\(\int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx\) [60]

Optimal result

Integrand size = 36, antiderivative size = 826 \[ \int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac {\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac {\left (4 a^4 C d^2 f^2+8 a^3 b d f (B d f-2 C (d e+c f))-b^4 \left (15 A d^2 e^2-2 c d e (9 B e-7 A f)+3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-44 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-29 B e f+22 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-34 c d e f+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}+\frac {\left (b^3 \left (5 A d^3 e^3-3 c d^2 e^2 (2 B e-A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^3 e^2 (B e-18 A f)-c^3 f^2 (4 C e-B f)-c d^2 e \left (4 C e^2-23 B e f+12 A f^2\right )-c^2 d f \left (40 C e^2-23 B e f+18 A f^2\right )\right )-2 a^3 d f \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d e f+c^2 f^2\right )\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{8 (b c-a d)^{7/2} (b e-a f)^{7/2}} \]

[Out]

1/8*(b^3*(5*A*d^3*e^3-3*c*d^2*e^2*(-A*f+2*B*e)+c^2*d*e*(3*A*f^2-4*B*e*f+8*C*e^2)+c^3*f*(5*A*f^2-6*B*e*f+8*C*e^
2))+a*b^2*(d^3*e^2*(-18*A*f+B*e)-c^3*f^2*(-B*f+4*C*e)-c*d^2*e*(12*A*f^2-23*B*e*f+4*C*e^2)-c^2*d*f*(18*A*f^2-23
*B*e*f+40*C*e^2))-2*a^3*d*f*(C*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2)+4*d*f*(2*A*d*f-B*(c*f+d*e)))+a^2*b*(C*(c^3*f^3+
23*c^2*d*e*f^2+23*c*d^2*e^2*f+d^3*e^3)+4*d*f*(6*A*d*f*(c*f+d*e)-B*(c^2*f^2+10*c*d*e*f+d^2*e^2))))*arctanh((-a*
f+b*e)^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2)/(f*x+e)^(1/2))/(-a*d+b*c)^(7/2)/(-a*f+b*e)^(7/2)-1/3*(A*b^2-a*(B*b
-C*a))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^3+1/12*(2*a^3*C*d*f+a*b^2*(-10*A*d*f+B*c*f+
B*d*e+12*C*c*e)-b^3*(6*B*c*e-5*A*(c*f+d*e))+a^2*b*(4*B*d*f-7*C*(c*f+d*e)))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/(-a*d
+b*c)^2/(-a*f+b*e)^2/(b*x+a)^2+1/24*(4*a^4*C*d^2*f^2+8*a^3*b*d*f*(B*d*f-2*C*(c*f+d*e))-b^4*(15*A*d^2*e^2-2*c*d
*e*(-7*A*f+9*B*e)+3*c^2*(5*A*f^2-6*B*e*f+8*C*e^2))-a*b^3*(d^2*e*(-44*A*f+3*B*e)-3*c^2*f*(-B*f+4*C*e)-2*c*d*(22
*A*f^2-29*B*e*f+6*C*e^2))-a^2*b^2*(C*(3*c^2*f^2-34*c*d*e*f+3*d^2*e^2)+2*d*f*(22*A*d*f-5*B*(c*f+d*e))))*(d*x+c)
^(1/2)*(f*x+e)^(1/2)/b/(-a*d+b*c)^3/(-a*f+b*e)^3/(b*x+a)

Rubi [A] (verified)

Time = 1.60 (sec) , antiderivative size = 826, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {1627, 156, 12, 95, 214} \[ \int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac {\left (-2 d f \left (C \left (3 d^2 e^2+2 c d f e+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) a^3+b \left (C \left (d^3 e^3+23 c d^2 f e^2+23 c^2 d f^2 e+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d f e+c^2 f^2\right )\right )\right ) a^2+b^2 \left (-f^2 (4 C e-B f) c^3-d f \left (40 C e^2-23 B f e+18 A f^2\right ) c^2-d^2 e \left (4 C e^2-23 B f e+12 A f^2\right ) c+d^3 e^2 (B e-18 A f)\right ) a+b^3 \left (f \left (8 C e^2-6 B f e+5 A f^2\right ) c^3+d e \left (8 C e^2-4 B f e+3 A f^2\right ) c^2-3 d^2 e^2 (2 B e-A f) c+5 A d^3 e^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{8 (b c-a d)^{7/2} (b e-a f)^{7/2}}+\frac {\left (4 C d^2 f^2 a^4+8 b d f (B d f-2 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-34 c d f e+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right ) a^2-b^3 \left (-3 f (4 C e-B f) c^2-2 d \left (6 C e^2-29 B f e+22 A f^2\right ) c+d^2 e (3 B e-44 A f)\right ) a-b^4 \left (3 \left (8 C e^2-6 B f e+5 A f^2\right ) c^2-2 d e (9 B e-7 A f) c+15 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}+\frac {\left (2 C d f a^3+b (4 B d f-7 C (d e+c f)) a^2+b^2 (12 c C e+B d e+B c f-10 A d f) a-b^3 (6 B c e-5 A (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2} \]

[In]

Int[(A + B*x + C*x^2)/((a + b*x)^4*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

-1/3*((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^3) + ((2*a^3*C
*d*f + a*b^2*(12*c*C*e + B*d*e + B*c*f - 10*A*d*f) - b^3*(6*B*c*e - 5*A*(d*e + c*f)) + a^2*b*(4*B*d*f - 7*C*(d
*e + c*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(12*b*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)^2) + ((4*a^4*C*d^2*f^2 +
8*a^3*b*d*f*(B*d*f - 2*C*(d*e + c*f)) - b^4*(15*A*d^2*e^2 - 2*c*d*e*(9*B*e - 7*A*f) + 3*c^2*(8*C*e^2 - 6*B*e*f
+ 5*A*f^2)) - a*b^3*(d^2*e*(3*B*e - 44*A*f) - 3*c^2*f*(4*C*e - B*f) - 2*c*d*(6*C*e^2 - 29*B*e*f + 22*A*f^2))
- a^2*b^2*(C*(3*d^2*e^2 - 34*c*d*e*f + 3*c^2*f^2) + 2*d*f*(22*A*d*f - 5*B*(d*e + c*f))))*Sqrt[c + d*x]*Sqrt[e
+ f*x])/(24*b*(b*c - a*d)^3*(b*e - a*f)^3*(a + b*x)) + ((b^3*(5*A*d^3*e^3 - 3*c*d^2*e^2*(2*B*e - A*f) + c^2*d*
e*(8*C*e^2 - 4*B*e*f + 3*A*f^2) + c^3*f*(8*C*e^2 - 6*B*e*f + 5*A*f^2)) + a*b^2*(d^3*e^2*(B*e - 18*A*f) - c^3*f
^2*(4*C*e - B*f) - c*d^2*e*(4*C*e^2 - 23*B*e*f + 12*A*f^2) - c^2*d*f*(40*C*e^2 - 23*B*e*f + 18*A*f^2)) - 2*a^3
*d*f*(C*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 4*d*f*(2*A*d*f - B*(d*e + c*f))) + a^2*b*(C*(d^3*e^3 + 23*c*d^2*
e^2*f + 23*c^2*d*e*f^2 + c^3*f^3) + 4*d*f*(6*A*d*f*(d*e + c*f) - B*(d^2*e^2 + 10*c*d*e*f + c^2*f^2))))*ArcTanh
[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(8*(b*c - a*d)^(7/2)*(b*e - a*f)^(7/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 1627

