3.16.81 \(\int (A+B x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)
Optimal. Leaf size=308 \[ -\frac {2 b^5 (d+e x)^{19/2} (-6 a B e-A b e+7 b B d)}{19 e^8}+\frac {6 b^4 (d+e x)^{17/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{17 e^8}-\frac {2 b^3 (d+e x)^{15/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac {10 b^2 (d+e x)^{13/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{13 e^8}-\frac {6 b (d+e x)^{11/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{11 e^8}+\frac {2 (d+e x)^{9/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{9 e^8}-\frac {2 (d+e x)^{7/2} (b d-a e)^6 (B d-A e)}{7 e^8}+\frac {2 b^6 B (d+e x)^{21/2}}{21 e^8} \]
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Rubi [A] time = 0.15, antiderivative size = 308, 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)^{19/2} (-6 a B e-A b e+7 b B d)}{19 e^8}+\frac {6 b^4 (d+e x)^{17/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{17 e^8}-\frac {2 b^3 (d+e x)^{15/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac {10 b^2 (d+e x)^{13/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{13 e^8}-\frac {6 b (d+e x)^{11/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{11 e^8}+\frac {2 (d+e x)^{9/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{9 e^8}-\frac {2 (d+e x)^{7/2} (b d-a e)^6 (B d-A e)}{7 e^8}+\frac {2 b^6 B (d+e x)^{21/2}}{21 e^8} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(-2*(b*d - a*e)^6*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^8) + (2*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x
)^(9/2))/(9*e^8) - (6*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^(11/2))/(11*e^8) + (10*b^2*(b*d
- a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^(13/2))/(13*e^8) - (2*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e -
4*a*B*e)*(d + e*x)^(15/2))/(3*e^8) + (6*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(17/2))/(17*e^
8) - (2*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^(19/2))/(19*e^8) + (2*b^6*B*(d + e*x)^(21/2))/(21*e^8)
Rule 27
Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (A+B x) (d+e x)^{5/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^6 (-B d+A e) (d+e x)^{5/2}}{e^7}+\frac {(-b d+a e)^5 (-7 b B d+6 A b e+a B e) (d+e x)^{7/2}}{e^7}+\frac {3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e) (d+e x)^{9/2}}{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)^{11/2}}{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)^{13/2}}{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)^{15/2}}{e^7}+\frac {b^5 (-7 b B d+A b e+6 a B e) (d+e x)^{17/2}}{e^7}+\frac {b^6 B (d+e x)^{19/2}}{e^7}\right ) \, dx\\ &=-\frac {2 (b d-a e)^6 (B d-A e) (d+e x)^{7/2}}{7 e^8}+\frac {2 (b d-a e)^5 (7 b B d-6 A b e-a B e) (d+e x)^{9/2}}{9 e^8}-\frac {6 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) (d+e x)^{11/2}}{11 e^8}+\frac {10 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) (d+e x)^{13/2}}{13 e^8}-\frac {2 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^{15/2}}{3 e^8}+\frac {6 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^{17/2}}{17 e^8}-\frac {2 b^5 (7 b B d-A b e-6 a B e) (d+e x)^{19/2}}{19 e^8}+\frac {2 b^6 B (d+e x)^{21/2}}{21 e^8}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 259, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (-153153 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+513513 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-969969 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+1119195 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)-793611 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)+323323 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)-415701 (b d-a e)^6 (B d-A e)+138567 b^6 B (d+e x)^7\right )}{2909907 e^8} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(d + e*x)^(7/2)*(-415701*(b*d - a*e)^6*(B*d - A*e) + 323323*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d +
e*x) - 793611*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^2 + 1119195*b^2*(b*d - a*e)^3*(7*b*B*d -
4*A*b*e - 3*a*B*e)*(d + e*x)^3 - 969969*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^4 + 513513*
b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^5 - 153153*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^6
+ 138567*b^6*B*(d + e*x)^7))/(2909907*e^8)
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IntegrateAlgebraic [B] time = 0.44, size = 1069, normalized size = 3.47 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (-415701 b^6 B d^7+415701 A b^6 e d^6+2494206 a b^5 B e d^6+2263261 b^6 B (d+e x) d^6-2494206 a A b^5 e^2 d^5-6235515 a^2 b^4 B e^2 d^5-5555277 b^6 B (d+e x)^2 d^5-1939938 A b^6 e (d+e x) d^5-11639628 a b^5 B e (d+e x) d^5+6235515 a^2 A b^4 e^3 d^4+8314020 a^3 b^3 B e^3 d^4+7834365 b^6 B (d+e x)^3 d^4+3968055 A b^6 e (d+e x)^2 d^4+23808330 a b^5 B e (d+e x)^2 d^4+9699690 a A b^5 e^2 (d+e x) d^4+24249225 a^2 b^4 B e^2 (d+e x) d^4-8314020 a^3 A b^3 e^4 d^3-6235515 a^4 b^2 B e^4 d^3-6789783 b^6 B (d+e x)^4 d^3-4476780 A b^6 e (d+e x)^3 d^3-26860680 a b^5 B e (d+e x)^3 d^3-15872220 a A b^5 e^2 (d+e x)^2 d^3-39680550 a^2 b^4 B e^2 (d+e x)^2 d^3-19399380 a^2 A b^4 e^3 (d+e x) d^3-25865840 a^3 b^3 B e^3 (d+e x) d^3+6235515 a^4 A b^2 e^5 d^2+2494206 a^5 b B e^5 d^2+3594591 b^6 B (d+e x)^5 d^2+2909907 A b^6 e (d+e x)^4 d^2+17459442 a b^5 B e (d+e x)^4 d^2+13430340 a A b^5 e^2 (d+e x)^3 d^2+33575850 a^2 b^4 B e^2 (d+e x)^3 d^2+23808330 a^2 A b^4 e^3 (d+e x)^2 d^2+31744440 a^3 b^3 B e^3 (d+e x)^2 d^2+19399380 a^3 A b^3 e^4 (d+e x) d^2+14549535 a^4 b^2 B e^4 (d+e x) d^2-2494206 a^5 A b e^6 d-415701 a^6 B e^6 d-1072071 b^6 B (d+e x)^6 d-1027026 A b^6 e (d+e x)^5 d-6162156 a b^5 B e (d+e x)^5 d-5819814 a A b^5 e^2 (d+e x)^4 d-14549535 a^2 b^4 B e^2 (d+e x)^4 d-13430340 a^2 A b^4 e^3 (d+e x)^3 d-17907120 a^3 b^3 B e^3 (d+e x)^3 d-15872220 a^3 A b^3 e^4 (d+e x)^2 d-11904165 a^4 b^2 B e^4 (d+e x)^2 d-9699690 a^4 A b^2 e^5 (d+e x) d-3879876 a^5 b B e^5 (d+e x) d+415701 a^6 A e^7+138567 b^6 B (d+e x)^7+153153 A b^6 e (d+e x)^6+918918 a b^5 B e (d+e x)^6+1027026 a A b^5 e^2 (d+e x)^5+2567565 a^2 b^4 B e^2 (d+e x)^5+2909907 a^2 A b^4 e^3 (d+e x)^4+3879876 a^3 b^3 B e^3 (d+e x)^4+4476780 a^3 A b^3 e^4 (d+e x)^3+3357585 a^4 b^2 B e^4 (d+e x)^3+3968055 a^4 A b^2 e^5 (d+e x)^2+1587222 a^5 b B e^5 (d+e x)^2+1939938 a^5 A b e^6 (d+e x)+323323 a^6 B e^6 (d+e x)\right )}{2909907 e^8} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(d + e*x)^(7/2)*(-415701*b^6*B*d^7 + 415701*A*b^6*d^6*e + 2494206*a*b^5*B*d^6*e - 2494206*a*A*b^5*d^5*e^2 -
6235515*a^2*b^4*B*d^5*e^2 + 6235515*a^2*A*b^4*d^4*e^3 + 8314020*a^3*b^3*B*d^4*e^3 - 8314020*a^3*A*b^3*d^3*e^4
- 6235515*a^4*b^2*B*d^3*e^4 + 6235515*a^4*A*b^2*d^2*e^5 + 2494206*a^5*b*B*d^2*e^5 - 2494206*a^5*A*b*d*e^6 - 4
15701*a^6*B*d*e^6 + 415701*a^6*A*e^7 + 2263261*b^6*B*d^6*(d + e*x) - 1939938*A*b^6*d^5*e*(d + e*x) - 11639628*
a*b^5*B*d^5*e*(d + e*x) + 9699690*a*A*b^5*d^4*e^2*(d + e*x) + 24249225*a^2*b^4*B*d^4*e^2*(d + e*x) - 19399380*
a^2*A*b^4*d^3*e^3*(d + e*x) - 25865840*a^3*b^3*B*d^3*e^3*(d + e*x) + 19399380*a^3*A*b^3*d^2*e^4*(d + e*x) + 14
549535*a^4*b^2*B*d^2*e^4*(d + e*x) - 9699690*a^4*A*b^2*d*e^5*(d + e*x) - 3879876*a^5*b*B*d*e^5*(d + e*x) + 193
9938*a^5*A*b*e^6*(d + e*x) + 323323*a^6*B*e^6*(d + e*x) - 5555277*b^6*B*d^5*(d + e*x)^2 + 3968055*A*b^6*d^4*e*
(d + e*x)^2 + 23808330*a*b^5*B*d^4*e*(d + e*x)^2 - 15872220*a*A*b^5*d^3*e^2*(d + e*x)^2 - 39680550*a^2*b^4*B*d
^3*e^2*(d + e*x)^2 + 23808330*a^2*A*b^4*d^2*e^3*(d + e*x)^2 + 31744440*a^3*b^3*B*d^2*e^3*(d + e*x)^2 - 1587222
0*a^3*A*b^3*d*e^4*(d + e*x)^2 - 11904165*a^4*b^2*B*d*e^4*(d + e*x)^2 + 3968055*a^4*A*b^2*e^5*(d + e*x)^2 + 158
7222*a^5*b*B*e^5*(d + e*x)^2 + 7834365*b^6*B*d^4*(d + e*x)^3 - 4476780*A*b^6*d^3*e*(d + e*x)^3 - 26860680*a*b^
5*B*d^3*e*(d + e*x)^3 + 13430340*a*A*b^5*d^2*e^2*(d + e*x)^3 + 33575850*a^2*b^4*B*d^2*e^2*(d + e*x)^3 - 134303
40*a^2*A*b^4*d*e^3*(d + e*x)^3 - 17907120*a^3*b^3*B*d*e^3*(d + e*x)^3 + 4476780*a^3*A*b^3*e^4*(d + e*x)^3 + 33
57585*a^4*b^2*B*e^4*(d + e*x)^3 - 6789783*b^6*B*d^3*(d + e*x)^4 + 2909907*A*b^6*d^2*e*(d + e*x)^4 + 17459442*a
*b^5*B*d^2*e*(d + e*x)^4 - 5819814*a*A*b^5*d*e^2*(d + e*x)^4 - 14549535*a^2*b^4*B*d*e^2*(d + e*x)^4 + 2909907*
a^2*A*b^4*e^3*(d + e*x)^4 + 3879876*a^3*b^3*B*e^3*(d + e*x)^4 + 3594591*b^6*B*d^2*(d + e*x)^5 - 1027026*A*b^6*
