\(\int \frac {(A+B x) (d+e x)^{7/2}}{(b x+c x^2)^3} \, dx\) [1248]

Optimal result

Integrand size = 26, antiderivative size = 363 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )+\left (24 A c^4 d^3-3 b^4 B e^3-A b^3 c e^3-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+14 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac {d^{3/2} \left (48 A c^2 d^2+7 b^2 e (4 B d+5 A e)-12 b c d (2 B d+7 A e)\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {(c d-b e)^{3/2} \left (48 A c^3 d^2-3 b^3 B e^2-12 b c^2 d (2 B d+A e)-b^2 c e (8 B d+A e)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{5/2}} \]

[Out]

-1/2*(e*x+d)^(5/2)*(A*b*c*d+(2*A*c^2*d+b^2*B*e-b*c*(A*e+B*d))*x)/b^2/c/(c*x^2+b*x)^2-1/4*d^(3/2)*(48*A*c^2*d^2
+7*b^2*e*(5*A*e+4*B*d)-12*b*c*d*(7*A*e+2*B*d))*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5+1/4*(-b*e+c*d)^(3/2)*(48*A*c
^3*d^2-3*b^3*B*e^2-12*b*c^2*d*(A*e+2*B*d)-b^2*c*e*(A*e+8*B*d))*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))
/b^5/c^(5/2)+1/4*(b*c*d^2*(12*A*c^2*d+2*b^2*B*e-b*c*(11*A*e+6*B*d))+(24*A*c^4*d^3-3*b^4*B*e^3-A*b^3*c*e^3-12*b
*c^3*d^2*(3*A*e+B*d)+b^2*c^2*d*e*(14*A*e+11*B*d))*x)*(e*x+d)^(1/2)/b^4/c^2/(c*x^2+b*x)

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {832, 840, 1180, 214} \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=-\frac {d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (7 b^2 e (5 A e+4 B d)-12 b c d (7 A e+2 B d)+48 A c^2 d^2\right )}{4 b^5}+\frac {(c d-b e)^{3/2} \left (-b^2 c e (A e+8 B d)-12 b c^2 d (A e+2 B d)+48 A c^3 d^2-3 b^3 B e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{5/2}}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 \left (-b c (11 A e+6 B d)+12 A c^2 d+2 b^2 B e\right )+x \left (-A b^3 c e^3+b^2 c^2 d e (14 A e+11 B d)-12 b c^3 d^2 (3 A e+B d)+24 A c^4 d^3-3 b^4 B e^3\right )\right )}{4 b^4 c^2 \left (b x+c x^2\right )} \]

[In]

Int[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^3,x]

[Out]

-1/2*((d + e*x)^(5/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(b^2*c*(b*x + c*x^2)^2) + (Sqrt[d
+ e*x]*(b*c*d^2*(12*A*c^2*d + 2*b^2*B*e - b*c*(6*B*d + 11*A*e)) + (24*A*c^4*d^3 - 3*b^4*B*e^3 - A*b^3*c*e^3 -
12*b*c^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(11*B*d + 14*A*e))*x))/(4*b^4*c^2*(b*x + c*x^2)) - (d^(3/2)*(48*A*c^
2*d^2 + 7*b^2*e*(4*B*d + 5*A*e) - 12*b*c*d*(2*B*d + 7*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + ((c*d -
b*e)^(3/2)*(48*A*c^3*d^2 - 3*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + A*e) - b^2*c*e*(8*B*d + A*e))*ArcTanh[(Sqrt[c]*Sq
rt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*c^(5/2))

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 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2)^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 840

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = -\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\int \frac {(d+e x)^{3/2} \left (-\frac {1}{2} d \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )-\frac {1}{2} e \left (2 A c^2 d-3 b^2 B e-b c (B d+A e)\right ) x\right )}{\left (b x+c x^2\right )^2} \, dx}{2 b^2 c} \\ & = -\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )+\left (24 A c^4 d^3-3 b^4 B e^3-A b^3 c e^3-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+14 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} c^2 d^2 \left (48 A c^2 d^2+7 b^2 e (4 B d+5 A e)-12 b c d (2 B d+7 A e)\right )+\frac {1}{4} e \left (24 A c^4 d^3+3 b^4 B e^3+b^3 c e^2 (2 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+10 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 c^2} \\ & = -\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )+\left (24 A c^4 d^3-3 b^4 B e^3-A b^3 c e^3-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+14 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {\frac {1}{4} c^2 d^2 e \left (48 A c^2 d^2+7 b^2 e (4 B d+5 A e)-12 b c d (2 B d+7 A e)\right )-\frac {1}{4} d e \left (24 A c^4 d^3+3 b^4 B e^3+b^3 c e^2 (2 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+10 A e)\right )+\frac {1}{4} e \left (24 A c^4 d^3+3 b^4 B e^3+b^3 c e^2 (2 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+10 A e)\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^4 c^2} \\ & = -\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )+\left (24 A c^4 d^3-3 b^4 B e^3-A b^3 c e^3-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+14 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac {\left ((c d-b e)^2 \left (48 A c^3 d^2-3 b^3 B e^2-12 b c^2 d (2 B d+A e)-b^2 c e (8 B d+A e)\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 c^2}+\frac {\left (c d^2 \left (48 A c^2 d^2+7 b^2 e (4 B d+5 A e)-12 b c d (2 B d+7 A e)\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5} \\ & = -\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )+\left (24 A c^4 d^3-3 b^4 B e^3-A b^3 c e^3-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+14 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac {d^{3/2} \left (48 A c^2 d^2+7 b^2 e (4 B d+5 A e)-12 b c d (2 B d+7 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {(c d-b e)^{3/2} \left (48 A c^3 d^2-3 b^3 B e^2-12 b c^2 d (2 B d+A e)-b^2 c e (8 B d+A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.01 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=-\frac {\frac {b \sqrt {d+e x} \left (b B x \left (3 b^4 e^3 x+12 c^4 d^3 x^2+b c^3 d^2 x (18 d-11 e x)+b^3 c e^2 x (4 d+5 e x)+b^2 c^2 d \left (4 d^2-17 d e x-2 e^2 x^2\right )\right )+A c \left (b^4 e^3 x^2-24 c^4 d^3 x^3-36 b c^3 d^2 x^2 (d-e x)+b^2 c^2 d x \left (-8 d^2+55 d e x-10 e^2 x^2\right )+b^3 c \left (2 d^3+13 d^2 e x-16 d e^2 x^2-e^3 x^3\right )\right )\right )}{c^2 x^2 (b+c x)^2}+\frac {(-c d+b e)^{3/2} \left (48 A c^3 d^2-3 b^3 B e^2-12 b c^2 d (2 B d+A e)-b^2 c e (8 B d+A e)\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{5/2}}-d^{3/2} \left (-48 A c^2 d^2-7 b^2 e (4 B d+5 A e)+12 b c d (2 B d+7 A e)\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5} \]

