Optimal. Leaf size=521 \[ -\frac {\left (B (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-A c e \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )\right ) x}{e^7}-\frac {c \left (A c e (4 c d-3 b e)-B \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )\right ) x^2}{2 e^6}-\frac {c^2 (4 B c d-3 b B e-A c e) x^3}{3 e^5}+\frac {B c^3 x^4}{4 e^4}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^3}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{2 e^8 (d+e x)^2}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{e^8 (d+e x)}-\frac {\left (A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) \log (d+e x)}{e^8} \]
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Rubi [A]
time = 0.66, antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {785}
\begin {gather*} -\frac {\log (d+e x) \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{e^8}-\frac {x \left (B (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-A c e \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )\right )}{e^7}-\frac {c x^2 \left (A c e (4 c d-3 b e)-B \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )\right )}{2 e^6}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8 (d+e x)}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{2 e^8 (d+e x)^2}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^3}-\frac {c^2 x^3 (-A c e-3 b B e+4 B c d)}{3 e^5}+\frac {B c^3 x^4}{4 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 785
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx &=\int \left (\frac {-B (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )+A c e \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )}{e^7}+\frac {c \left (-A c e (4 c d-3 b e)+B \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )\right ) x}{e^6}+\frac {c^2 (-4 B c d+3 b B e+A c e) x^2}{e^5}+\frac {B c^3 x^3}{e^4}+\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)^4}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{e^7 (d+e x)^3}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (-B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )+A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{e^7 (d+e x)^2}+\frac {-A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )+B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{e^7 (d+e x)}\right ) \, dx\\ &=-\frac {\left (B (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-A c e \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )\right ) x}{e^7}-\frac {c \left (A c e (4 c d-3 b e)-B \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )\right ) x^2}{2 e^6}-\frac {c^2 (4 B c d-3 b B e-A c e) x^3}{3 e^5}+\frac {B c^3 x^4}{4 e^4}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^3}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{2 e^8 (d+e x)^2}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{e^8 (d+e x)}-\frac {\left (A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) \log (d+e x)}{e^8}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 488, normalized size = 0.94 \begin {gather*} \frac {12 e \left (A c e \left (10 c^2 d^2+3 b^2 e^2+3 c e (-4 b d+a e)\right )-B (2 c d-b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right )\right ) x+6 c e^2 \left (A c e (-4 c d+3 b e)+B \left (10 c^2 d^2+3 b^2 e^2+3 c e (-4 b d+a e)\right )\right ) x^2+4 c^2 e^3 (-4 B c d+3 b B e+A c e) x^3+3 B c^3 e^4 x^4+\frac {4 (B d-A e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^3}-\frac {6 \left (c d^2+e (-b d+a e)\right )^2 \left (7 B c d^2+B e (-4 b d+a e)+3 A e (-2 c d+b e)\right )}{(d+e x)^2}+\frac {36 \left (c d^2+e (-b d+a e)\right ) \left (-A e \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right )+B \left (7 c^2 d^3+b e^2 (2 b d-a e)+c d e (-8 b d+3 a e)\right )\right )}{d+e x}+12 \left (A e (-2 c d+b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right )+B \left (35 c^3 d^4+30 c^2 d^2 e (-2 b d+a e)+b^2 e^3 (-4 b d+3 a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) \log (d+e x)}{12 e^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1088\) vs.
\(2(511)=1022\).
