Optimal. Leaf size=426 \[ -\frac {c \left (35 c^3 d^3-4 b^3 e^3+6 b c e^2 (5 b d-2 a e)-20 c^2 d e (3 b d-a e)\right ) x}{e^8}+\frac {c^2 \left (15 c^2 d^2+6 b^2 e^2-4 c e (5 b d-a e)\right ) x^2}{2 e^7}-\frac {c^3 (5 c d-4 b e) x^3}{3 e^6}+\frac {c^4 x^4}{4 e^5}-\frac {\left (c d^2-b d e+a e^2\right )^4}{4 e^9 (d+e x)^4}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^9 (d+e x)^3}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^9 (d+e x)^2}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \log (d+e x)}{e^9} \]
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Rubi [A]
time = 0.43, antiderivative size = 426, 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 = {712}
\begin {gather*} \frac {\log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^9}-\frac {c x \left (-20 c^2 d e (3 b d-a e)+6 b c e^2 (5 b d-2 a e)-4 b^3 e^3+35 c^3 d^3\right )}{e^8}-\frac {\left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )^2}{e^9 (d+e x)^2}+\frac {4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^9 (d+e x)}+\frac {c^2 x^2 \left (-4 c e (5 b d-a e)+6 b^2 e^2+15 c^2 d^2\right )}{2 e^7}-\frac {\left (a e^2-b d e+c d^2\right )^4}{4 e^9 (d+e x)^4}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9 (d+e x)^3}-\frac {c^3 x^3 (5 c d-4 b e)}{3 e^6}+\frac {c^4 x^4}{4 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx &=\int \left (\frac {c \left (-35 c^3 d^3+4 b^3 e^3-6 b c e^2 (5 b d-2 a e)+20 c^2 d e (3 b d-a e)\right )}{e^8}+\frac {c^2 \left (15 c^2 d^2+6 b^2 e^2-4 c e (5 b d-a e)\right ) x}{e^7}-\frac {c^3 (5 c d-4 b e) x^2}{e^6}+\frac {c^4 x^3}{e^5}+\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^5}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^4}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^8 (d+e x)^3}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right )}{e^8 (d+e x)^2}+\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{e^8 (d+e x)}\right ) \, dx\\ &=-\frac {c \left (35 c^3 d^3-4 b^3 e^3+6 b c e^2 (5 b d-2 a e)-20 c^2 d e (3 b d-a e)\right ) x}{e^8}+\frac {c^2 \left (15 c^2 d^2+6 b^2 e^2-4 c e (5 b d-a e)\right ) x^2}{2 e^7}-\frac {c^3 (5 c d-4 b e) x^3}{3 e^6}+\frac {c^4 x^4}{4 e^5}-\frac {\left (c d^2-b d e+a e^2\right )^4}{4 e^9 (d+e x)^4}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^9 (d+e x)^3}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^9 (d+e x)^2}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \log (d+e x)}{e^9}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 430, normalized size = 1.01 \begin {gather*} \frac {12 c e \left (-35 c^3 d^3+4 b^3 e^3-6 b c e^2 (5 b d-2 a e)+20 c^2 d e (3 b d-a e)\right ) x+6 c^2 e^2 \left (15 c^2 d^2+6 b^2 e^2+4 c e (-5 b d+a e)\right ) x^2+4 c^3 e^3 (-5 c d+4 b e) x^3+3 c^4 e^4 x^4-\frac {3 \left (c d^2+e (-b d+a e)\right )^4}{(d+e x)^4}+\frac {16 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^3}-\frac {12 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2}{(d+e x)^2}+\frac {48 (2 c d-b e) \left (7 c^3 d^4-2 c^2 d^2 e (7 b d-5 a e)+b^2 e^3 (-b d+a e)+c e^2 \left (8 b^2 d^2-10 a b d e+3 a^2 e^2\right )\right )}{d+e x}+12 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \log (d+e x)}{12 e^9} \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. \(911\) vs.
\(2(416)=832\).
time = 0.72, size = 912, normalized size = 2.14 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. 858 vs.