[In]

Integrate[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^3,x]

[Out]

-1/4*((b*Sqrt[d + e*x]*(b*B*x*(3*b^4*e^3*x + 12*c^4*d^3*x^2 + b*c^3*d^2*x*(18*d - 11*e*x) + b^3*c*e^2*x*(4*d +
5*e*x) + b^2*c^2*d*(4*d^2 - 17*d*e*x - 2*e^2*x^2)) + A*c*(b^4*e^3*x^2 - 24*c^4*d^3*x^3 - 36*b*c^3*d^2*x^2*(d
- e*x) + b^2*c^2*d*x*(-8*d^2 + 55*d*e*x - 10*e^2*x^2) + b^3*c*(2*d^3 + 13*d^2*e*x - 16*d*e^2*x^2 - e^3*x^3))))
/(c^2*x^2*(b + c*x)^2) + ((-(c*d) + b*e)^(3/2)*(48*A*c^3*d^2 - 3*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + A*e) - b^2*c*
e*(8*B*d + A*e))*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/c^(5/2) - d^(3/2)*(-48*A*c^2*d^2 - 7*b^2*
e*(4*B*d + 5*A*e) + 12*b*c*d*(2*B*d + 7*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^5

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(2 e^{4} \left (\frac {\left (b e -c d \right )^{2} \left (\frac {\frac {b e \left (A b c e +12 A \,c^{2} d -5 b^{2} B e -8 B b c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 c}-\frac {b e \left (A \,b^{2} c \,e^{2}-13 A b \,c^{2} d e +12 A \,c^{3} d^{2}+3 b^{3} B \,e^{2}+5 B \,b^{2} c d e -8 B b \,c^{2} d^{2}\right ) \sqrt {e x +d}}{8 c^{2}}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (A \,b^{2} c \,e^{2}+12 A b \,c^{2} d e -48 A \,c^{3} d^{2}+3 b^{3} B \,e^{2}+8 B \,b^{2} c d e +24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 c^{2} \sqrt {\left (b e -c d \right ) c}}\right )}{e^{4} b^{5}}-\frac {d^{2} \left (\frac {\left (\frac {13}{8} A \,b^{2} e^{2}-\frac {3}{2} A b c d e +\frac {1}{2} B \,b^{2} d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {11}{8} A \,b^{2} d \,e^{2}+\frac {3}{2} A b c \,d^{2} e -\frac {1}{2} B \,b^{2} d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (35 A \,b^{2} e^{2}-84 A b c d e +48 A \,c^{2} d^{2}+28 B \,b^{2} d e -24 c \,d^{2} B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{4}}\right )\)	\(399\)
default	\(2 e^{4} \left (\frac {\left (b e -c d \right )^{2} \left (\frac {\frac {b e \left (A b c e +12 A \,c^{2} d -5 b^{2} B e -8 B b c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 c}-\frac {b e \left (A \,b^{2} c \,e^{2}-13 A b \,c^{2} d e +12 A \,c^{3} d^{2}+3 b^{3} B \,e^{2}+5 B \,b^{2} c d e -8 B b \,c^{2} d^{2}\right ) \sqrt {e x +d}}{8 c^{2}}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (A \,b^{2} c \,e^{2}+12 A b \,c^{2} d e -48 A \,c^{3} d^{2}+3 b^{3} B \,e^{2}+8 B \,b^{2} c d e +24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 c^{2} \sqrt {\left (b e -c d \right ) c}}\right )}{e^{4} b^{5}}-\frac {d^{2} \left (\frac {\left (\frac {13}{8} A \,b^{2} e^{2}-\frac {3}{2} A b c d e +\frac {1}{2} B \,b^{2} d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {11}{8} A \,b^{2} d \,e^{2}+\frac {3}{2} A b c \,d^{2} e -\frac {1}{2} B \,b^{2} d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (35 A \,b^{2} e^{2}-84 A b c d e +48 A \,c^{2} d^{2}+28 B \,b^{2} d e -24 c \,d^{2} B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{4}}\right )\)	\(399\)
