3.1283 \(\int \frac{A+B x}{(d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=706 \[ -\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (c d-b e)}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (b^3 \left (-e^2\right ) (3 B d-4 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}+\frac{2 \left (b (c d-b e) \left (b^2 e (3 B d-4 A e)-b c d (3 A e+4 B d)+8 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (3 B d-4 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (2 b^4 e^3 (3 B d-4 A e)-b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)-8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}+\frac{2 e \sqrt{b x+c x^2} \left (b^4 \left (6 B d e^3-8 A e^4\right )-b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)-8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right )}{3 b^4 d^3 \sqrt{d+e x} (c d-b e)^3} \]

[Out]

(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(3*b^2*d*(c*d - b*e)*Sqrt
[d + e*x]*(b*x + c*x^2)^(3/2)) + (2*(b*(c*d - b*e)*(8*A*c^2*d^2 + b^2*e*(3*B*d -
4*A*e) - b*c*d*(4*B*d + 3*A*e)) + c*(16*A*c^3*d^3 + 15*b^2*B*c*d^2*e - b^3*e^2*
(3*B*d - 4*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e))*x))/(3*b^4*d^2*(c*d - b*e)^2*Sqrt[d
+ e*x]*Sqrt[b*x + c*x^2]) + (2*e*(16*A*c^4*d^4 - b^3*c*d*e^2*(9*B*d - 7*A*e) -
8*b*c^3*d^3*(B*d + 4*A*e) + b^2*c^2*d^2*e*(19*B*d + 9*A*e) + b^4*(6*B*d*e^3 - 8*
A*e^4))*Sqrt[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) - (2*Sqrt[c]*
(16*A*c^4*d^4 - b^3*c*d*e^2*(9*B*d - 7*A*e) + 2*b^4*e^3*(3*B*d - 4*A*e) - 8*b*c^
3*d^3*(B*d + 4*A*e) + b^2*c^2*d^2*e*(19*B*d + 9*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*
Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b
)^(7/2)*d^3*(c*d - b*e)^3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[c]*(16*
A*c^3*d^3 + 15*b^2*B*c*d^2*e - b^3*e^2*(3*B*d - 4*A*e) - 8*b*c^2*d^2*(B*d + 3*A*
e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x
])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*d^2*(c*d - b*e)^2*Sqrt[d + e*x]*Sqrt[b
*x + c*x^2])

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Rubi [A]  time = 3.06396, antiderivative size = 706, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (c d-b e)}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (b^3 \left (-e^2\right ) (3 B d-4 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}+\frac{2 \left (b (c d-b e) \left (b^2 e (3 B d-4 A e)-b c d (3 A e+4 B d)+8 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (3 B d-4 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}+\frac{2 e \sqrt{b x+c x^2} \left (2 b^4 e^3 (3 B d-4 A e)-b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)-8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right )}{3 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (2 b^4 e^3 (3 B d-4 A e)-b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)-8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(3*b^2*d*(c*d - b*e)*Sqrt
[d + e*x]*(b*x + c*x^2)^(3/2)) + (2*(b*(c*d - b*e)*(8*A*c^2*d^2 + b^2*e*(3*B*d -
4*A*e) - b*c*d*(4*B*d + 3*A*e)) + c*(16*A*c^3*d^3 + 15*b^2*B*c*d^2*e - b^3*e^2*
(3*B*d - 4*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e))*x))/(3*b^4*d^2*(c*d - b*e)^2*Sqrt[d
+ e*x]*Sqrt[b*x + c*x^2]) + (2*e*(16*A*c^4*d^4 - b^3*c*d*e^2*(9*B*d - 7*A*e) +
2*b^4*e^3*(3*B*d - 4*A*e) - 8*b*c^3*d^3*(B*d + 4*A*e) + b^2*c^2*d^2*e*(19*B*d +
9*A*e))*Sqrt[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) - (2*Sqrt[c]*
(16*A*c^4*d^4 - b^3*c*d*e^2*(9*B*d - 7*A*e) + 2*b^4*e^3*(3*B*d - 4*A*e) - 8*b*c^
3*d^3*(B*d + 4*A*e) + b^2*c^2*d^2*e*(19*B*d + 9*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*
Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b
)^(7/2)*d^3*(c*d - b*e)^3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[c]*(16*
A*c^3*d^3 + 15*b^2*B*c*d^2*e - b^3*e^2*(3*B*d - 4*A*e) - 8*b*c^2*d^2*(B*d + 3*A*
e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x
])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*d^2*(c*d - b*e)^2*Sqrt[d + e*x]*Sqrt[b
*x + c*x^2])

