3.1282 \(\int \frac{A+B x}{\sqrt{d+e x} \left (b x+c x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=543 \[ \frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (b^2 e (9 B d-A e)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^3 \left (-e^2\right ) (3 B d-2 A e)+b^2 c d e (4 A e+13 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}+\frac{2 \sqrt{d+e x} \left (b (c d-b e) \left (b^2 e (3 B d-2 A e)-b c d (5 A e+4 B d)+8 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (3 B d-2 A e)+b^2 c d e (4 A e+13 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2} \]
[Out]
(-2*Sqrt[d + e*x]*(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)*(b*x + c*x^2)^(3/2)) + (2*Sqrt[d + e*x]*(b*(c*d - b*e)*(8*A*c^2*d^2 +
b^2*e*(3*B*d - 2*A*e) - b*c*d*(4*B*d + 5*A*e)) + c*(16*A*c^3*d^3 - b^3*e^2*(3*B*
d - 2*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(13*B*d + 4*A*e))*x))/(3*b^4*
d^2*(c*d - b*e)^2*Sqrt[b*x + c*x^2]) - (2*Sqrt[c]*(16*A*c^3*d^3 - b^3*e^2*(3*B*d
- 2*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(13*B*d + 4*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^2*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (
2*Sqrt[c]*(16*A*c^2*d^2 + b^2*e*(9*B*d - A*e) - 8*b*c*d*(B*d + 2*A*e))*Sqrt[x]*S
qrt[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*(c*d - b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])
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Rubi [A] time = 1.9712, antiderivative size = 543, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25
\[ \frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (b^2 e (9 B d-A e)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^3 \left (-e^2\right ) (3 B d-2 A e)+b^2 c d e (4 A e+13 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}+\frac{2 \sqrt{d+e x} \left (b (c d-b e) \left (b^2 e (3 B d-2 A e)-b c d (5 A e+4 B d)+8 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (3 B d-2 A e)+b^2 c d e (4 A e+13 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(Sqrt[d + e*x]*(b*x + c*x^2)^(5/2)),x]
[Out]
(-2*Sqrt[d + e*x]*(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)*(b*x + c*x^2)^(3/2)) + (2*Sqrt[d + e*x]*(b*(c*d - b*e)*(8*A*c^2*d^2 +
b^2*e*(3*B*d - 2*A*e) - b*c*d*(4*B*d + 5*A*e)) + c*(16*A*c^3*d^3 - b^3*e^2*(3*B*
d - 2*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(13*B*d + 4*A*e))*x))/(3*b^4*
d^2*(c*d - b*e)^2*Sqrt[b*x + c*x^2]) - (2*Sqrt[c]*(16*A*c^3*d^3 - b^3*e^2*(3*B*d
- 2*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(13*B*d + 4*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^2*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (
2*Sqrt[c]*(16*A*c^2*d^2 + b^2*e*(9*B*d - A*e) - 8*b*c*d*(B*d + 2*A*e))*Sqrt[x]*S
qrt[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*(c*d - b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])
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Rubi in Sympy [A] time = 144.785, size = 558, normalized size = 1.03 \[ \frac{2 \sqrt{c} \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (A b^{2} e^{2} + 16 A b c d e - 16 A c^{2} d^{2} - 9 B b^{2} d e + 8 B b c d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 d \left (- b\right )^{\frac{7}{2}} \sqrt{d + e x} \left (b e - c d\right ) \sqrt{b x + c x^{2}}} - \frac{2 \sqrt{c} \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (3 b c d e \left (A b e - 2 A c d + B b d\right ) + \left (A \left (2 b^{2} e^{2} + 5 b c d e - 8 c^{2} d^{2}\right ) - B b d \left (3 b e - 4 c d\right )\right ) \left (b e - 2 c d\right )\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 d^{2} \left (- b\right )^{\frac{7}{2}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right )^{2} \sqrt{b x + c x^{2}}} - \frac{2 \sqrt{d + e x} \left (A b \left (b e - c d\right ) + c x \left (A b e - 2 A c d + B b d\right )\right )}{3 b^{2} d \left (b e - c d\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \sqrt{d + e x} \left (\frac{b \left (A \left (2 b^{2} e^{2} + 5 b c d e - 8 c^{2} d^{2}\right ) - B b d \left (3 b e - 4 c d\right )\right ) \left (b e - c d\right )}{2} + \frac{c x \left (3 b c d e \left (A b e - 2 A c d + B b d\right ) + \left (A \left (2 b^{2} e^{2} + 5 b c d e - 8 c^{2} d^{2}\right ) - B b d \left (3 b e - 4 c d\right )\right ) \left (b e - 2 c d\right )\right )}{2}\right )}{3 b^{4} d^{2} \left (b e - c d\right )^{2} \sqrt{b x + c x^{2}}} \]