pseudoelliptic	\(-\frac {12 \left (\left (-\frac {b^{4} e^{4} \left (A c +3 B b \right ) \sqrt {d}}{48}+c \left (-\frac {b^{4} B \,e^{3}}{24}-\frac {5 c \left (A e +\frac {11 B d}{10}\right ) e^{2} b^{3}}{24}+\frac {71 c^{2} \left (A e +\frac {40 B d}{71}\right ) d e \,b^{2}}{48}-\frac {9 c^{3} \left (A e +\frac {2 B d}{9}\right ) d^{2} b}{4}+A \,c^{4} d^{3}\right ) d^{\frac {3}{2}}\right ) x^{2} \left (c x +b \right )^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )+\frac {\sqrt {\left (b e -c d \right ) c}\, \left (\frac {35 c^{2} x^{2} \left (c x +b \right )^{2} d^{2} \left (\frac {48 A \,c^{2} d^{2}}{35}-\frac {12 d \left (A e +\frac {2 B d}{7}\right ) b c}{5}+b^{2} e \left (A e +\frac {4 B d}{5}\right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2}+\sqrt {e x +d}\, \left (\frac {x^{2} \left (\left (-A \,c^{2}+5 B b c \right ) x +A b c +3 B \,b^{2}\right ) e^{3} b^{3} \sqrt {d}}{2}+c \,d^{\frac {3}{2}} \left (-5 c \left (\frac {b^{3} B \,e^{2}}{5}+c e \left (A e +\frac {11 B d}{10}\right ) b^{2}-\frac {18 c^{2} d \left (A e +\frac {B d}{3}\right ) b}{5}+\frac {12 A \,c^{3} d^{2}}{5}\right ) x^{3}-8 \left (-\frac {b^{3} B \,e^{2}}{4}+c e \left (A e +\frac {17 B d}{16}\right ) b^{2}-\frac {55 c^{2} d \left (A e +\frac {18 B d}{55}\right ) b}{16}+\frac {9 A \,c^{3} d^{2}}{4}\right ) b \,x^{2}+\left (\left (\frac {13}{2} A d e +2 B \,d^{2}\right ) c \,b^{3}-4 A \,b^{2} c^{2} d^{2}\right ) x +A \,b^{3} c \,d^{2}\right )\right ) b \right )}{24}\right )}{\sqrt {\left (b e -c d \right ) c}\, \sqrt {d}\, \left (c x +b \right )^{2} b^{5} c^{2} x^{2}}\)	\(448\)
risch	\(-\frac {d^{2} \sqrt {e x +d}\, \left (13 A b e x -12 A c d x +4 B b d x +2 A b d \right )}{4 b^{4} x^{2}}-\frac {e \left (\frac {\frac {8 \left (-\frac {b e \left (A \,b^{3} c \,e^{3}+10 A \,b^{2} c^{2} d \,e^{2}-23 A b \,c^{3} d^{2} e +12 A \,c^{4} d^{3}-5 b^{4} B \,e^{3}+2 b^{3} B c d \,e^{2}+11 b^{2} B \,c^{2} d^{2} e -8 B b \,c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 c}+\frac {b e \left (A \,b^{4} c \,e^{4}-15 A \,b^{3} c^{2} d \,e^{3}+39 A \,b^{2} c^{3} d^{2} e^{2}-37 A b \,c^{4} d^{3} e +12 d^{4} A \,c^{5}+3 b^{5} B \,e^{4}-B \,b^{4} c d \,e^{3}-15 B \,b^{3} c^{2} d^{2} e^{2}+21 B \,b^{2} c^{3} d^{3} e -8 B b \,c^{4} d^{4}\right ) \sqrt {e x +d}}{8 c^{2}}\right )}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}-\frac {\left (A \,b^{4} c \,e^{4}+10 A \,b^{3} c^{2} d \,e^{3}-71 A \,b^{2} c^{3} d^{2} e^{2}+108 A b \,c^{4} d^{3} e -48 d^{4} A \,c^{5}+3 b^{5} B \,e^{4}+2 B \,b^{4} c d \,e^{3}+11 B \,b^{3} c^{2} d^{2} e^{2}-40 B \,b^{2} c^{3} d^{3} e +24 B b \,c^{4} d^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{c^{2} \sqrt {\left (b e -c d \right ) c}}}{b e}+\frac {d^{\frac {3}{2}} \left (35 A \,b^{2} e^{2}-84 A b c d e +48 A \,c^{2} d^{2}+28 B \,b^{2} d e -24 c \,d^{2} B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e}\right )}{4 b^{4}}\)	\(528\)

[In]

int((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^4*((b*e-c*d)^2/e^4/b^5*((1/8*b*e*(A*b*c*e+12*A*c^2*d-5*B*b^2*e-8*B*b*c*d)/c*(e*x+d)^(3/2)-1/8*b/c^2*e*(A*b
^2*c*e^2-13*A*b*c^2*d*e+12*A*c^3*d^2+3*B*b^3*e^2+5*B*b^2*c*d*e-8*B*b*c^2*d^2)*(e*x+d)^(1/2))/(c*(e*x+d)+b*e-c*
d)^2+1/8*(A*b^2*c*e^2+12*A*b*c^2*d*e-48*A*c^3*d^2+3*B*b^3*e^2+8*B*b^2*c*d*e+24*B*b*c^2*d^2)/c^2/((b*e-c*d)*c)^
(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)))-d^2/b^5/e^4*(((13/8*A*b^2*e^2-3/2*A*b*c*d*e+1/2*B*b^2*d*e)*
(e*x+d)^(3/2)+(-11/8*A*b^2*d*e^2+3/2*A*b*c*d^2*e-1/2*B*b^2*d^2*e)*(e*x+d)^(1/2))/e^2/x^2+1/8*(35*A*b^2*e^2-84*
A*b*c*d*e+48*A*c^2*d^2+28*B*b^2*d*e-24*B*b*c*d^2)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 836 vs. \(2 (335) = 670\).

