3.1800 \(\int (A+B x) (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)
Optimal. Leaf size=308 \[ -\frac{2 b^5 (d+e x)^{21/2} (-6 a B e-A b e+7 b B d)}{21 e^8}+\frac{6 b^4 (d+e x)^{19/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{19 e^8}-\frac{10 b^3 (d+e x)^{17/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{17 e^8}+\frac{2 b^2 (d+e x)^{15/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{3 e^8}-\frac{6 b (d+e x)^{13/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{13 e^8}+\frac{2 (d+e x)^{11/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{11 e^8}-\frac{2 (d+e x)^{9/2} (b d-a e)^6 (B d-A e)}{9 e^8}+\frac{2 b^6 B (d+e x)^{23/2}}{23 e^8} \]
[Out]
(-2*(b*d - a*e)^6*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^8) + (2*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x
)^(11/2))/(11*e^8) - (6*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^(13/2))/(13*e^8) + (2*b^2*(b*d
- a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^(15/2))/(3*e^8) - (10*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e -
4*a*B*e)*(d + e*x)^(17/2))/(17*e^8) + (6*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(19/2))/(19*
e^8) - (2*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^(21/2))/(21*e^8) + (2*b^6*B*(d + e*x)^(23/2))/(23*e^8)
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Rubi [A] time = 0.208937, antiderivative size = 308, normalized size of antiderivative = 1.,
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} \[ -\frac{2 b^5 (d+e x)^{21/2} (-6 a B e-A b e+7 b B d)}{21 e^8}+\frac{6 b^4 (d+e x)^{19/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{19 e^8}-\frac{10 b^3 (d+e x)^{17/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{17 e^8}+\frac{2 b^2 (d+e x)^{15/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{3 e^8}-\frac{6 b (d+e x)^{13/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{13 e^8}+\frac{2 (d+e x)^{11/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{11 e^8}-\frac{2 (d+e x)^{9/2} (b d-a e)^6 (B d-A e)}{9 e^8}+\frac{2 b^6 B (d+e x)^{23/2}}{23 e^8} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^(7/2)*(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)^(9/2))/(9*e^8) + (2*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x
)^(11/2))/(11*e^8) - (6*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^(13/2))/(13*e^8) + (2*b^2*(b*d
- a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^(15/2))/(3*e^8) - (10*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e -
4*a*B*e)*(d + e*x)^(17/2))/(17*e^8) + (6*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(19/2))/(19*
e^8) - (2*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^(21/2))/(21*e^8) + (2*b^6*B*(d + e*x)^(23/2))/(23*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)^{7/2} \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)^{7/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^6 (-B d+A e) (d+e x)^{7/2}}{e^7}+\frac{(-b d+a e)^5 (-7 b B d+6 A b e+a B e) (d+e x)^{9/2}}{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)^{11/2}}{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)^{13/2}}{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)^{15/2}}{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)^{17/2}}{e^7}+\frac{b^5 (-7 b B d+A b e+6 a B e) (d+e x)^{19/2}}{e^7}+\frac{b^6 B (d+e x)^{21/2}}{e^7}\right ) \, dx\\ &=-\frac{2 (b d-a e)^6 (B d-A e) (d+e x)^{9/2}}{9 e^8}+\frac{2 (b d-a e)^5 (7 b B d-6 A b e-a B e) (d+e x)^{11/2}}{11 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)^{13/2}}{13 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)^{15/2}}{3 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)^{17/2}}{17 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)^{19/2}}{19 e^8}-\frac{2 b^5 (7 b B d-A b e-6 a B e) (d+e x)^{21/2}}{21 e^8}+\frac{2 b^6 B (d+e x)^{23/2}}{23 e^8}\\ \end{align*}