\(2 (419) = 838\).
time = 0.30, size = 858, normalized size = 2.01 \begin {gather*} {\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4} + 30 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{2} - 20 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d\right )} e^{\left (-9\right )} \log \left (x e + d\right ) + \frac {1}{12} \, {\left (3 \, c^{4} x^{4} e^{3} - 4 \, {\left (5 \, c^{4} d e^{2} - 4 \, b c^{3} e^{3}\right )} x^{3} + 6 \, {\left (15 \, c^{4} d^{2} e - 20 \, b c^{3} d e^{2} + 6 \, b^{2} c^{2} e^{3} + 4 \, a c^{3} e^{3}\right )} x^{2} - 12 \, {\left (35 \, c^{4} d^{3} - 60 \, b c^{3} d^{2} e - 4 \, b^{3} c e^{3} - 12 \, a b c^{2} e^{3} + 10 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d\right )} x\right )} e^{\left (-8\right )} + \frac {533 \, c^{4} d^{8} - 1276 \, b c^{3} d^{7} e + 342 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{6} - 308 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{5} - 4 \, a^{3} b d e^{7} + 25 \, {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d^{4} - 3 \, a^{4} e^{8} - 12 \, {\left (a b^{3} e^{5} + 3 \, a^{2} b c e^{5}\right )} d^{3} + 48 \, {\left (14 \, c^{4} d^{5} e^{3} - 35 \, b c^{3} d^{4} e^{4} - a b^{3} e^{8} - 3 \, a^{2} b c e^{8} + 10 \, {\left (3 \, b^{2} c^{2} e^{5} + 2 \, a c^{3} e^{5}\right )} d^{3} - 10 \, {\left (b^{3} c e^{6} + 3 \, a b c^{2} e^{6}\right )} d^{2} + {\left (b^{4} e^{7} + 12 \, a b^{2} c e^{7} + 6 \, a^{2} c^{2} e^{7}\right )} d\right )} x^{3} - 2 \, {\left (3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6}\right )} d^{2} + 12 \, {\left (154 \, c^{4} d^{6} e^{2} - 378 \, b c^{3} d^{5} e^{3} + 105 \, {\left (3 \, b^{2} c^{2} e^{4} + 2 \, a c^{3} e^{4}\right )} d^{4} - 3 \, a^{2} b^{2} e^{8} - 2 \, a^{3} c e^{8} - 100 \, {\left (b^{3} c e^{5} + 3 \, a b c^{2} e^{5}\right )} d^{3} + 9 \, {\left (b^{4} e^{6} + 12 \, a b^{2} c e^{6} + 6 \, a^{2} c^{2} e^{6}\right )} d^{2} - 6 \, {\left (a b^{3} e^{7} + 3 \, a^{2} b c e^{7}\right )} d\right )} x^{2} + 8 \, {\left (214 \, c^{4} d^{7} e - 518 \, b c^{3} d^{6} e^{2} + 141 \, {\left (3 \, b^{2} c^{2} e^{3} + 2 \, a c^{3} e^{3}\right )} d^{5} - 130 \, {\left (b^{3} c e^{4} + 3 \, a b c^{2} e^{4}\right )} d^{4} - 2 \, a^{3} b e^{8} + 11 \, {\left (b^{4} e^{5} + 12 \, a b^{2} c e^{5} + 6 \, a^{2} c^{2} e^{5}\right )} d^{3} - 6 \, {\left (a b^{3} e^{6} + 3 \, a^{2} b c e^{6}\right )} d^{2} - {\left (3 \, a^{2} b^{2} e^{7} + 2 \, a^{3} c e^{7}\right )} d\right )} x}{12 \, {\left (x^{4} e^{13} + 4 \, d x^{3} e^{12} + 6 \, d^{2} x^{2} e^{11} + 4 \, d^{3} x e^{10} + d^{4} e^{9}\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. 1259 vs.
\(2 (419) = 838\).