d*e*(d + e*x)^5 - 6162156*a*b^5*B*d*e*(d + e*x)^5 + 1027026*a*A*b^5*e^2*(d + e*x)^5 + 2567565*a^2*b^4*B*e^2*(d
+ e*x)^5 - 1072071*b^6*B*d*(d + e*x)^6 + 153153*A*b^6*e*(d + e*x)^6 + 918918*a*b^5*B*e*(d + e*x)^6 + 138567*b
^6*B*(d + e*x)^7))/(2909907*e^8)
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fricas [B] time = 0.43, size = 1293, normalized size = 4.20
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
[Out]
2/2909907*(138567*B*b^6*e^10*x^10 - 2048*B*b^6*d^10 + 415701*A*a^6*d^3*e^7 + 3072*(6*B*a*b^5 + A*b^6)*d^9*e -
14592*(5*B*a^2*b^4 + 2*A*a*b^5)*d^8*e^2 + 41344*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^7*e^3 - 77520*(3*B*a^4*b^2 + 4*A
*a^3*b^3)*d^6*e^4 + 100776*(2*B*a^5*b + 5*A*a^4*b^2)*d^5*e^5 - 92378*(B*a^6 + 6*A*a^5*b)*d^4*e^6 + 7293*(43*B*
b^6*d*e^9 + 21*(6*B*a*b^5 + A*b^6)*e^10)*x^9 + 3861*(47*B*b^6*d^2*e^8 + 91*(6*B*a*b^5 + A*b^6)*d*e^9 + 133*(5*
B*a^2*b^4 + 2*A*a*b^5)*e^10)*x^8 + 429*(B*b^6*d^3*e^7 + 483*(6*B*a*b^5 + A*b^6)*d^2*e^8 + 2793*(5*B*a^2*b^4 +
2*A*a*b^5)*d*e^9 + 2261*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^10)*x^7 - 231*(2*B*b^6*d^4*e^6 - 3*(6*B*a*b^5 + A*b^6)*d
^3*e^7 - 3135*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^8 - 10013*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^9 - 4845*(3*B*a^4*b^2
+ 4*A*a^3*b^3)*e^10)*x^6 + 63*(8*B*b^6*d^5*e^5 - 12*(6*B*a*b^5 + A*b^6)*d^4*e^6 + 57*(5*B*a^2*b^4 + 2*A*a*b^5)
*d^3*e^7 + 22933*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^8 + 43605*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^9 + 12597*(2*B*a^
5*b + 5*A*a^4*b^2)*e^10)*x^5 - 7*(80*B*b^6*d^6*e^4 - 120*(6*B*a*b^5 + A*b^6)*d^5*e^5 + 570*(5*B*a^2*b^4 + 2*A*
a*b^5)*d^4*e^6 - 1615*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^7 - 256785*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^8 - 28973
1*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^9 - 46189*(B*a^6 + 6*A*a^5*b)*e^10)*x^4 + (640*B*b^6*d^7*e^3 + 415701*A*a^6*e^
10 - 960*(6*B*a*b^5 + A*b^6)*d^6*e^4 + 4560*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^5 - 12920*(4*B*a^3*b^3 + 3*A*a^2*b
^4)*d^4*e^6 + 24225*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^7 + 1423461*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^8 + 877591*(
B*a^6 + 6*A*a^5*b)*d*e^9)*x^3 - 3*(256*B*b^6*d^8*e^2 - 415701*A*a^6*d*e^9 - 384*(6*B*a*b^5 + A*b^6)*d^7*e^3 +
1824*(5*B*a^2*b^4 + 2*A*a*b^5)*d^6*e^4 - 5168*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^5*e^5 + 9690*(3*B*a^4*b^2 + 4*A*a^
3*b^3)*d^4*e^6 - 12597*(2*B*a^5*b + 5*A*a^4*b^2)*d^3*e^7 - 230945*(B*a^6 + 6*A*a^5*b)*d^2*e^8)*x^2 + (1024*B*b
^6*d^9*e + 1247103*A*a^6*d^2*e^8 - 1536*(6*B*a*b^5 + A*b^6)*d^8*e^2 + 7296*(5*B*a^2*b^4 + 2*A*a*b^5)*d^7*e^3 -
20672*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^6*e^4 + 38760*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^5*e^5 - 50388*(2*B*a^5*b + 5*
A*a^4*b^2)*d^4*e^6 + 46189*(B*a^6 + 6*A*a^5*b)*d^3*e^7)*x)*sqrt(e*x + d)/e^8
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giac [B] time = 0.51, size = 4768, normalized size = 15.48
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
[Out]
2/14549535*(4849845*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^6*d^3*e^(-1) + 29099070*((x*e + d)^(3/2) - 3*sqr
t(x*e + d)*d)*A*a^5*b*d^3*e^(-1) + 5819814*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B
*a^5*b*d^3*e^(-2) + 14549535*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^4*b^2*d^3*e
^(-2) + 6235515*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a
^4*b^2*d^3*e^(-3) + 8314020*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e +
d)*d^3)*A*a^3*b^3*d^3*e^(-3) + 923780*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 -
420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a^3*b^3*d^3*e^(-4) + 692835*(35*(x*e + d)^(9/2) - 180*(x*e
+ d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a^2*b^4*d^3*e^(-4
) + 314925*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 +
1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a^2*b^4*d^3*e^(-5) + 125970*(63*(x*e + d)^(11/2) - 385*(x
