Optimal. Leaf size=278 \[ -\frac {b^5 \log (d+e x) (-6 a B e-A b e+7 b B d)}{e^8}-\frac {3 b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^8 (d+e x)}+\frac {5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{2 e^8 (d+e x)^2}-\frac {5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{3 e^8 (d+e x)^3}+\frac {3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{4 e^8 (d+e x)^4}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{5 e^8 (d+e x)^5}+\frac {(b d-a e)^6 (B d-A e)}{6 e^8 (d+e x)^6}+\frac {b^6 B x}{e^7} \]
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Rubi [A] time = 0.33, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} -\frac {3 b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^8 (d+e x)}+\frac {5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{2 e^8 (d+e x)^2}-\frac {5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{3 e^8 (d+e x)^3}-\frac {b^5 \log (d+e x) (-6 a B e-A b e+7 b B d)}{e^8}+\frac {3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{4 e^8 (d+e x)^4}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{5 e^8 (d+e x)^5}+\frac {(b d-a e)^6 (B d-A e)}{6 e^8 (d+e x)^6}+\frac {b^6 B x}{e^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^7} \, dx &=\int \left (\frac {b^6 B}{e^7}+\frac {(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)^7}+\frac {(-b d+a e)^5 (-7 b B d+6 A b e+a B e)}{e^7 (d+e x)^6}+\frac {3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e)}{e^7 (d+e x)^5}-\frac {5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e)}{e^7 (d+e x)^4}+\frac {5 b^3 (b d-a e)^2 (-7 b B d+3 A b e+4 a B e)}{e^7 (d+e x)^3}-\frac {3 b^4 (b d-a e) (-7 b B d+2 A b e+5 a B e)}{e^7 (d+e x)^2}+\frac {b^5 (-7 b B d+A b e+6 a B e)}{e^7 (d+e x)}\right ) \, dx\\ &=\frac {b^6 B x}{e^7}+\frac {(b d-a e)^6 (B d-A e)}{6 e^8 (d+e x)^6}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{5 e^8 (d+e x)^5}+\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{4 e^8 (d+e x)^4}-\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{3 e^8 (d+e x)^3}+\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{2 e^8 (d+e x)^2}-\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e)}{e^8 (d+e x)}-\frac {b^5 (7 b B d-A b e-6 a B e) \log (d+e x)}{e^8}\\ \end {align*}
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Mathematica [B] time = 0.35, size = 619, normalized size = 2.23 \begin {gather*} -\frac {2 a^6 e^6 (5 A e+B (d+6 e x))+6 a^5 b e^5 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+15 a^4 b^2 e^4 \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+20 a^3 b^3 e^3 \left (A e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )+30 a^2 b^4 e^2 \left (A e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+5 B \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )-6 a b^5 e \left (B d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )-10 A e \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )+60 b^5 (d+e x)^6 \log (d+e x) (-6 a B e-A b e+7 b B d)-\left (b^6 \left (A d e \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )-B \left (669 d^7+3594 d^6 e x+7725 d^5 e^2 x^2+8200 d^4 e^3 x^3+4050 d^3 e^4 x^4+360 d^2 e^5 x^5-360 d e^6 x^6-60 e^7 x^7\right )\right )\right )}{60 e^8 (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^7} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.41, size = 1063, normalized size = 3.82 \begin {gather*} \frac {60 \, B b^{6} e^{7} x^{7} + 360 \, B b^{6} d e^{6} x^{6} - 669 \, B b^{6} d^{7} - 10 \, A a^{6} e^{7} + 147 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e - 30 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} - 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} - 2 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} - 