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

[Out]

Timed out

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Mathematica [C]  time = 10.4058, size = 628, normalized size = 0.89 \[ \frac{2 \left (b \left (3 b^4 e^4 x^2 (b+c x)^2 (B d-A e)+c^3 d^3 x^2 (b+c x) (d+e x) \left (-b c (13 A e+5 B d)+8 A c^2 d+10 b^2 B e\right )+b c^3 d^3 x^2 (d+e x) (b B-A c) (b e-c d)+x (b+c x)^2 (d+e x) (c d-b e)^3 (5 A b e+8 A c d-3 b B d)+A b d (b+c x)^2 (d+e x) (b e-c d)^3\right )-c x \sqrt{\frac{b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (b^3 \left (8 A e^3-6 B d e^2\right )+3 b^2 c d e (2 B d-A e)-b c^2 d^2 (9 A e+4 B d)+8 A c^3 d^3\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (7 A e-9 B d)+b^2 c^2 d^2 e (9 A e+19 B d)-8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (7 A e-9 B d)+b^2 c^2 d^2 e (9 A e+19 B d)-8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right )\right )\right )}{3 b^5 d^3 (x (b+c x))^{3/2} \sqrt{d+e x} (c d-b e)^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*(3*b^4*e^4*(B*d - A*e)*x^2*(b + c*x)^2 + b*c^3*(b*B - A*c)*d^3*(-(c*d) + b
*e)*x^2*(d + e*x) + c^3*d^3*(8*A*c^2*d + 10*b^2*B*e - b*c*(5*B*d + 13*A*e))*x^2*
(b + c*x)*(d + e*x) + A*b*d*(-(c*d) + b*e)^3*(b + c*x)^2*(d + e*x) + (c*d - b*e)
^3*(-3*b*B*d + 8*A*c*d + 5*A*b*e)*x*(b + c*x)^2*(d + e*x)) - Sqrt[b/c]*c*x*(b +
c*x)*(Sqrt[b/c]*(16*A*c^4*d^4 + 2*b^4*e^3*(3*B*d - 4*A*e) - 8*b*c^3*d^3*(B*d + 4
*A*e) + b^3*c*d*e^2*(-9*B*d + 7*A*e) + b^2*c^2*d^2*e*(19*B*d + 9*A*e))*(b + c*x)
*(d + e*x) + I*b*e*(16*A*c^4*d^4 + 2*b^4*e^3*(3*B*d - 4*A*e) - 8*b*c^3*d^3*(B*d
+ 4*A*e) + b^3*c*d*e^2*(-9*B*d + 7*A*e) + b^2*c^2*d^2*e*(19*B*d + 9*A*e))*Sqrt[1
+ b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c
*d)/(b*e)] - I*b*e*(c*d - b*e)*(8*A*c^3*d^3 + 3*b^2*c*d*e*(2*B*d - A*e) - b*c^2*
d^2*(4*B*d + 9*A*e) + b^3*(-6*B*d*e^2 + 8*A*e^3))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(
e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^5*d^3
*(c*d - b*e)^3*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])