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[{
Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[b*R*(a + b*x)^(m + 1)*
(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e
- a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f
*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x], x]] /; FreeQ[{a, b,
c, d, e, f, n, p}, x] && PolyQ[Px, x] && ILtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}-\frac {\int \frac {-\frac {a^2 C (d e+c f)-a b (6 c C e+B d e+B c f-6 A d f)+b^2 (6 B c e-5 A (d e+c f))}{2 b}+\left (-3 b c C e+3 a C d e+3 a c C f+2 A b d f-2 a B d f-\frac {a^2 C d f}{b}\right ) x}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}} \, dx}{3 (b c-a d) (b e-a f)} \\ & = -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac {\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac {\int \frac {\frac {2 a^3 C d f (d e+c f)+b^3 \left (15 A d^2 e^2-2 c d e (9 B e-7 A f)+3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^2 e (3 B e-34 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-23 B e f+17 A f^2\right )\right )+a^2 b \left (C \left (3 d^2 e^2-10 c d e f+3 c^2 f^2\right )+8 d f (3 A d f-B (d e+c f))\right )}{4 b}+\frac {d f \left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) x}{2 b}}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}} \, dx}{6 (b c-a d)^2 (b e-a f)^2} \\ & = -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac {\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac {\left (4 a^4 C d^2 f^2+8 a^3 b d f (B d f-2 C (d e+c f))-b^4 \left (15 A d^2 e^2-2 c d e (9 B e-7 A f)+3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-44 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-29 B e f+22 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-34 c d e f+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}-\frac {\int \frac {3 \left (b^3 \left (5 A d^3 e^3-3 c d^2 e^2 (2 B e-A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^3 e^2 (B e-18 A f)-c^3 f^2 (4 C e-B f)-c d^2 e \left (4 C e^2-23 B e f+12 A f^2\right )-c^2 d f \left (40 C e^2-23 B e f+18 A f^2\right )\right )-2 a^3 d f \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d e f+c^2 f^2\right )\right )\right )\right )}{8 (a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{6 (b c-a d)^3 (b e-a f)^3} \\ & = -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac {\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac {\left (4 a^4 C d^2 f^2+8 a^3 b d f (B d f-2 C (d e+c f))-b^4 \left (15 A d^2 e^2-2 c d e (9 B e-7 A f)+3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-44 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-29 B e f+22 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-34 c d e f+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}-\frac {\left (b^3 \left (5 A d^3 e^3-3 c d^2 e^2 (2 B e-A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^3 e^2 (B e-18 A f)-c^3 f^2 (4 C e-B f)-c d^2 e \left (4 C e^2-23 B e f+12 A f^2\right )-c^2 d f \left (40 C e^2-23 B e f+18 A f^2\right )\right )-2 a^3 d f \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d e f+c^2 f^2\right )\right )\right )\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{16 (b c-a d)^3 (b e-a f)^3} \\ & = -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac {\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac {\left (4 a^4 C d^2 f^2+8 a^3 b d f (B d f-2 C (d e+c f))-b^4 \left (15 A d^2 e^2-2 c d e (9 B e-7 A f)+3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-44 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-29 B e f+22 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-34 c d e f+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}-\frac {\left (b^3 \left (5 A d^3 e^3-3 c d^2 e^2 (2 B e-A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^3 e^2 (B e-18 A f)-c^3 f^2 (4 C e-B f)-c d^2 e \left (4 C e^2-23 B e f+12 A f^2\right )-c^2 d f \left (40 C e^2-23 B e f+18 A f^2\right )\right )-2 a^3 d f \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d e f+c^2 f^2\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-b c+a d-(-b e+a f) x^2} \, dx,x,\frac {\sqrt {c+d x}}{\sqrt {e+f x}}\right )}{8 (b c-a d)^3 (b e-a f)^3} \\ & = -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac {\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac {\left (4 a^4 C d^2 f^2+8 a^3 b d f (B d f-2 C (d e+c f))-b^4 \left (15 A d^2 e^2-2 c d e (9 B e-7 A f)+3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-44 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-29 B e f+22 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-34 c d e f+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}+\frac {\left (b^3 \left (5 A d^3 e^3-3 c d^2 e^2 (2 B e-A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^3 e^2 (B e-18 A f)-c^3 f^2 (4 C e-B f)-c d^2 e \left (4 C e^2-23 B e f+12 A f^2\right )-c^2 d f \left (40 C e^2-23 B e f+18 A f^2\right )\right )-2 a^3 d f \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d e f+c^2 f^2\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{8 (b c-a d)^{7/2} (b e-a f)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.96 (sec) , antiderivative size = 1036, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (6 b^5 c e x (4 c C e x+B (2 c e-3 d e x-3 c f x))+6 a^5 d f (-4 B d f+C (3 d e+3 c f-2 d f x))+a b^4 \left (-12 c C e x (-2 c e+d e x+c f x)+B \left (3 d^2 e^2 x^2+2 c d e x (-25 e+29 f x)+c^2 \left (4 e^2-50 e f x+3 f^2 x^2\right )\right )\right )+a^4 b \left (12 B d f (c f+d (e-2 f x))-C \left (3 c^2 f^2+2 c d f (29 e-25 f x)+d^2 \left (3 e^2-50 e f x+4 f^2 x^2\right )\right )\right )+a^2 b^3 \left (d^2 e x (8 B e+3 C e x-10 B f x)+c^2 \left (8 B f (-2 e+f x)+C \left (8 e^2+14 e f x+3 f^2 x^2\right )\right )-2 c d \left (C e x (-7 e+17 f x)+B \left (8 e^2-62 e f x+5 f^2 x^2\right )\right )\right )+a^3 b^2 \left (c^2 f (10 C e-3 B f-8 C f x)+2 c d \left (B f (17 e-7 f x)+C \left (5 e^2-62 e f x+8 f^2 x^2\right )\right )-d^2 \left (8 C e x (e-2 f x)+B \left (3 e^2+14 e f x+8 f^2 x^2\right )\right )\right )+A b \left (72 a^4 d^2 f^2+18 a^3 b d f (-5 d e-5 c f+6 d f x)+b^4 \left (15 d^2 e^2 x^2+2 c d e x (-5 e+7 f x)+c^2 \left (8 e^2-10 e f x+15 f^2 x^2\right )\right )-2 a b^3 \left (c^2 f (13 e-20 f x)+2 d^2 e x (-10 e+11 f x)+c d \left (13 e^2-34 e f x+22 f^2 x^2\right )\right )+a^2 b^2 \left (33 c^2 f^2+2 c d f (43 e-59 f x)+d^2 \left (33 e^2-118 e f x+44 f^2 x^2\right )\right )\right )\right )}{24 (b c-a d)^3 (b e-a f)^3 (a+b x)^3}+\frac {\left (a b^2 \left (d^3 e^2 (B e-18 A f)+c^3 f^2 (-4 C e+B f)+c^2 d f \left (-40 C e^2+23 B e f-18 A f^2\right )+c d^2 e \left (-4 C e^2+23 B e f-12 A f^2\right )\right )+b^3 \left (5 A d^3 e^3+3 c d^2 e^2 (-2 B e+A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )-2 a^3 d f \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d e f+c^2 f^2\right )\right )\right )\right ) \arctan \left (\frac {\sqrt {b c-a d} \sqrt {e+f x}}{\sqrt {-b e+a f} \sqrt {c+d x}}\right )}{8 (b c-a d)^{7/2} (-b e+a f)^{7/2}} \]

[In]

Integrate[(A + B*x + C*x^2)/((a + b*x)^4*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