Time = 24.00 (sec) , antiderivative size = 3378, normalized size of antiderivative = 9.31 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[-1/8*(((24*(B*b*c^5 - 2*A*c^6)*d^3 - 4*(4*B*b^2*c^4 - 15*A*b*c^5)*d^2*e - (5*B*b^3*c^3 + 11*A*b^2*c^4)*d*e^2
- (3*B*b^4*c^2 + A*b^3*c^3)*e^3)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 4*(4*B*b^3*c^3 - 15*A*b^2*c^4)*d^2*
e - (5*B*b^4*c^2 + 11*A*b^3*c^3)*d*e^2 - (3*B*b^5*c + A*b^4*c^2)*e^3)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3
- 4*(4*B*b^4*c^2 - 15*A*b^3*c^3)*d^2*e - (5*B*b^5*c + 11*A*b^4*c^2)*d*e^2 - (3*B*b^6 + A*b^5*c)*e^3)*x^2)*sqrt
((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - ((35*A*b^2*c^4*
d*e^2 - 24*(B*b*c^5 - 2*A*c^6)*d^3 + 28*(B*b^2*c^4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(35*A*b^3*c^3*d*e^2 - 24*(B*b^2
*c^4 - 2*A*b*c^5)*d^3 + 28*(B*b^3*c^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (35*A*b^4*c^2*d*e^2 - 24*(B*b^3*c^3 - 2*A*b^
2*c^4)*d^3 + 28*(B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2
*(2*A*b^4*c^2*d^3 + (12*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - (11*B*b^3*c^3 - 36*A*b^2*c^4)*d^2*e - 2*(B*b^4*c^2 + 5*A
*b^3*c^3)*d*e^2 + (5*B*b^5*c - A*b^4*c^2)*e^3)*x^3 + (18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - (17*B*b^4*c^2 - 55*A*
b^3*c^3)*d^2*e + 4*(B*b^5*c - 4*A*b^4*c^2)*d*e^2 + (3*B*b^6 + A*b^5*c)*e^3)*x^2 + (13*A*b^4*c^2*d^2*e + 4*(B*b
^4*c^2 - 2*A*b^3*c^3)*d^3)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2), -1/8*(2*((24*(B*b*c^
5 - 2*A*c^6)*d^3 - 4*(4*B*b^2*c^4 - 15*A*b*c^5)*d^2*e - (5*B*b^3*c^3 + 11*A*b^2*c^4)*d*e^2 - (3*B*b^4*c^2 + A*
b^3*c^3)*e^3)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 4*(4*B*b^3*c^3 - 15*A*b^2*c^4)*d^2*e - (5*B*b^4*c^2 +
11*A*b^3*c^3)*d*e^2 - (3*B*b^5*c + A*b^4*c^2)*e^3)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - 4*(4*B*b^4*c^2 -
15*A*b^3*c^3)*d^2*e - (5*B*b^5*c + 11*A*b^4*c^2)*d*e^2 - (3*B*b^6 + A*b^5*c)*e^3)*x^2)*sqrt(-(c*d - b*e)/c)*ar
ctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) - ((35*A*b^2*c^4*d*e^2 - 24*(B*b*c^5 - 2*A*c^6)*d^3 +
28*(B*b^2*c^4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(35*A*b^3*c^3*d*e^2 - 24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + 28*(B*b^3*c^3
- 3*A*b^2*c^4)*d^2*e)*x^3 + (35*A*b^4*c^2*d*e^2 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 + 28*(B*b^4*c^2 - 3*A*b^3*
c^3)*d^2*e)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c^2*d^3 + (12*(B*b^2*c^4 -
2*A*b*c^5)*d^3 - (11*B*b^3*c^3 - 36*A*b^2*c^4)*d^2*e - 2*(B*b^4*c^2 + 5*A*b^3*c^3)*d*e^2 + (5*B*b^5*c - A*b^4*
c^2)*e^3)*x^3 + (18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - (17*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e + 4*(B*b^5*c - 4*A*b^4
*c^2)*d*e^2 + (3*B*b^6 + A*b^5*c)*e^3)*x^2 + (13*A*b^4*c^2*d^2*e + 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^3)*x)*sqrt(e*
x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2), 1/8*(2*((35*A*b^2*c^4*d*e^2 - 24*(B*b*c^5 - 2*A*c^6)*d^3
+ 28*(B*b^2*c^4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(35*A*b^3*c^3*d*e^2 - 24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + 28*(B*b^3*c
^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (35*A*b^4*c^2*d*e^2 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 + 28*(B*b^4*c^2 - 3*A*b^
3*c^3)*d^2*e)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - ((24*(B*b*c^5 - 2*A*c^6)*d^3 - 4*(4*B*b^2*c^4 -
15*A*b*c^5)*d^2*e - (5*B*b^3*c^3 + 11*A*b^2*c^4)*d*e^2 - (3*B*b^4*c^2 + A*b^3*c^3)*e^3)*x^4 + 2*(24*(B*b^2*c^
4 - 2*A*b*c^5)*d^3 - 4*(4*B*b^3*c^3 - 15*A*b^2*c^4)*d^2*e - (5*B*b^4*c^2 + 11*A*b^3*c^3)*d*e^2 - (3*B*b^5*c +
A*b^4*c^2)*e^3)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - 4*(4*B*b^4*c^2 - 15*A*b^3*c^3)*d^2*e - (5*B*b^5*c +
11*A*b^4*c^2)*d*e^2 - (3*B*b^6 + A*b^5*c)*e^3)*x^2)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x
+ d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - 2*(2*A*b^4*c^2*d^3 + (12*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - (11*B*b^3*c^3
- 36*A*b^2*c^4)*d^2*e - 2*(B*b^4*c^2 + 5*A*b^3*c^3)*d*e^2 + (5*B*b^5*c - A*b^4*c^2)*e^3)*x^3 + (18*(B*b^3*c^3
- 2*A*b^2*c^4)*d^3 - (17*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e + 4*(B*b^5*c - 4*A*b^4*c^2)*d*e^2 + (3*B*b^6 + A*b^5*
c)*e^3)*x^2 + (13*A*b^4*c^2*d^2*e + 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^3)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^
3*x^3 + b^7*c^2*x^2), -1/4*(((24*(B*b*c^5 - 2*A*c^6)*d^3 - 4*(4*B*b^2*c^4 - 15*A*b*c^5)*d^2*e - (5*B*b^3*c^3 +
11*A*b^2*c^4)*d*e^2 - (3*B*b^4*c^2 + A*b^3*c^3)*e^3)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 4*(4*B*b^3*c^3
- 15*A*b^2*c^4)*d^2*e - (5*B*b^4*c^2 + 11*A*b^3*c^3)*d*e^2 - (3*B*b^5*c + A*b^4*c^2)*e^3)*x^3 + (24*(B*b^3*c^
3 - 2*A*b^2*c^4)*d^3 - 4*(4*B*b^4*c^2 - 15*A*b^3*c^3)*d^2*e - (5*B*b^5*c + 11*A*b^4*c^2)*d*e^2 - (3*B*b^6 + A*
b^5*c)*e^3)*x^2)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) - ((35*A*b^2*c
^4*d*e^2 - 24*(B*b*c^5 - 2*A*c^6)*d^3 + 28*(B*b^2*c^4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(35*A*b^3*c^3*d*e^2 - 24*(B*
b^2*c^4 - 2*A*b*c^5)*d^3 + 28*(B*b^3*c^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (35*A*b^4*c^2*d*e^2 - 24*(B*b^3*c^3 - 2*A
*b^2*c^4)*d^3 + 28*(B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (2*A*b^4*
c^2*d^3 + (12*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - (11*B*b^3*c^3 - 36*A*b^2*c^4)*d^2*e - 2*(B*b^4*c^2 + 5*A*b^3*c^3)*
d*e^2 + (5*B*b^5*c - A*b^4*c^2)*e^3)*x^3 + (18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - (17*B*b^4*c^2 - 55*A*b^3*c^3)*d
^2*e + 4*(B*b^5*c - 4*A*b^4*c^2)*d*e^2 + (3*B*b^6 + A*b^5*c)*e^3)*x^2 + (13*A*b^4*c^2*d^2*e + 4*(B*b^4*c^2 - 2
*A*b^3*c^3)*d^3)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((B*x+A)*(e*x+d)**(7/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^3,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(b*e-c*d>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1025 vs. \(2 (335) = 670\).

