3.1607 \(\int (b+2 c x) (d+e x)^{5/2} (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=427 \[ \frac{2 (d+e x)^{13/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{13 e^8}+\frac{6 c^2 (d+e x)^{17/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{17 e^8}-\frac{2 c (d+e x)^{15/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^8}-\frac{6 (d+e x)^{11/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{11 e^8}+\frac{2 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{9 e^8}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{7 e^8}-\frac{14 c^3 (d+e x)^{19/2} (2 c d-b e)}{19 e^8}+\frac{4 c^4 (d+e x)^{21/2}}{21 e^8} \]

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(7/2))/(7*e^8) + (2*(c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^
2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(9/2))/(9*e^8) - (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^
2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(11/2))/(11*e^8) + (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*
b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(13/2))
/(13*e^8) - (2*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(15/2))/(3*e^8) + (6*c^2*
(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(17/2))/(17*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(1
9/2))/(19*e^8) + (4*c^4*(d + e*x)^(21/2))/(21*e^8)

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Rubi [A]  time = 0.303318, antiderivative size = 427, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {771} \[ \frac{2 (d+e x)^{13/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{13 e^8}+\frac{6 c^2 (d+e x)^{17/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{17 e^8}-\frac{2 c (d+e x)^{15/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^8}-\frac{6 (d+e x)^{11/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{11 e^8}+\frac{2 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{9 e^8}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{7 e^8}-\frac{14 c^3 (d+e x)^{19/2} (2 c d-b e)}{19 e^8}+\frac{4 c^4 (d+e x)^{21/2}}{21 e^8} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x]

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(7/2))/(7*e^8) + (2*(c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^
2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(9/2))/(9*e^8) - (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^
2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(11/2))/(11*e^8) + (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*
b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(13/2))
/(13*e^8) - (2*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(15/2))/(3*e^8) + (6*c^2*
(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(17/2))/(17*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(1
9/2))/(19*e^8) + (4*c^4*(d + e*x)^(21/2))/(21*e^8)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{5/2}}{e^7}+\frac{\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{7/2}}{e^7}+\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right ) (d+e x)^{9/2}}{e^7}+\frac{\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{11/2}}{e^7}+\frac{5 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right ) (d+e x)^{13/2}}{e^7}+\frac{3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{15/2}}{e^7}-\frac{7 c^3 (2 c d-b e) (d+e x)^{17/2}}{e^7}+\frac{2 c^4 (d+e x)^{19/2}}{e^7}\right ) \, dx\\ &=-\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{7/2}}{7 e^8}+\frac{2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{9/2}}{9 e^8}-\frac{6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^8}+\frac{2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{13/2}}{13 e^8}-\frac{2 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{15/2}}{3 e^8}+\frac{6 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{17/2}}{17 e^8}-\frac{14 c^3 (2 c d-b e) (d+e x)^{19/2}}{19 e^8}+\frac{4 c^4 (d+e x)^{21/2}}{21 e^8}\\ \end{align*}

Mathematica [A]  time = 0.657616, size = 600, normalized size = 1.41 \[ \frac{2 (d+e x)^{7/2} \left (-57 c^2 e^2 \left (102 a^2 e^2 \left (-56 d^2 e x+16 d^3+126 d e^2 x^2-231 e^3 x^3\right )-17 a b e \left (1008 d^2 e^2 x^2-448 d^3 e x+128 d^4-1848 d e^3 x^3+3003 e^4 x^4\right )+3 b^2 \left (2016 d^3 e^2 x^2-3696 d^2 e^3 x^3-896 d^4 e x+256 d^5+6006 d e^4 x^4-9009 e^5 x^5\right )\right )+323 c e^3 \left (117 a^2 b e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+286 a^3 e^3 (7 e x-2 d)+36 a b^2 e \left (56 d^2 e x-16 d^3-126 d e^2 x^2+231 e^3 x^3\right )+b^3 \left (1008 d^2 e^2 x^2-448 d^3 e x+128 d^4-1848 d e^3 x^3+3003 e^4 x^4\right )\right )+969 b e^4 \left (143 a^2 b e^2 (7 e x-2 d)+429 a^3 e^3+13 a b^2 e \left (8 d^2-28 d e x+63 e^2 x^2\right )+b^3 \left (56 d^2 e x-16 d^3-126 d e^2 x^2+231 e^3 x^3\right )\right )+3 c^3 e \left (38 a e \left (-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3+896 d^4 e x-256 d^5-6006 d e^4 x^4+9009 e^5 x^5\right )+7 b \left (8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-3584 d^5 e x+1024 d^6-36036 d e^5 x^5+51051 e^6 x^6\right )\right )-2 c^4 \left (16128 d^5 e^2 x^2-29568 d^4 e^3 x^3+48048 d^3 e^4 x^4-72072 d^2 e^5 x^5-7168 d^6 e x+2048 d^7+102102 d e^6 x^6-138567 e^7 x^7\right )\right )}{2909907 e^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(7/2)*(-2*c^4*(2048*d^7 - 7168*d^6*e*x + 16128*d^5*e^2*x^2 - 29568*d^4*e^3*x^3 + 48048*d^3*e^4*x^
4 - 72072*d^2*e^5*x^5 + 102102*d*e^6*x^6 - 138567*e^7*x^7) + 969*b*e^4*(429*a^3*e^3 + 143*a^2*b*e^2*(-2*d + 7*
e*x) + 13*a*b^2*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + b^3*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3))
+ 323*c*e^3*(286*a^3*e^3*(-2*d + 7*e*x) + 117*a^2*b*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 36*a*b^2*e*(-16*d^3
+ 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + b^3*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 +
3003*e^4*x^4)) - 57*c^2*e^2*(102*a^2*e^2*(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^3*x^3) - 17*a*b*e*(128*
d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + 3*b^2*(256*d^5 - 896*d^4*e*x + 2016*d^
3*e^2*x^2 - 3696*d^2*e^3*x^3 + 6006*d*e^4*x^4 - 9009*e^5*x^5)) + 3*c^3*e*(38*a*e*(-256*d^5 + 896*d^4*e*x - 201
6*d^3*e^2*x^2 + 3696*d^2*e^3*x^3 - 6006*d*e^4*x^4 + 9009*e^5*x^5) + 7*b*(1024*d^6 - 3584*d^5*e*x + 8064*d^4*e^
2*x^2 - 14784*d^3*e^3*x^3 + 24024*d^2*e^4*x^4 - 36036*d*e^5*x^5 + 51051*e^6*x^6))))/(2909907*e^8)

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Maple [B]  time = 0.008, size = 795, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x)

[Out]