-1/24*(Sqrt[c + d*x]*Sqrt[e + f*x]*(6*b^5*c*e*x*(4*c*C*e*x + B*(2*c*e - 3*d*e*x - 3*c*f*x)) + 6*a^5*d*f*(-4*B*
d*f + C*(3*d*e + 3*c*f - 2*d*f*x)) + a*b^4*(-12*c*C*e*x*(-2*c*e + d*e*x + c*f*x) + B*(3*d^2*e^2*x^2 + 2*c*d*e*
x*(-25*e + 29*f*x) + c^2*(4*e^2 - 50*e*f*x + 3*f^2*x^2))) + a^4*b*(12*B*d*f*(c*f + d*(e - 2*f*x)) - C*(3*c^2*f
^2 + 2*c*d*f*(29*e - 25*f*x) + d^2*(3*e^2 - 50*e*f*x + 4*f^2*x^2))) + a^2*b^3*(d^2*e*x*(8*B*e + 3*C*e*x - 10*B
*f*x) + c^2*(8*B*f*(-2*e + f*x) + C*(8*e^2 + 14*e*f*x + 3*f^2*x^2)) - 2*c*d*(C*e*x*(-7*e + 17*f*x) + B*(8*e^2
- 62*e*f*x + 5*f^2*x^2))) + a^3*b^2*(c^2*f*(10*C*e - 3*B*f - 8*C*f*x) + 2*c*d*(B*f*(17*e - 7*f*x) + C*(5*e^2 -
62*e*f*x + 8*f^2*x^2)) - d^2*(8*C*e*x*(e - 2*f*x) + B*(3*e^2 + 14*e*f*x + 8*f^2*x^2))) + A*b*(72*a^4*d^2*f^2
+ 18*a^3*b*d*f*(-5*d*e - 5*c*f + 6*d*f*x) + b^4*(15*d^2*e^2*x^2 + 2*c*d*e*x*(-5*e + 7*f*x) + c^2*(8*e^2 - 10*e
*f*x + 15*f^2*x^2)) - 2*a*b^3*(c^2*f*(13*e - 20*f*x) + 2*d^2*e*x*(-10*e + 11*f*x) + c*d*(13*e^2 - 34*e*f*x + 2
2*f^2*x^2)) + a^2*b^2*(33*c^2*f^2 + 2*c*d*f*(43*e - 59*f*x) + d^2*(33*e^2 - 118*e*f*x + 44*f^2*x^2)))))/((b*c
- a*d)^3*(b*e - a*f)^3*(a + b*x)^3) + ((a*b^2*(d^3*e^2*(B*e - 18*A*f) + c^3*f^2*(-4*C*e + B*f) + c^2*d*f*(-40*
C*e^2 + 23*B*e*f - 18*A*f^2) + c*d^2*e*(-4*C*e^2 + 23*B*e*f - 12*A*f^2)) + b^3*(5*A*d^3*e^3 + 3*c*d^2*e^2*(-2*
B*e + A*f) + c^2*d*e*(8*C*e^2 - 4*B*e*f + 3*A*f^2) + c^3*f*(8*C*e^2 - 6*B*e*f + 5*A*f^2)) - 2*a^3*d*f*(C*(3*d^
2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 4*d*f*(2*A*d*f - B*(d*e + c*f))) + a^2*b*(C*(d^3*e^3 + 23*c*d^2*e^2*f + 23*c^
2*d*e*f^2 + c^3*f^3) + 4*d*f*(6*A*d*f*(d*e + c*f) - B*(d^2*e^2 + 10*c*d*e*f + c^2*f^2))))*ArcTan[(Sqrt[b*c - a
*d]*Sqrt[e + f*x])/(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])])/(8*(b*c - a*d)^(7/2)*(-(b*e) + a*f)^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(18801\) vs. \(2(794)=1588\).

Time = 1.70 (sec) , antiderivative size = 18802, normalized size of antiderivative = 22.76

method	result	size
default	\(\text {Expression too large to display}\)	\(18802\)

[In]

int((C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(1/2)/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Timed out} \]

[In]

integrate((C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Timed out} \]

[In]

integrate((C*x**2+B*x+A)/(b*x+a)**4/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume((a*d-b*c)>0)', see `assume?` f
or more deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 25632 vs. \(2 (796) = 1592\).

Time = 58.08 (sec) , antiderivative size = 25632, normalized size of antiderivative = 31.03 \[ \int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Too large to display} \]

[In]

integrate((C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="giac")

[Out]