Time = 0.31 (sec) , antiderivative size = 1025, normalized size of antiderivative = 2.82 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-1/4*(24*B*b*c*d^4 - 48*A*c^2*d^4 - 28*B*b^2*d^3*e + 84*A*b*c*d^3*e - 35*A*b^2*d^2*e^2)*arctan(sqrt(e*x + d)/s
qrt(-d))/(b^5*sqrt(-d)) + 1/4*(24*B*b*c^4*d^4 - 48*A*c^5*d^4 - 40*B*b^2*c^3*d^3*e + 108*A*b*c^4*d^3*e + 11*B*b
^3*c^2*d^2*e^2 - 71*A*b^2*c^3*d^2*e^2 + 2*B*b^4*c*d*e^3 + 10*A*b^3*c^2*d*e^3 + 3*B*b^5*e^4 + A*b^4*c*e^4)*arct
an(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^5*c^2) - 1/4*(12*(e*x + d)^(7/2)*B*b*c^4*d^3*
e - 24*(e*x + d)^(7/2)*A*c^5*d^3*e - 36*(e*x + d)^(5/2)*B*b*c^4*d^4*e + 72*(e*x + d)^(5/2)*A*c^5*d^4*e + 36*(e
*x + d)^(3/2)*B*b*c^4*d^5*e - 72*(e*x + d)^(3/2)*A*c^5*d^5*e - 12*sqrt(e*x + d)*B*b*c^4*d^6*e + 24*sqrt(e*x +
d)*A*c^5*d^6*e - 11*(e*x + d)^(7/2)*B*b^2*c^3*d^2*e^2 + 36*(e*x + d)^(7/2)*A*b*c^4*d^2*e^2 + 51*(e*x + d)^(5/2
)*B*b^2*c^3*d^3*e^2 - 144*(e*x + d)^(5/2)*A*b*c^4*d^3*e^2 - 69*(e*x + d)^(3/2)*B*b^2*c^3*d^4*e^2 + 180*(e*x +
d)^(3/2)*A*b*c^4*d^4*e^2 + 29*sqrt(e*x + d)*B*b^2*c^3*d^5*e^2 - 72*sqrt(e*x + d)*A*b*c^4*d^5*e^2 - 2*(e*x + d)
^(7/2)*B*b^3*c^2*d*e^3 - 10*(e*x + d)^(7/2)*A*b^2*c^3*d*e^3 - 11*(e*x + d)^(5/2)*B*b^3*c^2*d^2*e^3 + 85*(e*x +
d)^(5/2)*A*b^2*c^3*d^2*e^3 + 32*(e*x + d)^(3/2)*B*b^3*c^2*d^3*e^3 - 148*(e*x + d)^(3/2)*A*b^2*c^3*d^3*e^3 - 1
9*sqrt(e*x + d)*B*b^3*c^2*d^4*e^3 + 73*sqrt(e*x + d)*A*b^2*c^3*d^4*e^3 + 5*(e*x + d)^(7/2)*B*b^4*c*e^4 - (e*x
+ d)^(7/2)*A*b^3*c^2*e^4 - 11*(e*x + d)^(5/2)*B*b^4*c*d*e^4 - 13*(e*x + d)^(5/2)*A*b^3*c^2*d*e^4 + 7*(e*x + d)
^(3/2)*B*b^4*c*d^2*e^4 + 42*(e*x + d)^(3/2)*A*b^3*c^2*d^2*e^4 - sqrt(e*x + d)*B*b^4*c*d^3*e^4 - 26*sqrt(e*x +
d)*A*b^3*c^2*d^3*e^4 + 3*(e*x + d)^(5/2)*B*b^5*e^5 + (e*x + d)^(5/2)*A*b^4*c*e^5 - 6*(e*x + d)^(3/2)*B*b^5*d*e
^5 - 2*(e*x + d)^(3/2)*A*b^4*c*d*e^5 + 3*sqrt(e*x + d)*B*b^5*d^2*e^5 + sqrt(e*x + d)*A*b^4*c*d^2*e^5)/(((e*x +
d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e)^2*b^4*c^2)