time = 0.07, size = 1089, normalized size = 2.09 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 946, normalized size = 1.82 \begin {gather*} {\left (35 \, B c^{3} d^{4} + 3 \, B a b^{2} e^{4} + A b^{3} e^{4} - 20 \, {\left (3 \, B b c^{2} e + A c^{3} e\right )} d^{3} + 30 \, {\left (B b^{2} c e^{2} + {\left (B a e^{2} + A b e^{2}\right )} c^{2}\right )} d^{2} + 3 \, {\left (B a^{2} e^{4} + 2 \, A a b e^{4}\right )} c - 4 \, {\left (B b^{3} e^{3} + 3 \, A a c^{2} e^{3} + 3 \, {\left (2 \, B a b e^{3} + A b^{2} e^{3}\right )} c\right )} d\right )} e^{\left (-8\right )} \log \left (x e + d\right ) + \frac {1}{12} \, {\left (3 \, B c^{3} x^{4} e^{3} - 4 \, {\left (4 \, B c^{3} d e^{2} - 3 \, B b c^{2} e^{3} - A c^{3} e^{3}\right )} x^{3} + 6 \, {\left (10 \, B c^{3} d^{2} e + 3 \, B b^{2} c e^{3} + 3 \, {\left (B a e^{3} + A b e^{3}\right )} c^{2} - 4 \, {\left (3 \, B b c^{2} e^{2} + A c^{3} e^{2}\right )} d\right )} x^{2} - 12 \, {\left (20 \, B c^{3} d^{3} - B b^{3} e^{3} - 3 \, A a c^{2} e^{3} - 10 \, {\left (3 \, B b c^{2} e + A c^{3} e\right )} d^{2} - 3 \, {\left (2 \, B a b e^{3} + A b^{2} e^{3}\right )} c + 12 \, {\left (B b^{2} c e^{2} + {\left (B a e^{2} + A b e^{2}\right )} c^{2}\right )} d\right )} x\right )} e^{\left (-7\right )} + \frac {107 \, B c^{3} d^{7} - 74 \, {\left (3 \, B b c^{2} e + A c^{3} e\right )} d^{6} + 141 \, {\left (B b^{2} c e^{2} + {\left (B a e^{2} + A b e^{2}\right )} c^{2}\right )} d^{5} - 26 \, {\left (B b^{3} e^{3} + 3 \, A a c^{2} e^{3} + 3 \, {\left (2 \, B a b e^{3} + A b^{2} e^{3}\right )} c\right )} d^{4} - 2 \, A a^{3} e^{7} + 11 \, {\left (3 \, B a b^{2} e^{4} + A b^{3} e^{4} + 3 \, {\left (B a^{2} e^{4} + 2 \, A a b e^{4}\right )} c\right )} d^{3} - 6 \, {\left (B a^{2} b e^{5} + A a b^{2} e^{5} + A a^{2} c e^{5}\right )} d^{2} + 18 \, {\left (7 \, B c^{3} d^{5} e^{2} - 5 \, {\left (3 \, B b c^{2} e^{3} + A c^{3} e^{3}\right )} d^{4} - B a^{2} b e^{7} - A a b^{2} e^{7} - A a^{2} c e^{7} + 10 \, {\left (B b^{2} c e^{4} + {\left (B a e^{4} + A b e^{4}\right )} c^{2}\right )} d^{3} - 2 \, {\left (B b^{3} e^{5} + 3 \, A a c^{2} e^{5} + 3 \, {\left (2 \, B a b e^{5} + A b^{2} e^{5}\right )} c\right )} d^{2} + {\left (3 \, B a b^{2} e^{6} + A b^{3} e^{6} + 3 \, {\left (B a^{2} e^{6} + 2 \, A a b e^{6}\right )} c\right )} d\right )} x^{2} - {\left (B a^{3} e^{6} + 3 \, A a^{2} b e^{6}\right )} d + 3 \, {\left (77 \, B c^{3} d^{6} e - 54 \, {\left (3 \, B b c^{2} e^{2} + A c^{3} e^{2}\right )} d^{5} + 105 \, {\left (B b^{2} c e^{3} + {\left (B a e^{3} + A b e^{3}\right )} c^{2}\right )} d^{4} - B a^{3} e^{7} - 3 \, A a^{2} b e^{7} - 20 \, {\left (B b^{3} e^{4} + 3 \, A a c^{2} e^{4} + 3 \, {\left (2 \, B a b e^{4} + A b^{2} e^{4}\right )} c\right )} d^{3} + 9 \, {\left (3 \, B a b^{2} e^{5} + A b^{3} e^{5} + 3 \, {\left (B a^{2} e^{5} + 2 \, A a b e^{5}\right )} c\right )} d^{2} - 6 \, {\left (B a^{2} b e^{6} + A a b^{2} e^{6} + A a^{2} c e^{6}\right )} d\right )} x}{6 \, {\left (x^{3} e^{11} + 3 \, d x^{2} e^{10} + 3 \, d^{2} x e^{9} + d^{3} e^{8}\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. 1332 vs.
