3} d e^{13}}{\left (a e - b d\right )^{8}} + \frac {756 a^{7} b^{4} d^{2} e^{12}}{\left (a e - b d\right )^{8}} - \frac {1764 a^{6} b^{5} d^{3} e^{11}}{\left (a e - b d\right )^{8}} + \frac {2646 a^{5} b^{6} d^{4} e^{10}}{\left (a e - b d\right )^{8}} - \frac {2646 a^{4} b^{7} d^{5} e^{9}}{\left (a e - b d\right )^{8}} + \frac {1764 a^{3} b^{8} d^{6} e^{8}}{\left (a e - b d\right )^{8}} - \frac {756 a^{2} b^{9} d^{7} e^{7}}{\left (a e - b d\right )^{8}} + \frac {189 a b^{10} d^{8} e^{6}}{\left (a e - b d\right )^{8}} + 21 a b^{2} e^{6} - \frac {21 b^{11} d^{9} e^{5}}{\left (a e - b d\right )^{8}} + 21 b^{3} d e^{5}}{42 b^{3} e^{6}} \right )}}{\left (a e - b d\right )^{8}} + \frac {- 10 a^{6} e^{6} + 130 a^{5} b d e^{5} + 459 a^{4} b^{2} d^{2} e^{4} - 241 a^{3} b^{3} d^{3} e^{3} + 109 a^{2} b^{4} d^{4} e^{2} - 31 a b^{5} d^{5} e + 4 b^{6} d^{6} + 420 b^{6} e^{6} x^{6} + x^{5} \cdot \left (1890 a b^{5} e^{6} + 630 b^{6} d e^{5}\right ) + x^{4} \cdot \left (3290 a^{2} b^{4} e^{6} + 2870 a b^{5} d e^{5} + 140 b^{6} d^{2} e^{4}\right ) + x^{3} \cdot \left (2695 a^{3} b^{3} e^{6} + 5075 a^{2} b^{4} d e^{5} + 665 a b^{5} d^{2} e^{4} - 35 b^{6} d^{3} e^{3}\right ) + x^{2} \cdot \left (959 a^{4} b^{2} e^{6} + 4249 a^{3} b^{3} d e^{5} + 1239 a^{2} b^{4} d^{2} e^{4} - 161 a b^{5} d^{3} e^{3} + 14 b^{6} d^{4} e^{2}\right ) + x \left (70 a^{5} b e^{6} + 1568 a^{4} b^{2} d e^{5} + 1113 a^{3} b^{3} d^{2} e^{4} - 287 a^{2} b^{4} d^{3} e^{3} + 63 a b^{5} d^{4} e^{2} - 7 b^{6} d^{5} e\right )}{20 a^{12} d^{2} e^{7} - 140 a^{11} b d^{3} e^{6} + 420 a^{10} b^{2} d^{4} e^{5} - 700 a^{9} b^{3} d^{5} e^{4} + 700 a^{8} b^{4} d^{6} e^{3} - 420 a^{7} b^{5} d^{7} e^{2} + 140 a^{6} b^{6} d^{8} e - 20 a^{5} b^{7} d^{9} + x^{7} \cdot \left (20 a^{7} b^{5} e^{9} - 140 a^{6} b^{6} d e^{8} + 420 a^{5} b^{7} d^{2} e^{7} - 700 a^{4} b^{8} d^{3} e^{6} + 700 a^{3} b^{9} d^{4} e^{5} - 420 a^{2} b^{10} d^{5} e^{4} + 140 a b^{11} d^{6} e^{3} - 20 b^{12} d^{7} e^{2}\right ) + x^{6} \cdot \left (100 a^{8} b^{4} e^{9} - 660 a^{7} b^{5} d e^{8} + 1820 a^{6} b^{6} d^{2} e^{7} - 2660 a^{5} b^{7} d^{3} e^{6} + 2100 a^{4} b^{8} d^{4} e^{5} - 700 a^{3} b^{9} d^{5} e^{4} - 140 a^{2} b^{10} d^{6} e^{3} + 180 a b^{11} d^{7} e^{2} - 40 b^{12} d^{8} e\right ) + x^{5} \cdot \left (200 a^{9} b^{3} e^{9} - 1200 a^{8} b^{4} d e^{8} + 2820 a^{7} b^{5} d^{2} e^{7} - 2940 a^{6} b^{6} d^{3} e^{6} + 420 a^{5} b^{7} d^{4} e^{5} + 2100 a^{4} b^{8} d^{5} e^{4} - 2100 a^{3} b^{9} d^{6} e^{3} + 780 a^{2} b^{10} d^{7} e^{2} - 60 a b^{11} d^{8} e - 20 b^{12} d^{9}\right ) + x^{4} \cdot \left (200 a^{10} b^{2} e^{9} - 