1/8*(8*sqrt(d*f)*C*b^3*c^2*d^3*e^3 - 4*sqrt(d*f)*C*a*b^2*c*d^4*e^3 - 6*sqrt(d*f)*B*b^3*c*d^4*e^3 + sqrt(d*f)*C
*a^2*b*d^5*e^3 + sqrt(d*f)*B*a*b^2*d^5*e^3 + 5*sqrt(d*f)*A*b^3*d^5*e^3 + 8*sqrt(d*f)*C*b^3*c^3*d^2*e^2*f - 40*
sqrt(d*f)*C*a*b^2*c^2*d^3*e^2*f - 4*sqrt(d*f)*B*b^3*c^2*d^3*e^2*f + 23*sqrt(d*f)*C*a^2*b*c*d^4*e^2*f + 23*sqrt
(d*f)*B*a*b^2*c*d^4*e^2*f + 3*sqrt(d*f)*A*b^3*c*d^4*e^2*f - 6*sqrt(d*f)*C*a^3*d^5*e^2*f - 4*sqrt(d*f)*B*a^2*b*
d^5*e^2*f - 18*sqrt(d*f)*A*a*b^2*d^5*e^2*f - 4*sqrt(d*f)*C*a*b^2*c^3*d^2*e*f^2 - 6*sqrt(d*f)*B*b^3*c^3*d^2*e*f
^2 + 23*sqrt(d*f)*C*a^2*b*c^2*d^3*e*f^2 + 23*sqrt(d*f)*B*a*b^2*c^2*d^3*e*f^2 + 3*sqrt(d*f)*A*b^3*c^2*d^3*e*f^2
- 4*sqrt(d*f)*C*a^3*c*d^4*e*f^2 - 40*sqrt(d*f)*B*a^2*b*c*d^4*e*f^2 - 12*sqrt(d*f)*A*a*b^2*c*d^4*e*f^2 + 8*sqr
t(d*f)*B*a^3*d^5*e*f^2 + 24*sqrt(d*f)*A*a^2*b*d^5*e*f^2 + sqrt(d*f)*C*a^2*b*c^3*d^2*f^3 + sqrt(d*f)*B*a*b^2*c^
3*d^2*f^3 + 5*sqrt(d*f)*A*b^3*c^3*d^2*f^3 - 6*sqrt(d*f)*C*a^3*c^2*d^3*f^3 - 4*sqrt(d*f)*B*a^2*b*c^2*d^3*f^3 -
18*sqrt(d*f)*A*a*b^2*c^2*d^3*f^3 + 8*sqrt(d*f)*B*a^3*c*d^4*f^3 + 24*sqrt(d*f)*A*a^2*b*c*d^4*f^3 - 16*sqrt(d*f)
*A*a^3*d^5*f^3)*arctan(-1/2*(b*d^2*e + b*c*d*f - 2*a*d^2*f - (sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)
*d*f - c*d*f))^2*b)/(sqrt(-b^2*c*d*e*f + a*b*d^2*e*f + a*b*c*d*f^2 - a^2*d^2*f^2)*d))/((b^6*c^3*e^3*abs(d) - 3
*a*b^5*c^2*d*e^3*abs(d) + 3*a^2*b^4*c*d^2*e^3*abs(d) - a^3*b^3*d^3*e^3*abs(d) - 3*a*b^5*c^3*e^2*f*abs(d) + 9*a
^2*b^4*c^2*d*e^2*f*abs(d) - 9*a^3*b^3*c*d^2*e^2*f*abs(d) + 3*a^4*b^2*d^3*e^2*f*abs(d) + 3*a^2*b^4*c^3*e*f^2*ab
s(d) - 9*a^3*b^3*c^2*d*e*f^2*abs(d) + 9*a^4*b^2*c*d^2*e*f^2*abs(d) - 3*a^5*b*d^3*e*f^2*abs(d) - a^3*b^3*c^3*f^
3*abs(d) + 3*a^4*b^2*c^2*d*f^3*abs(d) - 3*a^5*b*c*d^2*f^3*abs(d) + a^6*d^3*f^3*abs(d))*sqrt(-b^2*c*d*e*f + a*b
*d^2*e*f + a*b*c*d*f^2 - a^2*d^2*f^2)*d) - 1/12*(24*sqrt(d*f)*C*b^7*c^2*d^13*e^8 - 12*sqrt(d*f)*C*a*b^6*c*d^14
*e^8 - 18*sqrt(d*f)*B*b^7*c*d^14*e^8 + 3*sqrt(d*f)*C*a^2*b^5*d^15*e^8 + 3*sqrt(d*f)*B*a*b^6*d^15*e^8 + 15*sqrt
(d*f)*A*b^7*d^15*e^8 - 144*sqrt(d*f)*C*b^7*c^3*d^12*e^7*f + 60*sqrt(d*f)*C*a*b^6*c^2*d^13*e^7*f + 90*sqrt(d*f)
*B*b^7*c^2*d^13*e^7*f - 52*sqrt(d*f)*C*a^2*b^5*c*d^14*e^7*f + 40*sqrt(d*f)*B*a*b^6*c*d^14*e^7*f - 76*sqrt(d*f)
*A*b^7*c*d^14*e^7*f + 16*sqrt(d*f)*C*a^3*b^4*d^15*e^7*f - 10*sqrt(d*f)*B*a^2*b^5*d^15*e^7*f - 44*sqrt(d*f)*A*a
*b^6*d^15*e^7*f + 360*sqrt(d*f)*C*b^7*c^4*d^11*e^6*f^2 - 108*sqrt(d*f)*C*a*b^6*c^3*d^12*e^6*f^2 - 162*sqrt(d*f
)*B*b^7*c^3*d^12*e^6*f^2 + 252*sqrt(d*f)*C*a^2*b^5*c^2*d^13*e^6*f^2 - 300*sqrt(d*f)*B*a*b^6*c^2*d^13*e^6*f^2 +
156*sqrt(d*f)*A*b^7*c^2*d^13*e^6*f^2 - 80*sqrt(d*f)*C*a^3*b^4*c*d^14*e^6*f^2 + 50*sqrt(d*f)*B*a^2*b^5*c*d^14*
e^6*f^2 + 220*sqrt(d*f)*A*a*b^6*c*d^14*e^6*f^2 - 4*sqrt(d*f)*C*a^4*b^3*d^15*e^6*f^2 - 8*sqrt(d*f)*B*a^3*b^4*d^
15*e^6*f^2 + 44*sqrt(d*f)*A*a^2*b^5*d^15*e^6*f^2 - 480*sqrt(d*f)*C*b^7*c^5*d^10*e^5*f^3 + 60*sqrt(d*f)*C*a*b^6
*c^4*d^11*e^5*f^3 + 90*sqrt(d*f)*B*b^7*c^4*d^11*e^5*f^3 - 588*sqrt(d*f)*C*a^2*b^5*c^3*d^12*e^5*f^3 + 792*sqrt(
d*f)*B*a*b^6*c^3*d^12*e^5*f^3 - 180*sqrt(d*f)*A*b^7*c^3*d^12*e^5*f^3 + 144*sqrt(d*f)*C*a^3*b^4*c^2*d^13*e^5*f^
3 - 90*sqrt(d*f)*B*a^2*b^5*c^2*d^13*e^5*f^3 - 396*sqrt(d*f)*A*a*b^6*c^2*d^13*e^5*f^3 + 24*sqrt(d*f)*C*a^4*b^3*
c*d^14*e^5*f^3 + 48*sqrt(d*f)*B*a^3*b^4*c*d^14*e^5*f^3 - 264*sqrt(d*f)*A*a^2*b^5*c*d^14*e^5*f^3 + 360*sqrt(d*f
)*C*b^7*c^6*d^9*e^4*f^4 + 60*sqrt(d*f)*C*a*b^6*c^5*d^10*e^4*f^4 + 90*sqrt(d*f)*B*b^7*c^5*d^10*e^4*f^4 + 770*sq
rt(d*f)*C*a^2*b^5*c^4*d^11*e^4*f^4 - 1070*sqrt(d*f)*B*a*b^6*c^4*d^11*e^4*f^4 + 170*sqrt(d*f)*A*b^7*c^4*d^11*e^
4*f^4 - 80*sqrt(d*f)*C*a^3*b^4*c^3*d^12*e^4*f^4 + 50*sqrt(d*f)*B*a^2*b^5*c^3*d^12*e^4*f^4 + 220*sqrt(d*f)*A*a*
b^6*c^3*d^12*e^4*f^4 - 60*sqrt(d*f)*C*a^4*b^3*c^2*d^13*e^4*f^4 - 120*sqrt(d*f)*B*a^3*b^4*c^2*d^13*e^4*f^4 + 66
0*sqrt(d*f)*A*a^2*b^5*c^2*d^13*e^4*f^4 - 144*sqrt(d*f)*C*b^7*c^7*d^8*e^3*f^5 - 108*sqrt(d*f)*C*a*b^6*c^6*d^9*e
^3*f^5 - 162*sqrt(d*f)*B*b^7*c^6*d^9*e^3*f^5 - 588*sqrt(d*f)*C*a^2*b^5*c^5*d^10*e^3*f^5 + 792*sqrt(d*f)*B*a*b^
6*c^5*d^10*e^3*f^5 - 180*sqrt(d*f)*A*b^7*c^5*d^10*e^3*f^5 - 80*sqrt(d*f)*C*a^3*b^4*c^4*d^11*e^3*f^5 + 50*sqrt(
d*f)*B*a^2*b^5*c^4*d^11*e^3*f^5 + 220*sqrt(d*f)*A*a*b^6*c^4*d^11*e^3*f^5 + 80*sqrt(d*f)*C*a^4*b^3*c^3*d^12*e^3
*f^5 + 160*sqrt(d*f)*B*a^3*b^4*c^3*d^12*e^3*f^5 - 880*sqrt(d*f)*A*a^2*b^5*c^3*d^12*e^3*f^5 + 24*sqrt(d*f)*C*b^
7*c^8*d^7*e^2*f^6 + 60*sqrt(d*f)*C*a*b^6*c^7*d^8*e^2*f^6 + 90*sqrt(d*f)*B*b^7*c^7*d^8*e^2*f^6 + 252*sqrt(d*f)*
C*a^2*b^5*c^6*d^9*e^2*f^6 - 300*sqrt(d*f)*B*a*b^6*c^6*d^9*e^2*f^6 + 156*sqrt(d*f)*A*b^7*c^6*d^9*e^2*f^6 + 144*
sqrt(d*f)*C*a^3*b^4*c^5*d^10*e^2*f^6 - 90*sqrt(d*f)*B*a^2*b^5*c^5*d^10*e^2*f^6 - 396*sqrt(d*f)*A*a*b^6*c^5*d^1
0*e^2*f^6 - 60*sqrt(d*f)*C*a^4*b^3*c^4*d^11*e^2*f^6 - 120*sqrt(d*f)*B*a^3*b^4*c^4*d^11*e^2*f^6 + 660*sqrt(d*f)
*A*a^2*b^5*c^4*d^11*e^2*f^6 - 12*sqrt(d*f)*C*a*b^6*c^8*d^7*e*f^7 - 18*sqrt(d*f)*B*b^7*c^8*d^7*e*f^7 - 52*sqrt(
d*f)*C*a^2*b^5*c^7*d^8*e*f^7 + 40*sqrt(d*f)*B*a*b^6*c^7*d^8*e*f^7 - 76*sqrt(d*f)*A*b^7*c^7*d^8*e*f^7 - 80*sqrt
(d*f)*C*a^3*b^4*c^6*d^9*e*f^7 + 50*sqrt(d*f)*B*a^2*b^5*c^6*d^9*e*f^7 + 220*sqrt(d*f)*A*a*b^6*c^6*d^9*e*f^7 + 2
4*sqrt(d*f)*C*a^4*b^3*c^5*d^10*e*f^7 + 48*sqrt(d*f)*B*a^3*b^4*c^5*d^10*e*f^7 - 264*sqrt(d*f)*A*a^2*b^5*c^5*d^1
0*e*f^7 + 3*sqrt(d*f)*C*a^2*b^5*c^8*d^7*f^8 + 3*sqrt(d*f)*B*a*b^6*c^8*d^7*f^8 + 15*sqrt(d*f)*A*b^7*c^8*d^7*f^8
+ 16*sqrt(d*f)*C*a^3*b^4*c^7*d^8*f^8 - 10*sqrt(d*f)*B*a^2*b^5*c^7*d^8*f^8 - 44*sqrt(d*f)*A*a*b^6*c^7*d^8*f^8
- 4*sqrt(d*f)*C*a^4*b^3*c^6*d^9*f^8 - 8*sqrt(d*f)*B*a^3*b^4*c^6*d^9*f^8 + 44*sqrt(d*f)*A*a^2*b^5*c^6*d^9*f^8 -
120*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*b^7*c^2*d^11*e^7 + 60*sqrt(
d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a*b^6*c*d^12*e^7 + 90*sqrt(d*f)*(sqrt
(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*b^7*c*d^12*e^7 - 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d
*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^2*b^5*d^13*e^7 - 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) -
sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a*b^6*d^13*e^7 - 75*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e +
(d*x + c)*d*f - c*d*f))^2*A*b^7*d^13*e^7 + 360*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*
f - c*d*f))^2*C*b^7*c^3*d^10*e^6*f + 72*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*
f))^2*C*a*b^6*c^2*d^11*e^6*f - 156*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2
*B*b^7*c^2*d^11*e^6*f + 171*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^2*
b^5*c*d^12*e^6*f - 453*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a*b^6*c*d
^12*e^6*f + 135*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*b^7*c*d^12*e^6*f
- 78*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^3*b^4*d^13*e^6*f + 84*sq
rt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a^2*b^5*d^13*e^6*f + 390*sqrt(d*f)
*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*a*b^6*d^13*e^6*f - 240*sqrt(d*f)*(sqrt(d*
f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*b^7*c^4*d^9*e^5*f^2 - 828*sqrt(d*f)*(sqrt(d*f)*sqr
t(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a*b^6*c^3*d^10*e^5*f^2 - 186*sqrt(d*f)*(sqrt(d*f)*sqrt(d
*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*b^7*c^3*d^10*e^5*f^2 - 423*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x +
c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^2*b^5*c^2*d^11*e^5*f^2 + 1449*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x +
c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a*b^6*c^2*d^11*e^5*f^2 + 45*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) -
sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*b^7*c^2*d^11*e^5*f^2 - 300*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt
(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^3*b^4*c*d^12*e^5*f^2 + 360*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^
2*e + (d*x + c)*d*f - c*d*f))^2*B*a^2*b^5*c*d^12*e^5*f^2 - 900*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e
+ (d*x + c)*d*f - c*d*f))^2*A*a*b^6*c*d^12*e^5*f^2 + 216*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d
*x + c)*d*f - c*d*f))^2*C*a^4*b^3*d^13*e^5*f^2 - 48*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c
)*d*f - c*d*f))^2*B*a^3*b^4*d^13*e^5*f^2 - 720*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f
- c*d*f))^2*A*a^2*b^5*d^13*e^5*f^2 - 240*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*
d*f))^2*C*b^7*c^5*d^8*e^4*f^3 + 1392*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))
^2*C*a*b^6*c^4*d^9*e^4*f^3 + 504*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B
*b^7*c^4*d^9*e^4*f^3 + 267*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^2*b
^5*c^3*d^10*e^4*f^3 - 981*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a*b^6*
c^3*d^10*e^4*f^3 - 105*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*b^7*c^3*d
^10*e^4*f^3 + 1902*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^3*b^4*c^2*d
^11*e^4*f^3 - 2196*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a^2*b^5*c^2*d
^11*e^4*f^3 + 90*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*a*b^6*c^2*d^11*
e^4*f^3 - 648*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^4*b^3*c*d^12*e^4
*f^3 + 144*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a^3*b^4*c*d^12*e^4*f^
3 + 2160*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*a^2*b^5*c*d^12*e^4*f^3
- 48*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^5*b^2*d^13*e^4*f^3 - 96*s
qrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a^4*b^3*d^13*e^4*f^3 + 480*sqrt(d
*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*a^3*b^4*d^13*e^4*f^3 + 360*sqrt(d*f)*(
sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*b^7*c^6*d^7*e^3*f^4 - 828*sqrt(d*f)*(sqrt(d
*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a*b^6*c^5*d^8*e^3*f^4 - 186*sqrt(d*f)*(sqrt(d*f)*
sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*b^7*c^5*d^8*e^3*f^4 + 267*sqrt(d*f)*(sqrt(d*f)*sqrt(d
*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^2*b^5*c^4*d^9*e^3*f^4 - 981*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x
+ c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a*b^6*c^4*d^9*e^3*f^4 - 105*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c
) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*b^7*c^4*d^9*e^3*f^4 - 3048*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - s
qrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^3*b^4*c^3*d^10*e^3*f^4 + 3504*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - s
qrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a^2*b^5*c^3*d^10*e^3*f^4 + 840*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sq
rt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*a*b^6*c^3*d^10*e^3*f^4 + 432*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(
d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^4*b^3*c^2*d^11*e^3*f^4 - 96*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^
2*e + (d*x + c)*d*f - c*d*f))^2*B*a^3*b^4*c^2*d^11*e^3*f^4 - 1440*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^
2*e + (d*x + c)*d*f - c*d*f))^2*A*a^2*b^5*c^2*d^11*e^3*f^4 + 192*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2
*e + (d*x + c)*d*f - c*d*f))^2*C*a^5*b^2*c*d^12*e^3*f^4 + 384*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e
+ (d*x + c)*d*f - c*d*f))^2*B*a^4*b^3*c*d^12*e^3*f^4 - 1920*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e +
(d*x + c)*d*f - c*d*f))^2*A*a^3*b^4*c*d^12*e^3*f^4 - 120*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*
x + c)*d*f - c*d*f))^2*C*b^7*c^7*d^6*e^2*f^5 + 72*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*
d*f - c*d*f))^2*C*a*b^6*c^6*d^7*e^2*f^5 - 156*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f
- c*d*f))^2*B*b^7*c^6*d^7*e^2*f^5 - 423*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*
f))^2*C*a^2*b^5*c^5*d^8*e^2*f^5 + 1449*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f
))^2*B*a*b^6*c^5*d^8*e^2*f^5 + 45*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*
A*b^7*c^5*d^8*e^2*f^5 + 1902*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^3
*b^4*c^4*d^9*e^2*f^5 - 2196*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a^2*
b^5*c^4*d^9*e^2*f^5 + 90*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*a*b^6*c
^4*d^9*e^2*f^5 + 432*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^4*b^3*c^3
*d^10*e^2*f^5 - 96*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a^3*b^4*c^3*d
^10*e^2*f^5 - 1440*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*a^2*b^5*c^3*d
^10*e^2*f^5 - 288*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^5*b^2*c^2*d^
11*e^2*f^5 - 576*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a^4*b^3*c^2*d^1
1*e^2*f^5 + 2880*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*a^3*b^4*c^2*d^1
1*e^2*f^5 + 60*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a*b^6*c^7*d^6*e*f
^6 + 90*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*b^7*c^7*d^6*e*f^6 + 171*
sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^2*b^5*c^6*d^7*e*f^6 - 453*sqrt
(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a*b^6*c^6*d^7*e*f^6 + 135*sqrt(d*f)*
(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*b^7*c^6*d^7*e*f^6 - 300*sqrt(d*f)*(sqrt(d*
f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^3*b^4*c^5*d^8*e*f^6 + 360*sqrt(d*f)*(sqrt(d*f)*s
qrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a^2*b^5*c^5*d^8*e*f^6 - 900*sqrt(d*f)*(sqrt(d*f)*sqrt(
d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*a*b^6*c^5*d^8*e*f^6 - 648*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x +
c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^4*b^3*c^4*d^9*e*f^6 + 144*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) -
sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a^3*b^4*c^4*d^9*e*f^6 + 2160*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sq
rt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*a^2*b^5*c^4*d^9*e*f^6 + 192*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d
^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^5*b^2*c^3*d^10*e*f^6 + 384*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*
e + (d*x + c)*d*f - c*d*f))^2*B*a^4*b^3*c^3*d^10*e*f^6 - 1920*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e
+ (d*x + c)*d*f - c*d*f))^2*A*a^3*b^4*c^3*d^10*e*f^6 - 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d
*x + c)*d*f - c*d*f))^2*C*a^2*b^5*c^7*d^6*f^7 - 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)
*d*f - c*d*f))^2*B*a*b^6*c^7*d^6*f^7 - 75*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*
d*f))^2*A*b^7*c^7*d^6*f^7 - 78*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a
^3*b^4*c^6*d^7*f^7 + 84*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a^2*b^5*
c^6*d^7*f^7 + 390*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*a*b^6*c^6*d^7*
f^7 + 216*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^4*b^3*c^5*d^8*f^7 -
48*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a^3*b^4*c^5*d^8*f^7 - 720*sqr
t(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*a^2*b^5*c^5*d^8*f^7 - 48*sqrt(d*f)*
(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^5*b^2*c^4*d^9*f^7 - 96*sqrt(d*f)*(sqrt(d
*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a^4*b^3*c^4*d^9*f^7 + 480*sqrt(d*f)*(sqrt(d*f)*sq
rt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*a^3*b^4*c^4*d^9*f^7 + 240*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x
+ c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*b^7*c^2*d^9*e^6 - 120*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sq
rt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a*b^6*c*d^10*e^6 - 180*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e
+ (d*x + c)*d*f - c*d*f))^4*B*b^7*c*d^10*e^6 + 30*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*
d*f - c*d*f))^4*C*a^2*b^5*d^11*e^6 + 30*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*
f))^4*B*a*b^6*d^11*e^6 + 150*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*b^7
*d^11*e^6 - 192*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*b^7*c^3*d^8*e^5*
f - 696*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a*b^6*c^2*d^9*e^5*f - 84
*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*b^7*c^2*d^9*e^5*f - 132*sqrt(d*
f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^2*b^5*c*d^10*e^5*f + 1188*sqrt(d*f)*(
sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a*b^6*c*d^10*e^5*f + 60*sqrt(d*f)*(sqrt(d*f
)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*b^7*c*d^10*e^5*f + 120*sqrt(d*f)*(sqrt(d*f)*sqrt(d*
x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^3*b^4*d^11*e^5*f - 204*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c)
- sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^2*b^5*d^11*e^5*f - 960*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(
d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a*b^6*d^11*e^5*f - 96*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d
*x + c)*d*f - c*d*f))^4*C*b^7*c^4*d^7*e^4*f^2 + 816*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c
)*d*f - c*d*f))^4*C*a*b^6*c^3*d^8*e^4*f^2 + 264*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*
f - c*d*f))^4*B*b^7*c^3*d^8*e^4*f^2 + 930*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*
d*f))^4*C*a^2*b^5*c^2*d^9*e^4*f^2 - 414*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*
f))^4*B*a*b^6*c^2*d^9*e^4*f^2 + 42*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4
*A*b^7*c^2*d^9*e^4*f^2 + 1176*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^
3*b^4*c*d^10*e^4*f^2 - 2364*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^2*
b^5*c*d^10*e^4*f^2 - 384*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a*b^6*c
*d^10*e^4*f^2 - 576*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^4*b^3*d^11
*e^4*f^2 + 264*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^3*b^4*d^11*e^4*
f^2 + 2592*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a^2*b^5*d^11*e^4*f^2
- 192*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*b^7*c^5*d^6*e^3*f^3 + 816*
sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a*b^6*c^4*d^7*e^3*f^3 + 264*sqrt
(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*b^7*c^4*d^7*e^3*f^3 - 1656*sqrt(d*f)
*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^2*b^5*c^3*d^8*e^3*f^3 - 1608*sqrt(d*f)*
(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a*b^6*c^3*d^8*e^3*f^3 - 504*sqrt(d*f)*(sqr
t(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*b^7*c^3*d^8*e^3*f^3 - 1296*sqrt(d*f)*(sqrt(d*f
)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^3*b^4*c^2*d^9*e^3*f^3 + 2568*sqrt(d*f)*(sqrt(d*f)
*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^2*b^5*c^2*d^9*e^3*f^3 + 1344*sqrt(d*f)*(sqrt(d*f)*
sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a*b^6*c^2*d^9*e^3*f^3 - 1344*sqrt(d*f)*(sqrt(d*f)*sqr
t(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^4*b^3*c*d^10*e^3*f^3 + 1632*sqrt(d*f)*(sqrt(d*f)*sqrt(
d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^3*b^4*c*d^10*e^3*f^3 - 576*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x
+ c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a^2*b^5*c*d^10*e^3*f^3 + 672*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x +
c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^5*b^2*d^11*e^3*f^3 + 144*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) -
sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^4*b^3*d^11*e^3*f^3 - 3264*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt
(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a^3*b^4*d^11*e^3*f^3 + 240*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*
e + (d*x + c)*d*f - c*d*f))^4*C*b^7*c^6*d^5*e^2*f^4 - 696*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d
*x + c)*d*f - c*d*f))^4*C*a*b^6*c^5*d^6*e^2*f^4 - 84*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x +
c)*d*f - c*d*f))^4*B*b^7*c^5*d^6*e^2*f^4 + 930*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f
- c*d*f))^4*C*a^2*b^5*c^4*d^7*e^2*f^4 - 414*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f -
c*d*f))^4*B*a*b^6*c^4*d^7*e^2*f^4 + 42*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*
f))^4*A*b^7*c^4*d^7*e^2*f^4 - 1296*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4
*C*a^3*b^4*c^3*d^8*e^2*f^4 + 2568*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*
B*a^2*b^5*c^3*d^8*e^2*f^4 + 1344*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A
*a*b^6*c^3*d^8*e^2*f^4 + 3840*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^
4*b^3*c^2*d^9*e^2*f^4 - 3792*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^3
*b^4*c^2*d^9*e^2*f^4 - 4032*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a^2*
b^5*c^2*d^9*e^2*f^4 - 672*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^5*b^
2*c*d^10*e^2*f^4 - 144*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^4*b^3*c
*d^10*e^2*f^4 + 3264*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a^3*b^4*c*d
^10*e^2*f^4 - 96*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^6*b*d^11*e^2*
f^4 - 384*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^5*b^2*d^11*e^2*f^4 +
1632*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a^4*b^3*d^11*e^2*f^4 - 120
*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a*b^6*c^6*d^5*e*f^5 - 180*sqrt(
d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*b^7*c^6*d^5*e*f^5 - 132*sqrt(d*f)*(sq
rt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^2*b^5*c^5*d^6*e*f^5 + 1188*sqrt(d*f)*(sqrt(
d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a*b^6*c^5*d^6*e*f^5 + 60*sqrt(d*f)*(sqrt(d*f)*sq
rt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*b^7*c^5*d^6*e*f^5 + 1176*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x
+ c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^3*b^4*c^4*d^7*e*f^5 - 2364*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c
) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^2*b^5*c^4*d^7*e*f^5 - 384*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) -
sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a*b^6*c^4*d^7*e*f^5 - 1344*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(
d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^4*b^3*c^3*d^8*e*f^5 + 1632*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2
*e + (d*x + c)*d*f - c*d*f))^4*B*a^3*b^4*c^3*d^8*e*f^5 - 576*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e +
(d*x + c)*d*f - c*d*f))^4*A*a^2*b^5*c^3*d^8*e*f^5 - 672*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*
x + c)*d*f - c*d*f))^4*C*a^5*b^2*c^2*d^9*e*f^5 - 144*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x +
c)*d*f - c*d*f))^4*B*a^4*b^3*c^2*d^9*e*f^5 + 3264*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*
d*f - c*d*f))^4*A*a^3*b^4*c^2*d^9*e*f^5 + 192*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f
- c*d*f))^4*C*a^6*b*c*d^10*e*f^5 + 768*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f
))^4*B*a^5*b^2*c*d^10*e*f^5 - 3264*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4
*A*a^4*b^3*c*d^10*e*f^5 + 30*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^2
*b^5*c^6*d^5*f^6 + 30*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a*b^6*c^6*
d^5*f^6 + 150*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*b^7*c^6*d^5*f^6 +
120*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^3*b^4*c^5*d^6*f^6 - 204*sq
rt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^2*b^5*c^5*d^6*f^6 - 960*sqrt(d*f
)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a*b^6*c^5*d^6*f^6 - 576*sqrt(d*f)*(sqrt(
d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^4*b^3*c^4*d^7*f^6 + 264*sqrt(d*f)*(sqrt(d*f)*s
qrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^3*b^4*c^4*d^7*f^6 + 2592*sqrt(d*f)*(sqrt(d*f)*sqrt(d
*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a^2*b^5*c^4*d^7*f^6 + 672*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c
) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^5*b^2*c^3*d^8*f^6 + 144*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sq
rt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^4*b^3*c^3*d^8*f^6 - 3264*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^
2*e + (d*x + c)*d*f - c*d*f))^4*A*a^3*b^4*c^3*d^8*f^6 - 96*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (
d*x + c)*d*f - c*d*f))^4*C*a^6*b*c^2*d^9*f^6 - 384*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)
*d*f - c*d*f))^4*B*a^5*b^2*c^2*d^9*f^6 + 1632*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f
- c*d*f))^4*A*a^4*b^3*c^2*d^9*f^6 - 240*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*
f))^6*C*b^7*c^2*d^7*e^5 + 120*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a*
b^6*c*d^8*e^5 + 180*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*b^7*c*d^8*e^
5 - 30*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^2*b^5*d^9*e^5 - 30*sqrt
(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a*b^6*d^9*e^5 - 150*sqrt(d*f)*(sqrt(
d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*b^7*d^9*e^5 - 144*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x