Mupad [B] (verification not implemented)

Time = 15.77 (sec) , antiderivative size = 11072, normalized size of antiderivative = 30.50 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

int(((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^3,x)

[Out]

atan(((((64*A*b^13*c^4*d*e^6 + 192*B*b^14*c^3*d*e^6 - 1536*A*b^10*c^7*d^4*e^3 + 3072*A*b^11*c^6*d^3*e^4 - 1600
*A*b^12*c^5*d^2*e^5 + 768*B*b^11*c^6*d^4*e^3 - 1088*B*b^12*c^5*d^3*e^4 + 128*B*b^13*c^4*d^2*e^5)/(64*b^12*c^3)
- ((64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(d + e*x)^(1/2)*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e
^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 2
1*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*
b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*
b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^
3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2))/(8*b^8*c^3))*(-(9*B^2*b^9*e^7 - 2
304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3
- 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*
B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*
e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^
5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2) - (
(d + e*x)^(1/2)*(9*B^2*b^10*e^10 + A^2*b^8*c^2*e^10 + 4608*A^2*c^10*d^8*e^2 + 28896*A^2*b^2*c^8*d^6*e^4 - 2217
6*A^2*b^3*c^7*d^5*e^5 + 8330*A^2*b^4*c^6*d^4*e^6 - 1204*A^2*b^5*c^5*d^3*e^7 - 42*A^2*b^6*c^4*d^2*e^8 + 1152*B^
2*b^2*c^8*d^8*e^2 - 3264*B^2*b^3*c^7*d^7*e^3 + 2912*B^2*b^4*c^6*d^6*e^4 - 784*B^2*b^5*c^5*d^5*e^5 + 105*B^2*b^
6*c^4*d^4*e^6 - 196*B^2*b^7*c^3*d^3*e^7 + 70*B^2*b^8*c^2*d^2*e^8 + 12*B^2*b^9*c*d*e^9 - 18432*A^2*b*c^9*d^7*e^
3 + 20*A^2*b^7*c^3*d*e^9 + 6*A*B*b^9*c*e^10 - 4608*A*B*b*c^9*d^8*e^2 + 64*A*B*b^8*c^2*d*e^9 + 15744*A*B*b^2*c^
8*d^7*e^3 - 19488*A*B*b^3*c^7*d^6*e^4 + 10304*A*B*b^4*c^6*d^5*e^5 - 2170*A*B*b^5*c^5*d^4*e^6 + 504*A*B*b^6*c^4
*d^3*e^7 - 364*A*B*b^7*c^3*d^2*e^8))/(8*b^8*c^3))*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*
B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5
*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6
*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7
- 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A
*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2)*1i - (((64*A*b^13*c^4*d*e^6 + 192*B*b^14*c^
3*d*e^6 - 1536*A*b^10*c^7*d^4*e^3 + 3072*A*b^11*c^6*d^3*e^4 - 1600*A*b^12*c^5*d^2*e^5 + 768*B*b^11*c^6*d^4*e^3
- 1088*B*b^12*c^5*d^3*e^4 + 128*B*b^13*c^4*d^2*e^5)/(64*b^12*c^3) + ((64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(
d + e*x)^(1/2)*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7
*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*
e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^
6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*
b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^
3*d^2*e^5)/(64*b^10*c^5))^(1/2))/(8*b^8*c^3))*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*
b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4
*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e +
21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6
720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b
^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2) + ((d + e*x)^(1/2)*(9*B^2*b^10*e^10 + A^2*b^8*c
^2*e^10 + 4608*A^2*c^10*d^8*e^2 + 28896*A^2*b^2*c^8*d^6*e^4 - 22176*A^2*b^3*c^7*d^5*e^5 + 8330*A^2*b^4*c^6*d^4
*e^6 - 1204*A^2*b^5*c^5*d^3*e^7 - 42*A^2*b^6*c^4*d^2*e^8 + 1152*B^2*b^2*c^8*d^8*e^2 - 3264*B^2*b^3*c^7*d^7*e^3
+ 2912*B^2*b^4*c^6*d^6*e^4 - 784*B^2*b^5*c^5*d^5*e^5 + 105*B^2*b^6*c^4*d^4*e^6 - 196*B^2*b^7*c^3*d^3*e^7 + 70
*B^2*b^8*c^2*d^2*e^8 + 12*B^2*b^9*c*d*e^9 - 18432*A^2*b*c^9*d^7*e^3 + 20*A^2*b^7*c^3*d*e^9 + 6*A*B*b^9*c*e^10
- 4608*A*B*b*c^9*d^8*e^2 + 64*A*B*b^8*c^2*d*e^9 + 15744*A*B*b^2*c^8*d^7*e^3 - 19488*A*B*b^3*c^7*d^6*e^4 + 1030
4*A*B*b^4*c^6*d^5*e^5 - 2170*A*B*b^5*c^5*d^4*e^6 + 504*A*B*b^6*c^4*d^3*e^7 - 364*A*B*b^7*c^3*d^2*e^8))/(8*b^8*
c^3))*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2
