3.1270 \(\int \frac{A+B x}{(d+e x)^{7/2} \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=510 \[ \frac{2 \sqrt{b x+c x^2} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right )}{15 d^3 \sqrt{d+e x} (c d-b e)^3}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^3 e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{2 \sqrt{b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{15 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (4 A e (2 c d-b e)-B d (b e+3 c d)) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^2 e \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}+\frac{2 \sqrt{b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)} \]

[Out]

(2*(B*d - A*e)*Sqrt[b*x + c*x^2])/(5*d*(c*d - b*e)*(d + e*x)^(5/2)) - (2*(4*A*e*
(2*c*d - b*e) - B*d*(3*c*d + b*e))*Sqrt[b*x + c*x^2])/(15*d^2*(c*d - b*e)^2*(d +
e*x)^(3/2)) + (2*(B*d*(3*c^2*d^2 + 7*b*c*d*e - 2*b^2*e^2) - A*e*(23*c^2*d^2 - 2
3*b*c*d*e + 8*b^2*e^2))*Sqrt[b*x + c*x^2])/(15*d^3*(c*d - b*e)^3*Sqrt[d + e*x])
- (2*Sqrt[-b]*Sqrt[c]*(B*d*(3*c^2*d^2 + 7*b*c*d*e - 2*b^2*e^2) - A*e*(23*c^2*d^2
- 23*b*c*d*e + 8*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[Ar
cSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*d^3*e*(c*d - b*e)^3*Sqrt[1 +
(e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*Sqrt[c]*(4*A*e*(2*c*d - b*e) - B*d*(3
*c*d + b*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)])/(15*d^2*e*(c*d - b*e)^2*Sqrt[d + e*x]*Sqrt[
b*x + c*x^2])

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Rubi [A]  time = 1.94816, antiderivative size = 510, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \sqrt{b x+c x^2} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right )}{15 d^3 \sqrt{d+e x} (c d-b e)^3}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^3 e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{2 \sqrt{b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{15 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (4 A e (2 c d-b e)-B d (b e+3 c d)) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^2 e \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}+\frac{2 \sqrt{b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((d + e*x)^(7/2)*Sqrt[b*x + c*x^2]),x]

[Out]

(2*(B*d - A*e)*Sqrt[b*x + c*x^2])/(5*d*(c*d - b*e)*(d + e*x)^(5/2)) - (2*(4*A*e*
(2*c*d - b*e) - B*d*(3*c*d + b*e))*Sqrt[b*x + c*x^2])/(15*d^2*(c*d - b*e)^2*(d +
e*x)^(3/2)) + (2*(B*d*(3*c^2*d^2 + 7*b*c*d*e - 2*b^2*e^2) - A*e*(23*c^2*d^2 - 2
3*b*c*d*e + 8*b^2*e^2))*Sqrt[b*x + c*x^2])/(15*d^3*(c*d - b*e)^3*Sqrt[d + e*x])
- (2*Sqrt[-b]*Sqrt[c]*(B*d*(3*c^2*d^2 + 7*b*c*d*e - 2*b^2*e^2) - A*e*(23*c^2*d^2
- 23*b*c*d*e + 8*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[Ar
cSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*d^3*e*(c*d - b*e)^3*Sqrt[1 +
(e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*Sqrt[c]*(4*A*e*(2*c*d - b*e) - B*d*(3
*c*d + b*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)])/(15*d^2*e*(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)**(7/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 5.21837, size = 506, normalized size = 0.99 \[ \frac{2 \left (b e x (b+c x) \left ((d+e x)^2 \left (A e \left (-8 b^2 e^2+23 b c d e-23 c^2 d^2\right )+B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )\right )+3 d^2 (B d-A e) (c d-b e)^2+d (d+e x) (c d-b e) (4 A e (b e-2 c d)+B d (b e+3 c d))\right )-c \sqrt{\frac{b}{c}} (d+e x)^2 \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )+B d \left (2 b^2 e^2-7 b c d e-3 c^2 d^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (A e \left (-8 b^2 e^2+23 b c d e-23 c^2 d^2\right )+B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )\right )-i e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (2 b^2 e (4 A e+B d)-b c d (19 A e+6 B d)+15 A c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )\right )\right )}{15 b d^3 e \sqrt{x (b+c x)} (d+e x)^{5/2} (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((d + e*x)^(7/2)*Sqrt[b*x + c*x^2]),x]

[Out]

(2*(b*e*x*(b + c*x)*(3*d^2*(B*d - A*e)*(c*d - b*e)^2 + d*(c*d - b*e)*(4*A*e*(-2*
c*d + b*e) + B*d*(3*c*d + b*e))*(d + e*x) + (A*e*(-23*c^2*d^2 + 23*b*c*d*e - 8*b
^2*e^2) + B*d*(3*c^2*d^2 + 7*b*c*d*e - 2*b^2*e^2))*(d + e*x)^2) - Sqrt[b/c]*c*(d
+ e*x)^2*(Sqrt[b/c]*(A*e*(-23*c^2*d^2 + 23*b*c*d*e - 8*b^2*e^2) + B*d*(3*c^2*d^
2 + 7*b*c*d*e - 2*b^2*e^2))*(b + c*x)*(d + e*x) - I*b*e*(B*d*(-3*c^2*d^2 - 7*b*c
*d*e + 2*b^2*e^2) + A*e*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2))*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*e*(c*d - b*e)*(15*A*c^2*d^2 + 2*b^2*e*(B*d + 4*A*e) - b*c*d*(6*B*d + 19*A*e)
)*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)])))/(15*b*d^3*e*(c*d - b*e)^3*Sqrt[x*(b + c*x)]*(d + e*x)^(5/2
))

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Maple [B]  time = 0.068, size = 3863, normalized size = 7.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x)^(1/2),x)

[Out]

2/15*(x*(c*x+b))^(1/2)*(2*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x
^2*b^4*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+16*A*
EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^4*d*e^5*((c*x+b)/b)^(1/2)
*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*B*EllipticE(((c*x+b)/b)^(1/2),(b*
e/(b*e-c*d))^(1/2))*x*b^4*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)
*(-c*x/b)^(1/2)+4*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^3
*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-12*A*Elliptic
F(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^4*e^2*((c*x+b)/b)^(1/2)*(-(
e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+8*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b
*e-c*d))^(1/2))*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x
/b)^(1/2)-31*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^3*e^3*
((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+46*A*EllipticE(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d
)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-23*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^
(1/2)+B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^4*e^2*((c*x+b
)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+2*B*EllipticF(((c*x+b)/b)
^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c
*d))^(1/2)*(-c*x/b)^(1/2)-9*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))
*b^3*c*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*B
*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^5*e*((c*x+b)/b)^(1
/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+15*A*x*b^3*c*d^2*e^4-23*A*x^4*b*
c^3*d*e^5-9*B*x^2*c^4*d^5*e+8*A*x^4*b^2*c^2*e^6+23*A*x^4*c^4*d^2*e^4-3*B*x^4*c^4
*d^3*e^3+8*A*x^3*b^3*c*e^6+54*A*x^3*c^4*d^3*e^3-9*B*x^3*c^4*d^4*e^2+34*A*x^2*c^4
*d^4*e^2+3*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b*c^3*d^4*e^
2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+8*A*EllipticF(((
c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+
d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-24*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-
c*d))^(1/2))*x*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-
c*x/b)^(1/2)+16*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^4
*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-62*A*Elliptic
E(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c*d^2*e^4*((c*x+b)/b)^(1/2)*(-(
e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+92*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(
b*e-c*d))^(1/2))*x*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)+4*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^
2*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-46*A*Ell
ipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^4*e^2*((c*x+b)/b)^(1/2
)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-6*B*EllipticF(((c*x+b)/b)^(1/2),(b
*e/(b*e-c*d))^(1/2))*x*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)+2*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c*
d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*A*Ellipt
icF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c*d*e^5*((c*x+b)/b)^(1/2)*(
-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-12*A*EllipticF(((c*x+b)/b)^(1/2),(b*e
/(b*e-c*d))^(1/2))*x^2*b^2*c^2*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^
(1/2)*(-c*x/b)^(1/2)+8*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*
b*c^3*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-31*A
*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c*d*e^5*((c*x+b)/b)^
(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+46*A*EllipticE(((c*x+b)/b)^(1/
2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^2*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e
-c*d))^(1/2)*(-c*x/b)^(1/2)-23*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/
2))*x^2*b*c^3*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1
/2)+B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c*d^2*e^4*((c*x
+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+2*B*EllipticF(((c*x+b)/
b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*
c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-9*B*x*b*c^3*d^5*e+2*B*x^4*b^2*c^2*d*e^5-7*B*x^
4*b*c^3*d^2*e^4-3*A*x^3*b^2*c^2*d*e^5-35*A*x^3*b*c^3*d^2*e^4+2*B*x^3*b^3*c*d*e^5
-2*B*x^3*b^2*c^2*d^2*e^4-15*B*x^3*b*c^3*d^3*e^3+20*A*x^2*b^3*c*d*e^5-43*A*x^2*b^
2*c^2*d^2*e^4+13*A*x^2*b*c^3*d^3*e^3-12*B*x^2*b^2*c^2*d^3*e^3-8*B*x^2*b*c^3*d^4*
e^2-41*A*x*b^2*c^2*d^3*e^3+34*A*x*b*c^3*d^4*e^2+B*x*b^2*c^2*d^4*e^2+5*B*x^2*b^3*
c*d^2*e^4-3*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b*c^3*d^4*e
^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-9*B*EllipticE((
(c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e
*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-18*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b
*e-c*d))^(1/2))*x*b^3*c*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(
-c*x/b)^(1/2)+8*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2*d
^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+6*B*Ellipti
cE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e
*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*
e-c*d))^(1/2))*x^2*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)+2*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*d^3*
e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+3*B*EllipticE(
((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c
/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+8*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))
^(1/2))*x^2*b^4*e^6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2
)+8*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*d^2*e^4*((c*x+b)/b)
^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-3*B*EllipticF(((c*x+b)/b)^(1/
2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/
2)*(-c*x/b)^(1/2))/(c*x+b)/x/(b*e-c*d)^3/(e*x+d)^(5/2)/c/d^3/e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^(7/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

integral((B*x + A)/((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^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)**(7/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^(7/2)), x)