3.1211 \(\int \frac{A+B x}{(d+e x) \left (b x+c x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=287 \[ -\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}+\frac{2 \left (b (c d-b e) \left (3 b^2 e (B d-A e)-4 b c d (A e+B d)+8 A c^2 d^2\right )+c x \left (-3 b^3 e^2 (B d-A e)+2 b^2 c d e (A e+7 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2}-\frac{e^3 (B d-A e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}} \]
[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)*(b*x
+ c*x^2)^(3/2)) + (2*(b*(c*d - b*e)*(8*A*c^2*d^2 + 3*b^2*e*(B*d - A*e) - 4*b*c*
d*(B*d + A*e)) + c*(16*A*c^3*d^3 - 3*b^3*e^2*(B*d - A*e) + 2*b^2*c*d*e*(7*B*d +
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[b*x + c*x^2]
) - (e^3*(B*d - A*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*
Sqrt[b*x + c*x^2])])/(d^(5/2)*(c*d - b*e)^(5/2))
_______________________________________________________________________________________
Rubi [A] time = 0.882512, antiderivative size = 287, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154
\[ -\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}+\frac{2 \left (b (c d-b e) \left (3 b^2 e (B d-A e)-4 b c d (A e+B d)+8 A c^2 d^2\right )+c x \left (-3 b^3 e^2 (B d-A e)+2 b^2 c d e (A e+7 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2}-\frac{e^3 (B d-A e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((d + e*x)*(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)*(b*x
+ c*x^2)^(3/2)) + (2*(b*(c*d - b*e)*(8*A*c^2*d^2 + 3*b^2*e*(B*d - A*e) - 4*b*c*
d*(B*d + A*e)) + c*(16*A*c^3*d^3 - 3*b^3*e^2*(B*d - A*e) + 2*b^2*c*d*e*(7*B*d +
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[b*x + c*x^2]
) - (e^3*(B*d - A*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*
Sqrt[b*x + c*x^2])])/(d^(5/2)*(c*d - b*e)^(5/2))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 104.563, size = 291, normalized size = 1.01 \[ \frac{e^{3} \left (A e - B d\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{d^{\frac{5}{2}} \left (b e - c d\right )^{\frac{5}{2}}} - \frac{2 \left (A b \left (b e - c d\right ) + c x \left (A b e - 2 A c d + B b d\right )\right )}{3 b^{2} d \left (b e - c d\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \left (\frac{b \left (A \left (3 b^{2} e^{2} + 4 b c d e - 8 c^{2} d^{2}\right ) - B b d \left (3 b e - 4 c d\right )\right ) \left (b e - c d\right )}{2} + \frac{c x \left (4 b c d e \left (A b e - 2 A c d + B b d\right ) + \left (A \left (3 b^{2} e^{2} + 4 b c d e - 8 c^{2} d^{2}\right ) - B b d \left (3 b e - 4 c d\right )\right ) \left (b e - 2 c d\right )\right )}{2}\right )}{3 b^{4} d^{2} \left (b e - c d\right )^{2} \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(e*x+d)/(c*x**2+b*x)**(5/2),x)
[Out]
e**3*(A*e - B*d)*atan((-b*d + x*(b*e - 2*c*d))/(2*sqrt(d)*sqrt(b*e - c*d)*sqrt(b
*x + c*x**2)))/(d**(5/2)*(b*e - c*d)**(5/2)) - 2*(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*(b*(A*(3*b**
2*e**2 + 4*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*(