+ 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105
*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e
^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d
*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5
)/(64*b^10*c^5))^(1/2)*1i)/((((64*A*b^13*c^4*d*e^6 + 192*B*b^14*c^3*d*e^6 - 1536*A*b^10*c^7*d^4*e^3 + 3072*A*b
^11*c^6*d^3*e^4 - 1600*A*b^12*c^5*d^2*e^5 + 768*B*b^11*c^6*d^4*e^3 - 1088*B*b^12*c^5*d^3*e^4 + 128*B*b^13*c^4*
d^2*e^5)/(64*b^12*c^3) - ((64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(d + e*x)^(1/2)*(-(9*B^2*b^9*e^7 - 2304*A^2*c
^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A
^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c
^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*
A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1
960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2))/(8*b^8*c^3)
)*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 58
80*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2
*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 +
1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6
+ 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(6
4*b^10*c^5))^(1/2) - ((d + e*x)^(1/2)*(9*B^2*b^10*e^10 + A^2*b^8*c^2*e^10 + 4608*A^2*c^10*d^8*e^2 + 28896*A^2*
b^2*c^8*d^6*e^4 - 22176*A^2*b^3*c^7*d^5*e^5 + 8330*A^2*b^4*c^6*d^4*e^6 - 1204*A^2*b^5*c^5*d^3*e^7 - 42*A^2*b^6
*c^4*d^2*e^8 + 1152*B^2*b^2*c^8*d^8*e^2 - 3264*B^2*b^3*c^7*d^7*e^3 + 2912*B^2*b^4*c^6*d^6*e^4 - 784*B^2*b^5*c^
5*d^5*e^5 + 105*B^2*b^6*c^4*d^4*e^6 - 196*B^2*b^7*c^3*d^3*e^7 + 70*B^2*b^8*c^2*d^2*e^8 + 12*B^2*b^9*c*d*e^9 -
18432*A^2*b*c^9*d^7*e^3 + 20*A^2*b^7*c^3*d*e^9 + 6*A*B*b^9*c*e^10 - 4608*A*B*b*c^9*d^8*e^2 + 64*A*B*b^8*c^2*d*
e^9 + 15744*A*B*b^2*c^8*d^7*e^3 - 19488*A*B*b^3*c^7*d^6*e^4 + 10304*A*B*b^4*c^6*d^5*e^5 - 2170*A*B*b^5*c^5*d^4
*e^6 + 504*A*B*b^6*c^4*d^3*e^7 - 364*A*B*b^7*c^3*d^2*e^8))/(8*b^8*c^3))*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 +
A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^
5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^
5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8
*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b
^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2) + (((64*A*b^13*c^4*d*
e^6 + 192*B*b^14*c^3*d*e^6 - 1536*A*b^10*c^7*d^4*e^3 + 3072*A*b^11*c^6*d^3*e^4 - 1600*A*b^12*c^5*d^2*e^5 + 768
*B*b^11*c^6*d^4*e^3 - 1088*B*b^12*c^5*d^3*e^4 + 128*B*b^13*c^4*d^2*e^5)/(64*b^12*c^3) + ((64*b^11*c^5*e^3 - 12
8*b^10*c^6*d*e^2)*(d + e*x)^(1/2)*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7
- 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 7
84*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*
c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*
c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e
^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2))/(8*b^8*c^3))*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7
*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e
^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 806
4*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 +
6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*
d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2) + ((d + e*x)^(1/2)*(9*B^2*b^
10*e^10 + A^2*b^8*c^2*e^10 + 4608*A^2*c^10*d^8*e^2 + 28896*A^2*b^2*c^8*d^6*e^4 - 22176*A^2*b^3*c^7*d^5*e^5 + 8
330*A^2*b^4*c^6*d^4*e^6 - 1204*A^2*b^5*c^5*d^3*e^7 - 42*A^2*b^6*c^4*d^2*e^8 + 1152*B^2*b^2*c^8*d^8*e^2 - 3264*
B^2*b^3*c^7*d^7*e^3 + 2912*B^2*b^4*c^6*d^6*e^4 - 784*B^2*b^5*c^5*d^5*e^5 + 105*B^2*b^6*c^4*d^4*e^6 - 196*B^2*b
^7*c^3*d^3*e^7 + 70*B^2*b^8*c^2*d^2*e^8 + 12*B^2*b^9*c*d*e^9 - 18432*A^2*b*c^9*d^7*e^3 + 20*A^2*b^7*c^3*d*e^9
+ 6*A*B*b^9*c*e^10 - 4608*A*B*b*c^9*d^8*e^2 + 64*A*B*b^8*c^2*d*e^9 + 15744*A*B*b^2*c^8*d^7*e^3 - 19488*A*B*b^3
*c^7*d^6*e^4 + 10304*A*B*b^4*c^6*d^5*e^5 - 2170*A*B*b^5*c^5*d^4*e^6 + 504*A*B*b^6*c^4*d^3*e^7 - 364*A*B*b^7*c^
3*d^2*e^8))/(8*b^8*c^3))*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A
^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^
