3.977 \(\int \frac{A+B x}{x^3 \left (a+b x+c x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=381 \[ -\frac{5 \left (-4 a A c-4 a b B+7 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 a^{9/2}}-\frac{2 \left (-A \left (40 a^2 c^2-42 a b^2 c+7 b^4\right )-c x \left (32 a^2 B c-36 a A b c-4 a b^2 B+7 A b^3\right )+4 a b B \left (b^2-6 a c\right )\right )}{3 a^2 x^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2} \left (4 a B \left (128 a^2 c^2-100 a b^2 c+15 b^4\right )-A \left (1296 a^2 b c^2-760 a b^3 c+105 b^5\right )\right )}{12 a^4 x \left (b^2-4 a c\right )^2}+\frac{\sqrt{a+b x+c x^2} \left (4 a b B \left (5 b^2-28 a c\right )-A \left (240 a^2 c^2-216 a b^2 c+35 b^4\right )\right )}{6 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac{2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
(2*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x))/(3*a*(b^2 - 4*a*c)*x^2*(a + b*
x + c*x^2)^(3/2)) - (2*(4*a*b*B*(b^2 - 6*a*c) - A*(7*b^4 - 42*a*b^2*c + 40*a^2*c
^2) - c*(7*A*b^3 - 4*a*b^2*B - 36*a*A*b*c + 32*a^2*B*c)*x))/(3*a^2*(b^2 - 4*a*c)
^2*x^2*Sqrt[a + b*x + c*x^2]) + ((4*a*b*B*(5*b^2 - 28*a*c) - A*(35*b^4 - 216*a*b
^2*c + 240*a^2*c^2))*Sqrt[a + b*x + c*x^2])/(6*a^3*(b^2 - 4*a*c)^2*x^2) - ((4*a*
B*(15*b^4 - 100*a*b^2*c + 128*a^2*c^2) - A*(105*b^5 - 760*a*b^3*c + 1296*a^2*b*c
^2))*Sqrt[a + b*x + c*x^2])/(12*a^4*(b^2 - 4*a*c)^2*x) - (5*(7*A*b^2 - 4*a*b*B -
4*a*A*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(9/2))
_______________________________________________________________________________________
Rubi [A] time = 1.12517, antiderivative size = 381, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217
\[ -\frac{5 \left (-4 a A c-4 a b B+7 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 a^{9/2}}-\frac{2 \left (-A \left (40 a^2 c^2-42 a b^2 c+7 b^4\right )-c x \left (32 a^2 B c-36 a A b c-4 a b^2 B+7 A b^3\right )+4 a b B \left (b^2-6 a c\right )\right )}{3 a^2 x^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2} \left (4 a B \left (128 a^2 c^2-100 a b^2 c+15 b^4\right )-A \left (1296 a^2 b c^2-760 a b^3 c+105 b^5\right )\right )}{12 a^4 x \left (b^2-4 a c\right )^2}+\frac{\sqrt{a+b x+c x^2} \left (4 a b B \left (5 b^2-28 a c\right )-A \left (240 a^2 c^2-216 a b^2 c+35 b^4\right )\right )}{6 a^3 x^2 \left (b^2-4 a c\right )^2}-\frac{2 \left (-A \left (b^2-2 a c\right )-c x (A b-2 a B)+a b B\right )}{3 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^3*(a + b*x + c*x^2)^(5/2)),x]
[Out]
(-2*(a*b*B - A*(b^2 - 2*a*c) - (A*b - 2*a*B)*c*x))/(3*a*(b^2 - 4*a*c)*x^2*(a + b
*x + c*x^2)^(3/2)) - (2*(4*a*b*B*(b^2 - 6*a*c) - A*(7*b^4 - 42*a*b^2*c + 40*a^2*
c^2) - c*(7*A*b^3 - 4*a*b^2*B - 36*a*A*b*c + 32*a^2*B*c)*x))/(3*a^2*(b^2 - 4*a*c
)^2*x^2*Sqrt[a + b*x + c*x^2]) + ((4*a*b*B*(5*b^2 - 28*a*c) - A*(35*b^4 - 216*a*
b^2*c + 240*a^2*c^2))*Sqrt[a + b*x + c*x^2])/(6*a^3*(b^2 - 4*a*c)^2*x^2) - ((4*a
*B*(15*b^4 - 100*a*b^2*c + 128*a^2*c^2) - A*(105*b^5 - 760*a*b^3*c + 1296*a^2*b*
c^2))*Sqrt[a + b*x + c*x^2])/(12*a^4*(b^2 - 4*a*c)^2*x) - (5*(7*A*b^2 - 4*a*b*B
- 4*a*A*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(9/2))
_______________________________________________________________________________________
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)/x**3/(c*x**2+b*x+a)**(5/2),x)
[Out]
Timed out
_______________________________________________________________________________________
Mathematica [A] time = 2.84433, size = 343, normalized size = 0.9 \[ \frac{2 \sqrt{a} \sqrt{a+x (b+c x)} \left (\frac{8 a \left (A \left (2 a^2 c^2-4 a b^2 c-3 a b c^2 x+b^4+b^3 c x\right )+a B \left (3 a b c+2 a c^2 x-b^3-b^2 c x\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}-\frac{8 \left (a B \left (68 a^2 b c^2+40 a^2 c^3 x-43 a b^3 c-38 a b^2 c^2 x+6 b^5+6 b^4 c x\right )+A \left (48 a^3 c^3-144 a^2 b^2 c^2-96 a^2 b c^3 x+70 a b^4 c+62 a b^3 c^2 x-9 b^6-9 b^5 c x\right )\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{33 A b-12 a B}{x}-\frac{6 a A}{x^2}\right )+15 \log (x) \left (-4 a A c-4 a b B+7 A b^2\right )+15 \left (4 a A c+4 a b B-7 A b^2\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{24 a^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^3*(a + b*x + c*x^2)^(5/2)),x]
[Out]
(2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*((-6*a*A)/x^2 + (33*A*b - 12*a*B)/x + (8*a*(a*B
*(-b^3 + 3*a*b*c - b^2*c*x + 2*a*c^2*x) + A*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c
*x - 3*a*b*c^2*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) - (8*(a*B*(6*b^5 - 43*a*
b^3*c + 68*a^2*b*c^2 + 6*b^4*c*x - 38*a*b^2*c^2*x + 40*a^2*c^3*x) + A*(-9*b^6 +
70*a*b^4*c - 144*a^2*b^2*c^2 + 48*a^3*c^3 - 9*b^5*c*x + 62*a*b^3*c^2*x - 96*a^2*
b*c^3*x)))/((b^2 - 4*a*c)^2*(a + x*(b + c*x)))) + 15*(7*A*b^2 - 4*a*b*B - 4*a*A*
c)*Log[x] + 15*(-7*A*b^2 + 4*a*b*B + 4*a*A*c)*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a +
x*(b + c*x)]])/(24*a^(9/2))
_______________________________________________________________________________________
Maple [B] time = 0.02, size = 1051, normalized size = 2.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^3/(c*x^2+b*x+a)^(5/2),x)
[Out]
-70/3*A*b^3/a^3*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-35/4*A*b^3/a^4/(4*a*c-b^
2)/(c*x^2+b*x+a)^(1/2)*c*x+11*A*b/a^2*c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+88*A
*b/a^2*c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+5*A/a^3*c^2*b/(4*a*c-b^2)/(c*x^2+
b*x+a)^(1/2)*x-35/12*A*b^3/a^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*c*x+5/3*B*b^2/a^2
/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*c*x+44*A*b^2/a^2*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a
)^(1/2)+5/2*A/a^3*c*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-8/3*B/a*c/(4*a*c-b^2)/(c
*x^2+b*x+a)^(3/2)*b-128/3*B/a*c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-16/3*B/a*c
^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-35/3*A*b^4/a^3*c/(4*a*c-b^2)^2/(c*x^2+b*x+a
)^(1/2)-64/3*B/a*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b+40/3*B*b^2/a^2*c^2/(4*a
*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+5*B*b^2/a^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*c*x+
7/4*A*b/a^2/x/(c*x^2+b*x+a)^(3/2)+5/2*B*b^3/a^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-
B/a/x/(c*x^2+b*x+a)^(3/2)-5/6*B*b/a^2/(c*x^2+b*x+a)^(3/2)-5/2*B*b/a^3/(c*x^2+b*x
+a)^(1/2)+5/2*B*b/a^(7/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/2*A/a/
x^2/(c*x^2+b*x+a)^(3/2)+35/24*A*b^2/a^3/(c*x^2+b*x+a)^(3/2)+35/8*A*b^2/a^4/(c*x^
2+b*x+a)^(1/2)-35/8*A*b^2/a^(9/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-
5/6*A/a^2*c/(c*x^2+b*x+a)^(3/2)-5/2*A/a^3*c/(c*x^2+b*x+a)^(1/2)+5/2*A/a^(7/2)*c*
ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+20/3*B*b^3/a^2*c/(4*a*c-b^2)^2/(c*
x^2+b*x+a)^(1/2)+11/2*A*b^2/a^2*c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-35/8*A*b^4/a^4
/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+5/6*B*b^3/a^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-3
5/24*A*b^4/a^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)
_______________________________________________________________________________________
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 + a)^(5/2)*x^3),x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.79366, 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 + a)^(5/2)*x^3),x, algorithm="fricas")
[Out]
[-1/48*(4*(6*A*a^3*b^4 - 48*A*a^4*b^2*c + 96*A*a^5*c^2 + (16*(32*B*a^3 - 81*A*a^
2*b)*c^4 - 40*(10*B*a^2*b^2 - 19*A*a*b^3)*c^3 + 15*(4*B*a*b^4 - 7*A*b^5)*c^2)*x^
5 + 6*(80*A*a^3*c^4 + 8*(26*B*a^3*b - 63*A*a^2*b^2)*c^3 - 5*(28*B*a^2*b^3 - 53*A
*a*b^4)*c^2 + 5*(4*B*a*b^5 - 7*A*b^6)*c)*x^4 + 3*(20*B*a*b^6 - 35*A*b^7 + 64*(4*
B*a^4 - 7*A*a^3*b)*c^3 + 8*(8*B*a^3*b^2 - 29*A*a^2*b^3)*c^2 - 10*(12*B*a^2*b^4 -
23*A*a*b^5)*c)*x^3 + 4*(20*B*a^2*b^5 - 35*A*a*b^6 + 160*A*a^4*c^3 + 4*(64*B*a^4
*b - 147*A*a^3*b^2)*c^2 - (148*B*a^3*b^3 - 279*A*a^2*b^4)*c)*x^2 + 3*(4*B*a^3*b^
4 - 7*A*a^2*b^5 + 16*(4*B*a^5 - 7*A*a^4*b)*c^2 - 8*(4*B*a^4*b^2 - 7*A*a^3*b^3)*c
)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a) - 15*((64*A*a^3*c^5 + 16*(4*B*a^3*b - 9*A*a^2
*b^2)*c^4 - 4*(8*B*a^2*b^3 - 15*A*a*b^4)*c^3 + (4*B*a*b^5 - 7*A*b^6)*c^2)*x^6 +
2*(64*A*a^3*b*c^4 + 16*(4*B*a^3*b^2 - 9*A*a^2*b^3)*c^3 - 4*(8*B*a^2*b^4 - 15*A*a
*b^5)*c^2 + (4*B*a*b^6 - 7*A*b^7)*c)*x^5 + (4*B*a*b^7 - 7*A*b^8 - 24*A*a^2*b^4*c
^2 + 128*A*a^4*c^4 + 32*(4*B*a^4*b - 7*A*a^3*b^2)*c^3 - 2*(12*B*a^2*b^5 - 23*A*a
*b^6)*c)*x^4 + 2*(4*B*a^2*b^6 - 7*A*a*b^7 + 64*A*a^4*b*c^3 + 16*(4*B*a^4*b^2 - 9
*A*a^3*b^3)*c^2 - 4*(8*B*a^3*b^4 - 15*A*a^2*b^5)*c)*x^3 + (4*B*a^3*b^5 - 7*A*a^2
*b^6 + 64*A*a^5*c^3 + 16*(4*B*a^5*b - 9*A*a^4*b^2)*c^2 - 4*(8*B*a^4*b^3 - 15*A*a
^3*b^4)*c)*x^2)*log(-(4*(a*b*x + 2*a^2)*sqrt(c*x^2 + b*x + a) + (8*a*b*x + (b^2
+ 4*a*c)*x^2 + 8*a^2)*sqrt(a))/x^2))/(((a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4
)*x^6 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*x^5 + (a^4*b^6 - 6*a^5*b^4*
c + 32*a^7*c^3)*x^4 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*x^3 + (a^6*b^4 -
8*a^7*b^2*c + 16*a^8*c^2)*x^2)*sqrt(a)), -1/24*(2*(6*A*a^3*b^4 - 48*A*a^4*b^2*c
+ 96*A*a^5*c^2 + (16*(32*B*a^3 - 81*A*a^2*b)*c^4 - 40*(10*B*a^2*b^2 - 19*A*a*b^3
)*c^3 + 15*(4*B*a*b^4 - 7*A*b^5)*c^2)*x^5 + 6*(80*A*a^3*c^4 + 8*(26*B*a^3*b - 63
*A*a^2*b^2)*c^3 - 5*(28*B*a^2*b^3 - 53*A*a*b^4)*c^2 + 5*(4*B*a*b^5 - 7*A*b^6)*c)
*x^4 + 3*(20*B*a*b^6 - 35*A*b^7 + 64*(4*B*a^4 - 7*A*a^3*b)*c^3 + 8*(8*B*a^3*b^2
- 29*A*a^2*b^3)*c^2 - 10*(12*B*a^2*b^4 - 23*A*a*b^5)*c)*x^3 + 4*(20*B*a^2*b^5 -
35*A*a*b^6 + 160*A*a^4*c^3 + 4*(64*B*a^4*b - 147*A*a^3*b^2)*c^2 - (148*B*a^3*b^3
- 279*A*a^2*b^4)*c)*x^2 + 3*(4*B*a^3*b^4 - 7*A*a^2*b^5 + 16*(4*B*a^5 - 7*A*a^4*
b)*c^2 - 8*(4*B*a^4*b^2 - 7*A*a^3*b^3)*c)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a) - 15
*((64*A*a^3*c^5 + 16*(4*B*a^3*b - 9*A*a^2*b^2)*c^4 - 4*(8*B*a^2*b^3 - 15*A*a*b^4
)*c^3 + (4*B*a*b^5 - 7*A*b^6)*c^2)*x^6 + 2*(64*A*a^3*b*c^4 + 16*(4*B*a^3*b^2 - 9
*A*a^2*b^3)*c^3 - 4*(8*B*a^2*b^4 - 15*A*a*b^5)*c^2 + (4*B*a*b^6 - 7*A*b^7)*c)*x^
5 + (4*B*a*b^7 - 7*A*b^8 - 24*A*a^2*b^4*c^2 + 128*A*a^4*c^4 + 32*(4*B*a^4*b - 7*
A*a^3*b^2)*c^3 - 2*(12*B*a^2*b^5 - 23*A*a*b^6)*c)*x^4 + 2*(4*B*a^2*b^6 - 7*A*a*b
^7 + 64*A*a^4*b*c^3 + 16*(4*B*a^4*b^2 - 9*A*a^3*b^3)*c^2 - 4*(8*B*a^3*b^4 - 15*A
*a^2*b^5)*c)*x^3 + (4*B*a^3*b^5 - 7*A*a^2*b^6 + 64*A*a^5*c^3 + 16*(4*B*a^5*b - 9
*A*a^4*b^2)*c^2 - 4*(8*B*a^4*b^3 - 15*A*a^3*b^4)*c)*x^2)*arctan(1/2*(b*x + 2*a)*
sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)))/(((a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4
)*x^6 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*x^5 + (a^4*b^6 - 6*a^5*b^4*
c + 32*a^7*c^3)*x^4 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*x^3 + (a^6*b^4 -
8*a^7*b^2*c + 16*a^8*c^2)*x^2)*sqrt(-a))]
_______________________________________________________________________________________
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)/x**3/(c*x**2+b*x+a)**(5/2),x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.292955, size = 1033, normalized size = 2.71 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x + a)^(5/2)*x^3),x, algorithm="giac")
[Out]
-1/3*((((6*B*a^12*b^4*c^2 - 9*A*a^11*b^5*c^2 - 38*B*a^13*b^2*c^3 + 62*A*a^12*b^3
*c^3 + 40*B*a^14*c^4 - 96*A*a^13*b*c^4)*x/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4) +
3*(4*B*a^12*b^5*c - 6*A*a^11*b^6*c - 27*B*a^13*b^3*c^2 + 44*A*a^12*b^4*c^2 + 36
*B*a^14*b*c^3 - 80*A*a^13*b^2*c^3 + 16*A*a^14*c^4)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a
^2*c^4))*x + 3*(2*B*a^12*b^6 - 3*A*a^11*b^7 - 12*B*a^13*b^4*c + 20*A*a^12*b^5*c
+ 8*B*a^14*b^2*c^2 - 25*A*a^13*b^3*c^2 + 16*B*a^15*c^3 - 20*A*a^14*b*c^3)/(b^4*c
^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + (7*B*a^13*b^5 - 10*A*a^12*b^6 - 50*B*a^14*b^
3*c + 78*A*a^13*b^4*c + 80*B*a^15*b*c^2 - 162*A*a^14*b^2*c^2 + 56*A*a^15*c^3)/(b
^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 + b*x + a)^(3/2) - 5/4*(4*B*a*b - 7*A
*b^2 + 4*A*a*c)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*
a^4) + 1/4*(4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*b - 11*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^3*A*b^2 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a*c + 8*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^2*sqrt(c) - 16*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^2*A*a*b*sqrt(c) - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^2*b + 13
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b^2 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*A*a^2*c - 8*B*a^3*sqrt(c) + 24*A*a^2*b*sqrt(c))/(((sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^2 - a)^2*a^4)