180 \, {\left (2 \, B b^{6} d^{2} e^{5} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} - 150 \, {\left (27 \, B b^{6} d^{3} e^{4} - 9 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} - 100 \, {\left (82 \, B b^{6} d^{4} e^{3} - 22 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 2 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} - 15 \, {\left (515 \, B b^{6} d^{5} e^{2} - 125 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 30 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} - 6 \, {\left (599 \, B b^{6} d^{6} e - 137 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 30 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 2 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x - 60 \, {\left (7 \, B b^{6} d^{7} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + {\left (7 \, B b^{6} d e^{6} - {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 6 \, {\left (7 \, B b^{6} d^{2} e^{5} - {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6}\right )} x^{5} + 15 \, {\left (7 \, B b^{6} d^{3} e^{4} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5}\right )} x^{4} + 20 \, {\left (7 \, B b^{6} d^{4} e^{3} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4}\right )} x^{3} + 15 \, {\left (7 \, B b^{6} d^{5} e^{2} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3}\right )} x^{2} + 6 \, {\left (7 \, B b^{6} d^{6} e - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2}\right )} x\right )} \log \left (e x + d\right )}{60 \, {\left (e^{14} x^{6} + 6 \, d e^{13} x^{5} + 15 \, d^{2} e^{12} x^{4} + 20 \, d^{3} e^{11} x^{3} + 15 \, d^{4} e^{10} x^{2} + 6 \, d^{5} e^{9} x + d^{6} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.28, size = 775, normalized size = 2.79 \begin {gather*} B b^{6} x e^{\left (-7\right )} - {\left (7 \, B b^{6} d - 6 \, B a b^{5} e - A b^{6} e\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (669 \, B b^{6} d^{7} - 882 \, B a b^{5} d^{6} e - 147 \, A b^{6} d^{6} e + 150 \, B a^{2} b^{4} d^{5} e^{2} + 60 \, A a b^{5} d^{5} e^{2} + 40 \, B a^{3} b^{3} d^{4} e^{3} + 30 \, A a^{2} b^{4} d^{4} e^{3} + 15 \, B a^{4} b^{2} d^{3} e^{4} + 20 \, A a^{3} b^{3} d^{3} e^{4} + 6 \, B a^{5} b d^{2} e^{5} + 15 \, A a^{4} b^{2} d^{2} e^{5} + 2 \, B a^{6} d e^{6} + 12 \, A a^{5} b d e^{6} + 10 \, A a^{6} e^{7} + 180 \, {\left (7 \, B b^{6} d^{2} e^{5} - 12 \, B a b^{5} d e^{6} - 2 \, A b^{6} d e^{6} + 5 \, B a^{2} b^{4} e^{7} + 2 \, A a b^{5} e^{7}\right )} x^{5} + 150 \, {\left (35 \, B b^{6} d^{3} e^{4} - 54 \, B a b^{5} d^{2} e^{5} - 9 \, A b^{6} d^{2} e^{5} + 15 \, B a^{2} b^{4} d e^{6} + 6 \, A a b^{5} d e^{6} + 4 \, B a^{3} b^{3} e^{7} + 3 \, A a^{2} b^{4} e^{7}\right )} x^{4} + 100 \, {\left (91 \, B b^{6} d^{4} e^{3} - 132 \, B a b^{5} d^{3} e^{4} - 22 \, A b^{6} d^{3} e^{4} + 30 \, B a^{2} b^{4} d^{2} e^{5} + 12 \, A a b^{5} d^{2} e^{5} + 8 \, B a^{3} b^{3} d e^{6} + 6 \, A a^{2} b^{4} d e^{6} + 3 \, B a^{4} b^{2} e^{7} + 4 \, A a^{3} b^{3} e^{7}\right )} x^{3} + 15 \, {\left (539 \, B b^{6} d^{5} e^{2} - 750 \, B a b^{5} d^{4} e^{3} - 125 \, A b^{6} d^{4} e^{3} + 150 \, B a^{2} b^{4} d^{3} e^{4} + 60 \, A a b^{5} d^{3} e^{4} + 40 \, B a^{3} b^{3} d^{2} e^{5} + 30 \, A a^{2} b^{4} d^{2} e^{5} + 15 \, B a^{4} b^{2} d e^{6} + 20 \, A a^{3} b^{3} d e^{6} + 6 \, B a^{5} b e^{7} + 15 \, A a^{4} b^{2} e^{7}\right )} x^{2} + 6 \, {\left (609 \, B b^{6} d^{6} e - 822 \, B a b^{5} d^{5} e^{2} - 137 \, A b^{6} d^{5} e^{2} + 150 \, B a^{2} b^{4} d^{4} e^{3} + 60 \, A a b^{5} d^{4} e^{3} + 40 \, B a^{3} b^{3} d^{3} e^{4} + 30 \, A a^{2} b^{4} d^{3} e^{4} + 15 \, B a^{4} b^{2} d^{2} e^{5} + 20 \, A a^{3} b^{3} d^{2} e^{5} + 6 \, B a^{5} b d e^{6} + 15 \, A a^{4} b^{2} d e^{6} + 2 \, B a^{6} e^{7} + 12 \, A a^{5} b e^{7}\right )} x\right )} e^{\left (-8\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 1217, normalized size = 4.38
