3.1187 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=574 \[ -\frac{\left (b x+c x^2\right )^{3/2} \left (6 c e x (-10 A c e-b B e+12 B c d)+10 A c e (8 c d-7 b e)-B \left (3 b^2 e^2-92 b c d e+96 c^2 d^2\right )\right )}{48 c e^4}-\frac{\sqrt{b x+c x^2} \left (-2 c e x \left (8 b c e (6 B d-5 A e) (2 c d-b e)-\left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right ) (-10 A c e-b B e+12 B c d)\right )+10 A c e \left (-b^3 e^3+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )-B \left (-3 b^4 e^4-20 b^3 c d e^3+656 b^2 c^2 d^2 e^2-1408 b c^3 d^3 e+768 c^4 d^4\right )\right )}{128 c^2 e^6}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (10 A c e \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )-B \left (-3 b^5 e^5-20 b^4 c d e^4-240 b^3 c^2 d^2 e^3+1920 b^2 c^3 d^3 e^2-3200 b c^4 d^4 e+1536 c^5 d^5\right )\right )}{128 c^{5/2} e^7}+\frac{d^{3/2} (c d-b e)^{3/2} (B d (12 c d-7 b e)-5 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 e^7}+\frac{\left (b x+c x^2\right )^{5/2} (-5 A e+6 B d+B e x)}{5 e^2 (d+e x)} \]

[Out]

-((10*A*c*e*(64*c^3*d^3 - 112*b*c^2*d^2*e + 48*b^2*c*d*e^2 - b^3*e^3) - B*(768*c
^4*d^4 - 1408*b*c^3*d^3*e + 656*b^2*c^2*d^2*e^2 - 20*b^3*c*d*e^3 - 3*b^4*e^4) -
2*c*e*(8*b*c*e*(6*B*d - 5*A*e)*(2*c*d - b*e) - (12*B*c*d - b*B*e - 10*A*c*e)*(16
*c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(128*c^2*e^6) - ((10*A*
c*e*(8*c*d - 7*b*e) - B*(96*c^2*d^2 - 92*b*c*d*e + 3*b^2*e^2) + 6*c*e*(12*B*c*d
- b*B*e - 10*A*c*e)*x)*(b*x + c*x^2)^(3/2))/(48*c*e^4) + ((6*B*d - 5*A*e + B*e*x
)*(b*x + c*x^2)^(5/2))/(5*e^2*(d + e*x)) + ((10*A*c*e*(128*c^4*d^4 - 256*b*c^3*d
^3*e + 144*b^2*c^2*d^2*e^2 - 16*b^3*c*d*e^3 - b^4*e^4) - B*(1536*c^5*d^5 - 3200*
b*c^4*d^4*e + 1920*b^2*c^3*d^3*e^2 - 240*b^3*c^2*d^2*e^3 - 20*b^4*c*d*e^4 - 3*b^
5*e^5))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(128*c^(5/2)*e^7) + (d^(3/2)*(c*
d - b*e)^(3/2)*(B*d*(12*c*d - 7*b*e) - 5*A*e*(2*c*d - b*e))*ArcTanh[(b*d + (2*c*
d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*e^7)

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Rubi [A]  time = 2.13895, antiderivative size = 574, 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 \[ -\frac{\left (b x+c x^2\right )^{3/2} \left (6 c e x (-10 A c e-b B e+12 B c d)+10 A c e (8 c d-7 b e)-B \left (3 b^2 e^2-92 b c d e+96 c^2 d^2\right )\right )}{48 c e^4}-\frac{\sqrt{b x+c x^2} \left (-2 c e x \left (8 b c e (6 B d-5 A e) (2 c d-b e)-\left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right ) (-10 A c e-b B e+12 B c d)\right )+10 A c e \left (-b^3 e^3+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )-B \left (-3 b^4 e^4-20 b^3 c d e^3+656 b^2 c^2 d^2 e^2-1408 b c^3 d^3 e+768 c^4 d^4\right )\right )}{128 c^2 e^6}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (10 A c e \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )-B \left (-3 b^5 e^5-20 b^4 c d e^4-240 b^3 c^2 d^2 e^3+1920 b^2 c^3 d^3 e^2-3200 b c^4 d^4 e+1536 c^5 d^5\right )\right )}{128 c^{5/2} e^7}+\frac{d^{3/2} (c d-b e)^{3/2} (B d (12 c d-7 b e)-5 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 e^7}+\frac{\left (b x+c x^2\right )^{5/2} (-5 A e+6 B d+B e x)}{5 e^2 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

-((10*A*c*e*(64*c^3*d^3 - 112*b*c^2*d^2*e + 48*b^2*c*d*e^2 - b^3*e^3) - B*(768*c
^4*d^4 - 1408*b*c^3*d^3*e + 656*b^2*c^2*d^2*e^2 - 20*b^3*c*d*e^3 - 3*b^4*e^4) -
2*c*e*(8*b*c*e*(6*B*d - 5*A*e)*(2*c*d - b*e) - (12*B*c*d - b*B*e - 10*A*c*e)*(16
*c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(128*c^2*e^6) - ((10*A*
c*e*(8*c*d - 7*b*e) - B*(96*c^2*d^2 - 92*b*c*d*e + 3*b^2*e^2) + 6*c*e*(12*B*c*d
- b*B*e - 10*A*c*e)*x)*(b*x + c*x^2)^(3/2))/(48*c*e^4) + ((6*B*d - 5*A*e + B*e*x
)*(b*x + c*x^2)^(5/2))/(5*e^2*(d + e*x)) + ((10*A*c*e*(128*c^4*d^4 - 256*b*c^3*d
^3*e + 144*b^2*c^2*d^2*e^2 - 16*b^3*c*d*e^3 - b^4*e^4) - B*(1536*c^5*d^5 - 3200*
b*c^4*d^4*e + 1920*b^2*c^3*d^3*e^2 - 240*b^3*c^2*d^2*e^3 - 20*b^4*c*d*e^4 - 3*b^
5*e^5))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(128*c^(5/2)*e^7) + (d^(3/2)*(c*
d - b*e)^(3/2)*(B*d*(12*c*d - 7*b*e) - 5*A*e*(2*c*d - b*e))*ArcTanh[(b*d + (2*c*
d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*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)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 2.96058, size = 617, normalized size = 1.07 \[ \frac{(x (b+c x))^{5/2} \left (\frac{e \sqrt{x} \left (10 A c e \left (15 b^3 e^3 (d+e x)+2 b^2 c e^2 \left (-360 d^2-205 d e x+59 e^2 x^2\right )+8 b c^2 e \left (210 d^3+110 d^2 e x-35 d e^2 x^2+17 e^3 x^3\right )-16 c^3 \left (60 d^4+30 d^3 e x-10 d^2 e^2 x^2+5 d e^3 x^3-3 e^4 x^4\right )\right )+B \left (-45 b^4 e^4 (d+e x)+30 b^3 c e^3 \left (-10 d^2-9 d e x+e^2 x^2\right )+8 b^2 c^2 e^2 \left (1230 d^3+695 d^2 e x-202 d e^2 x^2+93 e^3 x^3\right )+16 b c^3 e \left (-1320 d^4-690 d^3 e x+220 d^2 e^2 x^2-107 d e^3 x^3+63 e^4 x^4\right )+192 c^4 \left (60 d^5+30 d^4 e x-10 d^3 e^2 x^2+5 