+ c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*b^7*c^3*d^6*e^4*f + 1056*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) -
sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a*b^6*c^2*d^7*e^4*f + 288*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d
^2*e + (d*x + c)*d*f - c*d*f))^6*B*b^7*c^2*d^7*e^4*f - 110*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (
d*x + c)*d*f - c*d*f))^6*C*a^2*b^5*c*d^8*e^4*f - 1246*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x +
c)*d*f - c*d*f))^6*B*a*b^6*c*d^8*e^4*f - 230*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f
- c*d*f))^6*A*b^7*c*d^8*e^4*f - 52*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6
*C*a^3*b^4*d^9*e^4*f + 208*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a^2*b
^5*d^9*e^4*f + 980*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*a*b^6*d^9*e^4
*f - 144*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*b^7*c^4*d^5*e^3*f^2 + 7
20*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a*b^6*c^3*d^6*e^3*f^2 + 216*s
qrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*b^7*c^3*d^6*e^3*f^2 - 1716*sqrt(d
*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^2*b^5*c^2*d^7*e^3*f^2 - 1476*sqrt(d*
f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a*b^6*c^2*d^7*e^3*f^2 - 324*sqrt(d*f)*(
sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*b^7*c^2*d^7*e^3*f^2 - 832*sqrt(d*f)*(sqrt(d
*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^3*b^4*c*d^8*e^3*f^2 + 3184*sqrt(d*f)*(sqrt(d*f)
*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a^2*b^5*c*d^8*e^3*f^2 + 1568*sqrt(d*f)*(sqrt(d*f)*sq
rt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*a*b^6*c*d^8*e^3*f^2 + 472*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x
+ c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^4*b^3*d^9*e^3*f^2 - 424*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c)
- sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a^3*b^4*d^9*e^3*f^2 - 2744*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqr
t(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*a^2*b^5*d^9*e^3*f^2 - 240*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*
e + (d*x + c)*d*f - c*d*f))^6*C*b^7*c^5*d^4*e^2*f^3 + 1056*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (
d*x + c)*d*f - c*d*f))^6*C*a*b^6*c^4*d^5*e^2*f^3 + 288*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x
+ c)*d*f - c*d*f))^6*B*b^7*c^4*d^5*e^2*f^3 - 1716*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*
d*f - c*d*f))^6*C*a^2*b^5*c^3*d^6*e^2*f^3 - 1476*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d
*f - c*d*f))^6*B*a*b^6*c^3*d^6*e^2*f^3 - 324*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f -
c*d*f))^6*A*b^7*c^3*d^6*e^2*f^3 + 1128*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*
f))^6*C*a^3*b^4*c^2*d^7*e^2*f^3 + 3456*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f
))^6*B*a^2*b^5*c^2*d^7*e^2*f^3 + 1944*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f)
)^6*A*a*b^6*c^2*d^7*e^2*f^3 + 2088*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6
*C*a^4*b^3*c*d^8*e^2*f^3 - 4056*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*
a^3*b^4*c*d^8*e^2*f^3 - 4296*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*a^2
*b^5*c*d^8*e^2*f^3 - 816*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^5*b^2
*d^9*e^2*f^3 + 288*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a^4*b^3*d^9*e
^2*f^3 + 4176*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*a^3*b^4*d^9*e^2*f^
3 + 120*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a*b^6*c^5*d^4*e*f^4 + 18
0*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*b^7*c^5*d^4*e*f^4 - 110*sqrt(d
*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^2*b^5*c^4*d^5*e*f^4 - 1246*sqrt(d*f)
*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a*b^6*c^4*d^5*e*f^4 - 230*sqrt(d*f)*(sqrt
(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*b^7*c^4*d^5*e*f^4 - 832*sqrt(d*f)*(sqrt(d*f)*sq
rt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^3*b^4*c^3*d^6*e*f^4 + 3184*sqrt(d*f)*(sqrt(d*f)*sqrt(
d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a^2*b^5*c^3*d^6*e*f^4 + 1568*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x
+ c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*a*b^6*c^3*d^6*e*f^4 + 2088*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c)
- sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^4*b^3*c^2*d^7*e*f^4 - 4056*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) -
sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a^3*b^4*c^2*d^7*e*f^4 - 4296*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqr
t(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*a^2*b^5*c^2*d^7*e*f^4 - 2720*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d
^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^5*b^2*c*d^8*e*f^4 + 2624*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e
+ (d*x + c)*d*f - c*d*f))^6*B*a^4*b^3*c*d^8*e*f^4 + 5728*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*
x + c)*d*f - c*d*f))^6*A*a^3*b^4*c*d^8*e*f^4 + 704*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)
*d*f - c*d*f))^6*C*a^6*b*d^9*e*f^4 + 64*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*
f))^6*B*a^5*b^2*d^9*e*f^4 - 3520*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A
*a^4*b^3*d^9*e*f^4 - 30*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^2*b^5*
c^5*d^4*f^5 - 30*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a*b^6*c^5*d^4*f
^5 - 150*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*b^7*c^5*d^4*f^5 - 52*sq
rt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^3*b^4*c^4*d^5*f^5 + 208*sqrt(d*f
)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a^2*b^5*c^4*d^5*f^5 + 980*sqrt(d*f)*(sqr
t(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*a*b^6*c^4*d^5*f^5 + 472*sqrt(d*f)*(sqrt(d*f)*s
qrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^4*b^3*c^3*d^6*f^5 - 424*sqrt(d*f)*(sqrt(d*f)*sqrt(d*
x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a^3*b^4*c^3*d^6*f^5 - 2744*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c
) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*a^2*b^5*c^3*d^6*f^5 - 816*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sq
rt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^5*b^2*c^2*d^7*f^5 + 288*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2
*e + (d*x + c)*d*f - c*d*f))^6*B*a^4*b^3*c^2*d^7*f^5 + 4176*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e +
(d*x + c)*d*f - c*d*f))^6*A*a^3*b^4*c^2*d^7*f^5 + 704*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x +
c)*d*f - c*d*f))^6*C*a^6*b*c*d^8*f^5 + 64*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c
*d*f))^6*B*a^5*b^2*c*d^8*f^5 - 3520*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^
6*A*a^4*b^3*c*d^8*f^5 - 128*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^7*
d^9*f^5 - 256*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a^6*b*d^9*f^5 + 14
08*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*a^5*b^2*d^9*f^5 + 120*sqrt(d*
f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*C*b^7*c^2*d^5*e^4 - 60*sqrt(d*f)*(sqrt(d*
f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*C*a*b^6*c*d^6*e^4 - 90*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x
+ c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*B*b^7*c*d^6*e^4 + 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(
d^2*e + (d*x + c)*d*f - c*d*f))^8*C*a^2*b^5*d^7*e^4 + 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*
x + c)*d*f - c*d*f))^8*B*a*b^6*d^7*e^4 + 75*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f -
c*d*f))^8*A*b^7*d^7*e^4 + 144*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*C*b^
7*c^3*d^4*e^3*f - 612*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*C*a*b^6*c^2*
d^5*e^3*f - 150*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*B*b^7*c^2*d^5*e^3*
f + 192*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*C*a^2*b^5*c*d^6*e^3*f + 54
0*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*B*a*b^6*c*d^6*e^3*f + 120*sqrt(d
*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*A*b^7*c*d^6*e^3*f - 24*sqrt(d*f)*(sqrt(d
*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*C*a^3*b^4*d^7*e^3*f - 90*sqrt(d*f)*(sqrt(d*f)*sqrt(