4*b*c*d*e*(A*b*e - 2*A*c*d + B*b*d) + (A*(3*b**2*e**2 + 4*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))
_______________________________________________________________________________________
Mathematica [A] time = 1.90115, size = 223, normalized size = 0.78 \[ \frac{x^{5/2} \left (\frac{2 (b+c x)^3 \left (\frac{c^2 x^2 \left (-b c (11 A e+5 B d)+8 A c^2 d+8 b^2 B e\right )}{(b+c x) (c d-b e)^2}+\frac{b c^2 x^2 (b B-A c)}{(b+c x)^2 (b e-c d)}+\frac{x (3 A b e+8 A c d-3 b B d)}{d^2}-\frac{A b}{d}\right )}{3 b^4 x^{3/2}}+\frac{2 e^3 (b+c x)^{5/2} (A e-B d) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{5/2} (b e-c d)^{5/2}}\right )}{(x (b+c x))^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((d + e*x)*(b*x + c*x^2)^(5/2)),x]
[Out]
(x^(5/2)*((2*(b + c*x)^3*(-((A*b)/d) + ((-3*b*B*d + 8*A*c*d + 3*A*b*e)*x)/d^2 +
(b*c^2*(b*B - A*c)*x^2)/((-(c*d) + b*e)*(b + c*x)^2) + (c^2*(8*A*c^2*d + 8*b^2*B
*e - b*c*(5*B*d + 11*A*e))*x^2)/((c*d - b*e)^2*(b + c*x))))/(3*b^4*x^(3/2)) + (2
*e^3*(-(B*d) + A*e)*(b + c*x)^(5/2)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]
*Sqrt[b + c*x])])/(d^(5/2)*(-(c*d) + b*e)^(5/2))))/(x*(b + c*x))^(5/2)
_______________________________________________________________________________________
Maple [B] time = 0.017, size = 2000, normalized size = 7. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)/(c*x^2+b*x)^(5/2),x)
[Out]
-2/3*B/e/b/(c*x^2+b*x)^(3/2)+2*e^3/d^2/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d
/e+x)-d*(b*e-c*d)/e^2)^(1/2)*A-2/3*e/d/(b*e-c*d)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e
+x)-d*(b*e-c*d)/e^2)^(3/2)*A-2*e^2/d/(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d
/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c*B-2/3*e/d/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2*c*d)/
e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c*x*A-4/3/e/(b*e-c*d)/b^2/(c*(d/e+x)^2+(b*e-2*c
*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^2*x*B*d+16/3*e/d/(b*e-c*d)*c^2/b^3/(c*(d/
e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*A+32/3/e/(b*e-c*d)*c^3/b^4
/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*B*d+2*e^3/d^2/(b*e-
c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c*A-2/3/e/(
b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c*B*d+4*e/(
b*e-c*d)^2/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*B
+8/3*e/d/(b*e-c*d)*c/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/
2)*A+16/3/e/(b*e-c*d)*c^2/b^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2
)^(1/2)*B*d-2*e^2/d/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e
^2)^(1/2)*B+16/3*B/e*c/b^3/(c*x^2+b*x)^(1/2)-8/3/(b*e-c*d)*c/b^2/(c*(d/e+x)^2+(b
*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*B+2/3/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2
*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c*A-16/3/(b*e-c*d)*c^2/b^3/(c*(d/e+x)^2+(
b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*A+2/3/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-
2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c*x*B-16/3/(b*e-c*d)*c^2/b^3/(c*(d/e+x)^
2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*B+e^2/d/(b*e-c*d)^2/(-d*(b*e-c*
d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^
(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*B+2*e/
(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c*B+4/3/
(b*e-c*d)/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^2*x*A-
e^3/d^2/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/
e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c
*d)/e^2)^(1/2))/(d/e+x))*A-32/3/(b*e-c*d)*c^3/b^4/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/
e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*A-4/3*B/e/b^2/(c*x^2+b*x)^(3/2)*c*x+32/3*B/e*c^2/b
^4/(c*x^2+b*x)^(1/2)*x-4*e^2/d/(b*e-c*d)^2/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x
)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*A+2/3/(b*e-c*d)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x
)-d*(b*e-c*d)/e^2)^(3/2)*B-2*e^2/d/(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e
+x)-d*(b*e-c*d)/e^2)^(1/2)*c*A
_______________________________________________________________________________________
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((B*x + A)/((c*x^2 + b*x)^(5/2)*(e*x + d)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.298213, size = 1, normalized size = 0. \[ \text{result too large to display} \]
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)),x, algorithm="fricas")
[Out]
[-1/3*(3*((B*b^4*c*d*e^3 - A*b^4*c*e^4)*x^2 + (B*b^5*d*e^3 - A*b^5*e^4)*x)*sqrt(
c*x^2 + b*x)*log((2*(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) + sqrt(c*d^2 - b*d*e)*(b*d
+ (2*c*d - b*e)*x))/(e*x + d)) + 2*(A*b^3*c^2*d^3 - 2*A*b^4*c*d^2*e + A*b^5*d*e
^2 - (3*A*b^3*c^2*e^3 - 8*(B*b*c^4 - 2*A*c^5)*d^3 + 2*(7*B*b^2*c^3 - 12*A*b*c^4)
*d^2*e - (3*B*b^3*c^2 - 2*A*b^2*c^3)*d*e^2)*x^3 - 3*(2*A*b^4*c*e^3 - 4*(B*b^2*c^
3 - 2*A*b*c^4)*d^3 + (7*B*b^3*c^2 - 12*A*b^2*c^3)*d^2*e - (2*B*b^4*c - A*b^3*c^2
)*d*e^2)*x^2 + 3*(B*b^5*d*e^2 - A*b^5*e^3 + (B*b^3*c^2 - 2*A*b^2*c^3)*d^3 - (2*B
*b^4*c - 3*A*b^3*c^2)*d^2*e)*x)*sqrt(c*d^2 - b*d*e))/(sqrt(c*d^2 - b*d*e)*((b^4*
c^3*d^4 - 2*b^5*c^2*d^3*e + b^6*c*d^2*e^2)*x^2 + (b^5*c^2*d^4 - 2*b^6*c*d^3*e +
b^7*d^2*e^2)*x)*sqrt(c*x^2 + b*x)), 2/3*(3*((B*b^4*c*d*e^3 - A*b^4*c*e^4)*x^2 +
(B*b^5*d*e^3 - A*b^5*e^4)*x)*sqrt(c*x^2 + b*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt
(c*x^2 + b*x)/((c*d - b*e)*x)) - (A*b^3*c^2*d^3 - 2*A*b^4*c*d^2*e + A*b^5*d*e^2
- (3*A*b^3*c^2*e^3 - 8*(B*b*c^4 - 2*A*c^5)*d^3 + 2*(7*B*b^2*c^3 - 12*A*b*c^4)*d^
2*e - (3*B*b^3*c^2 - 2*A*b^2*c^3)*d*e^2)*x^3 - 3*(2*A*b^4*c*e^3 - 4*(B*b^2*c^3 -
2*A*b*c^4)*d^3 + (7*B*b^3*c^2 - 12*A*b^2*c^3)*d^2*e - (2*B*b^4*c - A*b^3*c^2)*d
*e^2)*x^2 + 3*(B*b^5*d*e^2 - A*b^5*e^3 + (B*b^3*c^2 - 2*A*b^2*c^3)*d^3 - (2*B*b^
4*c - 3*A*b^3*c^2)*d^2*e)*x)*sqrt(-c*d^2 + b*d*e))/(sqrt(-c*d^2 + b*d*e)*((b^4*c
^3*d^4 - 2*b^5*c^2*d^3*e + b^6*c*d^2*e^2)*x^2 + (b^5*c^2*d^4 - 2*b^6*c*d^3*e + b
^7*d^2*e^2)*x)*sqrt(c*x^2 + b*x))]
_______________________________________________________________________________________
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)/(c*x**2+b*x)**(5/2),x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.318456, size = 865, normalized size = 3.01 \[ \frac{2 \,{\left (B d e^{3} - A e^{4}\right )} \arctan \left (\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e}}\right )}{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt{-c d^{2} + b d e}} - \frac{{\left (x{\left (\frac{{\left (8 \, B b c^{6} d^{10} - 16 \, A c^{7} d^{10} - 30 \, B b^{2} c^{5} d^{9} e + 56 \, A b c^{6} d^{9} e + 39 \, B b^{3} c^{4} d^{8} e^{2} - 66 \, A b^{2} c^{5} d^{8} e^{2} - 20 \, B b^{4} c^{3} d^{7} e^{3} + 25 \, A b^{3} c^{4} d^{7} e^{3} + 3 \, B b^{5} c^{2} d^{6} e^{4} + 4 \, A b^{4} c^{3} d^{6} e^{4} - 3 \, A b^{5} c^{2} d^{5} e^{5}\right )} x}{b^{4} c^{2}} + \frac{3 \,{\left (4 \, B b^{2} c^{5} d^{10} - 8 \, A b c^{6} d^{10} - 15 \, B b^{3} c^{4} d^{9} e + 28 \, A b^{2} c^{5} d^{9} e + 20 \, B b^{4} c^{3} d^{8} e^{2} - 33 \, A b^{3} c^{4} d^{8} e^{2} - 11 \, B b^{5} c^{2} d^{7} e^{3} + 12 \, A b^{4} c^{3} d^{7} e^{3} + 2 \, B b^{6} c d^{6} e^{4} + 3 \, A b^{5} c^{2} d^{6} e^{4} - 2 \, A b^{6} c d^{5} e^{5}\right )}}{b^{4} c^{2}}\right )} + \frac{3 \,{\left (B b^{3} c^{4} d^{10} - 2 \, A b^{2} c^{5} d^{10} - 4 \, B b^{4} c^{3} d^{9} e + 7 \, A b^{3} c^{4} d^{9} e + 6 \, B b^{5} c^{2} d^{8} e^{2} - 8 \, A b^{4} c^{3} d^{8} e^{2} - 4 \, B b^{6} c d^{7} e^{3} + 2 \, A b^{5} c^{2} d^{7} e^{3} + B b^{7} d^{6} e^{4} + 2 \, A b^{6} c d^{6} e^{4} - A b^{7} d^{5} e^{5}\right )}}{b^{4} c^{2}}\right )} x + \frac{A b^{3} c^{4} d^{10} - 4 \, A b^{4} c^{3} d^{9} e + 6 \, A b^{5} c^{2} d^{8} e^{2} - 4 \, A b^{6} c d^{7} e^{3} + A b^{7} d^{6} e^{4}}{b^{4} c^{2}}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}}} \]
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)),x, algorithm="giac")
[Out]
2*(B*d*e^3 - A*e^4)*arctan(((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(
-c*d^2 + b*d*e))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*sqrt(-c*d^2 + b*d*e)) -
1/3*((x*((8*B*b*c^6*d^10 - 16*A*c^7*d^10 - 30*B*b^2*c^5*d^9*e + 56*A*b*c^6*d^9*e
+ 39*B*b^3*c^4*d^8*e^2 - 66*A*b^2*c^5*d^8*e^2 - 20*B*b^4*c^3*d^7*e^3 + 25*A*b^3
*c^4*d^7*e^3 + 3*B*b^5*c^2*d^6*e^4 + 4*A*b^4*c^3*d^6*e^4 - 3*A*b^5*c^2*d^5*e^5)*
x/(b^4*c^2) + 3*(4*B*b^2*c^5*d^10 - 8*A*b*c^6*d^10 - 15*B*b^3*c^4*d^9*e + 28*A*b
^2*c^5*d^9*e + 20*B*b^4*c^3*d^8*e^2 - 33*A*b^3*c^4*d^8*e^2 - 11*B*b^5*c^2*d^7*e^
3 + 12*A*b^4*c^3*d^7*e^3 + 2*B*b^6*c*d^6*e^4 + 3*A*b^5*c^2*d^6*e^4 - 2*A*b^6*c*d
^5*e^5)/(b^4*c^2)) + 3*(B*b^3*c^4*d^10 - 2*A*b^2*c^5*d^10 - 4*B*b^4*c^3*d^9*e +
7*A*b^3*c^4*d^9*e + 6*B*b^5*c^2*d^8*e^2 - 8*A*b^4*c^3*d^8*e^2 - 4*B*b^6*c*d^7*e^
3 + 2*A*b^5*c^2*d^7*e^3 + B*b^7*d^6*e^4 + 2*A*b^6*c*d^6*e^4 - A*b^7*d^5*e^5)/(b^
4*c^2))*x + (A*b^3*c^4*d^10 - 4*A*b^4*c^3*d^9*e + 6*A*b^5*c^2*d^8*e^2 - 4*A*b^6*
c*d^7*e^3 + A*b^7*d^6*e^4)/(b^4*c^2))/(c*x^2 + b*x)^(3/2)