*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e
+ d)*d^5)*A*a*b^5*d^3*e^(-5) + 29070*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d
^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*
B*a*b^5*d^3*e^(-6) + 4845*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x
*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*A*b^6*d^3*e^
(-6) + 2261*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/
2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d
)*d^7)*B*b^6*d^3*e^(-7) + 2909907*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^6*d^2*
e^(-1) + 17459442*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^5*b*d^2*e^(-1) + 74826
18*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a^5*b*d^2*e^(-
2) + 18706545*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^4
*b^2*d^2*e^(-2) + 2078505*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d
)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a^4*b^2*d^2*e^(-3) + 2771340*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*
d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a^3*b^3*d^2*e^(-3) + 1259700*
(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e
+ d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a^3*b^3*d^2*e^(-4) + 944775*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/
2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*
A*a^2*b^4*d^2*e^(-4) + 218025*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 858
0*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*a^2*b^
4*d^2*e^(-5) + 87210*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e +
d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*A*a*b^5*d^2*e^(-5
) + 40698*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)
*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*
d^7)*B*a*b^5*d^2*e^(-6) + 6783*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 -
25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6
- 6435*sqrt(x*e + d)*d^7)*A*b^6*d^2*e^(-6) + 399*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x
*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 6
12612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*B*b^6*d^2*e^(-7) + 14549535
*sqrt(x*e + d)*A*a^6*d^3 + 14549535*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a^6*d^2 + 1247103*(5*(x*e + d)^(7/
2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a^6*d*e^(-1) + 7482618*(5*(x*e +
d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^5*b*d*e^(-1) + 831402*(35
*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e +
d)*d^4)*B*a^5*b*d*e^(-2) + 2078505*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420
*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a^4*b^2*d*e^(-2) + 944775*(63*(x*e + d)^(11/2) - 385*(x*e + d)
^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d
^5)*B*a^4*b^2*d*e^(-3) + 1259700*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386
*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*a^3*b^3*d*e^(-3) + 290700*(231*(x*e
+ d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^
(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*a^3*b^3*d*e^(-4) + 218025*(231*(x*e + d)^(13/
2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4
- 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*A*a^2*b^4*d*e^(-4) + 101745*(429*(x*e + d)^(15/2) - 3465*
(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 2702
7*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*B*a^2*b^4*d*e^(-5) + 40698*(429*(x
*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e
+ d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*A*a*b^5*d*e^
(-5) + 2394*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)
^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x
*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*B*a*b^5*d*e^(-6) + 399*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^
(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*
e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*A*b^6*d
*e^(-6) + 189*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(17/2)*d + 554268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e
+ d)^(13/2)*d^3 + 2645370*(x*e + d)^(11/2)*d^4 - 3233230*(x*e + d)^(9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 -