d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*B*a^2*b^5*d^7*e^3*f - 420*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c)
- sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*A*a*b^6*d^7*e^3*f + 120*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^
2*e + (d*x + c)*d*f - c*d*f))^8*C*b^7*c^4*d^3*e^2*f^2 - 612*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e +
(d*x + c)*d*f - c*d*f))^8*C*a*b^6*c^3*d^4*e^2*f^2 - 150*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x
+ c)*d*f - c*d*f))^8*B*b^7*c^3*d^4*e^2*f^2 + 1026*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)
*d*f - c*d*f))^8*C*a^2*b^5*c^2*d^5*e^2*f^2 + 810*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d
*f - c*d*f))^8*B*a*b^6*c^2*d^5*e^2*f^2 + 90*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f -
c*d*f))^8*A*b^7*c^2*d^5*e^2*f^2 + 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))
^8*C*a^3*b^4*c*d^6*e^2*f^2 - 1350*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*
B*a^2*b^5*c*d^6*e^2*f^2 - 540*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*A*a*
b^6*c*d^6*e^2*f^2 - 108*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*C*a^4*b^3*
d^7*e^2*f^2 + 240*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*B*a^3*b^4*d^7*e^
2*f^2 + 900*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*A*a^2*b^5*d^7*e^2*f^2
- 60*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*C*a*b^6*c^4*d^3*e*f^3 - 90*sq
rt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*B*b^7*c^4*d^3*e*f^3 + 192*sqrt(d*f)*
(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*C*a^2*b^5*c^3*d^4*e*f^3 + 540*sqrt(d*f)*(sqr
t(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*B*a*b^6*c^3*d^4*e*f^3 + 120*sqrt(d*f)*(sqrt(d*f)
*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*A*b^7*c^3*d^4*e*f^3 + 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x
+ c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*C*a^3*b^4*c^2*d^5*e*f^3 - 1350*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x +
c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*B*a^2*b^5*c^2*d^5*e*f^3 - 540*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) -
sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*A*a*b^6*c^2*d^5*e*f^3 - 744*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(
d^2*e + (d*x + c)*d*f - c*d*f))^8*C*a^4*b^3*c*d^6*e*f^3 + 1440*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e
+ (d*x + c)*d*f - c*d*f))^8*B*a^3*b^4*c*d^6*e*f^3 + 1080*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d
*x + c)*d*f - c*d*f))^8*A*a^2*b^5*c*d^6*e*f^3 + 288*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c
)*d*f - c*d*f))^8*C*a^5*b^2*d^7*e*f^3 - 240*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f -
c*d*f))^8*B*a^4*b^3*d^7*e*f^3 - 960*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^
8*A*a^3*b^4*d^7*e*f^3 + 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*C*a^2*b
^5*c^4*d^3*f^4 + 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*B*a*b^6*c^4*d^
3*f^4 + 75*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*A*b^7*c^4*d^3*f^4 - 24*
sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*C*a^3*b^4*c^3*d^4*f^4 - 90*sqrt(d*
f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*B*a^2*b^5*c^3*d^4*f^4 - 420*sqrt(d*f)*(sq
rt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*A*a*b^6*c^3*d^4*f^4 - 108*sqrt(d*f)*(sqrt(d*f)*
sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*C*a^4*b^3*c^2*d^5*f^4 + 240*sqrt(d*f)*(sqrt(d*f)*sqrt(d
*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*B*a^3*b^4*c^2*d^5*f^4 + 900*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c
) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*A*a^2*b^5*c^2*d^5*f^4 + 288*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sq
rt(d^2*e + (d*x + c)*d*f - c*d*f))^8*C*a^5*b^2*c*d^6*f^4 - 240*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e
+ (d*x + c)*d*f - c*d*f))^8*B*a^4*b^3*c*d^6*f^4 - 960*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x
+ c)*d*f - c*d*f))^8*A*a^3*b^4*c*d^6*f^4 - 96*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f
- c*d*f))^8*C*a^6*b*d^7*f^4 + 480*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^8*
A*a^4*b^3*d^7*f^4 - 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*C*b^7*c^2*
d^3*e^3 + 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*C*a*b^6*c*d^4*e^3 +
18*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*B*b^7*c*d^4*e^3 - 3*sqrt(d*f)*
(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*C*a^2*b^5*d^5*e^3 - 3*sqrt(d*f)*(sqrt(d*f)*
sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*B*a*b^6*d^5*e^3 - 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c
) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*A*b^7*d^5*e^3 - 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e
+ (d*x + c)*d*f - c*d*f))^10*C*b^7*c^3*d^2*e^2*f + 120*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x
+ c)*d*f - c*d*f))^10*C*a*b^6*c^2*d^3*e^2*f + 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*
d*f - c*d*f))^10*B*b^7*c^2*d^3*e^2*f - 69*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*
d*f))^10*C*a^2*b^5*c*d^4*e^2*f - 69*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^
10*B*a*b^6*c*d^4*e^2*f - 9*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*A*b^7*
c*d^4*e^2*f + 18*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*C*a^3*b^4*d^5*e^
2*f + 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*B*a^2*b^5*d^5*e^2*f + 54
*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*A*a*b^6*d^5*e^2*f + 12*sqrt(d*f)
*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*C*a*b^6*c^3*d^2*e*f^2 + 18*sqrt(d*f)*(sqrt
(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*B*b^7*c^3*d^2*e*f^2 - 69*sqrt(d*f)*(sqrt(d*f)*sq
rt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*C*a^2*b^5*c^2*d^3*e*f^2 - 69*sqrt(d*f)*(sqrt(d*f)*sqrt(d
*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*B*a*b^6*c^2*d^3*e*f^2 - 9*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c)
- sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*A*b^7*c^2*d^3*e*f^2 + 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(
d^2*e + (d*x + c)*d*f - c*d*f))^10*C*a^3*b^4*c*d^4*e*f^2 + 120*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e
+ (d*x + c)*d*f - c*d*f))^10*B*a^2*b^5*c*d^4*e*f^2 + 36*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*
x + c)*d*f - c*d*f))^10*A*a*b^6*c*d^4*e*f^2 - 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d
*f - c*d*f))^10*B*a^3*b^4*d^5*e*f^2 - 72*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d
*f))^10*A*a^2*b^5*d^5*e*f^2 - 3*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*C
*a^2*b^5*c^3*d^2*f^3 - 3*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*B*a*b^6*
c^3*d^2*f^3 - 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*A*b^7*c^3*d^2*f^
3 + 18*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*C*a^3*b^4*c^2*d^3*f^3 + 12
*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*B*a^2*b^5*c^2*d^3*f^3 + 54*sqrt(
d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*A*a*b^6*c^2*d^3*f^3 - 24*sqrt(d*f)*(sq
rt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*B*a^3*b^4*c*d^4*f^3 - 72*sqrt(d*f)*(sqrt(d*f)*
sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*A*a^2*b^5*c*d^4*f^3 + 48*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x
+ c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^10*A*a^3*b^4*d^5*f^3)/((b^8*c^3*e^3*abs(d) - 3*a*b^7*c^2*d*e^3*ab
s(d) + 3*a^2*b^6*c*d^2*e^3*abs(d) - a^3*b^5*d^3*e^3*abs(d) - 3*a*b^7*c^3*e^2*f*abs(d) + 9*a^2*b^6*c^2*d*e^2*f*
abs(d) - 9*a^3*b^5*c*d^2*e^2*f*abs(d) + 3*a^4*b^4*d^3*e^2*f*abs(d) + 3*a^2*b^6*c^3*e*f^2*abs(d) - 9*a^3*b^5*c^
2*d*e*f^2*abs(d) + 9*a^4*b^4*c*d^2*e*f^2*abs(d) - 3*a^5*b^3*d^3*e*f^2*abs(d) - a^3*b^5*c^3*f^3*abs(d) + 3*a^4*
b^4*c^2*d*f^3*abs(d) - 3*a^5*b^3*c*d^2*f^3*abs(d) + a^6*b^2*d^3*f^3*abs(d))*(b*d^4*e^2 - 2*b*c*d^3*e*f + b*c^2
*d^2*f^2 - 2*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*b*d^2*e - 2*(sqrt(d*f)*sqrt(d*x
+ c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*b*c*d*f + 4*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d
*f - c*d*f))^2*a*d^2*f + (sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*b)^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Hanged} \]

[In]

int((A + B*x + C*x^2)/((e + f*x)^(1/2)*(a + b*x)^4*(c + d*x)^(1/2)),x)

[Out]

\text{Hanged}