3.1277 \(\int \frac{A+B x}{(d+e x)^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=570 \[ -\frac{2 e \sqrt{b x+c x^2} \left (b^2 (-e) (B d-4 A e)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (b^2 (-e) (B d-4 A e)-3 b c d (2 A e+B d)+6 A c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^{3/2} (c d-b e)}-\frac{2 e \sqrt{b x+c x^2} \left (2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{3 b^2 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3} \]

[Out]

(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*(d + e
*x)^(3/2)*Sqrt[b*x + c*x^2]) - (2*e*(6*A*c^2*d^2 - b^2*e*(B*d - 4*A*e) - 3*b*c*d
*(B*d + 2*A*e))*Sqrt[b*x + c*x^2])/(3*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)) - (
2*e*(6*A*c^3*d^3 - b^2*c*d*e*(7*B*d - 19*A*e) + 2*b^3*e^2*(B*d - 4*A*e) - 3*b*c^
2*d^2*(B*d + 3*A*e))*Sqrt[b*x + c*x^2])/(3*b^2*d^3*(c*d - b*e)^3*Sqrt[d + e*x])
+ (2*Sqrt[c]*(6*A*c^3*d^3 - b^2*c*d*e*(7*B*d - 19*A*e) + 2*b^3*e^2*(B*d - 4*A*e)
- 3*b*c^2*d^2*(B*d + 3*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)^(3/2)*d^3*(c*d - b*e)^
3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[c]*(6*A*c^2*d^2 - b^2*e*(B*d -
4*A*e) - 3*b*c*d*(B*d + 2*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Elli
pticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*d^2*(c*d -
b*e)^2*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 2.46625, antiderivative size = 570, 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 \[ -\frac{2 e \sqrt{b x+c x^2} \left (b^2 (-e) (B d-4 A e)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (b^2 (-e) (B d-4 A e)-3 b c d (2 A e+B d)+6 A c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^{3/2} (c d-b e)}-\frac{2 e \sqrt{b x+c x^2} \left (2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{3 b^2 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*(d + e
*x)^(3/2)*Sqrt[b*x + c*x^2]) - (2*e*(6*A*c^2*d^2 - b^2*e*(B*d - 4*A*e) - 3*b*c*d
*(B*d + 2*A*e))*Sqrt[b*x + c*x^2])/(3*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)) - (
2*e*(6*A*c^3*d^3 - b^2*c*d*e*(7*B*d - 19*A*e) + 2*b^3*e^2*(B*d - 4*A*e) - 3*b*c^
2*d^2*(B*d + 3*A*e))*Sqrt[b*x + c*x^2])/(3*b^2*d^3*(c*d - b*e)^3*Sqrt[d + e*x])
+ (2*Sqrt[c]*(6*A*c^3*d^3 - b^2*c*d*e*(7*B*d - 19*A*e) + 2*b^3*e^2*(B*d - 4*A*e)
- 3*b*c^2*d^2*(B*d + 3*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)^(3/2)*d^3*(c*d - b*e)^
3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[c]*(6*A*c^2*d^2 - b^2*e*(B*d -
4*A*e) - 3*b*c*d*(B*d + 2*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Elli
pticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*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)**(5/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 7.46072, size = 506, normalized size = 0.89 \[ \frac{2 \left (b \left (b^2 d e^2 x (b+c x) (B d-A e) (c d-b e)+b^2 e^2 x (b+c x) (d+e x) (5 A e (b e-2 c d)+B d (7 c d-2 b e))+3 c^3 d^3 x (d+e x)^2 (b B-A c)-3 A (b+c x) (d+e x)^2 (c d-b e)^3\right )+c \sqrt{\frac{b}{c}} (d+e x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (2 b^2 e (4 A e-B d)+3 b c d (2 B d-5 A e)+3 A c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (2 b^3 e^2 (B d-4 A e)+b^2 c d e (19 A e-7 B d)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (2 b^3 e^2 (B d-4 A e)+b^2 c d e (19 A e-7 B d)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )\right )\right )}{3 b^3 d^3 \sqrt{x (b+c x)} (d+e x)^{3/2} (c d-b e)^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*(b^2*d*e^2*(B*d - A*e)*(c*d - b*e)*x*(b + c*x) + b^2*e^2*(B*d*(7*c*d - 2*b
*e) + 5*A*e*(-2*c*d + b*e))*x*(b + c*x)*(d + e*x) + 3*c^3*(b*B - A*c)*d^3*x*(d +
e*x)^2 - 3*A*(c*d - b*e)^3*(b + c*x)*(d + e*x)^2) + Sqrt[b/c]*c*(d + e*x)*(Sqrt
[b/c]*(6*A*c^3*d^3 + 2*b^3*e^2*(B*d - 4*A*e) - 3*b*c^2*d^2*(B*d + 3*A*e) + b^2*c
*d*e*(-7*B*d + 19*A*e))*(b + c*x)*(d + e*x) + I*b*e*(6*A*c^3*d^3 + 2*b^3*e^2*(B*
d - 4*A*e) - 3*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(-7*B*d + 19*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)*(3*A*c^2*d^2 + 3*b*c*d*(2*B*d - 5*A*e) + 2*b^2*e*(-(B
*d) + 4*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sq
rt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^3*d^3*(c*d - b*e)^3*Sqrt[x*(b + c*x)]*(d
+ e*x)^(3/2))

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Maple [B]  time = 0.074, size = 3024, normalized size = 5.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(x*(c*x+b))^(1/2)/x*(-12*A*x^2*c^5*d^4*e+8*A*x^3*b^3*c^2*e^5-6*A*x^3*c^5*d^
3*e^2+8*A*x^2*b^4*c*e^5+6*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*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^4*c*d*e^4-10*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^2*d^2*e^3-4*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^2*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)*EllipticE(((c*x
+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^4*e-2*B*x^3*b^3*c^2*d*e^4-3*A*b*c^
4*d^5+7*B*x^3*b^2*c^3*d^2*e^3+3*B*x^3*b*c^4*d^3*e^2-2*B*x^2*b^4*c*d*e^4+6*B*x^2*
b*c^4*d^4*e+9*A*x^3*b*c^4*d^2*e^3+15*A*x^2*b*c^4*d^3*e^2-7*A*x^2*b^3*c^2*d*e^4-2
0*A*x^2*b^2*c^3*d^2*e^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^4*c*d^2*e^3-2*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^2*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^2*c^3*d^4*e+12*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^3*d^3*e^2-6
*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*c^4*d^4*e+9*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*d^2*e^3-27*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*d*e^4+28*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^2*d^2*e^3-15*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^2*c^3*d^3*e^2+3*B*x*b*c^4*d^5+6*A*((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^4*d^5-6*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elli
pticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^5-2*B*((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*d^2*e^3-3*B*((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^3*d^5+3*B*(
(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^3*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^5*e^5-2*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ell
ipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^4*e-2*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*d*e^4-27*A*((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*d^2*e
^3+28*A*((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^2*d^3*e^2-15*A*((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^3*d^4*e+4*A*((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*d^2*e^3-
10*A*((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^2*d^3*e^2+12*A*((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^3*d^4*e+9*B*((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*d^3*e^2-4*B
*((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^2*d^4*e-B*((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*d^3*e^2+3*A*b^4*c*d^2*e^3-9*A*b^3*c^2*d^3*e^2+9*A*b^2*c^3*d^4*e+4*B*x^2
*b^3*c^2*d^2*e^3+9*A*x*b^2*c^3*d^3*e^2+3*A*x*b*c^4*d^4*e+12*A*x*b^4*c*d*e^4-26*A
*x*b^3*c^2*d^2*e^3-3*B*x*b^4*c*d^2*e^3-19*A*x^3*b^2*c^3*d*e^4+8*B*x^2*b^2*c^3*d^
3*e^2+8*B*x*b^3*c^2*d^3*e^2-6*A*x*c^5*d^5+8*A*((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*d*e^4)/b^2/d^3/c/(b*e-c*d)^3/(e*x+d)^(3/2)/(c*x+b)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x + A)/((c*e^2*x^4 + b*d^2*x + (2*c*d*e + b*e^2)*x^3 + (c*d^2 + 2*b*
d*e)*x^2)*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)**(5/2)/(c*x**2+b*x)**(3/2),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)^(3/2)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError