3.16 \(\int \frac{A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{3/2}} \, dx\)
Optimal. Leaf size=350 \[ -\frac{A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 \sqrt{c+d x} (b c-a d)}-\frac{a^3 d^3 D-a^2 b C d^3+a b^2 B d^3+b^3 \left (-\left (5 A d^3-4 B c d^2-4 c^3 D+4 c^2 C d\right )\right )}{2 b^3 d \sqrt{c+d x} (b c-a d)^3}-\frac{\sqrt{c+d x} \left (-7 a^3 d D+3 a^2 b (4 c D+C d)-a b^2 (8 c C-B d)+b^3 (4 B c-5 A d)\right )}{4 b^2 (a+b x) (b c-a d)^3}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (-3 a^3 d^2 D-a^2 b d (C d-12 c D)+a b^2 \left (-3 B d^2-24 c^2 D+8 c C d\right )+b^3 \left (15 A d^2-12 B c d+8 c^2 C\right )\right )}{4 b^{5/2} (b c-a d)^{7/2}} \]
[Out]
-(a*b^2*B*d^3 - a^2*b*C*d^3 + a^3*d^3*D - b^3*(4*c^2*C*d - 4*B*c*d^2 + 5*A*d^3 -
4*c^3*D))/(2*b^3*d*(b*c - a*d)^3*Sqrt[c + d*x]) - (A*b^3 - a*(b^2*B - a*b*C + a
^2*D))/(2*b^3*(b*c - a*d)*(a + b*x)^2*Sqrt[c + d*x]) - ((b^3*(4*B*c - 5*A*d) - a
*b^2*(8*c*C - B*d) - 7*a^3*d*D + 3*a^2*b*(C*d + 4*c*D))*Sqrt[c + d*x])/(4*b^2*(b
*c - a*d)^3*(a + b*x)) - ((b^3*(8*c^2*C - 12*B*c*d + 15*A*d^2) - 3*a^3*d^2*D - a
^2*b*d*(C*d - 12*c*D) + a*b^2*(8*c*C*d - 3*B*d^2 - 24*c^2*D))*ArcTanh[(Sqrt[b]*S
qrt[c + d*x])/Sqrt[b*c - a*d]])/(4*b^(5/2)*(b*c - a*d)^(7/2))
_______________________________________________________________________________________
Rubi [A] time = 1.94552, antiderivative size = 350, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156
\[ -\frac{A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 \sqrt{c+d x} (b c-a d)}-\frac{a^3 d^3 D-a^2 b C d^3+a b^2 B d^3+b^3 \left (-\left (5 A d^3-4 B c d^2-4 c^3 D+4 c^2 C d\right )\right )}{2 b^3 d \sqrt{c+d x} (b c-a d)^3}-\frac{\sqrt{c+d x} \left (-7 a^3 d D+3 a^2 b (4 c D+C d)-a b^2 (8 c C-B d)+b^3 (4 B c-5 A d)\right )}{4 b^2 (a+b x) (b c-a d)^3}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (-3 a^3 d^2 D-a^2 b d (C d-12 c D)+a b^2 \left (-3 B d^2-24 c^2 D+8 c C d\right )+b^3 \left (15 A d^2-12 B c d+8 c^2 C\right )\right )}{4 b^{5/2} (b c-a d)^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^3*(c + d*x)^(3/2)),x]
[Out]
-(a*b^2*B*d^3 - a^2*b*C*d^3 + a^3*d^3*D - b^3*(4*c^2*C*d - 4*B*c*d^2 + 5*A*d^3 -
4*c^3*D))/(2*b^3*d*(b*c - a*d)^3*Sqrt[c + d*x]) - (A*b^3 - a*(b^2*B - a*b*C + a
^2*D))/(2*b^3*(b*c - a*d)*(a + b*x)^2*Sqrt[c + d*x]) - ((b^3*(4*B*c - 5*A*d) - a
