3.1188 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=508 \[ \frac{5 \sqrt{d} \sqrt{c d-b e} \left (A e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-28 b c d e+24 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 e^7}+\frac{5 \sqrt{b x+c x^2} \left (-2 c e x \left (16 A c e (2 c d-b e)-B \left (b^2 e^2-32 b c d e+48 c^2 d^2\right )\right )+8 A c e \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )-B \left (-b^3 e^3+88 b^2 c d e^2-272 b c^2 d^2 e+192 c^3 d^3\right )\right )}{64 c e^6}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (8 A c e \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right )-B \left (-b^4 e^4-24 b^3 c d e^3+288 b^2 c^2 d^2 e^2-640 b c^3 d^3 e+384 c^4 d^4\right )\right )}{64 c^{3/2} e^7}-\frac{5 \left (b x+c x^2\right )^{3/2} (e x (-4 A c e-b B e+6 B c d)-2 A e (8 c d-3 b e)+B d (24 c d-13 b e))}{24 e^4 (d+e x)}+\frac{\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2} \]

[Out]

(5*(8*A*c*e*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2) - B*(192*c^3*d^3 - 272*b*c^2*d
^2*e + 88*b^2*c*d*e^2 - b^3*e^3) - 2*c*e*(16*A*c*e*(2*c*d - b*e) - B*(48*c^2*d^2
- 32*b*c*d*e + b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(64*c*e^6) - (5*(B*d*(24*c*d - 1
3*b*e) - 2*A*e*(8*c*d - 3*b*e) + e*(6*B*c*d - b*B*e - 4*A*c*e)*x)*(b*x + c*x^2)^
(3/2))/(24*e^4*(d + e*x)) + ((3*B*d - 2*A*e + B*e*x)*(b*x + c*x^2)^(5/2))/(4*e^2
*(d + e*x)^2) - (5*(8*A*c*e*(32*c^3*d^3 - 48*b*c^2*d^2*e + 18*b^2*c*d*e^2 - b^3*
e^3) - B*(384*c^4*d^4 - 640*b*c^3*d^3*e + 288*b^2*c^2*d^2*e^2 - 24*b^3*c*d*e^3 -
b^4*e^4))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(3/2)*e^7) + (5*Sqrt[d]
*Sqrt[c*d - b*e]*(A*e*(16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2) - B*d*(24*c^2*d^2 -
28*b*c*d*e + 7*b^2*e^2))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b
*e]*Sqrt[b*x + c*x^2])])/(8*e^7)

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Rubi [A]  time = 1.67431, antiderivative size = 508, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{5 \sqrt{d} \sqrt{c d-b e} \left (A e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-28 b c d e+24 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 e^7}+\frac{5 \sqrt{b x+c x^2} \left (-2 c e x \left (16 A c e (2 c d-b e)-B \left (b^2 e^2-32 b c d e+48 c^2 d^2\right )\right )+8 A c e \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )-B \left (-b^3 e^3+88 b^2 c d e^2-272 b c^2 d^2 e+192 c^3 d^3\right )\right )}{64 c e^6}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (8 A c e \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right )-B \left (-b^4 e^4-24 b^3 c d e^3+288 b^2 c^2 d^2 e^2-640 b c^3 d^3 e+384 c^4 d^4\right )\right )}{64 c^{3/2} e^7}-\frac{5 \left (b x+c x^2\right )^{3/2} (e x (-4 A c e-b B e+6 B c d)-2 A e (8 c d-3 b e)+B d (24 c d-13 b e))}{24 e^4 (d+e x)}+\frac{\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(5*(8*A*c*e*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2) - B*(192*c^3*d^3 - 272*b*c^2*d