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Maple [B]  time = 0.085, size = 4001, normalized size = 5.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(-6*B*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipt
icE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^7*d*e^4-8*B*x*((c*x+b)/b)^(1/2)*(
-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e
-c*d))^(1/2))*b^3*c^4*d^5+8*B*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(
-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^4*d^5+8*A
*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^6*c*e^5+16*A*x^2*((c*x+b)/b)^(1/2)*(-(e*
x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d
))^(1/2))*b*c^6*d^5-16*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*
x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^6*d^5-8*B*x^2*
((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)
/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^5*d^5+8*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)
*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(
1/2))*b^2*c^5*d^5-2*A*x^2*b^3*c^4*d^3*e^2-3*B*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b
*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))
*b^6*c*d^2*e^3+18*B*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1
/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c^2*d^3*e^2-23*B*x*((
c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b
)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^3*d^4*e+15*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d
)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^
(1/2))*b^5*c^2*d^2*e^3-28*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(
-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^3*d^3*e^2
-23*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elliptic
F(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^4*d^4*e+12*B*x^2*b^2*c^5*d^5-6*
A*x*b^2*c^5*d^5+8*A*x^2*b^6*c*e^5+8*B*x^3*b*c^6*d^5-24*A*x^2*b*c^6*d^5+3*B*x*b^3
*c^4*d^5+8*A*x^4*b^4*c^3*e^5-16*A*x^4*c^7*d^4*e+16*A*x*((c*x+b)/b)^(1/2)*(-(e*x+
d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))
^(1/2))*b^2*c^5*d^5-16*A*x^3*c^7*d^5-15*A*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c
*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^6
*c*d*e^4-2*A*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ell
ipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c^2*d^2*e^3+41*A*x*((c*x+b)/
b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2)
,(b*e/(b*e-c*d))^(1/2))*b^4*c^3*d^3*e^2-48*A*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*
e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*
b^3*c^4*d^4*e+4*A*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2
)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^6*c*d*e^4+3*A*b^5*c^2*d^3
*e^2-3*A*b^4*c^3*d^4*e-A*b^6*c*d^2*e^3+A*b^3*c^4*d^5-26*B*x^2*b^3*c^4*d^4*e+4*A*
x*b^6*c*d*e^4+9*B*x^4*b^3*c^4*d^2*e^3-20*B*x^3*b^3*c^4*d^3*e^2-7*B*x^3*b^2*c^5*d
^4*e+32*A*x^4*b*c^6*d^3*e^2-10*A*x^3*b^4*c^3*d*e^4-9*A*x^4*b^2*c^5*d^2*e^3+8*A*x
^3*b*c^6*d^4*e-22*A*x^3*b^3*c^4*d^2*e^3+A*x^2*b^5*c^2*d*e^4-7*A*x^4*b^3*c^4*d*e^
4+40*A*x^3*b^2*c^5*d^3*e^2-6*B*x^4*b^4*c^3*d*e^4-3*B*x*b^6*c*d^2*e^3+9*B*x*b^5*c
^2*d^3*e^2-9*B*x*b^4*c^3*d^4*e-6*A*x*b^5*c^2*d^2*e^3-6*A*x*b^4*c^3*d^3*e^2+14*A*
x*b^3*c^4*d^4*e-12*B*x^3*b^5*c^2*d*e^4+15*B*x^3*b^4*c^3*d^2*e^3+43*A*x^2*b^2*c^5
*d^4*e-6*B*x^2*b^6*c*d*e^4+3*B*x^2*b^5*c^2*d^2*e^3+9*B*x^2*b^4*c^3*d^3*e^2-19*B*
x^4*b^2*c^5*d^3*e^2+8*B*x^4*b*c^6*d^4*e-18*A*x^2*b^4*c^3*d^2*e^3+27*B*x^2*((c*x+
b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1
/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^4*d^4*e-3*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(
b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2)
)*b^5*c^2*d^2*e^3+18*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/
b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^3*d^3*e^2+16*A
*x^3*b^5*c^2*e^5-4*A*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(
1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c^2*d^2*e^3-24*A*x*(
(c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/
b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^3*d^3*e^2+40*A*x*((c*x+b)/b)^(1/2)*(-(e*x+
d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))
^(1/2))*b^3*c^4*d^4*e-15*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-
c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c^2*d*e^4-2*
A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((
c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^3*d^2*e^3+41*A*x^2*((c*x+b)/b)^(1/2
)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(
b*e-c*d))^(1/2))*b^3*c^4*d^3*e^2-48*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d
))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c
^5*d^4*e+4*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*E
llipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c^2*d*e^4-4*A*x^2*((c*x+b)
/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2
),(b*e/(b*e-c*d))^(1/2))*b^4*c^3*d^2*e^3-24*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/
(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2
))*b^3*c^4*d^3*e^2+40*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x
/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^5*d^4*e-6*B*x
^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x
+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^6*c*d*e^4+15*B*x*((c*x+b)/b)^(1/2)*(-(e*x+
d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))
^(1/2))*b^6*c*d^2*e^3-28*B*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*
x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c^2*d^3*e^2+27
*B*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^3*d^4*e-16*A*x*((c*x+b)/b)^(1/2)*(-(
e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*b^2*c^5*d^5+8*A*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c
*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^7*e^5)/x^2*(x*(
c*x+b))^(1/2)/b^4/d^3/c/(b*e-c*d)^3/(c*x+b)^2/(e*x+d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

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)^(3/2)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x + A}{{\left (c^{2} e x^{5} + b^{2} d x^{2} +{\left (c^{2} d + 2 \, b c e\right )} x^{4} +{\left (2 \, b c d + b^{2} e\right )} x^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{e x + d}}, x\right ) \]

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)^(3/2)),x, algorithm="fricas")

[Out]

integral((B*x + A)/((c^2*e*x^5 + b^2*d*x^2 + (c^2*d + 2*b*c*e)*x^4 + (2*b*c*d +
b^2*e)*x^3)*sqrt(c*x^2 + b*x)*sqrt(e*x + d)), x)

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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)^(3/2)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError