3.1630 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=181 \[ -\frac{12 b^5 (d+e x)^{11/2} (b d-a e)}{11 e^7}+\frac{10 b^4 (d+e x)^{9/2} (b d-a e)^2}{3 e^7}-\frac{40 b^3 (d+e x)^{7/2} (b d-a e)^3}{7 e^7}+\frac{6 b^2 (d+e x)^{5/2} (b d-a e)^4}{e^7}-\frac{4 b (d+e x)^{3/2} (b d-a e)^5}{e^7}+\frac{2 \sqrt{d+e x} (b d-a e)^6}{e^7}+\frac{2 b^6 (d+e x)^{13/2}}{13 e^7} \]

[Out]

(2*(b*d - a*e)^6*Sqrt[d + e*x])/e^7 - (4*b*(b*d - a*e)^5*(d + e*x)^(3/2))/e^7 +
(6*b^2*(b*d - a*e)^4*(d + e*x)^(5/2))/e^7 - (40*b^3*(b*d - a*e)^3*(d + e*x)^(7/2
))/(7*e^7) + (10*b^4*(b*d - a*e)^2*(d + e*x)^(9/2))/(3*e^7) - (12*b^5*(b*d - a*e
)*(d + e*x)^(11/2))/(11*e^7) + (2*b^6*(d + e*x)^(13/2))/(13*e^7)

_______________________________________________________________________________________

Rubi [A]  time = 0.174857, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{12 b^5 (d+e x)^{11/2} (b d-a e)}{11 e^7}+\frac{10 b^4 (d+e x)^{9/2} (b d-a e)^2}{3 e^7}-\frac{40 b^3 (d+e x)^{7/2} (b d-a e)^3}{7 e^7}+\frac{6 b^2 (d+e x)^{5/2} (b d-a e)^4}{e^7}-\frac{4 b (d+e x)^{3/2} (b d-a e)^5}{e^7}+\frac{2 \sqrt{d+e x} (b d-a e)^6}{e^7}+\frac{2 b^6 (d+e x)^{13/2}}{13 e^7} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)^6*Sqrt[d + e*x])/e^7 - (4*b*(b*d - a*e)^5*(d + e*x)^(3/2))/e^7 +
(6*b^2*(b*d - a*e)^4*(d + e*x)^(5/2))/e^7 - (40*b^3*(b*d - a*e)^3*(d + e*x)^(7/2
))/(7*e^7) + (10*b^4*(b*d - a*e)^2*(d + e*x)^(9/2))/(3*e^7) - (12*b^5*(b*d - a*e
)*(d + e*x)^(11/2))/(11*e^7) + (2*b^6*(d + e*x)^(13/2))/(13*e^7)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 77.9443, size = 168, normalized size = 0.93 \[ \frac{2 b^{6} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{7}} + \frac{12 b^{5} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )}{11 e^{7}} + \frac{10 b^{4} \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{2}}{3 e^{7}} + \frac{40 b^{3} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{3}}{7 e^{7}} + \frac{6 b^{2} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{4}}{e^{7}} + \frac{4 b \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{5}}{e^{7}} + \frac{2 \sqrt{d + e x} \left (a e - b d\right )^{6}}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(1/2),x)

[Out]

2*b**6*(d + e*x)**(13/2)/(13*e**7) + 12*b**5*(d + e*x)**(11/2)*(a*e - b*d)/(11*e
**7) + 10*b**4*(d + e*x)**(9/2)*(a*e - b*d)**2/(3*e**7) + 40*b**3*(d + e*x)**(7/
2)*(a*e - b*d)**3/(7*e**7) + 6*b**2*(d + e*x)**(5/2)*(a*e - b*d)**4/e**7 + 4*b*(
d + e*x)**(3/2)*(a*e - b*d)**5/e**7 + 2*sqrt(d + e*x)*(a*e - b*d)**6/e**7

_______________________________________________________________________________________

Mathematica [A]  time = 0.218784, size = 290, normalized size = 1.6 \[ \frac{2 \sqrt{d+e x} \left (3003 a^6 e^6+6006 a^5 b e^5 (e x-2 d)+3003 a^4 b^2 e^4 \left (8 d^2-4 d e x+3 e^2 x^2\right )+1716 a^3 b^3 e^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+143 a^2 b^4 e^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+26 a b^5 e \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )+b^6 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )}{3003 e^7} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(3003*a^6*e^6 + 6006*a^5*b*e^5*(-2*d + e*x) + 3003*a^4*b^2*e^4*
(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 1716*a^3*b^3*e^3*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*
x^2 + 5*e^3*x^3) + 143*a^2*b^4*e^2*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d
*e^3*x^3 + 35*e^4*x^4) + 26*a*b^5*e*(-256*d^5 + 128*d^4*e*x - 96*d^3*e^2*x^2 + 8
0*d^2*e^3*x^3 - 70*d*e^4*x^4 + 63*e^5*x^5) + b^6*(1024*d^6 - 512*d^5*e*x + 384*d
^4*e^2*x^2 - 320*d^3*e^3*x^3 + 280*d^2*e^4*x^4 - 252*d*e^5*x^5 + 231*e^6*x^6)))/
(3003*e^7)

_______________________________________________________________________________________

Maple [B]  time = 0.014, size = 377, normalized size = 2.1 \[{\frac{462\,{x}^{6}{b}^{6}{e}^{6}+3276\,{x}^{5}a{b}^{5}{e}^{6}-504\,{x}^{5}{b}^{6}d{e}^{5}+10010\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-3640\,{x}^{4}a{b}^{5}d{e}^{5}+560\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+17160\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-11440\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+4160\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-640\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+18018\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-20592\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+13728\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-4992\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+768\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+12012\,x{a}^{5}b{e}^{6}-24024\,x{a}^{4}{b}^{2}d{e}^{5}+27456\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-18304\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+6656\,xa{b}^{5}{d}^{4}{e}^{2}-1024\,x{b}^{6}{d}^{5}e+6006\,{a}^{6}{e}^{6}-24024\,{a}^{5}bd{e}^{5}+48048\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-54912\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+36608\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-13312\,{d}^{5}a{b}^{5}e+2048\,{b}^{6}{d}^{6}}{3003\,{e}^{7}}\sqrt{ex+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x)

[Out]

2/3003*(231*b^6*e^6*x^6+1638*a*b^5*e^6*x^5-252*b^6*d*e^5*x^5+5005*a^2*b^4*e^6*x^
4-1820*a*b^5*d*e^5*x^4+280*b^6*d^2*e^4*x^4+8580*a^3*b^3*e^6*x^3-5720*a^2*b^4*d*e
^5*x^3+2080*a*b^5*d^2*e^4*x^3-320*b^6*d^3*e^3*x^3+9009*a^4*b^2*e^6*x^2-10296*a^3
*b^3*d*e^5*x^2+6864*a^2*b^4*d^2*e^4*x^2-2496*a*b^5*d^3*e^3*x^2+384*b^6*d^4*e^2*x
^2+6006*a^5*b*e^6*x-12012*a^4*b^2*d*e^5*x+13728*a^3*b^3*d^2*e^4*x-9152*a^2*b^4*d
^3*e^3*x+3328*a*b^5*d^4*e^2*x-512*b^6*d^5*e*x+3003*a^6*e^6-12012*a^5*b*d*e^5+240
24*a^4*b^2*d^2*e^4-27456*a^3*b^3*d^3*e^3+18304*a^2*b^4*d^4*e^2-6656*a*b^5*d^5*e+
1024*b^6*d^6)*(e*x+d)^(1/2)/e^7

_______________________________________________________________________________________

Maxima [A]  time = 0.74646, size = 729, normalized size = 4.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/15015*(15015*sqrt(e*x + d)*a^6 + 3003*(10*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d
)*a*b/e + (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^2/
e^2)*a^4 + 3432*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d
^2 - 35*sqrt(e*x + d)*d^3)*a^3*b^3/e^3 + 143*(84*(3*(e*x + d)^(5/2) - 10*(e*x +
d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*b^2/e^2 + 36*(5*(e*x + d)^(7/2) - 21*(e*x
+ d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a*b^3/e^3 + (35*(
e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)
^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^4/e^4)*a^2 + 572*(35*(e*x + d)^(9/2) - 180
*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqr
t(e*x + d)*d^4)*a^2*b^4/e^4 + 130*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d +
990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 -
693*sqrt(e*x + d)*d^5)*a*b^5/e^5 + 5*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11
/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/
2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*b^6/e^6)/e

_______________________________________________________________________________________

Fricas [A]  time = 0.207241, size = 481, normalized size = 2.66 \[ \frac{2 \,{\left (231 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 6656 \, a b^{5} d^{5} e + 18304 \, a^{2} b^{4} d^{4} e^{2} - 27456 \, a^{3} b^{3} d^{3} e^{3} + 24024 \, a^{4} b^{2} d^{2} e^{4} - 12012 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} - 126 \,{\left (2 \, b^{6} d e^{5} - 13 \, a b^{5} e^{6}\right )} x^{5} + 35 \,{\left (8 \, b^{6} d^{2} e^{4} - 52 \, a b^{5} d e^{5} + 143 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \,{\left (16 \, b^{6} d^{3} e^{3} - 104 \, a b^{5} d^{2} e^{4} + 286 \, a^{2} b^{4} d e^{5} - 429 \, a^{3} b^{3} e^{6}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{4} e^{2} - 832 \, a b^{5} d^{3} e^{3} + 2288 \, a^{2} b^{4} d^{2} e^{4} - 3432 \, a^{3} b^{3} d e^{5} + 3003 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \,{\left (256 \, b^{6} d^{5} e - 1664 \, a b^{5} d^{4} e^{2} + 4576 \, a^{2} b^{4} d^{3} e^{3} - 6864 \, a^{3} b^{3} d^{2} e^{4} + 6006 \, a^{4} b^{2} d e^{5} - 3003 \, a^{5} b e^{6}\right )} x\right )} \sqrt{e x + d}}{3003 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/sqrt(e*x + d),x, algorithm="fricas")

[Out]

2/3003*(231*b^6*e^6*x^6 + 1024*b^6*d^6 - 6656*a*b^5*d^5*e + 18304*a^2*b^4*d^4*e^
2 - 27456*a^3*b^3*d^3*e^3 + 24024*a^4*b^2*d^2*e^4 - 12012*a^5*b*d*e^5 + 3003*a^6
*e^6 - 126*(2*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 + 35*(8*b^6*d^2*e^4 - 52*a*b^5*d*e^5
+ 143*a^2*b^4*e^6)*x^4 - 20*(16*b^6*d^3*e^3 - 104*a*b^5*d^2*e^4 + 286*a^2*b^4*d
*e^5 - 429*a^3*b^3*e^6)*x^3 + 3*(128*b^6*d^4*e^2 - 832*a*b^5*d^3*e^3 + 2288*a^2*
b^4*d^2*e^4 - 3432*a^3*b^3*d*e^5 + 3003*a^4*b^2*e^6)*x^2 - 2*(256*b^6*d^5*e - 16
64*a*b^5*d^4*e^2 + 4576*a^2*b^4*d^3*e^3 - 6864*a^3*b^3*d^2*e^4 + 6006*a^4*b^2*d*
e^5 - 3003*a^5*b*e^6)*x)*sqrt(e*x + d)/e^7