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.74, size = 824, normalized size = 2.96 \begin {gather*} \frac {B b^{6} x}{e^{7}} - \frac {669 \, B b^{6} d^{7} + 10 \, A a^{6} e^{7} - 147 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 30 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 2 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 180 \, {\left (7 \, B b^{6} d^{2} e^{5} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 150 \, {\left (35 \, B b^{6} d^{3} e^{4} - 9 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 100 \, {\left (91 \, B b^{6} d^{4} e^{3} - 22 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 2 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 15 \, {\left (539 \, B b^{6} d^{5} e^{2} - 125 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 30 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 6 \, {\left (609 \, B b^{6} d^{6} e - 137 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 30 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 2 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{60 \, {\left (e^{14} x^{6} + 6 \, d e^{13} x^{5} + 15 \, d^{2} e^{12} x^{4} + 20 \, d^{3} e^{11} x^{3} + 15 \, d^{4} e^{10} x^{2} + 6 \, d^{5} e^{9} x + d^{6} e^{8}\right )}} - \frac {{\left (7 \, B b^{6} d - {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} \log \left (e x + d\right )}{e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 875, normalized size = 3.15 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (A\,b^6\,e-7\,B\,b^6\,d+6\,B\,a\,b^5\,e\right )}{e^8}-\frac {x^3\,\left (5\,B\,a^4\,b^2\,e^6+\frac {40\,B\,a^3\,b^3\,d\,e^5}{3}+\frac {20\,A\,a^3\,b^3\,e^6}{3}+50\,B\,a^2\,b^4\,d^2\,e^4+10\,A\,a^2\,b^4\,d\,e^5-220\,B\,a\,b^5\,d^3\,e^3+20\,A\,a\,b^5\,d^2\,e^4+\frac {455\,B\,b^6\,d^4\,e^2}{3}-\frac {110\,A\,b^6\,d^3\,e^3}{3}\right )+\frac {2\,B\,a^6\,d\,e^6+10\,A\,a^6\,e^7+6\,B\,a^5\,b\,d^2\,e^5+12\,A\,a^5\,b\,d\,e^6+15\,B\,a^4\,b^2\,d^3\,e^4+15\,A\,a^4\,b^2\,d^2\,e^5+40\,B\,a^3\,b^3\,d^4\,e^3+20\,A\,a^3\,b^3\,d^3\,e^4+150\,B\,a^2\,b^4\,d^5\,e^2+30\,A\,a^2\,b^4\,d^4\,e^3-882\,B\,a\,b^5\,d^6\,e+60\,A\,a\,b^5\,d^5\,e^2+669\,B\,b^6\,d^7-147\,A\,b^6\,d^6\,e}{60\,e}+x\,\left (\frac {B\,a^6\,e^6}{5}+\frac {3\,B\,a^5\,b\,d\,e^5}{5}+\frac {6\,A\,a^5\,b\,e^6}{5}+\frac {3\,B\,a^4\,b^2\,d^2\,e^4}{2}+\frac {3\,A\,a^4\,b^2\,d\,e^5}{2}+4\,B\,a^3\,b^3\,d^3\,e^3+2\,A\,a^3\,b^3\,d^2\,e^4+15\,B\,a^2\,b^4\,d^4\,e^2+3\,A\,a^2\,b^4\,d^3\,e^3-\frac {411\,B\,a\,b^5\,d^5\,e}{5}+6\,A\,a\,b^5\,d^4\,e^2+\frac {609\,B\,b^6\,d^6}{10}-\frac {137\,A\,b^6\,d^5\,e}{10}\right )+x^5\,\left (15\,B\,a^2\,b^4\,e^6-36\,B\,a\,b^5\,d\,e^5+6\,A\,a\,b^5\,e^6+21\,B\,b^6\,d^2\,e^4-6\,A\,b^6\,d\,e^5\right )+x^2\,\left (\frac {3\,B\,a^5\,b\,e^6}{2}+\frac {15\,B\,a^4\,b^2\,d\,e^5}{4}+\frac {15\,A\,a^4\,b^2\,e^6}{4}+10\,B\,a^3\,b^3\,d^2\,e^4+5\,A\,a^3\,b^3\,d\,e^5+\frac {75\,B\,a^2\,b^4\,d^3\,e^3}{2}+\frac {15\,A\,a^2\,b^4\,d^2\,e^4}{2}-\frac {375\,B\,a\,b^5\,d^4\,e^2}{2}+15\,A\,a\,b^5\,d^3\,e^3+\frac {539\,B\,b^6\,d^5\,e}{4}-\frac {125\,A\,b^6\,d^4\,e^2}{4}\right )+x^4\,\left (10\,B\,a^3\,b^3\,e^6+\frac {75\,B\,a^2\,b^4\,d\,e^5}{2}+\frac {15\,A\,a^2\,b^4\,e^6}{2}-135\,B\,a\,b^5\,d^2\,e^4+15\,A\,a\,b^5\,d\,e^5+\frac {175\,B\,b^6\,d^3\,e^3}{2}-\frac {45\,A\,b^6\,d^2\,e^4}{2}\right )}{d^6\,e^7+6\,d^5\,e^8\,x+15\,d^4\,e^9\,x^2+20\,d^3\,e^{10}\,x^3+15\,d^2\,e^{11}\,x^4+6\,d\,e^{12}\,x^5+e^{13}\,x^6}+\frac {B\,b^6\,x}{e^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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