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)**(1/2),x)
[Out]
2*sqrt(c)*sqrt(x)*sqrt(1 + c*x/b)*sqrt(1 + e*x/d)*(A*b**2*e**2 + 16*A*b*c*d*e -
16*A*c**2*d**2 - 9*B*b**2*d*e + 8*B*b*c*d**2)*elliptic_f(asin(sqrt(c)*sqrt(x)/sq
rt(-b)), b*e/(c*d))/(3*d*(-b)**(7/2)*sqrt(d + e*x)*(b*e - c*d)*sqrt(b*x + c*x**2
)) - 2*sqrt(c)*sqrt(x)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(3*b*c*d*e*(A*b*e - 2*A*c*d
+ B*b*d) + (A*(2*b**2*e**2 + 5*b*c*d*e - 8*c**2*d**2) - B*b*d*(3*b*e - 4*c*d))*
(b*e - 2*c*d))*elliptic_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(3*d**2*(-b
)**(7/2)*sqrt(1 + e*x/d)*(b*e - c*d)**2*sqrt(b*x + c*x**2)) - 2*sqrt(d + e*x)*(A
*b*(b*e - c*d) + c*x*(A*b*e - 2*A*c*d + B*b*d))/(3*b**2*d*(b*e - c*d)*(b*x + c*x
**2)**(3/2)) + 4*sqrt(d + e*x)*(b*(A*(2*b**2*e**2 + 5*b*c*d*e - 8*c**2*d**2) - B
*b*d*(3*b*e - 4*c*d))*(b*e - c*d)/2 + c*x*(3*b*c*d*e*(A*b*e - 2*A*c*d + B*b*d) +
(A*(2*b**2*e**2 + 5*b*c*d*e - 8*c**2*d**2) - B*b*d*(3*b*e - 4*c*d))*(b*e - 2*c*
d))/2)/(3*b**4*d**2*(b*e - c*d)**2*sqrt(b*x + c*x**2))
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Mathematica [C] time = 8.46933, size = 514, normalized size = 0.95 \[ -\frac{2 \left (b (d+e x) \left (c^2 d^2 x^2 (b+c x) \left (5 b c (2 A e+B d)-8 A c^2 d-7 b^2 B e\right )+b c^2 d^2 x^2 (b B-A c) (c d-b e)+x (b+c x)^2 (c d-b e)^2 (-2 A b e-8 A c d+3 b B d)+A b d (b+c x)^2 (c d-b e)^2\right )+c x \sqrt{\frac{b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (b^2 e (3 B d-2 A e)-b c d (5 A e+4 B d)+8 A c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^3 e^2 (2 A e-3 B d)+b^2 c d e (4 A e+13 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^3 e^2 (2 A e-3 B d)+b^2 c d e (4 A e+13 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )\right )\right )}{3 b^5 d^2 (x (b+c x))^{3/2} \sqrt{d+e x} (c d-b e)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(Sqrt[d + e*x]*(b*x + c*x^2)^(5/2)),x]
[Out]
(-2*(b*(d + e*x)*(b*c^2*(b*B - A*c)*d^2*(c*d - b*e)*x^2 + c^2*d^2*(-8*A*c^2*d -
7*b^2*B*e + 5*b*c*(B*d + 2*A*e))*x^2*(b + c*x) + A*b*d*(c*d - b*e)^2*(b + c*x)^2
+ (c*d - b*e)^2*(3*b*B*d - 8*A*c*d - 2*A*b*e)*x*(b + c*x)^2) + Sqrt[b/c]*c*x*(b
+ c*x)*(Sqrt[b/c]*(16*A*c^3*d^3 + b^3*e^2*(-3*B*d + 2*A*e) - 8*b*c^2*d^2*(B*d +
3*A*e) + b^2*c*d*e*(13*B*d + 4*A*e))*(b + c*x)*(d + e*x) + I*b*e*(16*A*c^3*d^3
+ b^3*e^2*(-3*B*d + 2*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(13*B*d + 4*A
*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/S
qrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(8*A*c^2*d^2 + b^2*e*(3*B*d - 2*A*e) -
b*c*d*(4*B*d + 5*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)])))/(3*b^5*d^2*(c*d - b*e)^2*(x*(b + c*x
))^(3/2)*Sqrt[d + e*x])
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Maple [B] time = 0.073, size = 3140, normalized size = 5.8 \[ \text{output too large to display} \]
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)^(1/2),x)
[Out]
2/3*(-A*b^5*c*d^2*e^2+16*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*d^2*e^2-21*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^2*d^3*e+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)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*b^2*c^4*d^3*e-9*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^2*d^2*e
^2+17*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipt
icF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^3*d^3*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)*EllipticE(((c*x+b)/b)^(1/2),(b*
e/(b*e-c*d))^(1/2))*b^5*c*d*e^3-16*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^4*
d^4+16*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellip
ticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^2*d^2*e^2-21*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^3*d^3*e+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
*d*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)*Elli
pticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^2*d^2*e^2-32*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^3*d^3*e+2*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*
c*d*e^3-28*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^4*c^2*d^2*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)*EllipticE(((c*x+b)/b)^(1/2)
,(b*e/(b*e-c*d))^(1/2))*b^3*c^3*d^3*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)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5
*c*d^2*e^2+17*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^2*d^3*e+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^2*d*e^3+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)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*
b^3*c^3*d^2*e^2-32*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^4*d^3*e+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^2*d*e^3-28*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^3*d^2*e^2+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^4*d^4+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^4*d^4-33*A*x^3*b^2*c^4*d^2*e^2+17*B*
x^3*b^3*c^3*d^2*e^2+B*x^3*b^2*c^4*d^3*e+6*A*x^2*b^4*c^2*d*e^3-3*A*x^2*b^3*c^3*d^
2*e^2-3*B*x*b^5*c*d^2*e^2+6*B*x*b^4*c^2*d^3*e-24*A*x^4*b*c^5*d^2*e^2-3*B*x^4*b^3
*c^3*d*e^3+A*x*b^5*c*d*e^3+13*B*x^4*b^2*c^4*d^2*e^2-8*B*x^4*b*c^5*d^3*e-6*B*x^3*
b^4*c^2*d*e^3+4*A*x^4*b^2*c^4*d*e^3-3*B*x^2*b^5*c*d*e^3+4*A*x*b^4*c^2*d^2*e^2-31
*A*x^2*b^2*c^4*d^3*e+17*B*x^2*b^3*c^3*d^3*e-11*A*x*b^3*c^3*d^3*e+4*A*x^3*b^4*c^2
*e^4+16*A*x^4*c^6*d^3*e-8*B*x^3*b*c^5*d^4+2*A*x^2*b^5*c*e^4+24*A*x^2*b*c^5*d^4-3
*B*x*b^3*c^3*d^4+2*A*x^4*b^3*c^3*e^4-12*B*x^2*b^2*c^4*d^4+6*A*x*b^2*c^4*d^4-A*b^
3*c^3*d^4-8*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^4*d^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)*EllipticF(((c*x+b)/b)^(1/2),(
b*e/(b*e-c*d))^(1/2))*b^3*c^3*d^4-3*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^6*d*e
^3+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^3*d^4+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^5*d^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^5*c*e^4-16*
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^5*d^4+2*A*b^4*c^2*d^3*e+16*A*x^3*c^6*
d^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)*Elliptic
E(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^6*e^4+9*A*x^3*b^3*c^3*d*e^3)/x^2*(x
*(c*x+b))^(1/2)/d^2/b^4/c/(b*e-c*d)^2/(c*x+b)^2/(e*x+d)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}} \sqrt{e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*sqrt(e*x + d)),x, algorithm="maxima")
[Out]
integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*sqrt(e*x + d)), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x + A}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{e x + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*sqrt(e*x + d)),x, algorithm="fricas")
[Out]
integral((B*x + A)/((c^2*x^4 + 2*b*c*x^3 + b^2*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)/(c*x**2+b*x)**(5/2)/(e*x+d)**(1/2),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)*sqrt(e*x + d)),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError