3.1225 \(\int \frac{(A+B x) \left (b x+c x^2\right )^2}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=265 \[ -\frac{2 (d+e x)^{7/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{7 e^6}+\frac{2 (d+e x)^{5/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{5 e^6}-\frac{2 d^2 \sqrt{d+e x} (B d-A e) (c d-b e)^2}{e^6}-\frac{2 c (d+e x)^{9/2} (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac{2 d (d+e x)^{3/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6}+\frac{2 B c^2 (d+e x)^{11/2}}{11 e^6} \]

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*Sqrt[d + e*x])/e^6 + (2*d*(c*d - b*e)*(B*d*(5*
c*d - 3*b*e) - 2*A*e*(2*c*d - b*e))*(d + e*x)^(3/2))/(3*e^6) + (2*(A*e*(6*c^2*d^
2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))*(d + e*x)^
(5/2))/(5*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2
))*(d + e*x)^(7/2))/(7*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(9/2))/
(9*e^6) + (2*B*c^2*(d + e*x)^(11/2))/(11*e^6)

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Rubi [A]  time = 0.448912, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 (d+e x)^{7/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{7 e^6}+\frac{2 (d+e x)^{5/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{5 e^6}-\frac{2 d^2 \sqrt{d+e x} (B d-A e) (c d-b e)^2}{e^6}-\frac{2 c (d+e x)^{9/2} (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac{2 d (d+e x)^{3/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6}+\frac{2 B c^2 (d+e x)^{11/2}}{11 e^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^2)/Sqrt[d + e*x],x]

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*Sqrt[d + e*x])/e^6 + (2*d*(c*d - b*e)*(B*d*(5*
c*d - 3*b*e) - 2*A*e*(2*c*d - b*e))*(d + e*x)^(3/2))/(3*e^6) + (2*(A*e*(6*c^2*d^
2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))*(d + e*x)^
(5/2))/(5*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2
))*(d + e*x)^(7/2))/(7*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(9/2))/
(9*e^6) + (2*B*c^2*(d + e*x)^(11/2))/(11*e^6)

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Rubi in Sympy [A]  time = 101.697, size = 291, normalized size = 1.1 \[ \frac{2 B c^{2} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{6}} + \frac{2 c \left (d + e x\right )^{\frac{9}{2}} \left (A c e + 2 B b e - 5 B c d\right )}{9 e^{6}} + \frac{2 d^{2} \sqrt{d + e x} \left (A e - B d\right ) \left (b e - c d\right )^{2}}{e^{6}} - \frac{2 d \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right ) \left (2 A b e^{2} - 4 A c d e - 3 B b d e + 5 B c d^{2}\right )}{3 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{7 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{5 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**(1/2),x)

[Out]

2*B*c**2*(d + e*x)**(11/2)/(11*e**6) + 2*c*(d + e*x)**(9/2)*(A*c*e + 2*B*b*e - 5
*B*c*d)/(9*e**6) + 2*d**2*sqrt(d + e*x)*(A*e - B*d)*(b*e - c*d)**2/e**6 - 2*d*(d
+ e*x)**(3/2)*(b*e - c*d)*(2*A*b*e**2 - 4*A*c*d*e - 3*B*b*d*e + 5*B*c*d**2)/(3*
e**6) + 2*(d + e*x)**(7/2)*(2*A*b*c*e**2 - 4*A*c**2*d*e + B*b**2*e**2 - 8*B*b*c*
d*e + 10*B*c**2*d**2)/(7*e**6) + 2*(d + e*x)**(5/2)*(A*b**2*e**3 - 6*A*b*c*d*e**
2 + 6*A*c**2*d**2*e - 3*B*b**2*d*e**2 + 12*B*b*c*d**2*e - 10*B*c**2*d**3)/(5*e**
6)

