Optimal. Leaf size=486 \[ \frac {\left (a^3 C e^6+c^3 d^4 \left (7 C d^2-e (6 B d-5 A e)\right )+3 a c^2 d^2 e^2 \left (5 C d^2-e (4 B d-3 A e)\right )+3 a^2 c e^4 \left (3 C d^2-e (2 B d-A e)\right )\right ) x}{e^8}-\frac {c \left (3 a^2 e^4 (2 C d-B e)+c^2 d^3 \left (6 C d^2-e (5 B d-4 A e)\right )+3 a c d e^2 \left (4 C d^2-e (3 B d-2 A e)\right )\right ) x^2}{2 e^7}+\frac {c \left (3 a^2 C e^4+c^2 d^2 \left (5 C d^2-e (4 B d-3 A e)\right )+3 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )\right ) x^3}{3 e^6}-\frac {c^2 \left (3 a e^2 (2 C d-B e)+c d \left (4 C d^2-e (3 B d-2 A e)\right )\right ) x^4}{4 e^5}+\frac {c^2 \left (3 a C e^2+c \left (3 C d^2-e (2 B d-A e)\right )\right ) x^5}{5 e^4}-\frac {c^3 (2 C d-B e) x^6}{6 e^3}+\frac {c^3 C x^7}{7 e^2}-\frac {\left (c d^2+a e^2\right )^3 \left (C d^2-B d e+A e^2\right )}{e^9 (d+e x)}-\frac {\left (c d^2+a e^2\right )^2 \left (a e^2 (2 C d-B e)+c d \left (8 C d^2-e (7 B d-6 A e)\right )\right ) \log (d+e x)}{e^9} \]
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Rubi [A]
time = 0.59, antiderivative size = 483, normalized size of antiderivative = 0.99, number of steps
used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {1642}
\begin {gather*} \frac {c x^3 \left (3 a^2 C e^4+3 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )+c^2 \left (5 C d^4-d^2 e (4 B d-3 A e)\right )\right )}{3 e^6}-\frac {c x^2 \left (3 a^2 e^4 (2 C d-B e)+3 a c d e^2 \left (4 C d^2-e (3 B d-2 A e)\right )+c^2 \left (6 C d^5-d^3 e (5 B d-4 A e)\right )\right )}{2 e^7}+\frac {x \left (a^3 C e^6+3 a^2 c e^4 \left (3 C d^2-e (2 B d-A e)\right )+3 a c^2 d^2 e^2 \left (5 C d^2-e (4 B d-3 A e)\right )+c^3 \left (7 C d^6-d^4 e (6 B d-5 A e)\right )\right )}{e^8}-\frac {c^2 x^4 \left (3 a e^2 (2 C d-B e)-c d e (3 B d-2 A e)+4 c C d^3\right )}{4 e^5}+\frac {c^2 x^5 \left (3 a C e^2-c e (2 B d-A e)+3 c C d^2\right )}{5 e^4}-\frac {\left (a e^2+c d^2\right )^3 \left (A e^2-B d e+C d^2\right )}{e^9 (d+e x)}-\frac {\left (a e^2+c d^2\right )^2 \log (d+e x) \left (a e^2 (2 C d-B e)-c d e (7 B d-6 A e)+8 c C d^3\right )}{e^9}-\frac {c^3 x^6 (2 C d-B e)}{6 e^3}+\frac {c^3 C x^7}{7 e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 1642
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{(d+e x)^2} \, dx &=\int \left (\frac {a^3 C e^6+c^3 \left (7 C d^6-d^4 e (6 B d-5 A e)\right )+3 a c^2 d^2 e^2 \left (5 C d^2-e (4 B d-3 A e)\right )+3 a^2 c e^4 \left (3 C d^2-e (2 B d-A e)\right )}{e^8}+\frac {c \left (-3 a^2 e^4 (2 C d-B e)-c^2 \left (6 C d^5-d^3 e (5 B d-4 A e)\right )-3 a c d e^2 \left (4 C d^2-e (3 B d-2 A e)\right )\right ) x}{e^7}+\frac {c \left (3 a^2 C e^4+c^2 \left (5 C d^4-d^2 e (4 B d-3 A e)\right )+3 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )\right ) x^2}{e^6}+\frac {c^2 \left (-4 c C d^3+c d e (3 B d-2 A e)-3 a e^2 (2 C d-B e)\right ) x^3}{e^5}+\frac {c^2 \left (3 c C d^2+3 a C e^2-c e (2 B d-A e)\right ) x^4}{e^4}+\frac {c^3 (-2 C d+B e) x^5}{e^3}+\frac {c^3 C x^6}{e^2}+\frac {\left (c d^2+a e^2\right )^3 \left (C d^2-B d e+A e^2\right )}{e^8 (d+e x)^2}+\frac {\left (c d^2+a e^2\right )^2 \left (-8 c C d^3+c d e (7 B d-6 A e)-a e^2 (2 C d-B e)\right )}{e^8 (d+e x)}\right ) \, dx\\ &=\frac {\left (a^3 C e^6+c^3 \left (7 C d^6-d^4 e (6 B d-5 A e)\right )+3 a c^2 d^2 e^2 \left (5 C d^2-e (4 B d-3 A e)\right )+3 a^2 c e^4 \left (3 C d^2-e (2 B d-A e)\right )\right ) x}{e^8}-\frac {c \left (3 a^2 e^4 (2 C d-B e)+c^2 \left (6 C d^5-d^3 e (5 B d-4 A e)\right )+3 a c d e^2 \left (4 C d^2-e (3 B d-2 A e)\right )\right ) x^2}{2 e^7}+\frac {c \left (3 a^2 C e^4+c^2 \left (5 C d^4-d^2 e (4 B d-3 A e)\right )+3 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )\right ) x^3}{3 e^6}-\frac {c^2 \left (4 c C d^3-c d e (3 B d-2 A e)+3 a e^2 (2 C d-B e)\right ) x^4}{4 e^5}+\frac {c^2 \left (3 c C d^2+3 a C e^2-c e (2 B d-A e)\right ) x^5}{5 e^4}-\frac {c^3 (2 C d-B e) x^6}{6 e^3}+\frac {c^3 C x^7}{7 e^2}-\frac {\left (c d^2+a e^2\right )^3 \left (C d^2-B d e+A e^2\right )}{e^9 (d+e x)}-\frac {\left (c d^2+a e^2\right )^2 \left (8 c C d^3-c d e (7 B d-6 A e)+a e^2 (2 C d-B e)\right ) \log (d+e x)}{e^9}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 641, normalized size = 1.32 \begin {gather*} \frac {420 a^3 e^6 \left (e (B d-A e)+C \left (-d^2+d e x+e^2 x^2\right )\right )+210 a^2 c e^4 \left (2 C \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+3 e \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )\right )\right )+21 a c^2 e^2 \left (-6 C \left (10 d^6-50 d^5 e x-30 d^4 e^2 x^2+10 d^3 e^3 x^3-5 d^2 e^4 x^4+3 d e^5 x^5-2 e^6 x^6\right )+5 e \left (4 A e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+B \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )\right )\right )+c^3 \left (-4 C \left (105 d^8-735 d^7 e x-420 d^6 e^2 x^2+140 d^5 e^3 x^3-70 d^4 e^4 x^4+42 d^3 e^5 x^5-28 d^2 e^6 x^6+20 d e^7 x^7-15 e^8 x^8\right )+7 e \left (6 A e \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )+B \left (60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right )\right )\right )-420 \left (c d^2+a e^2\right )^2 \left (8 c C d^3+c d e (-7 B d+6 A e)+a e^2 (2 C d-B e)\right ) (d+e x) \log (d+e x)}{420 e^9 (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 788, normalized size = 1.62 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 679, normalized size = 1.40 \begin {gather*} -{\left (8 \, C c^{3} d^{7} - 7 \, B c^{3} d^{6} e - 15 \, B a c^{2} d^{4} e^{3} - 9 \, B a^{2} c d^{2} e^{5} + 6 \, {\left (3 \, C a c^{2} e^{2} + A c^{3} e^{2}\right )} d^{5} - B a^{3} e^{7} + 12 \, {\left (C a^{2} c e^{4} + A a c^{2} e^{4}\right )} d^{3} + 2 \, {\left (C a^{3} e^{6} + 3 \, A a^{2} c e^{6}\right )} d\right )} e^{\left (-9\right )} \log \left (x e + d\right ) + \frac {1}{420} \, {\left (60 \, C c^{3} x^{7} e^{6} - 70 \, {\left (2 \, C c^{3} d e^{5} - B c^{3} e^{6}\right )} x^{6} + 84 \, {\left (3 \, C c^{3} d^{2} e^{4} - 2 \, B c^{3} d e^{5} + 3 \, C a c^{2} e^{6} + A c^{3} e^{6}\right )} x^{5} - 105 \, {\left (4 \, C c^{3} d^{3} e^{3} - 3 \, B c^{3} d^{2} e^{4} - 3 \, B a c^{2} e^{6} + 2 \, {\left (3 \, C a c^{2} e^{5} + A c^{3} e^{5}\right )} d\right )} x^{4} + 140 \, {\left (5 \, C c^{3} d^{4} e^{2} - 4 \, B c^{3} d^{3} e^{3} - 6 \, B a c^{2} d e^{5} + 3 \, C a^{2} c e^{6} + 3 \, A a c^{2} e^{6} + 3 \, {\left (3 \, C a c^{2} e^{4} + A c^{3} e^{4}\right )} d^{2}\right )} x^{3} - 210 \, {\left (6 \, C c^{3} d^{5} e - 5 \, B c^{3} d^{4} e^{2} - 9 \, B a c^{2} d^{2} e^{4} - 3 \, B a^{2} c e^{6} + 4 \, {\left (3 \, C a c^{2} e^{3} + A c^{3} e^{3}\right )} d^{3} + 6 \, {\left (C a^{2} c e^{5} + A a c^{2} e^{5}\right )} d\right )} x^{2} + 420 \, {\left (7 \, C c^{3} d^{6} - 6 \, B c^{3} d^{5} e - 12 \, B a c^{2} d^{3} e^{3} - 6 \, B a^{2} c d e^{5} + 5 \, {\left (3 \, C a c^{2} e^{2} + A c^{3} e^{2}\right )} d^{4} + C a^{3} e^{6} + 3 \, A a^{2} c e^{6} + 9 \, {\left (C a^{2} c e^{4} + A a c^{2} e^{4}\right )} d^{2}\right )} x\right )} e^{\left (-8\right )} - \frac {C c^{3} d^{8} - B c^{3} d^{7} e - 3 \, B a c^{2} d^{5} e^{3} - 3 \, B a^{2} c d^{3} e^{5} + {\left (3 \, C a c^{2} e^{2} + A c^{3} e^{2}\right )} d^{6} - B a^{3} d e^{7} + 3 \, {\left (C a^{2} c e^{4} + A a c^{2} e^{4}\right )} d^{4} + A a^{3} e^{8} + {\left (C a^{3} e^{6} + 3 \, A a^{2} c e^{6}\right )} d^{2}}{x e^{10} + d e^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 892, normalized size = 1.84 \begin {gather*} -\frac {420 \, C c^{3} d^{8} - {\left (60 \, C c^{3} x^{8} + 70 \, B c^{3} x^{7} + 315 \, B a c^{2} x^{5} + 630 \, B a^{2} c x^{3} + 84 \, {\left (3 \, C a c^{2} + A c^{3}\right )} x^{6} + 420 \, {\left (C a^{2} c + A a c^{2}\right )} x^{4} - 420 \, A a^{3} + 420 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} x^{2}\right )} e^{8} + {\left (80 \, C c^{3} d x^{7} + 98 \, B c^{3} d x^{6} + 525 \, B a c^{2} d x^{4} + 1890 \, B a^{2} c d x^{2} + 126 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d x^{5} - 420 \, B a^{3} d + 840 \, {\left (C a^{2} c + A a c^{2}\right )} d x^{3} - 420 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} d x\right )} e^{7} - 7 \, {\left (16 \, C c^{3} d^{2} x^{6} + 21 \, B c^{3} d^{2} x^{5} + 150 \, B a c^{2} d^{2} x^{3} - 360 \, B a^{2} c d^{2} x + 30 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} x^{4} + 360 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} x^{2} - 60 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2}\right )} e^{6} + 7 \, {\left (24 \, C c^{3} d^{3} x^{5} + 35 \, B c^{3} d^{3} x^{4} + 450 \, B a c^{2} d^{3} x^{2} - 180 \, B a^{2} c d^{3} + 60 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{3} x^{3} - 540 \, {\left (C a^{2} c + A a c^{2}\right )} d^{3} x\right )} e^{5} - 70 \, {\left (4 \, C c^{3} d^{4} x^{4} + 7 \, B c^{3} d^{4} x^{3} - 72 \, B a c^{2} d^{4} x + 18 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{4} x^{2} - 18 \, {\left (C a^{2} c + A a c^{2}\right )} d^{4}\right )} e^{4} + 70 \, {\left (8 \, C c^{3} d^{5} x^{3} + 21 \, B c^{3} d^{5} x^{2} - 18 \, B a c^{2} d^{5} - 30 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{5} x\right )} e^{3} - 420 \, {\left (4 \, C c^{3} d^{6} x^{2} - 6 \, B c^{3} d^{6} x - {\left (3 \, C a c^{2} + A c^{3}\right )} d^{6}\right )} e^{2} - 420 \, {\left (7 \, C c^{3} d^{7} x + B c^{3} d^{7}\right )} e + 420 \, {\left (8 \, C c^{3} d^{8} - B a^{3} x e^{8} - {\left (B a^{3} d - 2 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} d x\right )} e^{7} - {\left (9 \, B a^{2} c d^{2} x - 2 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2}\right )} e^{6} - 3 \, {\left (3 \, B a^{2} c d^{3} - 4 \, {\left (C a^{2} c + A a c^{2}\right )} d^{3} x\right )} e^{5} - 3 \, {\left (5 \, B a c^{2} d^{4} x - 4 \, {\left (C a^{2} c + A a c^{2}\right )} d^{4}\right )} e^{4} - 3 \, {\left (5 \, B a c^{2} d^{5} - 2 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{5} x\right )} e^{3} - {\left (7 \, B c^{3} d^{6} x - 6 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{6}\right )} e^{2} + {\left (8 \, C c^{3} d^{7} x - 7 \, B c^{3} d^{7}\right )} e\right )} \log \left (x e + d\right )}{420 \, {\left (x e^{10} + d e^{9}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.15, size = 748, normalized size = 1.54 \begin {gather*} \frac {C c^{3} x^{7}}{7 e^{2}} + x^{6} \left (\frac {B c^{3}}{6 e^{2}} - \frac {C c^{3} d}{3 e^{3}}\right ) + x^{5} \left (\frac {A c^{3}}{5 e^{2}} - \frac {2 B c^{3} d}{5 e^{3}} + \frac {3 C a c^{2}}{5 e^{2}} + \frac {3 C c^{3} d^{2}}{5 e^{4}}\right ) + x^{4} \left (- \frac {A c^{3} d}{2 e^{3}} + \frac {3 B a c^{2}}{4 e^{2}} + \frac {3 B c^{3} d^{2}}{4 e^{4}} - \frac {3 C a c^{2} d}{2 e^{3}} - \frac {C c^{3} d^{3}}{e^{5}}\right ) + x^{3} \left (\frac {A a c^{2}}{e^{2}} + \frac {A c^{3} d^{2}}{e^{4}} - \frac {2 B a c^{2} d}{e^{3}} - \frac {4 B c^{3} d^{3}}{3 e^{5}} + \frac {C a^{2} c}{e^{2}} + \frac {3 C a c^{2} d^{2}}{e^{4}} + \frac {5 C c^{3} d^{4}}{3 e^{6}}\right ) + x^{2} \left (- \frac {3 A a c^{2} d}{e^{3}} - \frac {2 A c^{3} d^{3}}{e^{5}} + \frac {3 B a^{2} c}{2 e^{2}} + \frac {9 B a c^{2} d^{2}}{2 e^{4}} + \frac {5 B c^{3} d^{4}}{2 e^{6}} - \frac {3 C a^{2} c d}{e^{3}} - \frac {6 C a c^{2} d^{3}}{e^{5}} - \frac {3 C c^{3} d^{5}}{e^{7}}\right ) + x \left (\frac {3 A a^{2} c}{e^{2}} + \frac {9 A a c^{2} d^{2}}{e^{4}} + \frac {5 A c^{3} d^{4}}{e^{6}} - \frac {6 B a^{2} c d}{e^{3}} - \frac {12 B a c^{2} d^{3}}{e^{5}} - \frac {6 B c^{3} d^{5}}{e^{7}} + \frac {C a^{3}}{e^{2}} + \frac {9 C a^{2} c d^{2}}{e^{4}} + \frac {15 C a c^{2} d^{4}}{e^{6}} + \frac {7 C c^{3} d^{6}}{e^{8}}\right ) + \frac {- A a^{3} e^{8} - 3 A a^{2} c d^{2} e^{6} - 3 A a c^{2} d^{4} e^{4} - A c^{3} d^{6} e^{2} + B a^{3} d e^{7} + 3 B a^{2} c d^{3} e^{5} + 3 B a c^{2} d^{5} e^{3} + B c^{3} d^{7} e - C a^{3} d^{2} e^{6} - 3 C a^{2} c d^{4} e^{4} - 3 C a c^{2} d^{6} e^{2} - C c^{3} d^{8}}{d e^{9} + e^{10} x} - \frac {\left (a e^{2} + c d^{2}\right )^{2} \cdot \left (6 A c d e^{2} - B a e^{3} - 7 B c d^{2} e + 2 C a d e^{2} + 8 C c d^{3}\right ) \log {\left (d + e x \right )}}{e^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.85, size = 838, normalized size = 1.72 \begin {gather*} \frac {1}{420} \, {\left (60 \, C c^{3} - \frac {70 \, {\left (8 \, C c^{3} d e - B c^{3} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {84 \, {\left (28 \, C c^{3} d^{2} e^{2} - 7 \, B c^{3} d e^{3} + 3 \, C a c^{2} e^{4} + A c^{3} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {105 \, {\left (56 \, C c^{3} d^{3} e^{3} - 21 \, B c^{3} d^{2} e^{4} + 18 \, C a c^{2} d e^{5} + 6 \, A c^{3} d e^{5} - 3 \, B a c^{2} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac {140 \, {\left (70 \, C c^{3} d^{4} e^{4} - 35 \, B c^{3} d^{3} e^{5} + 45 \, C a c^{2} d^{2} e^{6} + 15 \, A c^{3} d^{2} e^{6} - 15 \, B a c^{2} d e^{7} + 3 \, C a^{2} c e^{8} + 3 \, A a c^{2} e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} - \frac {210 \, {\left (56 \, C c^{3} d^{5} e^{5} - 35 \, B c^{3} d^{4} e^{6} + 60 \, C a c^{2} d^{3} e^{7} + 20 \, A c^{3} d^{3} e^{7} - 30 \, B a c^{2} d^{2} e^{8} + 12 \, C a^{2} c d e^{9} + 12 \, A a c^{2} d e^{9} - 3 \, B a^{2} c e^{10}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}} + \frac {420 \, {\left (28 \, C c^{3} d^{6} e^{6} - 21 \, B c^{3} d^{5} e^{7} + 45 \, C a c^{2} d^{4} e^{8} + 15 \, A c^{3} d^{4} e^{8} - 30 \, B a c^{2} d^{3} e^{9} + 18 \, C a^{2} c d^{2} e^{10} + 18 \, A a c^{2} d^{2} e^{10} - 9 \, B a^{2} c d e^{11} + C a^{3} e^{12} + 3 \, A a^{2} c e^{12}\right )} e^{\left (-6\right )}}{{\left (x e + d\right )}^{6}}\right )} {\left (x e + d\right )}^{7} e^{\left (-9\right )} + {\left (8 \, C c^{3} d^{7} - 7 \, B c^{3} d^{6} e + 18 \, C a c^{2} d^{5} e^{2} + 6 \, A c^{3} d^{5} e^{2} - 15 \, B a c^{2} d^{4} e^{3} + 12 \, C a^{2} c d^{3} e^{4} + 12 \, A a c^{2} d^{3} e^{4} - 9 \, B a^{2} c d^{2} e^{5} + 2 \, C a^{3} d e^{6} + 6 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} e^{\left (-9\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {C c^{3} d^{8} e^{7}}{x e + d} - \frac {B c^{3} d^{7} e^{8}}{x e + d} + \frac {3 \, C a c^{2} d^{6} e^{9}}{x e + d} + \frac {A c^{3} d^{6} e^{9}}{x e + d} - \frac {3 \, B a c^{2} d^{5} e^{10}}{x e + d} + \frac {3 \, C a^{2} c d^{4} e^{11}}{x e + d} + \frac {3 \, A a c^{2} d^{4} e^{11}}{x e + d} - \frac {3 \, B a^{2} c d^{3} e^{12}}{x e + d} + \frac {C a^{3} d^{2} e^{13}}{x e + d} + \frac {3 \, A a^{2} c d^{2} e^{13}}{x e + d} - \frac {B a^{3} d e^{14}}{x e + d} + \frac {A a^{3} e^{15}}{x e + d}\right )} e^{\left (-16\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.99, size = 1511, normalized size = 3.11 \begin {gather*} x\,\left (\frac {C\,a^3+3\,A\,c\,a^2}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e}-\frac {A\,c^3+3\,C\,a\,c^2}{e^2}+\frac {C\,c^3\,d^2}{e^4}\right )}{e^2}-\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e}-\frac {A\,c^3+3\,C\,a\,c^2}{e^2}+\frac {C\,c^3\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e^2}+\frac {3\,B\,a\,c^2}{e^2}\right )}{e}+\frac {3\,a\,c\,\left (A\,c+C\,a\right )}{e^2}\right )}{e}+\frac {d^2\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e}-\frac {A\,c^3+3\,C\,a\,c^2}{e^2}+\frac {C\,c^3\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e^2}+\frac {3\,B\,a\,c^2}{e^2}\right )}{e^2}-\frac {3\,B\,a^2\,c}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e}-\frac {A\,c^3+3\,C\,a\,c^2}{e^2}+\frac {C\,c^3\,d^2}{e^4}\right )}{e^2}-\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e}-\frac {A\,c^3+3\,C\,a\,c^2}{e^2}+\frac {C\,c^3\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e^2}+\frac {3\,B\,a\,c^2}{e^2}\right )}{e}+\frac {3\,a\,c\,\left (A\,c+C\,a\right )}{e^2}\right )}{e^2}\right )+x^4\,\left (\frac {d\,\left (\frac {2\,d\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e}-\frac {A\,c^3+3\,C\,a\,c^2}{e^2}+\frac {C\,c^3\,d^2}{e^4}\right )}{2\,e}-\frac {d^2\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{4\,e^2}+\frac {3\,B\,a\,c^2}{4\,e^2}\right )-x^2\,\left (\frac {d\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e}-\frac {A\,c^3+3\,C\,a\,c^2}{e^2}+\frac {C\,c^3\,d^2}{e^4}\right )}{e^2}-\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e}-\frac {A\,c^3+3\,C\,a\,c^2}{e^2}+\frac {C\,c^3\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e^2}+\frac {3\,B\,a\,c^2}{e^2}\right )}{e}+\frac {3\,a\,c\,\left (A\,c+C\,a\right )}{e^2}\right )}{e}+\frac {d^2\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e}-\frac {A\,c^3+3\,C\,a\,c^2}{e^2}+\frac {C\,c^3\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e^2}+\frac {3\,B\,a\,c^2}{e^2}\right )}{2\,e^2}-\frac {3\,B\,a^2\,c}{2\,e^2}\right )+x^6\,\left (\frac {B\,c^3}{6\,e^2}-\frac {C\,c^3\,d}{3\,e^3}\right )-x^5\,\left (\frac {2\,d\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{5\,e}-\frac {A\,c^3+3\,C\,a\,c^2}{5\,e^2}+\frac {C\,c^3\,d^2}{5\,e^4}\right )+x^3\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e}-\frac {A\,c^3+3\,C\,a\,c^2}{e^2}+\frac {C\,c^3\,d^2}{e^4}\right )}{3\,e^2}-\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e}-\frac {A\,c^3+3\,C\,a\,c^2}{e^2}+\frac {C\,c^3\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {B\,c^3}{e^2}-\frac {2\,C\,c^3\,d}{e^3}\right )}{e^2}+\frac {3\,B\,a\,c^2}{e^2}\right )}{3\,e}+\frac {a\,c\,\left (A\,c+C\,a\right )}{e^2}\right )-\frac {C\,a^3\,d^2\,e^6-B\,a^3\,d\,e^7+A\,a^3\,e^8+3\,C\,a^2\,c\,d^4\,e^4-3\,B\,a^2\,c\,d^3\,e^5+3\,A\,a^2\,c\,d^2\,e^6+3\,C\,a\,c^2\,d^6\,e^2-3\,B\,a\,c^2\,d^5\,e^3+3\,A\,a\,c^2\,d^4\,e^4+C\,c^3\,d^8-B\,c^3\,d^7\,e+A\,c^3\,d^6\,e^2}{e\,\left (x\,e^9+d\,e^8\right )}-\frac {\ln \left (d+e\,x\right )\,\left (2\,C\,a^3\,d\,e^6-B\,a^3\,e^7+12\,C\,a^2\,c\,d^3\,e^4-9\,B\,a^2\,c\,d^2\,e^5+6\,A\,a^2\,c\,d\,e^6+18\,C\,a\,c^2\,d^5\,e^2-15\,B\,a\,c^2\,d^4\,e^3+12\,A\,a\,c^2\,d^3\,e^4+8\,C\,c^3\,d^7-7\,B\,c^3\,d^6\,e+6\,A\,c^3\,d^5\,e^2\right )}{e^9}+\frac {C\,c^3\,x^7}{7\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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