Mathematica [A] time = 0.421191, size = 259, normalized size = 0.84 \[ \frac{2 (d+e x)^{9/2} \left (-3187041 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+10567557 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-19684665 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+22309287 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)-15444891 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)+6084351 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)-7436429 (b d-a e)^6 (B d-A e)+2909907 b^6 B (d+e x)^7\right )}{66927861 e^8} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(d + e*x)^(9/2)*(-7436429*(b*d - a*e)^6*(B*d - A*e) + 6084351*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d
+ e*x) - 15444891*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^2 + 22309287*b^2*(b*d - a*e)^3*(7*b*
B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^3 - 19684665*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^4 +
10567557*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^5 - 3187041*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(
d + e*x)^6 + 2909907*b^6*B*(d + e*x)^7))/(66927861*e^8)
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Maple [B] time = 0.01, size = 913, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
2/66927861*(e*x+d)^(9/2)*(2909907*B*b^6*e^7*x^7+3187041*A*b^6*e^7*x^6+19122246*B*a*b^5*e^7*x^6-1939938*B*b^6*d
*e^6*x^6+21135114*A*a*b^5*e^7*x^5-2012868*A*b^6*d*e^6*x^5+52837785*B*a^2*b^4*e^7*x^5-12077208*B*a*b^5*d*e^6*x^
5+1225224*B*b^6*d^2*e^5*x^5+59053995*A*a^2*b^4*e^7*x^4-12432420*A*a*b^5*d*e^6*x^4+1184040*A*b^6*d^2*e^5*x^4+78
738660*B*a^3*b^3*e^7*x^4-31081050*B*a^2*b^4*d*e^6*x^4+7104240*B*a*b^5*d^2*e^5*x^4-720720*B*b^6*d^3*e^4*x^4+892
37148*A*a^3*b^3*e^7*x^3-31495464*A*a^2*b^4*d*e^6*x^3+6630624*A*a*b^5*d^2*e^5*x^3-631488*A*b^6*d^3*e^4*x^3+6692
7861*B*a^4*b^2*e^7*x^3-41993952*B*a^3*b^3*d*e^6*x^3+16576560*B*a^2*b^4*d^2*e^5*x^3-3788928*B*a*b^5*d^3*e^4*x^3
+384384*B*b^6*d^4*e^3*x^3+77224455*A*a^4*b^2*e^7*x^2-41186376*A*a^3*b^3*d*e^6*x^2+14536368*A*a^2*b^4*d^2*e^5*x
^2-3060288*A*a*b^5*d^3*e^4*x^2+291456*A*b^6*d^4*e^3*x^2+30889782*B*a^5*b*e^7*x^2-30889782*B*a^4*b^2*d*e^6*x^2+
19381824*B*a^3*b^3*d^2*e^5*x^2-7650720*B*a^2*b^4*d^3*e^4*x^2+1748736*B*a*b^5*d^4*e^3*x^2-177408*B*b^6*d^5*e^2*
x^2+36506106*A*a^5*b*e^7*x-28081620*A*a^4*b^2*d*e^6*x+14976864*A*a^3*b^3*d^2*e^5*x-5285952*A*a^2*b^4*d^3*e^4*x
+1112832*A*a*b^5*d^4*e^3*x-105984*A*b^6*d^5*e^2*x+6084351*B*a^6*e^7*x-11232648*B*a^5*b*d*e^6*x+11232648*B*a^4*
b^2*d^2*e^5*x-7047936*B*a^3*b^3*d^3*e^4*x+2782080*B*a^2*b^4*d^4*e^3*x-635904*B*a*b^5*d^5*e^2*x+64512*B*b^6*d^6
*e*x+7436429*A*a^6*e^7-8112468*A*a^5*b*d*e^6+6240360*A*a^4*b^2*d^2*e^5-3328192*A*a^3*b^3*d^3*e^4+1174656*A*a^2
*b^4*d^4*e^3-247296*A*a*b^5*d^5*e^2+23552*A*b^6*d^6*e-1352078*B*a^6*d*e^6+2496144*B*a^5*b*d^2*e^5-2496144*B*a^
4*b^2*d^3*e^4+1566208*B*a^3*b^3*d^4*e^3-618240*B*a^2*b^4*d^5*e^2+141312*B*a*b^5*d^6*e-14336*B*b^6*d^7)/e^8
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Maxima [B] time = 0.998882, size = 1035, normalized size = 3.36 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
[Out]
2/66927861*(2909907*(e*x + d)^(23/2)*B*b^6 - 3187041*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*(e*x + d)^(21/2) + 10
567557*(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)^(19/2) - 19684665*(
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)^(17/2) + 22309287*(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)^(15/2) - 15444891*(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)^(13/2) + 6084351*(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)^(11/2) - 7436429*(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)^(9/2))/e^8
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Fricas [B] time = 1.45027, size = 3443, normalized size = 11.18 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
[Out]
2/66927861*(2909907*B*b^6*e^11*x^11 - 14336*B*b^6*d^11 + 7436429*A*a^6*d^4*e^7 + 23552*(6*B*a*b^5 + A*b^6)*d^1
0*e - 123648*(5*B*a^2*b^4 + 2*A*a*b^5)*d^9*e^2 + 391552*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^8*e^3 - 832048*(3*B*a^4*
b^2 + 4*A*a^3*b^3)*d^7*e^4 + 1248072*(2*B*a^5*b + 5*A*a^4*b^2)*d^6*e^5 - 1352078*(B*a^6 + 6*A*a^5*b)*d^5*e^6 +