time = 2.72, size = 1259, normalized size = 2.96 \begin {gather*} \frac {533 \, c^{4} d^{8} + {\left (3 \, c^{4} x^{8} + 16 \, b c^{3} x^{7} + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{6} + 48 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{5} - 16 \, a^{3} b x - 3 \, a^{4} - 48 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{3} - 12 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x^{2}\right )} e^{8} - 4 \, {\left (2 \, c^{4} d x^{7} + 14 \, b c^{3} d x^{6} + 18 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{5} - 48 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{4} + a^{3} b d - 12 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{3} + 18 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d x^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d x\right )} e^{7} + 2 \, {\left (14 \, c^{4} d^{2} x^{6} + 168 \, b c^{3} d^{2} x^{5} - 204 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{4} - 96 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{3} + 54 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} x^{2} - 24 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} x - {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2}\right )} e^{6} - 4 \, {\left (42 \, c^{4} d^{3} x^{5} - 556 \, b c^{3} d^{3} x^{4} + 48 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} x^{3} + 252 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} x^{2} - 22 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} x + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3}\right )} e^{5} - {\left (1217 \, c^{4} d^{4} x^{4} - 2176 \, b c^{3} d^{4} x^{3} - 792 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2} + 992 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} x - 25 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4}\right )} e^{4} - 4 \, {\left (377 \, c^{4} d^{5} x^{3} + 444 \, b c^{3} d^{5} x^{2} - 252 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} x + 77 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5}\right )} e^{3} + 2 \, {\left (129 \, c^{4} d^{6} x^{2} - 1712 \, b c^{3} d^{6} x + 171 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6}\right )} e^{2} + 4 \, {\left (323 \, c^{4} d^{7} x - 319 \, b c^{3} d^{7}\right )} e + 12 \, {\left (70 \, c^{4} d^{8} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{4} e^{8} - 4 \, {\left (5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{4} - {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{3}\right )} e^{7} + 2 \, {\left (15 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{4} - 40 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} x^{2}\right )} e^{6} - 4 \, {\left (35 \, b c^{3} d^{3} x^{4} - 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} x^{3} + 30 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} x^{2} - {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} x\right )} e^{5} + {\left (70 \, c^{4} d^{4} x^{4} - 560 \, b c^{3} d^{4} x^{3} + 180 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2} - 80 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} x + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4}\right )} e^{4} + 20 \, {\left (14 \, c^{4} d^{5} x^{3} - 42 \, b c^{3} d^{5} x^{2} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} x - {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5}\right )} e^{3} + 10 \, {\left (42 \, c^{4} d^{6} x^{2} - 56 \, b c^{3} d^{6} x + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6}\right )} e^{2} + 140 \, {\left (2 \, c^{4} d^{7} x - b c^{3} d^{7}\right )} e\right )} \log \left (x e + d\right )}{12 \, {\left (x^{4} e^{13} + 4 \, d x^{3} e^{12} + 6 \, d^{2} x^{2} e^{11} + 4 \, d^{3} x e^{10} + d^{4} e^{9}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1283 vs.