^2*e + 88*b^2*c*d*e^2 - b^3*e^3) - 2*c*e*(16*A*c*e*(2*c*d - b*e) - B*(48*c^2*d^2
- 32*b*c*d*e + b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(64*c*e^6) - (5*(B*d*(24*c*d - 1
3*b*e) - 2*A*e*(8*c*d - 3*b*e) + e*(6*B*c*d - b*B*e - 4*A*c*e)*x)*(b*x + c*x^2)^
(3/2))/(24*e^4*(d + e*x)) + ((3*B*d - 2*A*e + B*e*x)*(b*x + c*x^2)^(5/2))/(4*e^2
*(d + e*x)^2) - (5*(8*A*c*e*(32*c^3*d^3 - 48*b*c^2*d^2*e + 18*b^2*c*d*e^2 - b^3*
e^3) - B*(384*c^4*d^4 - 640*b*c^3*d^3*e + 288*b^2*c^2*d^2*e^2 - 24*b^3*c*d*e^3 -
b^4*e^4))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(3/2)*e^7) + (5*Sqrt[d]
*Sqrt[c*d - b*e]*(A*e*(16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2) - B*d*(24*c^2*d^2 -
28*b*c*d*e + 7*b^2*e^2))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b
*e]*Sqrt[b*x + c*x^2])])/(8*e^7)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(5/2)/(e*x+d)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 4.3065, size = 509, normalized size = 1. \[ \frac{(x (b+c x))^{5/2} \left (\frac{240 \sqrt{d} \sqrt{b e-c d} \left (A e \left (-3 b^2 e^2+16 b c d e-16 c^2 d^2\right )+B d \left (7 b^2 e^2-28 b c d e+24 c^2 d^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{(b+c x)^{5/2}}+\frac{e \sqrt{x} \left (2 e x \left (8 A c e (13 b e-18 c d)+B \left (59 b^2 e^2-312 b c d e+288 c^2 d^2\right )\right )+24 A e \left (11 b^2 e^2-54 b c d e+48 c^2 d^2\right )+\frac{96 d^2 (B d-A e) (c d-b e)^2}{(d+e x)^2}+8 c e^2 x^2 (8 A c e+17 b B e-24 B c d)-\frac{48 d (c d-b e) (9 A e (b e-2 c d)+B d (22 c d-13 b e))}{d+e x}+3 B \left (\frac{5 b^3 e^3}{c}-264 b^2 d e^2+864 b c d^2 e-640 c^2 d^3\right )+48 B c^2 e^3 x^3\right )}{(b+c x)^2}-\frac{15 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (8 A c e \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right )+B \left (b^4 e^4+24 b^3 c d e^3-288 b^2 c^2 d^2 e^2+640 b c^3 d^3 e-384 c^4 d^4\right )\right )}{c^{3/2} (b+c x)^{5/2}}\right )}{192 e^7 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((x*(b + c*x))^(5/2)*((e*Sqrt[x]*(24*A*e*(48*c^2*d^2 - 54*b*c*d*e + 11*b^2*e^2)
+ 3*B*(-640*c^2*d^3 + 864*b*c*d^2*e - 264*b^2*d*e^2 + (5*b^3*e^3)/c) + 2*e*(8*A*
c*e*(-18*c*d + 13*b*e) + B*(288*c^2*d^2 - 312*b*c*d*e + 59*b^2*e^2))*x + 8*c*e^2
*(-24*B*c*d + 17*b*B*e + 8*A*c*e)*x^2 + 48*B*c^2*e^3*x^3 + (96*d^2*(B*d - A*e)*(