1662804*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8 - 230945*sqrt(x*e + d)*d^9)*B*b^6*d*e^(-7) + 2909907*
(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^6*d + 46189*(35*(x*e + d)^(9/2) - 180*(x
*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a^6*e^(-1) + 27
7134*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqr
t(x*e + d)*d^4)*A*a^5*b*e^(-1) + 125970*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2
- 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a^5*b*e^(-2) + 314925*(63*(x
*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(
3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*a^4*b^2*e^(-2) + 72675*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5
005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 300
3*sqrt(x*e + d)*d^6)*B*a^4*b^2*e^(-3) + 96900*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)
^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e +
d)*d^6)*A*a^3*b^3*e^(-3) + 45220*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2
- 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^
6 - 6435*sqrt(x*e + d)*d^7)*B*a^3*b^3*e^(-4) + 33915*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(
x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 1501
5*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*A*a^2*b^4*e^(-4) + 1995*(6435*(x*e + d)^(17/2) - 58344*(x*e +
d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*
(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*B*a^
2*b^4*e^(-5) + 798*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x
*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 29
1720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*A*a*b^5*e^(-5) + 378*(12155*(x*e + d)^(19/2) - 122265*(x*
e + d)^(17/2)*d + 554268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e + d)^(13/2)*d^3 + 2645370*(x*e + d)^(11/2)*d^4 -
3233230*(x*e + d)^(9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 - 1662804*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/
2)*d^8 - 230945*sqrt(x*e + d)*d^9)*B*a*b^5*e^(-6) + 63*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(17/2)*d + 5
54268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e + d)^(13/2)*d^3 + 2645370*(x*e + d)^(11/2)*d^4 - 3233230*(x*e + d)^(
9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 - 1662804*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8 - 230945*sqr
t(x*e + d)*d^9)*A*b^6*e^(-6) + 15*(46189*(x*e + d)^(21/2) - 510510*(x*e + d)^(19/2)*d + 2567565*(x*e + d)^(17/
2)*d^2 - 7759752*(x*e + d)^(15/2)*d^3 + 15668730*(x*e + d)^(13/2)*d^4 - 22221108*(x*e + d)^(11/2)*d^5 + 226326
10*(x*e + d)^(9/2)*d^6 - 16628040*(x*e + d)^(7/2)*d^7 + 8729721*(x*e + d)^(5/2)*d^8 - 3233230*(x*e + d)^(3/2)*
d^9 + 969969*sqrt(x*e + d)*d^10)*B*b^6*e^(-7) + 415701*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d
)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^6)*e^(-1)
________________________________________________________________________________________
maple [B] time = 0.06, size = 913, normalized size = 2.96 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (138567 B \,b^{6} x^{7} e^{7}+153153 A \,b^{6} e^{7} x^{6}+918918 B a \,b^{5} e^{7} x^{6}-102102 B \,b^{6} d \,e^{6} x^{6}+1027026 A a \,b^{5} e^{7} x^{5}-108108 A \,b^{6} d \,e^{6} x^{5}+2567565 B \,a^{2} b^{4} e^{7} x^{5}-648648 B a \,b^{5} d \,e^{6} x^{5}+72072 B \,b^{6} d^{2} e^{5} x^{5}+2909907 A \,a^{2} b^{4} e^{7} x^{4}-684684 A a \,b^{5} d \,e^{6} x^{4}+72072 A \,b^{6} d^{2} e^{5} x^{4}+3879876 B \,a^{3} b^{3} e^{7} x^{4}-1711710 B \,a^{2} b^{4} d \,e^{6} x^{4}+432432 B a \,b^{5} d^{2} e^{5} x^{4}-48048 B \,b^{6} d^{3} e^{4} x^{4}+4476780 A \,a^{3} b^{3} e^{7} x^{3}-1790712 A \,a^{2} b^{4} d \,e^{6} x^{3}+421344 A a \,b^{5} d^{2} e^{5} x^{3}-44352 A \,b^{6} d^{3} e^{4} x^{3}+3357585 B \,a^{4} b^{2} e^{7} x^{3}-2387616 B \,a^{3} b^{3} d \,e^{6} x^{3}+1053360 B \,a^{2} b^{4} d^{2} e^{5} x^{3}-266112 B a \,b^{5} d^{3} e^{4} x^{3}+29568 B \,b^{6} d^{4} e^{3} x^{3}+3968055 A \,a^{4} b^{2} e^{7} x^{2}-2441880 A \,a^{3} b^{3} d \,e^{6} x^{2}+976752 A \,a^{2} b^{4} d^{2} e^{5} x^{2}-229824 A a \,b^{5} d^{3} e^{4} x^{2}+24192 A \,b^{6} d^{4} e^{3} x^{2}+1587222 B \,a^{5} b \,e^{7} x^{2}-1831410 B \,a^{4} b^{2} d \,e^{6} x^{2}+1302336 B \,a^{3} b^{3} d^{2} e^{5} x^{2}-574560 B \,a^{2} b^{4} d^{3} e^{4} x^{2}+145152 B a \,b^{5} d^{4} e^{3} x^{2}-16128 B \,b^{6} d^{5} e^{2} x^{2}+1939938 A \,a^{5} b \,e^{7} x -1763580 A \,a^{4} b^{2} d \,e^{6} x +1085280 A \,a^{3} b^{3} d^{2} e^{5} x -434112 A \,a^{2} b^{4} d^{3} e^{4} x +102144 A a \,b^{5} d^{4} e^{3} x -10752 A \,b^{6} d^{5} e^{2} x +323323 B \,a^{6} e^{7} x -705432 B \,a^{5} b d \,e^{6} x +813960 B \,a^{4} b^{2} d^{2} e^{5} x -578816 B \,a^{3} b^{3} d^{3} e^{4} x +255360 B \,a^{2} b^{4} d^{4} e^{3} x -64512 B a \,b^{5} d^{5} e^{2} x +7168 B \,b^{6} d^{6} e x +415701 A \,a^{6} e^{7}-554268 A \,a^{5} b d \,e^{6}+503880 A \,a^{4} b^{2} d^{2} e^{5}-310080 A \,a^{3} b^{3} d^{3} e^{4}+124032 A \,a^{2} b^{4} d^{4} e^{3}-29184 A a \,b^{5} d^{5} e^{2}+3072 A \,b^{6} d^{6} e -92378 B \,a^{6} d \,e^{6}+201552 B \,a^{5} b \,d^{2} e^{5}-232560 B \,a^{4} b^{2} d^{3} e^{4}+165376 B \,a^{3} b^{3} d^{4} e^{3}-72960 B \,a^{2} b^{4} d^{5} e^{2}+18432 B a \,b^{5} d^{6} e -2048 B \,b^{6} d^{7}\right )}{2909907 e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
2/2909907*(e*x+d)^(7/2)*(138567*B*b^6*e^7*x^7+153153*A*b^6*e^7*x^6+918918*B*a*b^5*e^7*x^6-102102*B*b^6*d*e^6*x
^6+1027026*A*a*b^5*e^7*x^5-108108*A*b^6*d*e^6*x^5+2567565*B*a^2*b^4*e^7*x^5-648648*B*a*b^5*d*e^6*x^5+72072*B*b
^6*d^2*e^5*x^5+2909907*A*a^2*b^4*e^7*x^4-684684*A*a*b^5*d*e^6*x^4+72072*A*b^6*d^2*e^5*x^4+3879876*B*a^3*b^3*e^
7*x^4-1711710*B*a^2*b^4*d*e^6*x^4+432432*B*a*b^5*d^2*e^5*x^4-48048*B*b^6*d^3*e^4*x^4+4476780*A*a^3*b^3*e^7*x^3
-1790712*A*a^2*b^4*d*e^6*x^3+421344*A*a*b^5*d^2*e^5*x^3-44352*A*b^6*d^3*e^4*x^3+3357585*B*a^4*b^2*e^7*x^3-2387
616*B*a^3*b^3*d*e^6*x^3+1053360*B*a^2*b^4*d^2*e^5*x^3-266112*B*a*b^5*d^3*e^4*x^3+29568*B*b^6*d^4*e^3*x^3+39680
55*A*a^4*b^2*e^7*x^2-2441880*A*a^3*b^3*d*e^6*x^2+976752*A*a^2*b^4*d^2*e^5*x^2-229824*A*a*b^5*d^3*e^4*x^2+24192
*A*b^6*d^4*e^3*x^2+1587222*B*a^5*b*e^7*x^2-1831410*B*a^4*b^2*d*e^6*x^2+1302336*B*a^3*b^3*d^2*e^5*x^2-574560*B*
a^2*b^4*d^3*e^4*x^2+145152*B*a*b^5*d^4*e^3*x^2-16128*B*b^6*d^5*e^2*x^2+1939938*A*a^5*b*e^7*x-1763580*A*a^4*b^2
*d*e^6*x+1085280*A*a^3*b^3*d^2*e^5*x-434112*A*a^2*b^4*d^3*e^4*x+102144*A*a*b^5*d^4*e^3*x-10752*A*b^6*d^5*e^2*x
+323323*B*a^6*e^7*x-705432*B*a^5*b*d*e^6*x+813960*B*a^4*b^2*d^2*e^5*x-578816*B*a^3*b^3*d^3*e^4*x+255360*B*a^2*
b^4*d^4*e^3*x-64512*B*a*b^5*d^5*e^2*x+7168*B*b^6*d^6*e*x+415701*A*a^6*e^7-554268*A*a^5*b*d*e^6+503880*A*a^4*b^
2*d^2*e^5-310080*A*a^3*b^3*d^3*e^4+124032*A*a^2*b^4*d^4*e^3-29184*A*a*b^5*d^5*e^2+3072*A*b^6*d^6*e-92378*B*a^6
*d*e^6+201552*B*a^5*b*d^2*e^5-232560*B*a^4*b^2*d^3*e^4+165376*B*a^3*b^3*d^4*e^3-72960*B*a^2*b^4*d^5*e^2+18432*
B*a*b^5*d^6*e-2048*B*b^6*d^7)/e^8
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maxima [B] time = 0.55, size = 767, normalized size = 2.49 \begin {gather*} \frac {2 \, {\left (138567 \, {\left (e x + d\right )}^{\frac {21}{2}} B b^{6} - 153153 \, {\left (7 \, B b^{6} d - {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} {\left (e x + d\right )}^{\frac {19}{2}} + 513513 \, {\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 {17}{2}} - 969969 \, {\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 {15}{2}} + 1119195 \, {\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 {13}{2}} - 793611 \, {\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 {11}{2}} + 323323 \, {\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 )} {\left (e x + d\right )}^{\frac {9}{2}} - 415701 \, {\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 )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{2909907 \, e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
[Out]
2/2909907*(138567*(e*x + d)^(21/2)*B*b^6 - 153153*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*(e*x + d)^(19/2) + 51351
3*(7*B*b^6*d^2 - 2*(6*B*a*b^5 + A*b^6)*d*e + (5*B*a^2*b^4 + 2*A*a*b^5)*e^2)*(e*x + d)^(17/2) - 969969*(7*B*b^6
*d^3 - 3*(6*B*a*b^5 + A*b^6)*d^2*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^2 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^3)*(e*x