*b^2*(8*c*C - B*d) - 7*a^3*d*D + 3*a^2*b*(C*d + 4*c*D))*Sqrt[c + d*x])/(4*b^2*(b
*c - a*d)^3*(a + b*x)) - ((b^3*(8*c^2*C - 12*B*c*d + 15*A*d^2) - 3*a^3*d^2*D - a
^2*b*d*(C*d - 12*c*D) + a*b^2*(8*c*C*d - 3*B*d^2 - 24*c^2*D))*ArcTanh[(Sqrt[b]*S
qrt[c + d*x])/Sqrt[b*c - a*d]])/(4*b^(5/2)*(b*c - a*d)^(7/2))
_______________________________________________________________________________________
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((D*x**3+C*x**2+B*x+A)/(b*x+a)**3/(d*x+c)**(3/2),x)
[Out]
Timed out
_______________________________________________________________________________________
Mathematica [A] time = 1.80251, size = 297, normalized size = 0.85 \[ \frac{1}{4} \left (\sqrt{c+d x} \left (\frac{2 \left (a \left (a^2 D-a b C+b^2 B\right )-A b^3\right )}{b^2 (a+b x)^2 (b c-a d)^2}+\frac{5 a^3 d D-a^2 b (12 c D+C d)+a b^2 (8 c C-3 B d)+b^3 (7 A d-4 B c)}{b^2 (a+b x) (b c-a d)^3}+\frac{8 \left (-A d^3+B c d^2+c^3 D-c^2 C d\right )}{d (c+d x) (a d-b c)^3}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (-3 a^3 d^2 D+a^2 b d (12 c D-C d)+a b^2 \left (-3 B d^2-24 c^2 D+8 c C d\right )+b^3 \left (15 A d^2-12 B c d+8 c^2 C\right )\right )}{b^{5/2} (b c-a d)^{7/2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^3*(c + d*x)^(3/2)),x]
[Out]
(Sqrt[c + d*x]*((2*(-(A*b^3) + a*(b^2*B - a*b*C + a^2*D)))/(b^2*(b*c - a*d)^2*(a
+ b*x)^2) + (b^3*(-4*B*c + 7*A*d) + a*b^2*(8*c*C - 3*B*d) + 5*a^3*d*D - a^2*b*(
C*d + 12*c*D))/(b^2*(b*c - a*d)^3*(a + b*x)) + (8*(-(c^2*C*d) + B*c*d^2 - A*d^3
+ c^3*D))/(d*(-(b*c) + a*d)^3*(c + d*x))) - ((b^3*(8*c^2*C - 12*B*c*d + 15*A*d^2
) - 3*a^3*d^2*D + a^2*b*d*(-(C*d) + 12*c*D) + a*b^2*(8*c*C*d - 3*B*d^2 - 24*c^2*
D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(5/2)*(b*c - a*d)^(7/2)
))/4
_______________________________________________________________________________________
Maple [B] time = 0.04, size = 1225, normalized size = 3.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((D*x^3+C*x^2+B*x+A)/(b*x+a)^3/(d*x+c)^(3/2),x)
[Out]
15/4*d^2/(a*d-b*c)^3/(b*d*x+a*d)^2/b*(d*x+c)^(1/2)*D*a^3*c-3*d/(a*d-b*c)^3/b/((a
*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*D*a^2*c-2*d/(a*d-b*
c)^3/(b*d*x+a*d)^2*b*(d*x+c)^(3/2)*C*a*c-1/4*d^2/(a*d-b*c)^3/(b*d*x+a*d)^2*b*(d*
x+c)^(1/2)*B*a*c+2*d/(a*d-b*c)^3/(b*d*x+a*d)^2*b*(d*x+c)^(1/2)*C*a*c^2+1/4*d^2/(
a*d-b*c)^3/b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*a^2
*C+3/4*d^2/(a*d-b*c)^3/b^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)
*b)^(1/2))*a^3*D-2*d/(a*d-b*c)^3/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*
d-b*c)*b)^(1/2))*C*a*c+3/4*d^2/(a*d-b*c)^3/(b*d*x+a*d)^2*b*(d*x+c)^(3/2)*B*a+d/(
a*d-b*c)^3/(b*d*x+a*d)^2*b^2*(d*x+c)^(3/2)*B*c-5/4*d^2/(a*d-b*c)^3/(b*d*x+a*d)^2
/b*(d*x+c)^(3/2)*a^3*D+3*d/(a*d-b*c)^3/(b*d*x+a*d)^2*(d*x+c)^(3/2)*D*a^2*c-9/4*d
^3/(a*d-b*c)^3/(b*d*x+a*d)^2*b*(d*x+c)^(1/2)*A*a+9/4*d^2/(a*d-b*c)^3/(b*d*x+a*d)
^2*b^2*(d*x+c)^(1/2)*A*c-d/(a*d-b*c)^3/(b*d*x+a*d)^2*b^2*(d*x+c)^(1/2)*B*c^2-1/4
*d^3/(a*d-b*c)^3/(b*d*x+a*d)^2/b*(d*x+c)^(1/2)*C*a^3-7/4*d^2/(a*d-b*c)^3/(b*d*x+
a*d)^2*(d*x+c)^(1/2)*C*a^2*c-3/4*d^3/(a*d-b*c)^3/(b*d*x+a*d)^2/b^2*(d*x+c)^(1/2)
*D*a^4-3*d/(a*d-b*c)^3/(b*d*x+a*d)^2*(d*x+c)^(1/2)*D*a^2*c^2+3*d/(a*d-b*c)^3*b/(
(a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*B*c-2*d^2/(a*d-b*
c)^3/(d*x+c)^(1/2)*A-2/(a*d-b*c)^3/(d*x+c)^(1/2)*C*c^2+6/(a*d-b*c)^3/((a*d-b*c)*
b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*D*a*c^2-2/(a*d-b*c)^3*b/((a
*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*C*c^2+3/4*d^2/(a*d-
b*c)^3/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*B*a-7/4*d
^2/(a*d-b*c)^3/(b*d*x+a*d)^2*b^2*(d*x+c)^(3/2)*A+1/4*d^2/(a*d-b*c)^3/(b*d*x+a*d)
^2*(d*x+c)^(3/2)*C*a^2+5/4*d^3/(a*d-b*c)^3/(b*d*x+a*d)^2*(d*x+c)^(1/2)*B*a^2-15/
4*d^2/(a*d-b*c)^3*b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/
2))*A+2*d/(a*d-b*c)^3/(d*x+c)^(1/2)*B*c+2/d/(a*d-b*c)^3/(d*x+c)^(1/2)*D*c^3
_______________________________________________________________________________________
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((D*x^3 + C*x^2 + B*x + A)/((b*x + a)^3*(d*x + c)^(3/2)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.248481, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)/((b*x + a)^3*(d*x + c)^(3/2)),x, algorithm="fricas")
[Out]
[1/8*((8*(3*D*a^3*b^2 - C*a^2*b^3)*c^2*d - 4*(3*D*a^4*b + 2*C*a^3*b^2 - 3*B*a^2*
b^3)*c*d^2 + (3*D*a^5 + C*a^4*b + 3*B*a^3*b^2 - 15*A*a^2*b^3)*d^3 + (8*(3*D*a*b^
4 - C*b^5)*c^2*d - 4*(3*D*a^2*b^3 + 2*C*a*b^4 - 3*B*b^5)*c*d^2 + (3*D*a^3*b^2 +
C*a^2*b^3 + 3*B*a*b^4 - 15*A*b^5)*d^3)*x^2 + 2*(8*(3*D*a^2*b^3 - C*a*b^4)*c^2*d
- 4*(3*D*a^3*b^2 + 2*C*a^2*b^3 - 3*B*a*b^4)*c*d^2 + (3*D*a^4*b + C*a^3*b^2 + 3*B
*a^2*b^3 - 15*A*a*b^4)*d^3)*x)*sqrt(d*x + c)*log((sqrt(b^2*c - a*b*d)*(b*d*x + 2