\(2 (528) = 1056\).
time = 3.89, size = 1332, normalized size = 2.56 \begin {gather*} \frac {214 \, B c^{3} d^{7} + {\left (3 \, B c^{3} x^{7} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 18 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 12 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 4 \, A a^{3} - 36 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} e^{7} - {\left (7 \, B c^{3} d x^{6} + 12 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d x^{5} + 90 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d x^{4} - 36 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d x^{3} - 36 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d x^{2} + 36 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d x + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} e^{6} + 3 \, {\left (7 \, B c^{3} d^{2} x^{5} + 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} x^{4} - 126 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} x^{3} - 12 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} x^{2} + 18 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} x - 4 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2}\right )} e^{5} - {\left (105 \, B c^{3} d^{3} x^{4} - 292 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} x^{3} + 54 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} x^{2} + 108 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} x - 22 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3}\right )} e^{4} - 2 \, {\left (278 \, B c^{3} d^{4} x^{3} - 78 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} x^{2} - 243 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} x + 26 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4}\right )} e^{3} - 6 \, {\left (68 \, B c^{3} d^{5} x^{2} + 34 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} x - 47 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5}\right )} e^{2} + 74 \, {\left (3 \, B c^{3} d^{6} x - 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6}\right )} e + 12 \, {\left (35 \, B c^{3} d^{7} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} e^{7} - {\left (4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d x^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d x^{2}\right )} e^{6} + 3 \, {\left (10 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} x^{3} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} x^{2} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} x\right )} e^{5} - {\left (20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} x^{3} - 90 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} x^{2} + 12 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} x - {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3}\right )} e^{4} + {\left (35 \, B c^{3} d^{4} x^{3} - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} x^{2} + 90 