+ d)^(15/2) + 1119195*(7*B*b^6*d^4 - 4*(6*B*a*b^5 + A*b^6)*d^3*e + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^2 - 4*(4
*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^4)*(e*x + d)^(13/2) - 793611*(7*B*b^6*d^5 - 5*
(6*B*a*b^5 + A*b^6)*d^4*e + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^2 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^3 + 5*
(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^4 - (2*B*a^5*b + 5*A*a^4*b^2)*e^5)*(e*x + d)^(11/2) + 323323*(7*B*b^6*d^6 - 6*
(6*B*a*b^5 + A*b^6)*d^5*e + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^2 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^3 + 15
*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^4 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)*e^6)*(e*x + d)^
(9/2) - 415701*(B*b^6*d^7 - A*a^6*e^7 - (6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 5*(4
*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^
5 + (B*a^6 + 6*A*a^5*b)*d*e^6)*(e*x + d)^(7/2))/e^8
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mupad [B] time = 1.93, size = 279, normalized size = 0.91 \begin {gather*} \frac {{\left (d+e\,x\right )}^{19/2}\,\left (2\,A\,b^6\,e-14\,B\,b^6\,d+12\,B\,a\,b^5\,e\right )}{19\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{9/2}\,\left (6\,A\,b\,e+B\,a\,e-7\,B\,b\,d\right )}{9\,e^8}+\frac {2\,B\,b^6\,{\left (d+e\,x\right )}^{21/2}}{21\,e^8}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{7/2}}{7\,e^8}+\frac {6\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{11/2}\,\left (5\,A\,b\,e+2\,B\,a\,e-7\,B\,b\,d\right )}{11\,e^8}+\frac {6\,b^4\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{17/2}\,\left (2\,A\,b\,e+5\,B\,a\,e-7\,B\,b\,d\right )}{17\,e^8}+\frac {10\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{13/2}\,\left (4\,A\,b\,e+3\,B\,a\,e-7\,B\,b\,d\right )}{13\,e^8}+\frac {2\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{15/2}\,\left (3\,A\,b\,e+4\,B\,a\,e-7\,B\,b\,d\right )}{3\,e^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)*(d + e*x)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
[Out]
((d + e*x)^(19/2)*(2*A*b^6*e - 14*B*b^6*d + 12*B*a*b^5*e))/(19*e^8) + (2*(a*e - b*d)^5*(d + e*x)^(9/2)*(6*A*b*
e + B*a*e - 7*B*b*d))/(9*e^8) + (2*B*b^6*(d + e*x)^(21/2))/(21*e^8) + (2*(A*e - B*d)*(a*e - b*d)^6*(d + e*x)^(
7/2))/(7*e^8) + (6*b*(a*e - b*d)^4*(d + e*x)^(11/2)*(5*A*b*e + 2*B*a*e - 7*B*b*d))/(11*e^8) + (6*b^4*(a*e - b*
d)*(d + e*x)^(17/2)*(2*A*b*e + 5*B*a*e - 7*B*b*d))/(17*e^8) + (10*b^2*(a*e - b*d)^3*(d + e*x)^(13/2)*(4*A*b*e
+ 3*B*a*e - 7*B*b*d))/(13*e^8) + (2*b^3*(a*e - b*d)^2*(d + e*x)^(15/2)*(3*A*b*e + 4*B*a*e - 7*B*b*d))/(3*e^8)
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sympy [A] time = 113.73, size = 3728, normalized size = 12.10
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
A*a**6*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*A*a**6*d*(-d*(d + e*x)**(3/
2)/3 + (d + e*x)**(5/2)/5)/e + 2*A*a**6*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7
)/e + 12*A*a**5*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 24*A*a**5*b*d*(d**2*(d + e*x)**(3/2
)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 12*A*a**5*b*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d +
e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**2 + 30*A*a**4*b**2*d**2*(d**2*(d + e*x)**(3/2)
/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 60*A*a**4*b**2*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(
d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 30*A*a**4*b**2*(d**4*(d + e*x)**(3/2)/
3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**
3 + 40*A*a**3*b**3*d**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d +
e*x)**(9/2)/9)/e**4 + 80*A*a**3*b**3*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)
**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 40*A*a**3*b**3*(-d**5*(d + e*x)**(3/2)/3 + d
**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d
+ e*x)**(13/2)/13)/e**4 + 30*A*a**2*b**4*d**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(