*b*c - a*d) + 2*(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) - 2*(8*D*a^2*b^2*c^3 -
8*A*a^2*b^2*d^3 + 2*(5*D*a^3*b - 7*C*a^2*b^2 + B*a*b^3 + A*b^4)*c^2*d - (3*D*a^
4 + C*a^3*b - 13*B*a^2*b^2 + 9*A*a*b^3)*c*d^2 + (8*D*b^4*c^3 - 8*C*b^4*c^2*d + 4
*(3*D*a^2*b^2 - 2*C*a*b^3 + 3*B*b^4)*c*d^2 - (5*D*a^3*b - C*a^2*b^2 - 3*B*a*b^3
+ 15*A*b^4)*d^3)*x^2 + (16*D*a*b^3*c^3 + 4*(3*D*a^2*b^2 - 6*C*a*b^3 + B*b^4)*c^2
*d + (5*D*a^3*b - 5*C*a^2*b^2 + 21*B*a*b^3 - 5*A*b^4)*c*d^2 - (3*D*a^4 + C*a^3*b
- 5*B*a^2*b^2 + 25*A*a*b^3)*d^3)*x)*sqrt(b^2*c - a*b*d))/((a^2*b^5*c^3*d - 3*a^
3*b^4*c^2*d^2 + 3*a^4*b^3*c*d^3 - a^5*b^2*d^4 + (b^7*c^3*d - 3*a*b^6*c^2*d^2 + 3
*a^2*b^5*c*d^3 - a^3*b^4*d^4)*x^2 + 2*(a*b^6*c^3*d - 3*a^2*b^5*c^2*d^2 + 3*a^3*b
^4*c*d^3 - a^4*b^3*d^4)*x)*sqrt(b^2*c - a*b*d)*sqrt(d*x + c)), 1/4*((8*(3*D*a^3*
b^2 - C*a^2*b^3)*c^2*d - 4*(3*D*a^4*b + 2*C*a^3*b^2 - 3*B*a^2*b^3)*c*d^2 + (3*D*
a^5 + C*a^4*b + 3*B*a^3*b^2 - 15*A*a^2*b^3)*d^3 + (8*(3*D*a*b^4 - C*b^5)*c^2*d -
4*(3*D*a^2*b^3 + 2*C*a*b^4 - 3*B*b^5)*c*d^2 + (3*D*a^3*b^2 + C*a^2*b^3 + 3*B*a*
b^4 - 15*A*b^5)*d^3)*x^2 + 2*(8*(3*D*a^2*b^3 - C*a*b^4)*c^2*d - 4*(3*D*a^3*b^2 +
2*C*a^2*b^3 - 3*B*a*b^4)*c*d^2 + (3*D*a^4*b + C*a^3*b^2 + 3*B*a^2*b^3 - 15*A*a*
b^4)*d^3)*x)*sqrt(d*x + c)*arctan(-(b*c - a*d)/(sqrt(-b^2*c + a*b*d)*sqrt(d*x +
c))) - (8*D*a^2*b^2*c^3 - 8*A*a^2*b^2*d^3 + 2*(5*D*a^3*b - 7*C*a^2*b^2 + B*a*b^3
+ A*b^4)*c^2*d - (3*D*a^4 + C*a^3*b - 13*B*a^2*b^2 + 9*A*a*b^3)*c*d^2 + (8*D*b^
4*c^3 - 8*C*b^4*c^2*d + 4*(3*D*a^2*b^2 - 2*C*a*b^3 + 3*B*b^4)*c*d^2 - (5*D*a^3*b
- C*a^2*b^2 - 3*B*a*b^3 + 15*A*b^4)*d^3)*x^2 + (16*D*a*b^3*c^3 + 4*(3*D*a^2*b^2
- 6*C*a*b^3 + B*b^4)*c^2*d + (5*D*a^3*b - 5*C*a^2*b^2 + 21*B*a*b^3 - 5*A*b^4)*c
*d^2 - (3*D*a^4 + C*a^3*b - 5*B*a^2*b^2 + 25*A*a*b^3)*d^3)*x)*sqrt(-b^2*c + a*b*
d))/((a^2*b^5*c^3*d - 3*a^3*b^4*c^2*d^2 + 3*a^4*b^3*c*d^3 - a^5*b^2*d^4 + (b^7*c
^3*d - 3*a*b^6*c^2*d^2 + 3*a^2*b^5*c*d^3 - a^3*b^4*d^4)*x^2 + 2*(a*b^6*c^3*d - 3
*a^2*b^5*c^2*d^2 + 3*a^3*b^4*c*d^3 - a^4*b^3*d^4)*x)*sqrt(-b^2*c + a*b*d)*sqrt(d
*x + c))]
_______________________________________________________________________________________
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((D*x**3+C*x**2+B*x+A)/(b*x+a)**3/(d*x+c)**(3/2),x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.230137, size = 833, normalized size = 2.38 \[ -\frac{{\left (24 \, D a b^{2} c^{2} - 8 \, C b^{3} c^{2} - 12 \, D a^{2} b c d - 8 \, C a b^{2} c d + 12 \, B b^{3} c d + 3 \, D a^{3} d^{2} + C a^{2} b d^{2} + 3 \, B a b^{2} d^{2} - 15 \, A b^{3} d^{2}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{2 \,{\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )}}{{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} \sqrt{d x + c}} - \frac{12 \,{\left (d x + c\right )}^{\frac{3}{2}} D a^{2} b^{2} c d - 8 \,{\left (d x + c\right )}^{\frac{3}{2}} C a b^{3} c d + 4 \,{\left (d x + c\right )}^{\frac{3}{2}} B b^{4} c d - 12 \, \sqrt{d x + c} D a^{2} b^{2} c^{2} d + 8 \, \sqrt{d x + c} C a b^{3} c^{2} d - 4 \, \sqrt{d x + c} B b^{4} c^{2} d - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} D a^{3} b d^{2} +{\left (d x + c\right )}^{\frac{3}{2}} C a^{2} b^{2} d^{2} + 3 \,{\left (d x + c\right )}^{\frac{3}{2}} B a b^{3} d^{2} - 7 \,{\left (d x + c\right )}^{\frac{3}{2}} A b^{4} d^{2} + 15 \, \sqrt{d x + c} D a^{3} b c d^{2} - 7 \, \sqrt{d x + c} C a^{2} b^{2} c d^{2} - \sqrt{d x + c} B a b^{3} c d^{2} + 9 \, \sqrt{d x + c} A b^{4} c d^{2} - 3 \, \sqrt{d x + c} D a^{4} d^{3} - \sqrt{d x + c} C a^{3} b d^{3} + 5 \, \sqrt{d x + c} B a^{2} b^{2} d^{3} - 9 \, \sqrt{d x + c} A a b^{3} d^{3}}{4 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)/((b*x + a)^3*(d*x + c)^(3/2)),x, algorithm="giac")
[Out]
-1/4*(24*D*a*b^2*c^2 - 8*C*b^3*c^2 - 12*D*a^2*b*c*d - 8*C*a*b^2*c*d + 12*B*b^3*c
*d + 3*D*a^3*d^2 + C*a^2*b*d^2 + 3*B*a*b^2*d^2 - 15*A*b^3*d^2)*arctan(sqrt(d*x +
c)*b/sqrt(-b^2*c + a*b*d))/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^
2*d^3)*sqrt(-b^2*c + a*b*d)) - 2*(D*c^3 - C*c^2*d + B*c*d^2 - A*d^3)/((b^3*c^3*d
- 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*sqrt(d*x + c)) - 1/4*(12*(d*x + c)
^(3/2)*D*a^2*b^2*c*d - 8*(d*x + c)^(3/2)*C*a*b^3*c*d + 4*(d*x + c)^(3/2)*B*b^4*c
*d - 12*sqrt(d*x + c)*D*a^2*b^2*c^2*d + 8*sqrt(d*x + c)*C*a*b^3*c^2*d - 4*sqrt(d
*x + c)*B*b^4*c^2*d - 5*(d*x + c)^(3/2)*D*a^3*b*d^2 + (d*x + c)^(3/2)*C*a^2*b^2*
d^2 + 3*(d*x + c)^(3/2)*B*a*b^3*d^2 - 7*(d*x + c)^(3/2)*A*b^4*d^2 + 15*sqrt(d*x
+ c)*D*a^3*b*c*d^2 - 7*sqrt(d*x + c)*C*a^2*b^2*c*d^2 - sqrt(d*x + c)*B*a*b^3*c*d
^2 + 9*sqrt(d*x + c)*A*b^4*c*d^2 - 3*sqrt(d*x + c)*D*a^4*d^3 - sqrt(d*x + c)*C*a
^3*b*d^3 + 5*sqrt(d*x + c)*B*a^2*b^2*d^3 - 9*sqrt(d*x + c)*A*a*b^3*d^3)/((b^5*c^
3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*((d*x + c)*b - b*c + a*d)^2)