2/2909907*(e*x+d)^(7/2)*(277134*c^4*e^7*x^7+1072071*b*c^3*e^7*x^6-204204*c^4*d*e^6*x^6+1027026*a*c^3*e^7*x^5+1
540539*b^2*c^2*e^7*x^5-756756*b*c^3*d*e^6*x^5+144144*c^4*d^2*e^5*x^5+2909907*a*b*c^2*e^7*x^4-684684*a*c^3*d*e^
6*x^4+969969*b^3*c*e^7*x^4-1027026*b^2*c^2*d*e^6*x^4+504504*b*c^3*d^2*e^5*x^4-96096*c^4*d^3*e^4*x^4+1343034*a^
2*c^2*e^7*x^3+2686068*a*b^2*c*e^7*x^3-1790712*a*b*c^2*d*e^6*x^3+421344*a*c^3*d^2*e^5*x^3+223839*b^4*e^7*x^3-59
6904*b^3*c*d*e^6*x^3+632016*b^2*c^2*d^2*e^5*x^3-310464*b*c^3*d^3*e^4*x^3+59136*c^4*d^4*e^3*x^3+2380833*a^2*b*c
*e^7*x^2-732564*a^2*c^2*d*e^6*x^2+793611*a*b^3*e^7*x^2-1465128*a*b^2*c*d*e^6*x^2+976752*a*b*c^2*d^2*e^5*x^2-22
9824*a*c^3*d^3*e^4*x^2-122094*b^4*d*e^6*x^2+325584*b^3*c*d^2*e^5*x^2-344736*b^2*c^2*d^3*e^4*x^2+169344*b*c^3*d
^4*e^3*x^2-32256*c^4*d^5*e^2*x^2+646646*a^3*c*e^7*x+969969*a^2*b^2*e^7*x-1058148*a^2*b*c*d*e^6*x+325584*a^2*c^
2*d^2*e^5*x-352716*a*b^3*d*e^6*x+651168*a*b^2*c*d^2*e^5*x-434112*a*b*c^2*d^3*e^4*x+102144*a*c^3*d^4*e^3*x+5426
4*b^4*d^2*e^5*x-144704*b^3*c*d^3*e^4*x+153216*b^2*c^2*d^4*e^3*x-75264*b*c^3*d^5*e^2*x+14336*c^4*d^6*e*x+415701
*a^3*b*e^7-184756*a^3*c*d*e^6-277134*a^2*b^2*d*e^6+302328*a^2*b*c*d^2*e^5-93024*a^2*c^2*d^3*e^4+100776*a*b^3*d
^2*e^5-186048*a*b^2*c*d^3*e^4+124032*a*b*c^2*d^4*e^3-29184*a*c^3*d^5*e^2-15504*b^4*d^3*e^4+41344*b^3*c*d^4*e^3
-43776*b^2*c^2*d^5*e^2+21504*b*c^3*d^6*e-4096*c^4*d^7)/e^8

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Maxima [A]  time = 1.08398, size = 871, normalized size = 2.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

2/2909907*(277134*(e*x + d)^(21/2)*c^4 - 1072071*(2*c^4*d - b*c^3*e)*(e*x + d)^(19/2) + 513513*(14*c^4*d^2 - 1
4*b*c^3*d*e + (3*b^2*c^2 + 2*a*c^3)*e^2)*(e*x + d)^(17/2) - 969969*(14*c^4*d^3 - 21*b*c^3*d^2*e + 3*(3*b^2*c^2
+ 2*a*c^3)*d*e^2 - (b^3*c + 3*a*b*c^2)*e^3)*(e*x + d)^(15/2) + 223839*(70*c^4*d^4 - 140*b*c^3*d^3*e + 30*(3*b
^2*c^2 + 2*a*c^3)*d^2*e^2 - 20*(b^3*c + 3*a*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*(e*x + d)^(13/2
) - 793611*(14*c^4*d^5 - 35*b*c^3*d^4*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 10*(b^3*c + 3*a*b*c^2)*d^2*e^3 +
(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^4 - (a*b^3 + 3*a^2*b*c)*e^5)*(e*x + d)^(11/2) + 323323*(14*c^4*d^6 - 42*b*c
^3*d^5*e + 15*(3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 20*(b^3*c + 3*a*b*c^2)*d^3*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2
)*d^2*e^4 - 6*(a*b^3 + 3*a^2*b*c)*d*e^5 + (3*a^2*b^2 + 2*a^3*c)*e^6)*(e*x + d)^(9/2) - 415701*(2*c^4*d^7 - 7*b
*c^3*d^6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c +
6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)*(e*x + d)^(7/2))/e^8

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Fricas [B]  time = 1.52517, size = 2577, normalized size = 6.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