c*d - b*e)^2)/(d + e*x)^2 - (48*d*(c*d - b*e)*(B*d*(22*c*d - 13*b*e) + 9*A*e*(-2
*c*d + b*e)))/(d + e*x)))/(b + c*x)^2 + (240*Sqrt[d]*Sqrt[-(c*d) + b*e]*(A*e*(-1
6*c^2*d^2 + 16*b*c*d*e - 3*b^2*e^2) + B*d*(24*c^2*d^2 - 28*b*c*d*e + 7*b^2*e^2))
*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(b + c*x)^(5/2) -
(15*(8*A*c*e*(32*c^3*d^3 - 48*b*c^2*d^2*e + 18*b^2*c*d*e^2 - b^3*e^3) + B*(-384
*c^4*d^4 + 640*b*c^3*d^3*e - 288*b^2*c^2*d^2*e^2 + 24*b^3*c*d*e^3 + b^4*e^4))*Lo
g[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(c^(3/2)*(b + c*x)^(5/2))))/(192*e^7*x^(5/
2))

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Maple [B]  time = 0.027, size = 11558, normalized size = 22.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(5/2)/(e*x+d)^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 11.0498, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/(e*x + d)^3,x, algorithm="fricas")

[Out]

[1/384*(240*(24*B*c^3*d^5 - 3*A*b^2*c*d^2*e^3 - 4*(7*B*b*c^2 + 4*A*c^3)*d^4*e +
(7*B*b^2*c + 16*A*b*c^2)*d^3*e^2 + (24*B*c^3*d^3*e^2 - 3*A*b^2*c*e^5 - 4*(7*B*b*
c^2 + 4*A*c^3)*d^2*e^3 + (7*B*b^2*c + 16*A*b*c^2)*d*e^4)*x^2 + 2*(24*B*c^3*d^4*e
- 3*A*b^2*c*d*e^4 - 4*(7*B*b*c^2 + 4*A*c^3)*d^3*e^2 + (7*B*b^2*c + 16*A*b*c^2)*
d^2*e^3)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c)*log((b*d + (2*c*d - b*e)*x - 2*sqrt(c*d^
2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + 2*(48*B*c^3*e^6*x^5 - 2880*B*c^3*d^5*
e + 240*(17*B*b*c^2 + 8*A*c^3)*d^4*e^2 - 120*(11*B*b^2*c + 20*A*b*c^2)*d^3*e^3 +
15*(B*b^3 + 40*A*b^2*c)*d^2*e^4 - 8*(12*B*c^3*d*e^5 - (17*B*b*c^2 + 8*A*c^3)*e^
6)*x^4 + 2*(120*B*c^3*d^2*e^4 - 16*(11*B*b*c^2 + 5*A*c^3)*d*e^5 + (59*B*b^2*c +
104*A*b*c^2)*e^6)*x^3 - (960*B*c^3*d^3*e^3 - 40*(37*B*b*c^2 + 16*A*c^3)*d^2*e^4
+ 4*(139*B*b^2*c + 220*A*b*c^2)*d*e^5 - 3*(5*B*b^3 + 88*A*b^2*c)*e^6)*x^2 - 10*(
432*B*c^3*d^4*e^2 - 48*(13*B*b*c^2 + 6*A*c^3)*d^3*e^3 + (209*B*b^2*c + 368*A*b*c
^2)*d^2*e^4 - 3*(B*b^3 + 32*A*b^2*c)*d*e^5)*x)*sqrt(c*x^2 + b*x)*sqrt(c) - 15*(3
84*B*c^4*d^6 - 128*(5*B*b*c^3 + 2*A*c^4)*d^5*e + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^
4*e^2 - 24*(B*b^3*c + 6*A*b^2*c^2)*d^3*e^3 - (B*b^4 - 8*A*b^3*c)*d^2*e^4 + (384*
B*c^4*d^4*e^2 - 128*(5*B*b*c^3 + 2*A*c^4)*d^3*e^3 + 96*(3*B*b^2*c^2 + 4*A*b*c^3)
*d^2*e^4 - 24*(B*b^3*c + 6*A*b^2*c^2)*d*e^5 - (B*b^4 - 8*A*b^3*c)*e^6)*x^2 + 2*(
384*B*c^4*d^5*e - 128*(5*B*b*c^3 + 2*A*c^4)*d^4*e^2 + 96*(3*B*b^2*c^2 + 4*A*b*c^
3)*d^3*e^3 - 24*(B*b^3*c + 6*A*b^2*c^2)*d^2*e^4 - (B*b^4 - 8*A*b^3*c)*d*e^5)*x)*
log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c))/((c*e^9*x^2 + 2*c*d*e^8*x + c*
d^2*e^7)*sqrt(c)), -1/384*(480*(24*B*c^3*d^5 - 3*A*b^2*c*d^2*e^3 - 4*(7*B*b*c^2