_______________________________________________________________________________________

Sympy [A]  time = 75.5769, size = 1003, normalized size = 5.54 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*a**6*d/sqrt(d + e*x) + 2*a**6*(-d/sqrt(d + e*x) - sqrt(d + e*x))
+ 12*a**5*b*d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e + 12*a**5*b*(d**2/sqrt(d + e*
x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 30*a**4*b**2*d*(d**2/sqrt(d + e
*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 30*a**4*b**2*(-d**3/sqrt(d
+ e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 +
40*a**3*b**3*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2)
- (d + e*x)**(5/2)/5)/e**3 + 40*a**3*b**3*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d +
e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e*
*3 + 30*a**2*b**4*d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x
)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 + 30*a**2*b**4*(-d*
*5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d
+ e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 + 12*a*b**5*d
*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d*
*2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**5 + 12*a*b
**5*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**
3*(d + e*x)**(5/2) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d +
e*x)**(11/2)/11)/e**5 + 2*b**6*d*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*
d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d + e*x)**(7/2)/7 + 2
*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**6 + 2*b**6*(-d**7/sqrt(d + e*x)
- 7*d**6*sqrt(d + e*x) + 7*d**5*(d + e*x)**(3/2) - 7*d**4*(d + e*x)**(5/2) + 5*
d**3*(d + e*x)**(7/2) - 7*d**2*(d + e*x)**(9/2)/3 + 7*d*(d + e*x)**(11/2)/11 - (
d + e*x)**(13/2)/13)/e**6)/e, Ne(e, 0)), ((a**6*x + 3*a**5*b*x**2 + 5*a**4*b**2*
x**3 + 5*a**3*b**3*x**4 + 3*a**2*b**4*x**5 + a*b**5*x**6 + b**6*x**7/7)/sqrt(d),
True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214778, size = 601, normalized size = 3.32 \[ \frac{2}{3003} \,{\left (6006 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{5} b e^{\left (-1\right )} + 3003 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} a^{4} b^{2} e^{\left (-10\right )} + 1716 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} a^{3} b^{3} e^{\left (-21\right )} + 143 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} a^{2} b^{4} e^{\left (-36\right )} + 26 \,{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{50} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{50} + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{50} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{50} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{50} - 693 \, \sqrt{x e + d} d^{5} e^{50}\right )} a b^{5} e^{\left (-55\right )} +{\left (231 \,{\left (x e + d\right )}^{\frac{13}{2}} e^{72} - 1638 \,{\left (x e + d\right )}^{\frac{11}{2}} d e^{72} + 5005 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} e^{72} - 8580 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} e^{72} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} e^{72} - 6006 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5} e^{72} + 3003 \, \sqrt{x e + d} d^{6} e^{72}\right )} b^{6} e^{\left (-78\right )} + 3003 \, \sqrt{x e + d} a^{6}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/sqrt(e*x + d),x, algorithm="giac")

[Out]

2/3003*(6006*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5*b*e^(-1) + 3003*(3*(x*e +
d)^(5/2)*e^8 - 10*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*a^4*b^2*e^(
-10) + 1716*(5*(x*e + d)^(7/2)*e^18 - 21*(x*e + d)^(5/2)*d*e^18 + 35*(x*e + d)^(
3/2)*d^2*e^18 - 35*sqrt(x*e + d)*d^3*e^18)*a^3*b^3*e^(-21) + 143*(35*(x*e + d)^(
9/2)*e^32 - 180*(x*e + d)^(7/2)*d*e^32 + 378*(x*e + d)^(5/2)*d^2*e^32 - 420*(x*e
+ d)^(3/2)*d^3*e^32 + 315*sqrt(x*e + d)*d^4*e^32)*a^2*b^4*e^(-36) + 26*(63*(x*e
+ d)^(11/2)*e^50 - 385*(x*e + d)^(9/2)*d*e^50 + 990*(x*e + d)^(7/2)*d^2*e^50 -
1386*(x*e + d)^(5/2)*d^3*e^50 + 1155*(x*e + d)^(3/2)*d^4*e^50 - 693*sqrt(x*e + d
)*d^5*e^50)*a*b^5*e^(-55) + (231*(x*e + d)^(13/2)*e^72 - 1638*(x*e + d)^(11/2)*d
*e^72 + 5005*(x*e + d)^(9/2)*d^2*e^72 - 8580*(x*e + d)^(7/2)*d^3*e^72 + 9009*(x*
e + d)^(5/2)*d^4*e^72 - 6006*(x*e + d)^(3/2)*d^5*e^72 + 3003*sqrt(x*e + d)*d^6*e
^72)*b^6*e^(-78) + 3003*sqrt(x*e + d)*a^6)*e^(-1)