2/2909907*(277134*c^4*e^10*x^10 - 4096*c^4*d^10 + 21504*b*c^3*d^9*e + 415701*a^3*b*d^3*e^7 - 14592*(3*b^2*c^2
+ 2*a*c^3)*d^8*e^2 + 41344*(b^3*c + 3*a*b*c^2)*d^7*e^3 - 15504*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^4 + 100776
*(a*b^3 + 3*a^2*b*c)*d^5*e^5 - 92378*(3*a^2*b^2 + 2*a^3*c)*d^4*e^6 + 7293*(86*c^4*d*e^9 + 147*b*c^3*e^10)*x^9
+ 3861*(94*c^4*d^2*e^8 + 637*b*c^3*d*e^9 + 133*(3*b^2*c^2 + 2*a*c^3)*e^10)*x^8 + 429*(2*c^4*d^3*e^7 + 3381*b*c
^3*d^2*e^8 + 2793*(3*b^2*c^2 + 2*a*c^3)*d*e^9 + 2261*(b^3*c + 3*a*b*c^2)*e^10)*x^7 - 231*(4*c^4*d^4*e^6 - 21*b
*c^3*d^3*e^7 - 3135*(3*b^2*c^2 + 2*a*c^3)*d^2*e^8 - 10013*(b^3*c + 3*a*b*c^2)*d*e^9 - 969*(b^4 + 12*a*b^2*c +
6*a^2*c^2)*e^10)*x^6 + 63*(16*c^4*d^5*e^5 - 84*b*c^3*d^4*e^6 + 57*(3*b^2*c^2 + 2*a*c^3)*d^3*e^7 + 22933*(b^3*c
+ 3*a*b*c^2)*d^2*e^8 + 8721*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^9 + 12597*(a*b^3 + 3*a^2*b*c)*e^10)*x^5 - 7*(1
60*c^4*d^6*e^4 - 840*b*c^3*d^5*e^5 + 570*(3*b^2*c^2 + 2*a*c^3)*d^4*e^6 - 1615*(b^3*c + 3*a*b*c^2)*d^3*e^7 - 51
357*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^8 - 289731*(a*b^3 + 3*a^2*b*c)*d*e^9 - 46189*(3*a^2*b^2 + 2*a^3*c)*e^
10)*x^4 + (1280*c^4*d^7*e^3 - 6720*b*c^3*d^6*e^4 + 415701*a^3*b*e^10 + 4560*(3*b^2*c^2 + 2*a*c^3)*d^5*e^5 - 12
920*(b^3*c + 3*a*b*c^2)*d^4*e^6 + 4845*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^7 + 1423461*(a*b^3 + 3*a^2*b*c)*d^
2*e^8 + 877591*(3*a^2*b^2 + 2*a^3*c)*d*e^9)*x^3 - 3*(512*c^4*d^8*e^2 - 2688*b*c^3*d^7*e^3 - 415701*a^3*b*d*e^9
+ 1824*(3*b^2*c^2 + 2*a*c^3)*d^6*e^4 - 5168*(b^3*c + 3*a*b*c^2)*d^5*e^5 + 1938*(b^4 + 12*a*b^2*c + 6*a^2*c^2)
*d^4*e^6 - 12597*(a*b^3 + 3*a^2*b*c)*d^3*e^7 - 230945*(3*a^2*b^2 + 2*a^3*c)*d^2*e^8)*x^2 + (2048*c^4*d^9*e - 1
0752*b*c^3*d^8*e^2 + 1247103*a^3*b*d^2*e^8 + 7296*(3*b^2*c^2 + 2*a*c^3)*d^7*e^3 - 20672*(b^3*c + 3*a*b*c^2)*d^
6*e^4 + 7752*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^5 - 50388*(a*b^3 + 3*a^2*b*c)*d^4*e^6 + 46189*(3*a^2*b^2 + 2
*a^3*c)*d^3*e^7)*x)*sqrt(e*x + d)/e^8

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Sympy [A]  time = 96.6579, size = 3529, normalized size = 8.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)**(5/2)*(c*x**2+b*x+a)**3,x)

[Out]

a**3*b*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**3*b*d*(-d*(d + e*x)**(3/
2)/3 + (d + e*x)**(5/2)/5)/e + 2*a**3*b*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7
)/e + 4*a**3*c*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 8*a**3*c*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 + 4*a**3*c*(-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 + 6*a**2*b**2*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)
**(5/2)/5)/e**2 + 12*a**2*b**2*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
+ 6*a**2*b**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**2 + 18*a**2*b*c*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 + 36
*a**2*b*c*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 + 18*a**2*b*c*(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*a**2*c**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 + 24*a**2*c**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 + 12*
a**2*c**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 + 6*a*b**3*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 + 12*a*b**3*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 + 6*a*b**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**3 + 24*a*b**2*c*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 +
48*a*b**2*c*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 + 24*a*b**2*c*(-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 + 3
0*a*b*c**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**5 + 60*a*b*c**2*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 1
0*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*b*c**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**5 + 12*a*
c**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**6 + 24*a*c**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)**(11/2)/11
- 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6 + 12*a*c**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**6 + 2*b**4*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 + 4*b**4*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 + 2*b**4*(-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 + 10*b**3*c*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 + 20*b**3*c*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 + 10*b**3*c*(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 + 18*b**2*c**2*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 + 36*b**2*c**2*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 + 18*b**2*c**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**6 + 14*b*c**3*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 + 28*b*c**3*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 + 14*b*c**3*(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
+ 4*c**4*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 + 8*c**4*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 + 4*c**4*(-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)/1
1 + 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 = 1.56596, size = 4224, normalized size = 9.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

2/14549535*(2909907*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*b^2*d^2*e^(-1) + 1939938*(3*(x*e + d)^(5/2)
- 5*(x*e + d)^(3/2)*d)*a^3*c*d^2*e^(-1) + 415701*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/
2)*d^2)*a*b^3*d^2*e^(-2) + 1247103*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*b*
c*d^2*e^(-2) + 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^4*d^2*e^(-3) + 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)*a*b^2*c*d^2*e^(-3) + 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^2*c^2*d^2*e^(-3) + 20995*(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^3*c*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*b*c^2*d^2*e^(-4) + 14535*(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^2*c^2*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*c^3*d^2*e^(-5) + 2261*(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 - 540
54*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*b*c^3*d^2*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 - 3281
85*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*c^4*d^2*e^(-7) + 4849845*(x*e
+ d)^(3/2)*a^3*b*d^2 + 831402*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*b^2*d*
e^(-1) + 554268*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^3*c*d*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*b^3*d*e^(-2)
+ 831402*(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^2
*b*c*d*e^(-2) + 8398*(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^4*d*e^(-3) + 100776*(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*b^2*c*d*e^(-3) + 50388*(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^2*c^2*d*e^(-3) + 16150*(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^3*c*d*e^(-4) + 4845
0*(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 + 90
09*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*a*b*c^2*d*e^(-4) + 5814*(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^2*c^2*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*c^3*d*e^(-5) + 1862*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(1
5/2)*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*c^3*d*e^(-6) + 28*(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)*c^4*d*e^(-7) + 1939938*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^3*b*d + 1385
67*(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^2*b^2*e^
(-1) + 92378*(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^3*c*e^(-1) + 12597*(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*b^3*e^(-2) + 37791*(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^2*b*c*e^(-2) + 1615*(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^4*e^(-3) + 19380*(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
*b^2*c*e^(-3) + 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^2*c^2*e^(-3) + 1615*(3003*(x*e + d)^(1
5/2) - 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^3*c*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 + 15015*(x*e + d)^(3/2)*d^6)*a*b*c^2*e^(-4) + 1197*(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^2*c^2*e^(-5) + 798*(64
35*(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 + 4
25425*(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*c^3*e^(-5) + 49*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 - 895
3560*(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*c^3*e^(-6) + 6*(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)*c^4*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^3*b)*e^(-1)