+ 4*A*c^3)*d^4*e + (7*B*b^2*c + 16*A*b*c^2)*d^3*e^2 + (24*B*c^3*d^3*e^2 - 3*A*b^
2*c*e^5 - 4*(7*B*b*c^2 + 4*A*c^3)*d^2*e^3 + (7*B*b^2*c + 16*A*b*c^2)*d*e^4)*x^2
+ 2*(24*B*c^3*d^4*e - 3*A*b^2*c*d*e^4 - 4*(7*B*b*c^2 + 4*A*c^3)*d^3*e^2 + (7*B*b
^2*c + 16*A*b*c^2)*d^2*e^3)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c)*arctan(sqrt(c*x^2 +
b*x)*d/(sqrt(-c*d^2 + b*d*e)*x)) - 2*(48*B*c^3*e^6*x^5 - 2880*B*c^3*d^5*e + 240*
(17*B*b*c^2 + 8*A*c^3)*d^4*e^2 - 120*(11*B*b^2*c + 20*A*b*c^2)*d^3*e^3 + 15*(B*b
^3 + 40*A*b^2*c)*d^2*e^4 - 8*(12*B*c^3*d*e^5 - (17*B*b*c^2 + 8*A*c^3)*e^6)*x^4 +
2*(120*B*c^3*d^2*e^4 - 16*(11*B*b*c^2 + 5*A*c^3)*d*e^5 + (59*B*b^2*c + 104*A*b*
c^2)*e^6)*x^3 - (960*B*c^3*d^3*e^3 - 40*(37*B*b*c^2 + 16*A*c^3)*d^2*e^4 + 4*(139
*B*b^2*c + 220*A*b*c^2)*d*e^5 - 3*(5*B*b^3 + 88*A*b^2*c)*e^6)*x^2 - 10*(432*B*c^
3*d^4*e^2 - 48*(13*B*b*c^2 + 6*A*c^3)*d^3*e^3 + (209*B*b^2*c + 368*A*b*c^2)*d^2*
e^4 - 3*(B*b^3 + 32*A*b^2*c)*d*e^5)*x)*sqrt(c*x^2 + b*x)*sqrt(c) + 15*(384*B*c^4
*d^6 - 128*(5*B*b*c^3 + 2*A*c^4)*d^5*e + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^4*e^2 -
24*(B*b^3*c + 6*A*b^2*c^2)*d^3*e^3 - (B*b^4 - 8*A*b^3*c)*d^2*e^4 + (384*B*c^4*d^
4*e^2 - 128*(5*B*b*c^3 + 2*A*c^4)*d^3*e^3 + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^2*e^4
- 24*(B*b^3*c + 6*A*b^2*c^2)*d*e^5 - (B*b^4 - 8*A*b^3*c)*e^6)*x^2 + 2*(384*B*c^
4*d^5*e - 128*(5*B*b*c^3 + 2*A*c^4)*d^4*e^2 + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^3*e
^3 - 24*(B*b^3*c + 6*A*b^2*c^2)*d^2*e^4 - (B*b^4 - 8*A*b^3*c)*d*e^5)*x)*log((2*c
*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c))/((c*e^9*x^2 + 2*c*d*e^8*x + c*d^2*e^7)
*sqrt(c)), 1/192*(120*(24*B*c^3*d^5 - 3*A*b^2*c*d^2*e^3 - 4*(7*B*b*c^2 + 4*A*c^3
)*d^4*e + (7*B*b^2*c + 16*A*b*c^2)*d^3*e^2 + (24*B*c^3*d^3*e^2 - 3*A*b^2*c*e^5 -
4*(7*B*b*c^2 + 4*A*c^3)*d^2*e^3 + (7*B*b^2*c + 16*A*b*c^2)*d*e^4)*x^2 + 2*(24*B
*c^3*d^4*e - 3*A*b^2*c*d*e^4 - 4*(7*B*b*c^2 + 4*A*c^3)*d^3*e^2 + (7*B*b^2*c + 16
*A*b*c^2)*d^2*e^3)*x)*sqrt(c*d^2 - b*d*e)*sqrt(-c)*log((b*d + (2*c*d - b*e)*x -
2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + (48*B*c^3*e^6*x^5 - 2880*B
*c^3*d^5*e + 240*(17*B*b*c^2 + 8*A*c^3)*d^4*e^2 - 120*(11*B*b^2*c + 20*A*b*c^2)*
d^3*e^3 + 15*(B*b^3 + 40*A*b^2*c)*d^2*e^4 - 8*(12*B*c^3*d*e^5 - (17*B*b*c^2 + 8*
A*c^3)*e^6)*x^4 + 2*(120*B*c^3*d^2*e^4 - 16*(11*B*b*c^2 + 5*A*c^3)*d*e^5 + (59*B
*b^2*c + 104*A*b*c^2)*e^6)*x^3 - (960*B*c^3*d^3*e^3 - 40*(37*B*b*c^2 + 16*A*c^3)
*d^2*e^4 + 4*(139*B*b^2*c + 220*A*b*c^2)*d*e^5 - 3*(5*B*b^3 + 88*A*b^2*c)*e^6)*x
^2 - 10*(432*B*c^3*d^4*e^2 - 48*(13*B*b*c^2 + 6*A*c^3)*d^3*e^3 + (209*B*b^2*c +