1000 a^{9} b^{3} d e^{8} + 1500 a^{8} b^{4} d^{2} e^{7} + 700 a^{7} b^{5} d^{3} e^{6} - 4900 a^{6} b^{6} d^{4} e^{5} + 6300 a^{5} b^{7} d^{5} e^{4} - 3500 a^{4} b^{8} d^{6} e^{3} + 500 a^{3} b^{9} d^{7} e^{2} + 300 a^{2} b^{10} d^{8} e - 100 a b^{11} d^{9}\right ) + x^{3} \cdot \left (100 a^{11} b e^{9} - 300 a^{10} b^{2} d e^{8} - 500 a^{9} b^{3} d^{2} e^{7} + 3500 a^{8} b^{4} d^{3} e^{6} - 6300 a^{7} b^{5} d^{4} e^{5} + 4900 a^{6} b^{6} d^{5} e^{4} - 700 a^{5} b^{7} d^{6} e^{3} - 1500 a^{4} b^{8} d^{7} e^{2} + 1000 a^{3} b^{9} d^{8} e - 200 a^{2} b^{10} d^{9}\right ) + x^{2} \cdot \left (20 a^{12} e^{9} + 60 a^{11} b d e^{8} - 780 a^{10} b^{2} d^{2} e^{7} + 2100 a^{9} b^{3} d^{3} e^{6} - 2100 a^{8} b^{4} d^{4} e^{5} - 420 a^{7} b^{5} d^{5} e^{4} + 2940 a^{6} b^{6} d^{6} e^{3} - 2820 a^{5} b^{7} d^{7} e^{2} + 1200 a^{4} b^{8} d^{8} e - 200 a^{3} b^{9} d^{9}\right ) + x \left (40 a^{12} d e^{8} - 180 a^{11} b d^{2} e^{7} + 140 a^{10} b^{2} d^{3} e^{6} + 700 a^{9} b^{3} d^{4} e^{5} - 2100 a^{8} b^{4} d^{5} e^{4} + 2660 a^{7} b^{5} d^{6} e^{3} - 1820 a^{6} b^{6} d^{7} e^{2} + 660 a^{5} b^{7} d^{8} e - 100 a^{4} b^{8} d^{9}\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. 672 vs.
\(2 (220) = 440\).
time = 1.78, size = 672, normalized size = 3.05 \begin {gather*} -\frac {21 \, b^{3} e^{5} \log \left ({\left | b x + a \right |}\right )}{b^{9} d^{8} - 8 \, a b^{8} d^{7} e + 28 \, a^{2} b^{7} d^{6} e^{2} - 56 \, a^{3} b^{6} d^{5} e^{3} + 70 \, a^{4} b^{5} d^{4} e^{4} - 56 \, a^{5} b^{4} d^{3} e^{5} + 28 \, a^{6} b^{3} d^{2} e^{6} - 8 \, a^{7} b^{2} d e^{7} + a^{8} b e^{8}} + \frac {21 \, b^{2} e^{6} \log \left ({\left | x e + d \right |}\right )}{b^{8} d^{8} e - 8 \, a b^{7} d^{7} e^{2} + 28 \, a^{2} b^{6} d^{6} e^{3} - 56 \, a^{3} b^{5} d^{5} e^{4} + 70 \, a^{4} b^{4} d^{4} e^{5} - 56 \, a^{5} b^{3} d^{3} e^{6} + 28 \, a^{6} b^{2} d^{2} e^{7} - 8 \, a^{7} b d e^{8} + a^{8} e^{9}} - \frac {4 \, b^{7} d^{7} - 35 \, a b^{6} d^{6} e + 140 \, a^{2} b^{5} d^{5} e^{2} - 350 \, a^{3} b^{4} d^{4} e^{3} + 700 \, a^{4} b^{3} d^{3} e^{4} - 329 \, a^{5} b^{2} d^{2} e^{5} - 140 \, a^{6} b d e^{6} + 10 \, a^{7} e^{7} + 420 \, {\left (b^{7} d e^{6} - a b^{6} e^{7}\right )} x^{6} + 630 \, {\left (b^{7} d^{2} e^{5} + 2 \, a b^{6} d e^{6} - 3 \, a^{2} b^{5} e^{7}\right )} x^{5} + 70 \, {\left (2 \, b^{7} d^{3} e^{4} + 39 \, a b^{6} d^{2} e^{5} + 6 \, a^{2} b^{5} d e^{6} - 47 \, a^{3} b^{4} e^{7}\right )} x^{4} - 35 \, {\left (b^{7} d^{4} e^{3} - 20 \, a b^{6} d^{3} e^{4} - 126 \, a^{2} b^{5} d^{2} e^{5} + 68 \, a^{3} b^{4} d e^{6} + 77 \, a^{4} b^{3} e^{7}\right )} x^{3} + 7 \, {\left (2 \, b^{7} d^{5} e^{2} - 25 \, a b^{6} d^{4} e^{3} + 200 \, a^{2} b^{5} d^{3} e^{4} + 430 \, a^{3} b^{4} d^{2} e^{5} - 470 \, a^{4} b^{3} d e^{6} - 137 \, a^{5} b^{2} e^{7}\right )} x^{2} - 7 \, {\left (b^{7} d^{6} e - 10 \, a b^{6} d^{5} e^{2} + 50 \, a^{2} b^{5} d^{4} e^{3} - 200 \, a^{3} b^{4} d^{3} e^{4} - 65 \, a^{4} b^{3} d^{2} e^{5} + 214 \, a^{5} b^{2} d e^{6} + 10 \, a^{6} b e^{7}\right )} x}{20 \, {\left (b d - a e\right )}^{8} {\left (b x + a\right )}^{5} {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.47, size = 1427, normalized size = 6.49 \begin {gather*} \frac {\frac {-10\,a^6\,e^6+130\,a^5\,b\,d\,e^5+459\,a^4\,b^2\,d^2\,e^4-241\,a^3\,b^3\,d^3\,e^3+109\,a^2\,b^4\,d^4\,e^2-31\,a\,b^5\,d^5\,e+4\,b^6\,d^6}{20\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}+\frac {7\,e^3\,x^3\,\left (77\,a^3\,b^3\,e^3+145\,a^2\,b^4\,d\,e^2+19\,a\,b^5\,d^2\,e-b^6\,d^3\right )}{4\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}+\frac {21\,b^6\,e^6\,x^6}{a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7}+\frac {7\,e^2\,x^2\,\left (137\,a^4\,b^2\,e^4+607\,a^3\,b^3\,d\,e^3+177\,a^2\,b^4\,d^2\,e^2-23\,a\,b^5\,d^3\,e+2\,b^6\,d^4\right )}{20\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}+\frac {7\,e^4\,x^4\,\left (47\,a^2\,b^4\,e^2+41\,a\,b^5\,d\,e+2\,b^6\,d^2\right )}{2\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}+\frac {7\,e\,x\,\left (10\,a^5\,b\,e^5+224\,a^4\,b^2\,d\,e^4+159\,a^3\,b^3\,d^2\,e^3-41\,a^2\,b^4\,d^3\,e^2+9\,a\,b^5\,d^4\,e-b^6\,d^5\right )}{20\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}+\frac {63\,b\,e^4\,x^5\,\left (d\,b^5\,e+3\,a\,b^4\,e^2\right )}{2\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}}{x^3\,\left (5\,a^4\,b\,e^2+20\,a^3\,b^2\,d\,e+10\,a^2\,b^3\,d^2\right )+x^4\,\left (10\,a^3\,b^2\,e^2+20\,a^2\,b^3\,d\,e+5\,a\,b^4\,d^2\right )+x\,\left (2\,e\,a^5\,d+5\,b\,a^4\,d^2\right )+x^2\,\left (a^5\,e^2+10\,a^4\,b\,d\,e+10\,a^3\,b^2\,d^2\right )+x^5\,\left (10\,a^2\,b^3\,e^2+10\,a\,b^4\,d\,e+b^5\,d^2\right )+x^6\,\left (2\,d\,b^5\,e+5\,a\,b^4\,e^2\right )+a^5\,d^2+b^5\,e^2\,x^7}-\frac {42\,b^2\,e^5\,\mathrm {atanh}\left (\frac {a^8\,e^8-6\,a^7\,b\,d\,e^7+14\,a^6\,b^2\,d^2\,e^6-14\,a^5\,b^3\,d^3\,e^5+14\,a^3\,b^5\,d^5\,e^3-14\,a^2\,b^6\,d^6\,e^2+6\,a\,b^7\,d^7\,e-b^8\,d^8}{{\left (a\,e-b\,d\right )}^8}+\frac {2\,b\,e\,x\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}{{\left (a\,e-b\,d\right )}^8}\right )}{{\left (a\,e-b\,d\right )}^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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