d^2 e^3 x^3-3 d e^4 x^4+2 e^5 x^5\right )\right )\right )}{c^2 (b+c x)^2 (d+e x)}+\frac{15 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (B \left (3 b^5 e^5+20 b^4 c d e^4+240 b^3 c^2 d^2 e^3-1920 b^2 c^3 d^3 e^2+3200 b c^4 d^4 e-1536 c^5 d^5\right )-10 A c e \left (b^4 e^4+16 b^3 c d e^3-144 b^2 c^2 d^2 e^2+256 b c^3 d^3 e-128 c^4 d^4\right )\right )}{c^{5/2} (b+c x)^{5/2}}+\frac{1920 d^{3/2} (b e-c d)^{3/2} (5 A e (b e-2 c d)+B d (12 c d-7 b e)) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{(b+c x)^{5/2}}\right )}{1920 e^7 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((x*(b + c*x))^(5/2)*((e*Sqrt[x]*(10*A*c*e*(15*b^3*e^3*(d + e*x) + 2*b^2*c*e^2*(
-360*d^2 - 205*d*e*x + 59*e^2*x^2) + 8*b*c^2*e*(210*d^3 + 110*d^2*e*x - 35*d*e^2
*x^2 + 17*e^3*x^3) - 16*c^3*(60*d^4 + 30*d^3*e*x - 10*d^2*e^2*x^2 + 5*d*e^3*x^3
- 3*e^4*x^4)) + B*(-45*b^4*e^4*(d + e*x) + 30*b^3*c*e^3*(-10*d^2 - 9*d*e*x + e^2
*x^2) + 8*b^2*c^2*e^2*(1230*d^3 + 695*d^2*e*x - 202*d*e^2*x^2 + 93*e^3*x^3) + 16
*b*c^3*e*(-1320*d^4 - 690*d^3*e*x + 220*d^2*e^2*x^2 - 107*d*e^3*x^3 + 63*e^4*x^4
) + 192*c^4*(60*d^5 + 30*d^4*e*x - 10*d^3*e^2*x^2 + 5*d^2*e^3*x^3 - 3*d*e^4*x^4
+ 2*e^5*x^5))))/(c^2*(b + c*x)^2*(d + e*x)) + (1920*d^(3/2)*(-(c*d) + b*e)^(3/2)
*(B*d*(12*c*d - 7*b*e) + 5*A*e*(-2*c*d + b*e))*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x
])/(Sqrt[d]*Sqrt[b + c*x])])/(b + c*x)^(5/2) + (15*(-10*A*c*e*(-128*c^4*d^4 + 25
6*b*c^3*d^3*e - 144*b^2*c^2*d^2*e^2 + 16*b^3*c*d*e^3 + b^4*e^4) + B*(-1536*c^5*d
^5 + 3200*b*c^4*d^4*e - 1920*b^2*c^3*d^3*e^2 + 240*b^3*c^2*d^2*e^3 + 20*b^4*c*d*
e^4 + 3*b^5*e^5))*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(c^(5/2)*(b + c*x)^(5/
2))))/(1920*e^7*x^(5/2))

