3.1801 \(\int (A+B x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=308 \[ -\frac{2 b^5 (d+e x)^{19/2} (-6 a B e-A b e+7 b B d)}{19 e^8}+\frac{6 b^4 (d+e x)^{17/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{17 e^8}-\frac{2 b^3 (d+e x)^{15/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac{10 b^2 (d+e x)^{13/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{13 e^8}-\frac{6 b (d+e x)^{11/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{11 e^8}+\frac{2 (d+e x)^{9/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{9 e^8}-\frac{2 (d+e x)^{7/2} (b d-a e)^6 (B d-A e)}{7 e^8}+\frac{2 b^6 B (d+e x)^{21/2}}{21 e^8} \]

[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)

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Rubi [A]  time = 0.146438, 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} \[ -\frac{2 b^5 (d+e x)^{19/2} (-6 a B e-A b e+7 b B d)}{19 e^8}+\frac{6 b^4 (d+e x)^{17/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{17 e^8}-\frac{2 b^3 (d+e x)^{15/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac{10 b^2 (d+e x)^{13/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{13 e^8}-\frac{6 b (d+e x)^{11/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{11 e^8}+\frac{2 (d+e x)^{9/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{9 e^8}-\frac{2 (d+e x)^{7/2} (b d-a e)^6 (B d-A e)}{7 e^8}+\frac{2 b^6 B (d+e x)^{21/2}}{21 e^8} \]

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*}

Mathematica [A]  time = 0.334436, size = 259, normalized size = 0.84 \[ \frac{2 (d+e x)^{7/2} \left (-153153 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+513513 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-969969 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+1119195 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)-793611 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)+323323 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)-415701 (b d-a e)^6 (B d-A e)+138567 b^6 B (d+e x)^7\right )}{2909907 e^8} \]

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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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)^(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 = 1.03141, 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)^(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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Fricas [B]  time = 1.36473, size = 2927, normalized size = 9.5 \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)^(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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Sympy [A]  time = 108.895, size = 3728, normalized size = 12.1 \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)**(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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Giac [B]  time = 1.44785, size = 4443, normalized size = 14.43 \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)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

2/14549535*(969969*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^6*d^2*e^(-1) + 5819814*(3*(x*e + d)^(5/2) - 5
*(x*e + d)^(3/2)*d)*A*a^5*b*d^2*e^(-1) + 831402*(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^2*e^(-2) + 2078505*(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^2*e^(-2) + 692835*(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^2*e^(-3) + 923780*(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^2*e^(-3) + 83980*(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)*B*a^3*b^3*d^2*e^(-4) + 62985*(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^2*e^(-4) + 24225*(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^2*e^(
-5) + 9690*(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*b^5*d^2*e^(-5) + 1938*(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*b^5*d^2*e^(-6) + 323*(3003*(x*e + d)^(15/2) - 2
0790*(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^2*e^(-6) + 133*(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^2*e^(-7) + 48498
45*(x*e + d)^(3/2)*A*a^6*d^2 + 277134*(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*e^(-1) + 1662804*(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*e^(-1) +
554268*(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*e^(-2) + 1385670*(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*e^(-2) + 125970*(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*e^(-3) + 167960*(315*(x*e + d)^(11/2) - 154
0*(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*e^(-3) + 64600*(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*e^(-4) + 48450*(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*e^(-4) + 9690*(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*e^(-5) + 3876*(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*e^(-5) + 1596*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 17
6715*(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*e^(-6) + 266*(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*e^(-6) + 14
*(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*b^6*d*e^(-7) + 1939938*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/
2)*d)*A*a^6*d + 46189*(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*e^(-1) + 277134*(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^5*b*e^(-1) + 25194*(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*e^(-2) + 62985*(315*(x*e + d)^(11/2) - 154
0*(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^4*b^
2*e^(-2) + 24225*(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*e^(-3) + 32300*(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*e^(-3) + 6460*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 6142
5*(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*e^(-4) + 4845*(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 + 150
15*(x*e + d)^(3/2)*d^6)*A*a^2*b^4*e^(-4) + 1995*(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 + 15
3153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*B*a^2*b^4*e^(-5) + 798*(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 - 32
8185*(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*e^(-5) + 42*(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*b^5*e^(-6) + 7*(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*
b^6*e^(-6) + 3*(230945*(x*e + d)^(21/2) - 2297295*(x*e + d)^(19/2)*d + 10270260*(x*e + d)^(17/2)*d^2 - 2715913
2*(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*e^(-
7) + 138567*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^6)*e^(-1)