138567*(70*B*b^6*d*e^10 + 23*(6*B*a*b^5 + A*b^6)*e^11)*x^10 + 7293*(1498*B*b^6*d^2*e^9 + 1472*(6*B*a*b^5 + A*
b^6)*d*e^10 + 1449*(5*B*a^2*b^4 + 2*A*a*b^5)*e^11)*x^9 + 1287*(3248*B*b^6*d^3*e^8 + 9522*(6*B*a*b^5 + A*b^6)*d
^2*e^9 + 28014*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^10 + 15295*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^11)*x^8 + 429*(7*B*b^6*d
^4*e^7 + 11132*(6*B*a*b^5 + A*b^6)*d^3*e^8 + 97566*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^9 + 159068*(4*B*a^3*b^3 + 3
*A*a^2*b^4)*d*e^10 + 52003*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^11)*x^7 - 231*(14*B*b^6*d^5*e^6 - 23*(6*B*a*b^5 + A*b
^6)*d^4*e^7 - 72312*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^8 - 350474*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^9 - 341734*(3
*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^10 - 66861*(2*B*a^5*b + 5*A*a^4*b^2)*e^11)*x^6 + 63*(56*B*b^6*d^6*e^5 - 92*(6*B*
a*b^5 + A*b^6)*d^5*e^6 + 483*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^7 + 529644*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^8 +
1530374*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^9 + 891480*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^10 + 96577*(B*a^6 + 6*A*a^5
*b)*e^11)*x^5 - 7*(560*B*b^6*d^7*e^4 - 1062347*A*a^6*e^11 - 920*(6*B*a*b^5 + A*b^6)*d^6*e^5 + 4830*(5*B*a^2*b^
4 + 2*A*a*b^5)*d^5*e^6 - 15295*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^7 - 5943200*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e
^8 - 10207446*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^9 - 3283618*(B*a^6 + 6*A*a^5*b)*d*e^10)*x^4 + (4480*B*b^6*d^8*e^
3 + 29745716*A*a^6*d*e^10 - 7360*(6*B*a*b^5 + A*b^6)*d^7*e^4 + 38640*(5*B*a^2*b^4 + 2*A*a*b^5)*d^6*e^5 - 12236
0*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^5*e^6 + 260015*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4*e^7 + 33073908*(2*B*a^5*b + 5*A
*a^4*b^2)*d^3*e^8 + 31097794*(B*a^6 + 6*A*a^5*b)*d^2*e^9)*x^3 - 3*(1792*B*b^6*d^9*e^2 - 14872858*A*a^6*d^2*e^9
- 2944*(6*B*a*b^5 + A*b^6)*d^8*e^3 + 15456*(5*B*a^2*b^4 + 2*A*a*b^5)*d^7*e^4 - 48944*(4*B*a^3*b^3 + 3*A*a^2*b
^4)*d^6*e^5 + 104006*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^5*e^6 - 156009*(2*B*a^5*b + 5*A*a^4*b^2)*d^4*e^7 - 5408312*
(B*a^6 + 6*A*a^5*b)*d^3*e^8)*x^2 + (7168*B*b^6*d^10*e + 29745716*A*a^6*d^3*e^8 - 11776*(6*B*a*b^5 + A*b^6)*d^9
*e^2 + 61824*(5*B*a^2*b^4 + 2*A*a*b^5)*d^8*e^3 - 195776*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^7*e^4 + 416024*(3*B*a^4*
b^2 + 4*A*a^3*b^3)*d^6*e^5 - 624036*(2*B*a^5*b + 5*A*a^4*b^2)*d^5*e^6 + 676039*(B*a^6 + 6*A*a^5*b)*d^4*e^7)*x)
*sqrt(e*x + d)/e^8
________________________________________________________________________________________
Sympy [A] time = 159.862, size = 5414, normalized size = 17.58 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
A*a**6*d**3*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 6*A*a**6*d**2*(-d*(d + e*x)**
(3/2)/3 + (d + e*x)**(5/2)/5)/e + 6*A*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*A*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 + 12*A*a**5*b*d**3*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 36*A*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**2 + 36*A*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**2 + 12*A*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**2 + 30*A*a**4*b**2*d**3*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 +
90*A*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**3 + 90*A*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**3 + 30*A*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**3 + 40*A*a**3*b**3*d**3*(-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 + 120*A*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**4 + 120*A*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**4 + 40*A*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**4 + 30*A*a**2*b**4*d**3*(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 + 90*A*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**5 + 90*A*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**5 + 30*A*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**5 + 12*A*a*b**5*d**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**6 + 36*A*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**6 + 36*A*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**6 + 12*A*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
**6 + 2*A*b**6*d**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**7 +
6*A*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**7 + 6*A*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**7 + 2*A*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**7 + 2*B*a**6*d**3*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2
+ 6*B*a**6*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**2 + 6*B*a**6*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**2 + 2*B*a
**6*(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**2 + 12*B*a**5*b*d**3*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)*
*(7/2)/7)/e**3 + 36*B*a**5*b*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**3 + 36*B*a**5*b*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**3 + 12*B*a**5*b*(-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**3 + 30*B*a**4*b**2*d**3*(-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 + 90*B*a**4*b**2*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**4 + 90*B*a**4*b**2
*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**4 + 30*B*a**4*b**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**4 + 40*B*a**3*b**3*d**3*(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 + 120*B*a**
3*b**3*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**5 + 120*B*a**3*b**3*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)**(1
1/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5 + 40*B*a**3*b**3*(-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**5 + 30*B*a**2*b**4*d**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**6 + 90*B*a**2*b**4*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**6 + 90*B*a**2*b**4*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**6 + 30*B*a**2*b**4*(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**6 + 12*B*a*b**5*d**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**7 + 36*B*a*b**5*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**7 + 36*B*a*b**5*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**7 + 12*B*a*b
**5*(-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**7 + 2*B*b**6*d**3*(-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 + 6*B*b**6
*d**2*(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 + 6*B*b**6*d*(-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 + 2*B*b**6*(d**10*(d + e*x)**(3/2)/3 - 2*d**9*(d + e*x)**(5/2) + 45*d**8*(d + e*x)**(7/2)/7
- 40*d**7*(d + e*x)**(9/2)/3 + 210*d**6*(d + e*x)**(11/2)/11 - 252*d**5*(d + e*x)**(13/2)/13 + 14*d**4*(d + e
*x)**(15/2) - 120*d**3*(d + e*x)**(17/2)/17 + 45*d**2*(d + e*x)**(19/2)/19 - 10*d*(d + e*x)**(21/2)/21 + (d +
e*x)**(23/2)/23)/e**8
________________________________________________________________________________________
Giac [B] time = 1.49195, size = 6445, normalized size = 20.93 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
[Out]
2/334639305*(22309287*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^6*d^3*e^(-1) + 133855722*(3*(x*e + d)^(5/2
) - 5*(x*e + d)^(3/2)*d)*A*a^5*b*d^3*e^(-1) + 19122246*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e +
d)^(3/2)*d^2)*B*a^5*b*d^3*e^(-2) + 47805615*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^
2)*A*a^4*b^2*d^3*e^(-2) + 15935205*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105
*(x*e + d)^(3/2)*d^3)*B*a^4*b^2*d^3*e^(-3) + 21246940*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e +
d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*A*a^3*b^3*d^3*e^(-3) + 1931540*(315*(x*e + d)^(11/2) - 1540*(x*e + d)
^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B*a^3*b^3*d^3*e^(-4
) + 1448655*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d
^3 + 1155*(x*e + d)^(3/2)*d^4)*A*a^2*b^4*d^3*e^(-4) + 557175*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d +
10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*
B*a^2*b^4*d^3*e^(-5) + 222870*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12
870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*A*a*b^5*d^3*e^(-5) + 44574*(300
3*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 9652
5*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*B*a*b^5*d^3*e^(-6) + 7429*(3003
*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525
*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*A*b^6*d^3*e^(-6) + 3059*(6435*(x
*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425425
*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*B*
b^6*d^3*e^(-7) + 111546435*(x*e + d)^(3/2)*A*a^6*d^3 + 9561123*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35
*(x*e + d)^(3/2)*d^2)*B*a^6*d^2*e^(-1) + 57366738*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3
/2)*d^2)*A*a^5*b*d^2*e^(-1) + 19122246*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 -
105*(x*e + d)^(3/2)*d^3)*B*a^5*b*d^2*e^(-2) + 47805615*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e
+ d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*A*a^4*b^2*d^2*e^(-2) + 4345965*(315*(x*e + d)^(11/2) - 1540*(x*e +
d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B*a^4*b^2*d^2*e^(
-3) + 5794620*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)
*d^3 + 1155*(x*e + d)^(3/2)*d^4)*A*a^3*b^3*d^2*e^(-3) + 2228700*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*
d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^
5)*B*a^3*b^3*d^2*e^(-4) + 1671525*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2
- 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*A*a^2*b^4*d^2*e^(-4) + 3343
05*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3
+ 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*B*a^2*b^4*d^2*e^(-5) + 1
33722*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*
d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*A*a*b^5*d^2*e^(-5) +
55062*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2
)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)
^(3/2)*d^7)*B*a*b^5*d^2*e^(-6) + 9177*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13
/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e
+ d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*A*b^6*d^2*e^(-6) + 483*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)
^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 - 8953560*(x*e + d)^(13/2)*d^3 + 13226850*(x*e + d)^(11/2)*d^4 - 1293
2920*(x*e + d)^(9/2)*d^5 + 8314020*(x*e + d)^(7/2)*d^6 - 3325608*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*
d^8)*B*b^6*d^2*e^(-7) + 66927861*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^6*d^2 + 3187041*(35*(x*e + d)^(
9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a^6*d*e^(-1) + 19122246*(3
5*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*A*a^5*b*d*e^(-1
) + 1738386*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d
^3 + 1155*(x*e + d)^(3/2)*d^4)*B*a^5*b*d*e^(-2) + 4345965*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 297
0*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*A*a^4*b^2*d*e^(-2) + 1671525*(693
*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*
e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*B*a^4*b^2*d*e^(-3) + 2228700*(693*(x*e + d)^(13/2) - 4095*(x*e +
d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d
)^(3/2)*d^5)*A*a^3*b^3*d*e^(-3) + 445740*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(
11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e +
d)^(3/2)*d^6)*B*a^3*b^3*d*e^(-4) + 334305*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^
(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e +
d)^(3/2)*d^6)*A*a^2*b^4*d*e^(-4) + 137655*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d
)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*
