Optimal. Leaf size=300 \[ -\frac {2 b^5 (d+e x)^{11/2} (-6 a B e-A b e+7 b B d)}{11 e^8}+\frac {2 b^4 (d+e x)^{9/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{3 e^8}-\frac {10 b^3 (d+e x)^{7/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{7 e^8}+\frac {2 b^2 (d+e x)^{5/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8}-\frac {2 b (d+e x)^{3/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8}+\frac {2 \sqrt {d+e x} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8}+\frac {2 (b d-a e)^6 (B d-A e)}{e^8 \sqrt {d+e x}}+\frac {2 b^6 B (d+e x)^{13/2}}{13 e^8} \]
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Rubi [A] time = 0.15, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {27, 77} \begin {gather*} -\frac {2 b^5 (d+e x)^{11/2} (-6 a B e-A b e+7 b B d)}{11 e^8}+\frac {2 b^4 (d+e x)^{9/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{3 e^8}-\frac {10 b^3 (d+e x)^{7/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{7 e^8}+\frac {2 b^2 (d+e x)^{5/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8}-\frac {2 b (d+e x)^{3/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8}+\frac {2 \sqrt {d+e x} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8}+\frac {2 (b d-a e)^6 (B d-A e)}{e^8 \sqrt {d+e x}}+\frac {2 b^6 B (d+e x)^{13/2}}{13 e^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx &=\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)^{3/2}}+\frac {(-b d+a e)^5 (-7 b B d+6 A b e+a B e)}{e^7 \sqrt {d+e x}}+\frac {3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e) \sqrt {d+e x}}{e^7}-\frac {5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e) (d+e x)^{3/2}}{e^7}+\frac {5 b^3 (b d-a e)^2 (-7 b B d+3 A b e+4 a B e) (d+e x)^{5/2}}{e^7}-\frac {3 b^4 (b d-a e) (-7 b B d+2 A b e+5 a B e) (d+e x)^{7/2}}{e^7}+\frac {b^5 (-7 b B d+A b e+6 a B e) (d+e x)^{9/2}}{e^7}+\frac {b^6 B (d+e x)^{11/2}}{e^7}\right ) \, dx\\ &=\frac {2 (b d-a e)^6 (B d-A e)}{e^8 \sqrt {d+e x}}+\frac {2 (b d-a e)^5 (7 b B d-6 A b e-a B e) \sqrt {d+e x}}{e^8}-\frac {2 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) (d+e x)^{3/2}}{e^8}+\frac {2 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) (d+e x)^{5/2}}{e^8}-\frac {10 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^{7/2}}{7 e^8}+\frac {2 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^{9/2}}{3 e^8}-\frac {2 b^5 (7 b B d-A b e-6 a B e) (d+e x)^{11/2}}{11 e^8}+\frac {2 b^6 B (d+e x)^{13/2}}{13 e^8}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 259, normalized size = 0.86 \begin {gather*} \frac {2 \left (-273 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+1001 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-2145 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+3003 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)-3003 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)+3003 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)+3003 (b d-a e)^6 (B d-A e)+231 b^6 B (d+e x)^7\right )}{3003 e^8 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.23, size = 1069, normalized size = 3.56 \begin {gather*} \frac {2 \left (3003 b^6 B d^7-3003 A b^6 e d^6-18018 a b^5 B e d^6+21021 b^6 B (d+e x) d^6+18018 a A b^5 e^2 d^5+45045 a^2 b^4 B e^2 d^5-21021 b^6 B (d+e x)^2 d^5-18018 A b^6 e (d+e x) d^5-108108 a b^5 B e (d+e x) d^5-45045 a^2 A b^4 e^3 d^4-60060 a^3 b^3 B e^3 d^4+21021 b^6 B (d+e x)^3 d^4+15015 A b^6 e (d+e x)^2 d^4+90090 a b^5 B e (d+e x)^2 d^4+90090 a A b^5 e^2 (d+e x) d^4+225225 a^2 b^4 B e^2 (d+e x) d^4+60060 a^3 A b^3 e^4 d^3+45045 a^4 b^2 B e^4 d^3-15015 b^6 B (d+e x)^4 d^3-12012 A b^6 e (d+e x)^3 d^3-72072 a b^5 B e (d+e x)^3 d^3-60060 a A b^5 e^2 (d+e x)^2 d^3-150150 a^2 b^4 B e^2 (d+e x)^2 d^3-180180 a^2 A b^4 e^3 (d+e x) d^3-240240 a^3 b^3 B e^3 (d+e x) d^3-45045 a^4 A b^2 e^5 d^2-18018 a^5 b B e^5 d^2+7007 b^6 B (d+e x)^5 d^2+6435 A b^6 e (d+e x)^4 d^2+38610 a b^5 B e (d+e x)^4 d^2+36036 a A b^5 e^2 (d+e x)^3 d^2+90090 a^2 b^4 B e^2 (d+e x)^3 d^2+90090 a^2 A b^4 e^3 (d+e x)^2 d^2+120120 a^3 b^3 B e^3 (d+e x)^2 d^2+180180 a^3 A b^3 e^4 (d+e x) d^2+135135 a^4 b^2 B e^4 (d+e x) d^2+18018 a^5 A b e^6 d+3003 a^6 B e^6 d-1911 b^6 B (d+e x)^6 d-2002 A b^6 e (d+e x)^5 d-12012 a b^5 B e (d+e x)^5 d-12870 a A b^5 e^2 (d+e x)^4 d-32175 a^2 b^4 B e^2 (d+e x)^4 d-36036 a^2 A b^4 e^3 (d+e x)^3 d-48048 a^3 b^3 B e^3 (d+e x)^3 d-60060 a^3 A b^3 e^4 (d+e x)^2 d-45045 a^4 b^2 B e^4 (d+e x)^2 d-90090 a^4 A b^2 e^5 (d+e x) d-36036 a^5 b B e^5 (d+e x) d-3003 a^6 A e^7+231 b^6 B (d+e x)^7+273 A b^6 e (d+e x)^6+1638 a b^5 B e (d+e x)^6+2002 a A b^5 e^2 (d+e x)^5+5005 a^2 b^4 B e^2 (d+e x)^5+6435 a^2 A b^4 e^3 (d+e x)^4+8580 a^3 b^3 B e^3 (d+e x)^4+12012 a^3 A b^3 e^4 (d+e x)^3+9009 a^4 b^2 B e^4 (d+e x)^3+15015 a^4 A b^2 e^5 (d+e x)^2+6006 a^5 b B e^5 (d+e x)^2+18018 a^5 A b e^6 (d+e x)+3003 a^6 B e^6 (d+e x)\right )}{3003 e^8 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 778, normalized size = 2.59 \begin {gather*} \frac {2 \, {\left (231 \, B b^{6} e^{7} x^{7} + 14336 \, B b^{6} d^{7} - 3003 \, A a^{6} e^{7} - 13312 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 36608 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 54912 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 48048 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 24024 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 6006 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} - 21 \, {\left (14 \, B b^{6} d e^{6} - 13 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 7 \, {\left (56 \, B b^{6} d^{2} e^{5} - 52 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 143 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} - 5 \, {\left (112 \, B b^{6} d^{3} e^{4} - 104 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 286 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} - 429 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + {\left (896 \, B b^{6} d^{4} e^{3} - 832 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 2288 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} - 3432 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 3003 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} - {\left (1792 \, B b^{6} d^{5} e^{2} - 1664 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 4576 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} - 6864 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 6006 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} - 3003 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + {\left (7168 \, B b^{6} d^{6} e - 6656 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 18304 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} - 27456 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 24024 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 12012 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 3003 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x\right )} \sqrt {e x + d}}{3003 \, {\left (e^{9} x + d e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 1131, normalized size = 3.77
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 913, normalized size = 3.04 \begin {gather*} -\frac {2 \left (-231 B \,b^{6} x^{7} e^{7}-273 A \,b^{6} e^{7} x^{6}-1638 B a \,b^{5} e^{7} x^{6}+294 B \,b^{6} d \,e^{6} x^{6}-2002 A a \,b^{5} e^{7} x^{5}+364 A \,b^{6} d \,e^{6} x^{5}-5005 B \,a^{2} b^{4} e^{7} x^{5}+2184 B a \,b^{5} d \,e^{6} x^{5}-392 B \,b^{6} d^{2} e^{5} x^{5}-6435 A \,a^{2} b^{4} e^{7} x^{4}+2860 A a \,b^{5} d \,e^{6} x^{4}-520 A \,b^{6} d^{2} e^{5} x^{4}-8580 B \,a^{3} b^{3} e^{7} x^{4}+7150 B \,a^{2} b^{4} d \,e^{6} x^{4}-3120 B a \,b^{5} d^{2} e^{5} x^{4}+560 B \,b^{6} d^{3} e^{4} x^{4}-12012 A \,a^{3} b^{3} e^{7} x^{3}+10296 A \,a^{2} b^{4} d \,e^{6} x^{3}-4576 A a \,b^{5} d^{2} e^{5} x^{3}+832 A \,b^{6} d^{3} e^{4} x^{3}-9009 B \,a^{4} b^{2} e^{7} x^{3}+13728 B \,a^{3} b^{3} d \,e^{6} x^{3}-11440 B \,a^{2} b^{4} d^{2} e^{5} x^{3}+4992 B a \,b^{5} d^{3} e^{4} x^{3}-896 B \,b^{6} d^{4} e^{3} x^{3}-15015 A \,a^{4} b^{2} e^{7} x^{2}+24024 A \,a^{3} b^{3} d \,e^{6} x^{2}-20592 A \,a^{2} b^{4} d^{2} e^{5} x^{2}+9152 A a \,b^{5} d^{3} e^{4} x^{2}-1664 A \,b^{6} d^{4} e^{3} x^{2}-6006 B \,a^{5} b \,e^{7} x^{2}+18018 B \,a^{4} b^{2} d \,e^{6} x^{2}-27456 B \,a^{3} b^{3} d^{2} e^{5} x^{2}+22880 B \,a^{2} b^{4} d^{3} e^{4} x^{2}-9984 B a \,b^{5} d^{4} e^{3} x^{2}+1792 B \,b^{6} d^{5} e^{2} x^{2}-18018 A \,a^{5} b \,e^{7} x +60060 A \,a^{4} b^{2} d \,e^{6} x -96096 A \,a^{3} b^{3} d^{2} e^{5} x +82368 A \,a^{2} b^{4} d^{3} e^{4} x -36608 A a \,b^{5} d^{4} e^{3} x +6656 A \,b^{6} d^{5} e^{2} x -3003 B \,a^{6} e^{7} x +24024 B \,a^{5} b d \,e^{6} x -72072 B \,a^{4} b^{2} d^{2} e^{5} x +109824 B \,a^{3} b^{3} d^{3} e^{4} x -91520 B \,a^{2} b^{4} d^{4} e^{3} x +39936 B a \,b^{5} d^{5} e^{2} x -7168 B \,b^{6} d^{6} e x +3003 A \,a^{6} e^{7}-36036 A \,a^{5} b d \,e^{6}+120120 A \,a^{4} b^{2} d^{2} e^{5}-192192 A \,a^{3} b^{3} d^{3} e^{4}+164736 A \,a^{2} b^{4} d^{4} e^{3}-73216 A a \,b^{5} d^{5} e^{2}+13312 A \,b^{6} d^{6} e -6006 B \,a^{6} d \,e^{6}+48048 B \,a^{5} b \,d^{2} e^{5}-144144 B \,a^{4} b^{2} d^{3} e^{4}+219648 B \,a^{3} b^{3} d^{4} e^{3}-183040 B \,a^{2} b^{4} d^{5} e^{2}+79872 B a \,b^{5} d^{6} e -14336 B \,b^{6} d^{7}\right )}{3003 \sqrt {e x +d}\, e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.77, size = 775, normalized size = 2.58 \begin {gather*} \frac {2 \, {\left (\frac {231 \, {\left (e x + d\right )}^{\frac {13}{2}} B b^{6} - 273 \, {\left (7 \, B b^{6} d - {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 1001 \, {\left (7 \, B b^{6} d^{2} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 2145 \, {\left (7 \, B b^{6} d^{3} - 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{2} - {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 3003 \, {\left (7 \, B b^{6} d^{4} - 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{2} - 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{3} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 3003 \, {\left (7 \, B b^{6} d^{5} - 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{4} - {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{5}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 3003 \, {\left (7 \, B b^{6} d^{6} - 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{2} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{3} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{4} - 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{6}\right )} \sqrt {e x + d}}{e^{7}} + \frac {3003 \, {\left (B b^{6} d^{7} - A a^{6} e^{7} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 5 \, {\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} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6}\right )}}{\sqrt {e x + d} e^{7}}\right )}}{3003 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.94, size = 438, normalized size = 1.46 \begin {gather*} \frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b^6\,e-14\,B\,b^6\,d+12\,B\,a\,b^5\,e\right )}{11\,e^8}-\frac {-2\,B\,a^6\,d\,e^6+2\,A\,a^6\,e^7+12\,B\,a^5\,b\,d^2\,e^5-12\,A\,a^5\,b\,d\,e^6-30\,B\,a^4\,b^2\,d^3\,e^4+30\,A\,a^4\,b^2\,d^2\,e^5+40\,B\,a^3\,b^3\,d^4\,e^3-40\,A\,a^3\,b^3\,d^3\,e^4-30\,B\,a^2\,b^4\,d^5\,e^2+30\,A\,a^2\,b^4\,d^4\,e^3+12\,B\,a\,b^5\,d^6\,e-12\,A\,a\,b^5\,d^5\,e^2-2\,B\,b^6\,d^7+2\,A\,b^6\,d^6\,e}{e^8\,\sqrt {d+e\,x}}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,\sqrt {d+e\,x}\,\left (6\,A\,b\,e+B\,a\,e-7\,B\,b\,d\right )}{e^8}+\frac {2\,B\,b^6\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {2\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}\,\left (5\,A\,b\,e+2\,B\,a\,e-7\,B\,b\,d\right )}{e^8}+\frac {2\,b^4\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e+5\,B\,a\,e-7\,B\,b\,d\right )}{3\,e^8}+\frac {2\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}\,\left (4\,A\,b\,e+3\,B\,a\,e-7\,B\,b\,d\right )}{e^8}+\frac {10\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}\,\left (3\,A\,b\,e+4\,B\,a\,e-7\,B\,b\,d\right )}{7\,e^8} \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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