368*A*b*c^2)*d^2*e^4 - 3*(B*b^3 + 32*A*b^2*c)*d*e^5)*x)*sqrt(c*x^2 + b*x)*sqrt(-
c) + 15*(384*B*c^4*d^6 - 128*(5*B*b*c^3 + 2*A*c^4)*d^5*e + 96*(3*B*b^2*c^2 + 4*A
*b*c^3)*d^4*e^2 - 24*(B*b^3*c + 6*A*b^2*c^2)*d^3*e^3 - (B*b^4 - 8*A*b^3*c)*d^2*e
^4 + (384*B*c^4*d^4*e^2 - 128*(5*B*b*c^3 + 2*A*c^4)*d^3*e^3 + 96*(3*B*b^2*c^2 +
4*A*b*c^3)*d^2*e^4 - 24*(B*b^3*c + 6*A*b^2*c^2)*d*e^5 - (B*b^4 - 8*A*b^3*c)*e^6)
*x^2 + 2*(384*B*c^4*d^5*e - 128*(5*B*b*c^3 + 2*A*c^4)*d^4*e^2 + 96*(3*B*b^2*c^2
+ 4*A*b*c^3)*d^3*e^3 - 24*(B*b^3*c + 6*A*b^2*c^2)*d^2*e^4 - (B*b^4 - 8*A*b^3*c)*
d*e^5)*x)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/((c*e^9*x^2 + 2*c*d*e^8*x +
c*d^2*e^7)*sqrt(-c)), -1/192*(240*(24*B*c^3*d^5 - 3*A*b^2*c*d^2*e^3 - 4*(7*B*b*c
^2 + 4*A*c^3)*d^4*e + (7*B*b^2*c + 16*A*b*c^2)*d^3*e^2 + (24*B*c^3*d^3*e^2 - 3*A
*b^2*c*e^5 - 4*(7*B*b*c^2 + 4*A*c^3)*d^2*e^3 + (7*B*b^2*c + 16*A*b*c^2)*d*e^4)*x
^2 + 2*(24*B*c^3*d^4*e - 3*A*b^2*c*d*e^4 - 4*(7*B*b*c^2 + 4*A*c^3)*d^3*e^2 + (7*
B*b^2*c + 16*A*b*c^2)*d^2*e^3)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(-c)*arctan(sqrt(c*x^
2 + b*x)*d/(sqrt(-c*d^2 + b*d*e)*x)) - (48*B*c^3*e^6*x^5 - 2880*B*c^3*d^5*e + 24
0*(17*B*b*c^2 + 8*A*c^3)*d^4*e^2 - 120*(11*B*b^2*c + 20*A*b*c^2)*d^3*e^3 + 15*(B
*b^3 + 40*A*b^2*c)*d^2*e^4 - 8*(12*B*c^3*d*e^5 - (17*B*b*c^2 + 8*A*c^3)*e^6)*x^4
+ 2*(120*B*c^3*d^2*e^4 - 16*(11*B*b*c^2 + 5*A*c^3)*d*e^5 + (59*B*b^2*c + 104*A*
b*c^2)*e^6)*x^3 - (960*B*c^3*d^3*e^3 - 40*(37*B*b*c^2 + 16*A*c^3)*d^2*e^4 + 4*(1
39*B*b^2*c + 220*A*b*c^2)*d*e^5 - 3*(5*B*b^3 + 88*A*b^2*c)*e^6)*x^2 - 10*(432*B*
c^3*d^4*e^2 - 48*(13*B*b*c^2 + 6*A*c^3)*d^3*e^3 + (209*B*b^2*c + 368*A*b*c^2)*d^
2*e^4 - 3*(B*b^3 + 32*A*b^2*c)*d*e^5)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 15*(384*B*
c^4*d^6 - 128*(5*B*b*c^3 + 2*A*c^4)*d^5*e + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^4*e^2
- 24*(B*b^3*c + 6*A*b^2*c^2)*d^3*e^3 - (B*b^4 - 8*A*b^3*c)*d^2*e^4 + (384*B*c^4
*d^4*e^2 - 128*(5*B*b*c^3 + 2*A*c^4)*d^3*e^3 + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^2*
e^4 - 24*(B*b^3*c + 6*A*b^2*c^2)*d*e^5 - (B*b^4 - 8*A*b^3*c)*e^6)*x^2 + 2*(384*B
*c^4*d^5*e - 128*(5*B*b*c^3 + 2*A*c^4)*d^4*e^2 + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^
3*e^3 - 24*(B*b^3*c + 6*A*b^2*c^2)*d^2*e^4 - (B*b^4 - 8*A*b^3*c)*d*e^5)*x)*arcta
n(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/((c*e^9*x^2 + 2*c*d*e^8*x + c*d^2*e^7)*sqrt
(-c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(5/2)/(e*x+d)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.66743, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/(e*x + d)^3,x, algorithm="giac")

[Out]

sage0*x