\(2 (419) = 838\).
time = 1.18, size = 1283, normalized size = 3.01 \begin {gather*} \frac {1}{12} \, {\left (3 \, c^{4} - \frac {16 \, {\left (2 \, c^{4} d e - b c^{3} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {12 \, {\left (14 \, c^{4} d^{2} e^{2} - 14 \, b c^{3} d e^{3} + 3 \, b^{2} c^{2} e^{4} + 2 \, a c^{3} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {48 \, {\left (14 \, c^{4} d^{3} e^{3} - 21 \, b c^{3} d^{2} e^{4} + 9 \, b^{2} c^{2} d e^{5} + 6 \, a c^{3} d e^{5} - b^{3} c e^{6} - 3 \, a b c^{2} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )} {\left (x e + d\right )}^{4} e^{\left (-9\right )} - {\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + 90 \, b^{2} c^{2} d^{2} e^{2} + 60 \, a c^{3} d^{2} e^{2} - 20 \, b^{3} c d e^{3} - 60 \, a b c^{2} d e^{3} + b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} e^{\left (-9\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac {1}{12} \, {\left (\frac {672 \, c^{4} d^{5} e^{43}}{x e + d} - \frac {168 \, c^{4} d^{6} e^{43}}{{\left (x e + d\right )}^{2}} + \frac {32 \, c^{4} d^{7} e^{43}}{{\left (x e + d\right )}^{3}} - \frac {3 \, c^{4} d^{8} e^{43}}{{\left (x e + d\right )}^{4}} - \frac {1680 \, b c^{3} d^{4} e^{44}}{x e + d} + \frac {504 \, b c^{3} d^{5} e^{44}}{{\left (x e + d\right )}^{2}} - \frac {112 \, b c^{3} d^{6} e^{44}}{{\left (x e + d\right )}^{3}} + \frac {12 \, b c^{3} d^{7} e^{44}}{{\left (x e + d\right )}^{4}} + \frac {1440 \, b^{2} c^{2} d^{3} e^{45}}{x e + d} + \frac {960 \, a c^{3} d^{3} e^{45}}{x e + d} - \frac {540 \, b^{2} c^{2} d^{4} e^{45}}{{\left (x e + d\right )}^{2}} - \frac {360 \, a c^{3} d^{4} e^{45}}{{\left (x e + d\right )}^{2}} + \frac {144 \, b^{2} c^{2} d^{5} e^{45}}{{\left (x e + d\right )}^{3}} + \frac {96 \, a c^{3} d^{5} e^{45}}{{\left (x e + d\right )}^{3}} - \frac {18 \, b^{2} c^{2} d^{6} e^{45}}{{\left (x e + d\right )}^{4}} - \frac {12 \, a c^{3} d^{6} e^{45}}{{\left (x e + d\right )}^{4}} - \frac {480 \, b^{3} c d^{2} e^{46}}{x e + d} - \frac {1440 \, a b c^{2} d^{2} e^{46}}{x e + d} + \frac {240 \, b^{3} c d^{3} e^{46}}{{\left (x e + d\right )}^{2}} + \frac {720 \, a b c^{2} d^{3} e^{46}}{{\left (x e + d\right )}^{2}} - \frac {80 \, b^{3} c d^{4} e^{46}}{{\left (x e + d\right )}^{3}} - \frac {240 \, a b c^{2} d^{4} e^{46}}{{\left (x e + d\right )}^{3}} + \frac {12 \, b^{3} c d^{5} e^{46}}{{\left (x e + d\right )}^{4}} + \frac {36 \, a b c^{2} d^{5} e^{46}}{{\left (x e + d\right )}^{4}} + \frac {48 \, b^{4} d e^{47}}{x e + d} + \frac {576 \, a b^{2} c d e^{47}}{x e + d} + \frac {288 \, a^{2} c^{2} d e^{47}}{x e + d} - \frac {36 \, b^{4} d^{2} e^{47}}{{\left (x e + d\right )}^{2}} - \frac {432 \, a b^{2} c d^{2} e^{47}}{{\left (x e + d\right )}^{2}} - \frac {216 \, a^{2} c^{2} d^{2} e^{47}}{{\left (x e + d\right )}^{2}} + \frac {16 \, b^{4} d^{3} e^{47}}{{\left (x e + d\right )}^{3}} + \frac {192 \, a b^{2} c d^{3} e^{47}}{{\left (x e + d\right )}^{3}} + \frac {96 \, a^{2} c^{2} d^{3} e^{47}}{{\left (x e + d\right )}^{3}} - \frac {3 \, b^{4} d^{4} e^{47}}{{\left (x e + d\right )}^{4}} - \frac {36 \, a b^{2} c d^{4} e^{47}}{{\left (x e + d\right )}^{4}} - \frac {18 \, a^{2} c^{2} d^{4} e^{47}}{{\left (x e + d\right )}^{4}} - \frac {48 \, a