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Maple [B]  time = 0.022, size = 7095, normalized size = 12.4 \[ \text{output too large to display} \]

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)^2,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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 81.3695, 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)^2,x, algorithm="fricas")

[Out]

[-1/3840*(1920*(12*B*c^4*d^5 - 5*A*b^2*c^2*d^2*e^3 - (19*B*b*c^3 + 10*A*c^4)*d^4
*e + (7*B*b^2*c^2 + 15*A*b*c^3)*d^3*e^2 + (12*B*c^4*d^4*e - 5*A*b^2*c^2*d*e^4 -
(19*B*b*c^3 + 10*A*c^4)*d^3*e^2 + (7*B*b^2*c^2 + 15*A*b*c^3)*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*(384*B*c^4*e^6*x^5 + 11520*B*c^4*d^5*e - 1920*(11*B*b*
c^3 + 5*A*c^4)*d^4*e^2 + 240*(41*B*b^2*c^2 + 70*A*b*c^3)*d^3*e^3 - 300*(B*b^3*c
+ 24*A*b^2*c^2)*d^2*e^4 - 15*(3*B*b^4 - 10*A*b^3*c)*d*e^5 - 48*(12*B*c^4*d*e^5 -
(21*B*b*c^3 + 10*A*c^4)*e^6)*x^4 + 8*(120*B*c^4*d^2*e^4 - 2*(107*B*b*c^3 + 50*A
*c^4)*d*e^5 + (93*B*b^2*c^2 + 170*A*b*c^3)*e^6)*x^3 - 2*(960*B*c^4*d^3*e^3 - 160
*(11*B*b*c^3 + 5*A*c^4)*d^2*e^4 + 8*(101*B*b^2*c^2 + 175*A*b*c^3)*d*e^5 - 5*(3*B
*b^3*c + 118*A*b^2*c^2)*e^6)*x^2 + 5*(1152*B*c^4*d^4*e^2 - 96*(23*B*b*c^3 + 10*A
*c^4)*d^3*e^3 + 8*(139*B*b^2*c^2 + 220*A*b*c^3)*d^2*e^4 - 2*(27*B*b^3*c + 410*A*
b^2*c^2)*d*e^5 - 3*(3*B*b^4 - 10*A*b^3*c)*e^6)*x)*sqrt(c*x^2 + b*x)*sqrt(c) - 15
*(1536*B*c^5*d^6 - 640*(5*B*b*c^4 + 2*A*c^5)*d^5*e + 640*(3*B*b^2*c^3 + 4*A*b*c^
4)*d^4*e^2 - 240*(B*b^3*c^2 + 6*A*b^2*c^3)*d^3*e^3 - 20*(B*b^4*c - 8*A*b^3*c^2)*
d^2*e^4 - (3*B*b^5 - 10*A*b^4*c)*d*e^5 + (1536*B*c^5*d^5*e - 640*(5*B*b*c^4 + 2*
A*c^5)*d^4*e^2 + 640*(3*B*b^2*c^3 + 4*A*b*c^4)*d^3*e^3 - 240*(B*b^3*c^2 + 6*A*b^
2*c^3)*d^2*e^4 - 20*(B*b^4*c - 8*A*b^3*c^2)*d*e^5 - (3*B*b^5 - 10*A*b^4*c)*e^6)*
x)*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c))/((c^2*e^8*x + c^2*d*e^7)*sq
rt(c)), 1/3840*(3840*(12*B*c^4*d^5 - 5*A*b^2*c^2*d^2*e^3 - (19*B*b*c^3 + 10*A*c^
4)*d^4*e + (7*B*b^2*c^2 + 15*A*b*c^3)*d^3*e^2 + (12*B*c^4*d^4*e - 5*A*b^2*c^2*d*
e^4 - (19*B*b*c^3 + 10*A*c^4)*d^3*e^2 + (7*B*b^2*c^2 + 15*A*b*c^3)*d^2*e^3)*x)*s
qrt(-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*(384*B*c^4*e^6*x^5 + 11520*B*c^4*d^5*e - 1920*(11*B*b*c^3 + 5*A*c^4)*d^4*e^
2 + 240*(41*B*b^2*c^2 + 70*A*b*c^3)*d^3*e^3 - 300*(B*b^3*c + 24*A*b^2*c^2)*d^2*e
^4 - 15*(3*B*b^4 - 10*A*b^3*c)*d*e^5 - 48*(12*B*c^4*d*e^5 - (21*B*b*c^3 + 10*A*c
^4)*e^6)*x^4 + 8*(120*B*c^4*d^2*e^4 - 2*(107*B*b*c^3 + 50*A*c^4)*d*e^5 + (93*B*b
^2*c^2 + 170*A*b*c^3)*e^6)*x^3 - 2*(960*B*c^4*d^3*e^3 - 160*(11*B*b*c^3 + 5*A*c^
4)*d^2*e^4 + 8*(101*B*b^2*c^2 + 175*A*b*c^3)*d*e^5 - 5*(3*B*b^3*c + 118*A*b^2*c^
2)*e^6)*x^2 + 5*(1152*B*c^4*d^4*e^2 - 96*(23*B*b*c^3 + 10*A*c^4)*d^3*e^3 + 8*(13
9*B*b^2*c^2 + 220*A*b*c^3)*d^2*e^4 - 2*(27*B*b^3*c + 410*A*b^2*c^2)*d*e^5 - 3*(3
*B*b^4 - 10*A*b^3*c)*e^6)*x)*sqrt(c*x^2 + b*x)*sqrt(c) + 15*(1536*B*c^5*d^6 - 64
0*(5*B*b*c^4 + 2*A*c^5)*d^5*e + 640*(3*B*b^2*c^3 + 4*A*b*c^4)*d^4*e^2 - 240*(B*b
^3*c^2 + 6*A*b^2*c^3)*d^3*e^3 - 20*(B*b^4*c - 8*A*b^3*c^2)*d^2*e^4 - (3*B*b^5 -
10*A*b^4*c)*d*e^5 + (1536*B*c^5*d^5*e - 640*(5*B*b*c^4 + 2*A*c^5)*d^4*e^2 + 640*
(3*B*b^2*c^3 + 4*A*b*c^4)*d^3*e^3 - 240*(B*b^3*c^2 + 6*A*b^2*c^3)*d^2*e^4 - 20*(
B*b^4*c - 8*A*b^3*c^2)*d*e^5 - (3*B*b^5 - 10*A*b^4*c)*e^6)*x)*log((2*c*x + b)*sq
rt(c) - 2*sqrt(c*x^2 + b*x)*c))/((c^2*e^8*x + c^2*d*e^7)*sqrt(c)), -1/1920*(960*
(12*B*c^4*d^5 - 5*A*b^2*c^2*d^2*e^3 - (19*B*b*c^3 + 10*A*c^4)*d^4*e + (7*B*b^2*c
^2 + 15*A*b*c^3)*d^3*e^2 + (12*B*c^4*d^4*e - 5*A*b^2*c^2*d*e^4 - (19*B*b*c^3 + 1
0*A*c^4)*d^3*e^2 + (7*B*b^2*c^2 + 15*A*b*c^3)*d^2*e^3)*x)*sqrt(c*d^2 - b*d*e)*sq
rt(-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)) - (384*B*c^4*e^6*x^5 + 11520*B*c^4*d^5*e - 1920*(11*B*b*c^3 + 5*A*c^4)*d
^4*e^2 + 240*(41*B*b^2*c^2 + 70*A*b*c^3)*d^3*e^3 - 300*(B*b^3*c + 24*A*b^2*c^2)*
d^2*e^4 - 15*(3*B*b^4 - 10*A*b^3*c)*d*e^5 - 48*(12*B*c^4*d*e^5 - (21*B*b*c^3 + 1
0*A*c^4)*e^6)*x^4 + 8*(120*B*c^4*d^2*e^4 - 2*(107*B*b*c^3 + 50*A*c^4)*d*e^5 + (9
3*B*b^2*c^2 + 170*A*b*c^3)*e^6)*x^3 - 2*(960*B*c^4*d^3*e^3 - 160*(11*B*b*c^3 + 5
*A*c^4)*d^2*e^4 + 8*(101*B*b^2*c^2 + 175*A*b*c^3)*d*e^5 - 5*(3*B*b^3*c + 118*A*b
^2*c^2)*e^6)*x^2 + 5*(1152*B*c^4*d^4*e^2 - 96*(23*B*b*c^3 + 10*A*c^4)*d^3*e^3 +
8*(139*B*b^2*c^2 + 220*A*b*c^3)*d^2*e^4 - 2*(27*B*b^3*c + 410*A*b^2*c^2)*d*e^5 -
3*(3*B*b^4 - 10*A*b^3*c)*e^6)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) + 15*(1536*B*c^5*d^
6 - 640*(5*B*b*c^4 + 2*A*c^5)*d^5*e + 640*(3*B*b^2*c^3 + 4*A*b*c^4)*d^4*e^2 - 24
0*(B*b^3*c^2 + 6*A*b^2*c^3)*d^3*e^3 - 20*(B*b^4*c - 8*A*b^3*c^2)*d^2*e^4 - (3*B*
b^5 - 10*A*b^4*c)*d*e^5 + (1536*B*c^5*d^5*e - 640*(5*B*b*c^4 + 2*A*c^5)*d^4*e^2
+ 640*(3*B*b^2*c^3 + 4*A*b*c^4)*d^3*e^3 - 240*(B*b^3*c^2 + 6*A*b^2*c^3)*d^2*e^4
- 20*(B*b^4*c - 8*A*b^3*c^2)*d*e^5 - (3*B*b^5 - 10*A*b^4*c)*e^6)*x)*arctan(sqrt(
c*x^2 + b*x)*sqrt(-c)/(c*x)))/((c^2*e^8*x + c^2*d*e^7)*sqrt(-c)), 1/1920*(1920*(
12*B*c^4*d^5 - 5*A*b^2*c^2*d^2*e^3 - (19*B*b*c^3 + 10*A*c^4)*d^4*e + (7*B*b^2*c^
2 + 15*A*b*c^3)*d^3*e^2 + (12*B*c^4*d^4*e - 5*A*b^2*c^2*d*e^4 - (19*B*b*c^3 + 10
*A*c^4)*d^3*e^2 + (7*B*b^2*c^2 + 15*A*b*c^3)*d^2*e^3)*x)*sqrt(-c*d^2 + b*d*e)*sq
rt(-c)*arctan(sqrt(c*x^2 + b*x)*d/(sqrt(-c*d^2 + b*d*e)*x)) + (384*B*c^4*e^6*x^5
+ 11520*B*c^4*d^5*e - 1920*(11*B*b*c^3 + 5*A*c^4)*d^4*e^2 + 240*(41*B*b^2*c^2 +
70*A*b*c^3)*d^3*e^3 - 300*(B*b^3*c + 24*A*b^2*c^2)*d^2*e^4 - 15*(3*B*b^4 - 10*A
*b^3*c)*d*e^5 - 48*(12*B*c^4*d*e^5 - (21*B*b*c^3 + 10*A*c^4)*e^6)*x^4 + 8*(120*B
*c^4*d^2*e^4 - 2*(107*B*b*c^3 + 50*A*c^4)*d*e^5 + (93*B*b^2*c^2 + 170*A*b*c^3)*e
^6)*x^3 - 2*(960*B*c^4*d^3*e^3 - 160*(11*B*b*c^3 + 5*A*c^4)*d^2*e^4 + 8*(101*B*b
^2*c^2 + 175*A*b*c^3)*d*e^5 - 5*(3*B*b^3*c + 118*A*b^2*c^2)*e^6)*x^2 + 5*(1152*B
*c^4*d^4*e^2 - 96*(23*B*b*c^3 + 10*A*c^4)*d^3*e^3 + 8*(139*B*b^2*c^2 + 220*A*b*c
^3)*d^2*e^4 - 2*(27*B*b^3*c + 410*A*b^2*c^2)*d*e^5 - 3*(3*B*b^4 - 10*A*b^3*c)*e^
6)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 15*(1536*B*c^5*d^6 - 640*(5*B*b*c^4 + 2*A*c^5
)*d^5*e + 640*(3*B*b^2*c^3 + 4*A*b*c^4)*d^4*e^2 - 240*(B*b^3*c^2 + 6*A*b^2*c^3)*
d^3*e^3 - 20*(B*b^4*c - 8*A*b^3*c^2)*d^2*e^4 - (3*B*b^5 - 10*A*b^4*c)*d*e^5 + (1
536*B*c^5*d^5*e - 640*(5*B*b*c^4 + 2*A*c^5)*d^4*e^2 + 640*(3*B*b^2*c^3 + 4*A*b*c
^4)*d^3*e^3 - 240*(B*b^3*c^2 + 6*A*b^2*c^3)*d^2*e^4 - 20*(B*b^4*c - 8*A*b^3*c^2)
*d*e^5 - (3*B*b^5 - 10*A*b^4*c)*e^6)*x)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x))
)/((c^2*e^8*x + c^2*d*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)**2,x)

[Out]

Timed out

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

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)^2,x, algorithm="giac")

[Out]

Timed out