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Mathematica [A]  time = 0.393456, size = 273, normalized size = 1.03 \[ \frac{2 \sqrt{d+e x} \left (11 A e \left (21 b^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+18 b c e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (99 b^2 e^2 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+22 b c e \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )-5 c^2 \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )\right )\right )}{3465 e^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^2)/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(11*A*e*(21*b^2*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 18*b*c*e*(-
16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + c^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)) + B*(99*b^2*e^2*(-16*d^3 + 8*d^2*e*x -
6*d*e^2*x^2 + 5*e^3*x^3) + 22*b*c*e*(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) - 5*c^2*(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))))/(3465*e^6)

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Maple [A]  time = 0.01, size = 341, normalized size = 1.3 \[{\frac{630\,B{c}^{2}{x}^{5}{e}^{5}+770\,A{c}^{2}{e}^{5}{x}^{4}+1540\,Bbc{e}^{5}{x}^{4}-700\,B{c}^{2}d{e}^{4}{x}^{4}+1980\,Abc{e}^{5}{x}^{3}-880\,A{c}^{2}d{e}^{4}{x}^{3}+990\,B{b}^{2}{e}^{5}{x}^{3}-1760\,Bbcd{e}^{4}{x}^{3}+800\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+1386\,A{b}^{2}{e}^{5}{x}^{2}-2376\,Abcd{e}^{4}{x}^{2}+1056\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-1188\,B{b}^{2}d{e}^{4}{x}^{2}+2112\,Bbc{d}^{2}{e}^{3}{x}^{2}-960\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}-1848\,A{b}^{2}d{e}^{4}x+3168\,Abc{d}^{2}{e}^{3}x-1408\,A{c}^{2}{d}^{3}{e}^{2}x+1584\,B{b}^{2}{d}^{2}{e}^{3}x-2816\,Bbc{d}^{3}{e}^{2}x+1280\,B{c}^{2}{d}^{4}ex+3696\,A{b}^{2}{d}^{2}{e}^{3}-6336\,Abc{d}^{3}{e}^{2}+2816\,A{c}^{2}{d}^{4}e-3168\,B{b}^{2}{d}^{3}{e}^{2}+5632\,Bbc{d}^{4}e-2560\,B{c}^{2}{d}^{5}}{3465\,{e}^{6}}\sqrt{ex+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x)

[Out]

2/3465*(315*B*c^2*e^5*x^5+385*A*c^2*e^5*x^4+770*B*b*c*e^5*x^4-350*B*c^2*d*e^4*x^
4+990*A*b*c*e^5*x^3-440*A*c^2*d*e^4*x^3+495*B*b^2*e^5*x^3-880*B*b*c*d*e^4*x^3+40
0*B*c^2*d^2*e^3*x^3+693*A*b^2*e^5*x^2-1188*A*b*c*d*e^4*x^2+528*A*c^2*d^2*e^3*x^2
-594*B*b^2*d*e^4*x^2+1056*B*b*c*d^2*e^3*x^2-480*B*c^2*d^3*e^2*x^2-924*A*b^2*d*e^
4*x+1584*A*b*c*d^2*e^3*x-704*A*c^2*d^3*e^2*x+792*B*b^2*d^2*e^3*x-1408*B*b*c*d^3*
e^2*x+640*B*c^2*d^4*e*x+1848*A*b^2*d^2*e^3-3168*A*b*c*d^3*e^2+1408*A*c^2*d^4*e-1
584*B*b^2*d^3*e^2+2816*B*b*c*d^4*e-1280*B*c^2*d^5)*(e*x+d)^(1/2)/e^6

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Maxima [A]  time = 0.700892, size = 393, normalized size = 1.48 \[ \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B c^{2} - 385 \,{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (10 \, B c^{2} d^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 693 \,{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 3465 \,{\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )} \sqrt{e x + d}\right )}}{3465 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*(e*x + d)^(11/2)*B*c^2 - 385*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x
+ d)^(9/2) + 495*(10*B*c^2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c)*e^2
)*(e*x + d)^(7/2) - 693*(10*B*c^2*d^3 - A*b^2*e^3 - 6*(2*B*b*c + A*c^2)*d^2*e +
3*(B*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^(5/2) + 1155*(5*B*c^2*d^4 - 2*A*b^2*d*e^3 -
4*(2*B*b*c + A*c^2)*d^3*e + 3*(B*b^2 + 2*A*b*c)*d^2*e^2)*(e*x + d)^(3/2) - 3465
*(B*c^2*d^5 - A*b^2*d^2*e^3 - (2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^
2)*sqrt(e*x + d))/e^6

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Fricas [A]  time = 0.273866, size = 392, normalized size = 1.48 \[ \frac{2 \,{\left (315 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 1848 \, A b^{2} d^{2} e^{3} + 1408 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e - 1584 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 35 \,{\left (10 \, B c^{2} d e^{4} - 11 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 5 \,{\left (80 \, B c^{2} d^{2} e^{3} - 88 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 99 \,{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 3 \,{\left (160 \, B c^{2} d^{3} e^{2} - 231 \, A b^{2} e^{5} - 176 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 198 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \,{\left (160 \, B c^{2} d^{4} e - 231 \, A b^{2} d e^{4} - 176 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 198 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*B*c^2*e^5*x^5 - 1280*B*c^2*d^5 + 1848*A*b^2*d^2*e^3 + 1408*(2*B*b*c
+ A*c^2)*d^4*e - 1584*(B*b^2 + 2*A*b*c)*d^3*e^2 - 35*(10*B*c^2*d*e^4 - 11*(2*B*b
*c + A*c^2)*e^5)*x^4 + 5*(80*B*c^2*d^2*e^3 - 88*(2*B*b*c + A*c^2)*d*e^4 + 99*(B*
b^2 + 2*A*b*c)*e^5)*x^3 - 3*(160*B*c^2*d^3*e^2 - 231*A*b^2*e^5 - 176*(2*B*b*c +
A*c^2)*d^2*e^3 + 198*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 4*(160*B*c^2*d^4*e - 231*A*b
^2*d*e^4 - 176*(2*B*b*c + A*c^2)*d^3*e^2 + 198*(B*b^2 + 2*A*b*c)*d^2*e^3)*x)*sqr
t(e*x + d)/e^6

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Sympy [A]  time = 74.4374, size = 944, normalized size = 3.56 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*A*b**2*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/
2)/3)/e**2 + 2*A*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 + 4*A*b*c*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 + 4*A*b*c*(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 + 2*A*c**2*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 + 2*A*c**2*(-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 + 2*B*b**2*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 + 2*B*b**2*(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 + 4*B*b*c*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 + 4*B*b*c*(-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 + 2*B*c**2*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 + 2*B*c**2*(d**6/sqr
t(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)/e, Ne(e, 0)), ((A*b**2*x**3/3 + B*c**2*x**6/6 + x**5*(A*c**2 + 2*B*b*c
)/5 + x**4*(2*A*b*c + B*b**2)/4)/sqrt(d), True))

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GIAC/XCAS [A]  time = 0.295763, size = 583, normalized size = 2.2 \[ \frac{2}{3465} \,{\left (231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} A b^{2} e^{\left (-10\right )} + 99 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} B b^{2} e^{\left (-21\right )} + 198 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} A b c e^{\left (-21\right )} + 22 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} B b c e^{\left (-36\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} A c^{2} e^{\left (-36\right )} + 5 \,{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{50} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{50} + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{50} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{50} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{50} - 693 \, \sqrt{x e + d} d^{5} e^{50}\right )} B c^{2} e^{\left (-55\right )}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(231*(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*b^2*e^(-10) + 99*(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)*B*b^2*e^(-21) + 198*
(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^1
8 - 35*sqrt(x*e + d)*d^3*e^18)*A*b*c*e^(-21) + 22*(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)*B*b*c*e^(-36) + 11*(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*c^2*e^(-36) + 5*(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)*B
*c^2*e^(-55))*e^(-1)