b^{3} e^{48}}{x e + d} - \frac {144 \, a^{2} b c e^{48}}{x e + d} + \frac {72 \, a b^{3} d e^{48}}{{\left (x e + d\right )}^{2}} + \frac {216 \, a^{2} b c d e^{48}}{{\left (x e + d\right )}^{2}} - \frac {48 \, a b^{3} d^{2} e^{48}}{{\left (x e + d\right )}^{3}} - \frac {144 \, a^{2} b c d^{2} e^{48}}{{\left (x e + d\right )}^{3}} + \frac {12 \, a b^{3} d^{3} e^{48}}{{\left (x e + d\right )}^{4}} + \frac {36 \, a^{2} b c d^{3} e^{48}}{{\left (x e + d\right )}^{4}} - \frac {36 \, a^{2} b^{2} e^{49}}{{\left (x e + d\right )}^{2}} - \frac {24 \, a^{3} c e^{49}}{{\left (x e + d\right )}^{2}} + \frac {48 \, a^{2} b^{2} d e^{49}}{{\left (x e + d\right )}^{3}} + \frac {32 \, a^{3} c d e^{49}}{{\left (x e + d\right )}^{3}} - \frac {18 \, a^{2} b^{2} d^{2} e^{49}}{{\left (x e + d\right )}^{4}} - \frac {12 \, a^{3} c d^{2} e^{49}}{{\left (x e + d\right )}^{4}} - \frac {16 \, a^{3} b e^{50}}{{\left (x e + d\right )}^{3}} + \frac {12 \, a^{3} b d e^{50}}{{\left (x e + d\right )}^{4}} - \frac {3 \, a^{4} e^{51}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-52\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.81, size = 1005, normalized size = 2.36 \begin {gather*} x^3\,\left (\frac {4\,b\,c^3}{3\,e^5}-\frac {5\,c^4\,d}{3\,e^6}\right )-\frac {x\,\left (\frac {4\,a^3\,b\,e^7}{3}+\frac {4\,a^3\,c\,d\,e^6}{3}+2\,a^2\,b^2\,d\,e^6+12\,a^2\,b\,c\,d^2\,e^5-44\,a^2\,c^2\,d^3\,e^4+4\,a\,b^3\,d^2\,e^5-88\,a\,b^2\,c\,d^3\,e^4+260\,a\,b\,c^2\,d^4\,e^3-188\,a\,c^3\,d^5\,e^2-\frac {22\,b^4\,d^3\,e^4}{3}+\frac {260\,b^3\,c\,d^4\,e^3}{3}-282\,b^2\,c^2\,d^5\,e^2+\frac {1036\,b\,c^3\,d^6\,e}{3}-\frac {428\,c^4\,d^7}{3}\right )-x^3\,\left (-12\,a^2\,b\,c\,e^7+24\,a^2\,c^2\,d\,e^6-4\,a\,b^3\,e^7+48\,a\,b^2\,c\,d\,e^6-120\,a\,b\,c^2\,d^2\,e^5+80\,a\,c^3\,d^3\,e^4+4\,b^4\,d\,e^6-40\,b^3\,c\,d^2\,e^5+120\,b^2\,c^2\,d^3\,e^4-140\,b\,c^3\,d^4\,e^3+56\,c^4\,d^5\,e^2\right )+\frac {3\,a^4\,e^8+4\,a^3\,b\,d\,e^7+4\,a^3\,c\,d^2\,e^6+6\,a^2\,b^2\,d^2\,e^6+36\,a^2\,b\,c\,d^3\,e^5-150\,a^2\,c^2\,d^4\,e^4+12\,a\,b^3\,d^3\,e^5-300\,a\,b^2\,c\,d^4\,e^4+924\,a\,b\,c^2\,d^5\,e^3-684\,a\,c^3\,d^6\,e^2-25\,b^4\,d^4\,e^4+308\,b^3\,c\,d^5\,e^3-1026\,b^2\,c^2\,d^6\,e^2+1276\,b\,c^3\,d^7\,e-533\,c^4\,d^8}{12\,e}+x^2\,\left (2\,a^3\,c\,e^7+3\,a^2\,b^2\,e^7+18\,a^2\,b\,c\,d\,e^6-54\,a^2\,c^2\,d^2\,e^5+6\,a\,b^3\,d\,e^6-108\,a\,b^2\,c\,d^2\,e^5+300\,a\,b\,c^2\,d^3\,e^4-210\,a\,c^3\,d^4\,e^3-9\,b^4\,d^2\,e^5+100\,b^3\,c\,d^3\,e^4-315\,b^2\,c^2\,d^4\,e^3+378\,b\,c^3\,d^5\,e^2-154\,c^4\,d^6\,e\right )}{d^4\,e^8+4\,d^3\,e^9\,x+6\,d^2\,e^{10}\,x^2+4\,d\,e^{11}\,x^3+e^{12}\,x^4}-x^2\,\left (\frac {5\,d\,\left (\frac {4\,b\,c^3}{e^5}-\frac {5\,c^4\,d}{e^6}\right )}{2\,e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{2\,e^5}+\frac {5\,c^4\,d^2}{e^7}\right )-x\,\left (\frac {10\,c^4\,d^3}{e^8}+\frac {10\,d^2\,\left (\frac {4\,b\,c^3}{e^5}-\frac {5\,c^4\,d}{e^6}\right )}{e^2}-\frac {5\,d\,\left (\frac {5\,d\,\left (\frac {4\,b\,c^3}{e^5}-\frac {5\,c^4\,d}{e^6}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^5}+\frac {10\,c^4\,d^2}{e^7}\right )}{e}-\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e^5}\right )+\frac {c^4\,x^4}{4\,e^5}+\frac {\ln \left (d+e\,x\right )\,\left (6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4-60\,a\,b\,c^2\,d\,e^3+60\,a\,c^3\,d^2\,e^2+b^4\,e^4-20\,b^3\,c\,d\,e^3+90\,b^2\,c^2\,d^2\,e^2-140\,b\,c^3\,d^3\,e+70\,c^4\,d^4\right )}{e^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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