d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 60*A*a**2*b**4*d*(-d**5*(d + e*x)**(
3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/
2)/11 + (d + e*x)**(13/2)/13)/e**5 + 30*A*a**2*b**4*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*
d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13
+ (d + e*x)**(15/2)/15)/e**5 + 12*A*a*b**5*d**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(
d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**6 + 24*A*
a*b**5*d*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)
**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6 + 12*A*a*b**5
*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9
- 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/1
7)/e**6 + 2*A*b**6*d**2*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20
*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**
7 + 4*A*b**6*d*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d +
e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d +
e*x)**(17/2)/17)/e**7 + 2*A*b**6*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2
) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*
x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**7 + 2*B*a**6*d**2*(-d*(d + e*x)**(3/2)/3 +
(d + e*x)**(5/2)/5)/e**2 + 4*B*a**6*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)
/e**2 + 2*B*a**6*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(
9/2)/9)/e**2 + 12*B*a**5*b*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 +
24*B*a**5*b*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/
2)/9)/e**3 + 12*B*a**5*b*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*
d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**3 + 30*B*a**4*b**2*d**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d
+ e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 60*B*a**4*b**2*d*(d**4*(d + e*x)**(3/2)/
3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**
4 + 30*B*a**4*b**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d
+ e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**4 + 40*B*a**3*b**3*d**2*(d**4*(d + e*x)
**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)
/11)/e**5 + 80*B*a**3*b**3*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 +
10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 40*B*a**3*b**3*(d**6*(d +
e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2
*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5 + 30*B*a**2*b**4*d**2*(-d**5*(d
+ e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e
*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**6 + 60*B*a**2*b**4*d*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5
/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)
**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6 + 30*B*a**2*b**4*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/
5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(
13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**6 + 12*B*a*b**5*d**2*(d**6*(d + e*x)**(3/2)/3 -
6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/
2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**7 + 24*B*a*b**5*d*(-d**7*(d + e*x)**(3/2)/3 + 7*d*
*6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 +
21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7 + 12*B*a*b**5*(d**8*(d +
e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d
+ e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (
d + e*x)**(19/2)/19)/e**7 + 2*B*b**6*d**2*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d +
e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(
d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**8 + 4*B*b**6*d*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/
2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)
**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**8 + 2*B*b**6*
(-d**9*(d + e*x)**(3/2)/3 + 9*d**8*(d + e*x)**(5/2)/5 - 36*d**7*(d + e*x)**(7/2)/7 + 28*d**6*(d + e*x)**(9/2)/
3 - 126*d**5*(d + e*x)**(11/2)/11 + 126*d**4*(d + e*x)**(13/2)/13 - 28*d**3*(d + e*x)**(15/2)/5 + 36*d**2*(d +
e*x)**(17/2)/17 - 9*d*(d + e*x)**(19/2)/19 + (d + e*x)**(21/2)/21)/e**8
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