(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*B*a^2*b^4*d*e^(-5) + 55062*(6435*(x*e + d)^(17/2) - 51051*(x*
e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328
185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*A*a*b^5*d*e^(-5) + 2898*(109
395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 - 8953560*(x*e + d)^(13/2)*d^3
+ 13226850*(x*e + d)^(11/2)*d^4 - 12932920*(x*e + d)^(9/2)*d^5 + 8314020*(x*e + d)^(7/2)*d^6 - 3325608*(x*e +
d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8)*B*a*b^5*d*e^(-6) + 483*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)
^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 - 8953560*(x*e + d)^(13/2)*d^3 + 13226850*(x*e + d)^(11/2)*d^4 - 1293
2920*(x*e + d)^(9/2)*d^5 + 8314020*(x*e + d)^(7/2)*d^6 - 3325608*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*
d^8)*A*b^6*d*e^(-6) + 207*(230945*(x*e + d)^(21/2) - 2297295*(x*e + d)^(19/2)*d + 10270260*(x*e + d)^(17/2)*d^
2 - 27159132*(x*e + d)^(15/2)*d^3 + 47006190*(x*e + d)^(13/2)*d^4 - 55552770*(x*e + d)^(11/2)*d^5 + 45265220*(
x*e + d)^(9/2)*d^6 - 24942060*(x*e + d)^(7/2)*d^7 + 8729721*(x*e + d)^(5/2)*d^8 - 1616615*(x*e + d)^(3/2)*d^9)
*B*b^6*d*e^(-7) + 9561123*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^6*d + 96577
*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(
x*e + d)^(3/2)*d^4)*B*a^6*e^(-1) + 579462*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2
)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*A*a^5*b*e^(-1) + 222870*(693*(x*e + d)^(13/2) - 4
095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 30
03*(x*e + d)^(3/2)*d^5)*B*a^5*b*e^(-2) + 557175*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e +
d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*A*a^4*b^2*e^(
-2) + 111435*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)
^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*B*a^4*b^2*e^(-
3) + 148580*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^
(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*A*a^3*b^3*e^(-3
) + 61180*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(
11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e
+ d)^(3/2)*d^7)*B*a^3*b^3*e^(-4) + 45885*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^
(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x
*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*A*a^2*b^4*e^(-4) + 2415*(109395*(x*e + d)^(19/2) - 978120*(x*e
+ d)^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 - 8953560*(x*e + d)^(13/2)*d^3 + 13226850*(x*e + d)^(11/2)*d^4 -
12932920*(x*e + d)^(9/2)*d^5 + 8314020*(x*e + d)^(7/2)*d^6 - 3325608*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3
/2)*d^8)*B*a^2*b^4*e^(-5) + 966*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(15/2
)*d^2 - 8953560*(x*e + d)^(13/2)*d^3 + 13226850*(x*e + d)^(11/2)*d^4 - 12932920*(x*e + d)^(9/2)*d^5 + 8314020*
(x*e + d)^(7/2)*d^6 - 3325608*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8)*A*a*b^5*e^(-5) + 414*(230945*(
x*e + d)^(21/2) - 2297295*(x*e + d)^(19/2)*d + 10270260*(x*e + d)^(17/2)*d^2 - 27159132*(x*e + d)^(15/2)*d^3 +
47006190*(x*e + d)^(13/2)*d^4 - 55552770*(x*e + d)^(11/2)*d^5 + 45265220*(x*e + d)^(9/2)*d^6 - 24942060*(x*e
+ d)^(7/2)*d^7 + 8729721*(x*e + d)^(5/2)*d^8 - 1616615*(x*e + d)^(3/2)*d^9)*B*a*b^5*e^(-6) + 69*(230945*(x*e +
d)^(21/2) - 2297295*(x*e + d)^(19/2)*d + 10270260*(x*e + d)^(17/2)*d^2 - 27159132*(x*e + d)^(15/2)*d^3 + 4700
6190*(x*e + d)^(13/2)*d^4 - 55552770*(x*e + d)^(11/2)*d^5 + 45265220*(x*e + d)^(9/2)*d^6 - 24942060*(x*e + d)^
(7/2)*d^7 + 8729721*(x*e + d)^(5/2)*d^8 - 1616615*(x*e + d)^(3/2)*d^9)*A*b^6*e^(-6) + 15*(969969*(x*e + d)^(23
/2) - 10623470*(x*e + d)^(21/2)*d + 52837785*(x*e + d)^(19/2)*d^2 - 157477320*(x*e + d)^(17/2)*d^3 + 312330018
*(x*e + d)^(15/2)*d^4 - 432456948*(x*e + d)^(13/2)*d^5 + 425904570*(x*e + d)^(11/2)*d^6 - 297457160*(x*e + d)^
(9/2)*d^7 + 143416845*(x*e + d)^(7/2)*d^8 - 44618574*(x*e + d)^(5/2)*d^9 + 7436429*(x*e + d)^(3/2)*d^10)*B*b^6
*e^(-7) + 1062347*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*
d^3)*A*a^6)*e^(-1)