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} x - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4}\right )} e^{3} + 15 \, {\left (7 \, B c^{3} d^{5} x^{2} - 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} x + 2 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5}\right )} e^{2} + 5 \, {\left (21 \, B c^{3} d^{6} x - 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6}\right )} e\right )} \log \left (x e + d\right )}{12 \, {\left (x^{3} e^{11} + 3 \, d x^{2} e^{10} + 3 \, d^{2} x e^{9} + d^{3} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 1.76, size = 1021, normalized size = 1.96 \begin {gather*} {\left (35 \, B c^{3} d^{4} - 60 \, B b c^{2} d^{3} e - 20 \, A c^{3} d^{3} e + 30 \, B b^{2} c d^{2} e^{2} + 30 \, B a c^{2} d^{2} e^{2} + 30 \, A b c^{2} d^{2} e^{2} - 4 \, B b^{3} d e^{3} - 24 \, B a b c d e^{3} - 12 \, A b^{2} c d e^{3} - 12 \, A a c^{2} d e^{3} + 3 \, B a b^{2} e^{4} + A b^{3} e^{4} + 3 \, B a^{2} c e^{4} + 6 \, A a b c e^{4}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (3 \, B c^{3} x^{4} e^{12} - 16 \, B c^{3} d x^{3} e^{11} + 60 \, B c^{3} d^{2} x^{2} e^{10} - 240 \, B c^{3} d^{3} x e^{9} + 12 \, B b c^{2} x^{3} e^{12} + 4 \, A c^{3} x^{3} e^{12} - 72 \, B b c^{2} d x^{2} e^{11} - 24 \, A c^{3} d x^{2} e^{11} + 360 \, B b c^{2} d^{2} x e^{10} + 120 \, A c^{3} d^{2} x e^{10} + 18 \, B b^{2} c x^{2} e^{12} + 18 \, B a c^{2} x^{2} e^{12} + 18 \, A b c^{2} x^{2} e^{12} - 144 \, B b^{2} c d x e^{11} - 144 \, B a c^{2} d x e^{11} - 144 \, A b c^{2} d x e^{11} + 12 \, B b^{3} x e^{12} + 72 \, B a b c x e^{12} + 36 \, A b^{2} c x e^{12} + 36 \, A a c^{2} x e^{12}\right )} e^{\left (-16\right )} + \frac {{\left (107 \, B c^{3} d^{7} - 222 \, B b c^{2} d^{6} e - 74 \, A c^{3} d^{6} e + 141 \, B b^{2} c d^{5} e^{2} + 141 \, B a c^{2} d^{5} e^{2} + 141 \, A b c^{2} d^{5} e^{2} - 26 \, B b^{3} d^{4} e^{3} - 156 \, B a b c d^{4} e^{3} - 78 \, A b^{2} c d^{4} e^{3} - 78 \, A a c^{2} d^{4} e^{3} + 33 \, B a b^{2} d^{3} e^{4} + 11 \, A b^{3} d^{3} e^{4} + 33 \, B a^{2} c d^{3} e^{4} + 66 \, A a b c d^{3} e^{4} - 6 \, B a^{2} b d^{2} e^{5} - 6 \, A a b^{2} d^{2} e^{5} - 6 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 3 \, A a^{2} b d e^{6} - 2 \, A a^{3} e^{7} + 18 \, {\left (7 \, B c^{3} d^{5} e^{2} - 15 \, B b c^{2} d^{4} e^{3} - 5 \, A c^{3} d^{4} e^{3} + 10 \, B b^{2} c d^{3} e^{4} + 10 \, B a c^{2} d^{3} e^{4} + 10 \, A b c^{2} d^{3} e^{4} - 2 \, B b^{3} d^{2} e^{5} - 12 \, B a b c d^{2} e^{5} - 6 \, A b^{2} c d^{2} e^{5} - 6 \, A a c^{2} d^{2} e^{5} + 3 \, B a b^{2} d e^{6} + A b^{3} d e^{6} + 3 \, B a^{2} c d e^{6} + 6 \, A a b c d e^{6} - B a^{2} b e^{7} - A a b^{2} e^{7} - A a^{2} c e^{7}\right )} x^{2} + 3 \, {\left (77 \, B c^{3} d^{6} e - 162 \, B b c^{2} d^{5} e^{2} - 54 \, A c^{3} d^{5} e^{2} + 105 \, B b^{2} c d^{4} e^{3} + 105 \, B a c^{2} d^{4} e^{3} + 105 \, A b c^{2} d^{4} e^{3} - 20 \, B b^{3} d^{3} e^{4} - 120 \, B a b c d^{3} e^{4} - 60 \, A b^{2} c d^{3} e^{4} - 60 \, A a c^{2} d^{3} e^{4} + 27 \, B a b^{2} d^{2} e^{5} + 9 \, A b^{3} d^{2} e^{5} + 27 \, B a^{2} c d^{2} e^{5} + 54 \, A a b c d^{2} e^{5} - 6 \, B a^{2} b d e^{6} - 6 \, A a b^{2} d e^{6} - 6 \, A a^{2} c d e^{6} - B a^{3} e^{7} - 3 \, A a^{2} b e^{7}\right )} x\right )} e^{\left (-8\right )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.53, size = 1151, normalized size = 2.21 \begin {gather*} x\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{e^4}-\frac {6\,d^2\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e^2}+\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e^4}+\frac {6\,B\,c^3\,d^2}{e^6}\right )}{e}-\frac {4\,B\,c^3\,d^3}{e^7}\right )-x^2\,\left (\frac {2\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{2\,e^4}+\frac {3\,B\,c^3\,d^2}{e^6}\right )-\frac {\frac {B\,a^3\,d\,e^6+2\,A\,a^3\,e^7+6\,B\,a^2\,b\,d^2\,e^5+3\,A\,a^2\,b\,d\,e^6-33\,B\,a^2\,c\,d^3\,e^4+6\,A\,a^2\,c\,d^2\,e^5-33\,B\,a\,b^2\,d^3\,e^4+6\,A\,a\,b^2\,d^2\,e^5+156\,B\,a\,b\,c\,d^4\,e^3-66\,A\,a\,b\,c\,d^3\,e^4-141\,B\,a\,c^2\,d^5\,e^2+78\,A\,a\,c^2\,d^4\,e^3+26\,B\,b^3\,d^4\,e^3-11\,A\,b^3\,d^3\,e^4-141\,B\,b^2\,c\,d^5\,e^2+78\,A\,b^2\,c\,d^4\,e^3+222\,B\,b\,c^2\,d^6\,e-141\,A\,b\,c^2\,d^5\,e^2-107\,B\,c^3\,d^7+74\,A\,c^3\,d^6\,e}{6\,e}+x\,\left (\frac {B\,a^3\,e^6}{2}+3\,B\,a^2\,b\,d\,e^5+\frac {3\,A\,a^2\,b\,e^6}{2}-\frac {27\,B\,a^2\,c\,d^2\,e^4}{2}+3\,A\,a^2\,c\,d\,e^5-\frac {27\,B\,a\,b^2\,d^2\,e^4}{2}+3\,A\,a\,b^2\,d\,e^5+60\,B\,a\,b\,c\,d^3\,e^3-27\,A\,a\,b\,c\,d^2\,e^4-\frac {105\,B\,a\,c^2\,d^4\,e^2}{2}+30\,A\,a\,c^2\,d^3\,e^3+10\,B\,b^3\,d^3\,e^3-\frac {9\,A\,b^3\,d^2\,e^4}{2}-\frac {105\,B\,b^2\,c\,d^4\,e^2}{2}+30\,A\,b^2\,c\,d^3\,e^3+81\,B\,b\,c^2\,d^5\,e-\frac {105\,A\,b\,c^2\,d^4\,e^2}{2}-\frac {77\,B\,c^3\,d^6}{2}+27\,A\,c^3\,d^5\,e\right )+x^2\,\left (3\,B\,a^2\,b\,e^6-9\,B\,a^2\,c\,d\,e^5+3\,A\,a^2\,c\,e^6-9\,B\,a\,b^2\,d\,e^5+3\,A\,a\,b^2\,e^6+36\,B\,a\,b\,c\,d^2\,e^4-18\,A\,a\,b\,c\,d\,e^5-30\,B\,a\,c^2\,d^3\,e^3+18\,A\,a\,c^2\,d^2\,e^4+6\,B\,b^3\,d^2\,e^4-3\,A\,b^3\,d\,e^5-30\,B\,b^2\,c\,d^3\,e^3+18\,A\,b^2\,c\,d^2\,e^4+45\,B\,b\,c^2\,d^4\,e^2-30\,A\,b\,c^2\,d^3\,e^3-21\,B\,c^3\,d^5\,e+15\,A\,c^3\,d^4\,e^2\right )}{d^3\,e^7+3\,d^2\,e^8\,x+3\,d\,e^9\,x^2+e^{10}\,x^3}+x^3\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{3\,e^4}-\frac {4\,B\,c^3\,d}{3\,e^5}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,B\,a^2\,c\,e^4+3\,B\,a\,b^2\,e^4-24\,B\,a\,b\,c\,d\,e^3+6\,A\,a\,b\,c\,e^4+30\,B\,a\,c^2\,d^2\,e^2-12\,A\,a\,c^2\,d\,e^3-4\,B\,b^3\,d\,e^3+A\,b^3\,e^4+30\,B\,b^2\,c\,d^2\,e^2-12\,A\,b^2\,c\,d\,e^3-60\,B\,b\,c^2\,d^3\,e+30\,A\,b\,c^2\,d^2\,e^2+35\,B\,c^3\,d^4-20\,A\,c^3\,d^3\,e\right )}{e^8}+\frac {B\,c^3\,x^4}{4\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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