4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 +
21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e
+ 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*
A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2) - (252*B^3*b^11*d^3*e^11 - 55296*A^3*c^11*d^11*e^3 - 694656*A^3*b^2*
c^9*d^9*e^5 + 844992*A^3*b^3*c^8*d^8*e^6 - 579096*A^3*b^4*c^7*d^7*e^7 + 212436*A^3*b^5*c^6*d^6*e^8 - 31282*A^3
*b^6*c^5*d^5*e^9 - 1877*A^3*b^7*c^4*d^4*e^10 + 616*A^3*b^8*c^3*d^3*e^11 + 35*A^3*b^9*c^2*d^2*e^12 + 6912*B^3*b
^3*c^8*d^11*e^3 - 25920*B^3*b^4*c^7*d^10*e^4 + 33408*B^3*b^5*c^6*d^9*e^5 - 16808*B^3*b^6*c^5*d^8*e^6 + 5180*B^
3*b^7*c^4*d^7*e^7 - 4816*B^3*b^8*c^3*d^6*e^8 + 1672*B^3*b^9*c^2*d^5*e^9 + 315*A*B^2*b^11*d^2*e^12 + 304128*A^3
*b*c^10*d^10*e^4 + 120*B^3*b^10*c*d^4*e^10 - 41472*A*B^2*b^2*c^9*d^11*e^3 + 179712*A*B^2*b^3*c^8*d^10*e^4 - 29
3184*A*B^2*b^4*c^7*d^9*e^5 + 220464*A*B^2*b^5*c^6*d^8*e^6 - 83076*A*B^2*b^6*c^5*d^7*e^7 + 31899*A*B^2*b^7*c^4*
d^6*e^8 - 18012*A*B^2*b^8*c^3*d^5*e^9 + 3522*A*B^2*b^9*c^2*d^4*e^10 - 407808*A^2*B*b^2*c^9*d^10*e^4 + 800064*A
^2*B*b^3*c^8*d^9*e^5 - 790704*A^2*B*b^4*c^7*d^8*e^6 + 413028*A^2*B*b^5*c^6*d^7*e^7 - 122283*A^2*B*b^6*c^5*d^6*
e^8 + 36402*A^2*B*b^7*c^4*d^5*e^9 - 13617*A^2*B*b^8*c^3*d^4*e^10 + 1764*A^2*B*b^9*c^2*d^3*e^11 - 168*A*B^2*b^1
0*c*d^3*e^11 + 82944*A^2*B*b*c^10*d^11*e^3 + 210*A^2*B*b^10*c*d^2*e^12)/(32*b^12*c^3)))*(-(9*B^2*b^9*e^7 - 230
4*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 -
1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^
2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e
+ 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*
e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2)*2i -
log((d^2*e^3*(b*e - c*d)^2*(35*A^3*b^7*c^2*e^7 - 55296*A^3*c^9*d^7 + 6912*B^3*b^3*c^6*d^7 + 315*A*B^2*b^9*e^7
+ 252*B^3*b^9*d*e^6 - 252288*A^3*b^2*c^7*d^5*e^2 + 146880*A^3*b^3*c^6*d^4*e^3 - 33048*A^3*b^4*c^5*d^3*e^4 - 54
0*A^3*b^5*c^4*d^2*e^5 + 2304*B^3*b^5*c^4*d^5*e^2 - 104*B^3*b^6*c^3*d^4*e^3 + 2668*B^3*b^7*c^2*d^3*e^4 + 82944*
A^2*B*b*c^8*d^7 + 210*A^2*B*b^8*c*e^7 + 193536*A^3*b*c^8*d^6*e - 41472*A*B^2*b^2*c^7*d^7 + 686*A^3*b^6*c^3*d*e
^6 - 12096*B^3*b^4*c^5*d^6*e + 624*B^3*b^8*c*d^2*e^5 - 58176*A*B^2*b^4*c^5*d^5*e^2 + 7344*A*B^2*b^5*c^4*d^4*e^
3 - 10212*A*B^2*b^6*c^3*d^3*e^4 + 4131*A*B^2*b^7*c^2*d^2*e^5 + 233280*A^2*B*b^3*c^6*d^5*e^2 - 82224*A^2*B*b^4*
c^5*d^4*e^3 + 15300*A^2*B*b^5*c^4*d^3*e^4 - 9459*A^2*B*b^6*c^3*d^2*e^5 + 462*A*B^2*b^8*c*d*e^6 + 96768*A*B^2*b
^3*c^6*d^6*e - 241920*A^2*B*b^2*c^7*d^6*e + 2184*A^2*B*b^7*c^2*d*e^6))/(64*b^12*c^3) - (((((d*e^3*(b*e - c*d)*
(24*A*c^3*d^2 + 3*B*b^3*e^2 + A*b^2*c*e^2 - 12*B*b*c^2*d^2 - 24*A*b*c^2*d*e + 5*B*b^2*c*d*e))/b^2 - b^2*c^2*e^
2*(b*e - 2*c*d)*(d + e*x)^(1/2)*((d^3*(35*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 28*B*b^2*d*e - 84*A*b*c*d*
e)^2)/b^10)^(1/2))*((d^3*(35*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 28*B*b^2*d*e - 84*A*b*c*d*e)^2)/b^10)^(
1/2))/8 - ((d + e*x)^(1/2)*(9*B^2*b^10*e^10 + A^2*b^8*c^2*e^10 + 4608*A^2*c^10*d^8*e^2 + 28896*A^2*b^2*c^8*d^6
*e^4 - 22176*A^2*b^3*c^7*d^5*e^5 + 8330*A^2*b^4*c^6*d^4*e^6 - 1204*A^2*b^5*c^5*d^3*e^7 - 42*A^2*b^6*c^4*d^2*e^
8 + 1152*B^2*b^2*c^8*d^8*e^2 - 3264*B^2*b^3*c^7*d^7*e^3 + 2912*B^2*b^4*c^6*d^6*e^4 - 784*B^2*b^5*c^5*d^5*e^5 +
105*B^2*b^6*c^4*d^4*e^6 - 196*B^2*b^7*c^3*d^3*e^7 + 70*B^2*b^8*c^2*d^2*e^8 + 12*B^2*b^9*c*d*e^9 - 18432*A^2*b
*c^9*d^7*e^3 + 20*A^2*b^7*c^3*d*e^9 + 6*A*B*b^9*c*e^10 - 4608*A*B*b*c^9*d^8*e^2 + 64*A*B*b^8*c^2*d*e^9 + 15744
*A*B*b^2*c^8*d^7*e^3 - 19488*A*B*b^3*c^7*d^6*e^4 + 10304*A*B*b^4*c^6*d^5*e^5 - 2170*A*B*b^5*c^5*d^4*e^6 + 504*
A*B*b^6*c^4*d^3*e^7 - 364*A*B*b^7*c^3*d^2*e^8))/(8*b^8*c^3))*((d^3*(35*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2
+ 28*B*b^2*d*e - 84*A*b*c*d*e)^2)/b^10)^(1/2))/8)*((36*A^2*c^4*d^7 + 9*B^2*b^2*c^2*d^7 + (1225*A^2*b^4*d^3*e^
4)/64 + (49*B^2*b^4*d^5*e^2)/4 + (651*A^2*b^2*c^2*d^5*e^2)/4 + (245*A*B*b^4*d^4*e^3)/8 - 126*A^2*b*c^3*d^6*e -
21*B^2*b^3*c*d^6*e - (735*A^2*b^3*c*d^4*e^3)/8 - 36*A*B*b*c^3*d^7 + 105*A*B*b^2*c^2*d^6*e - (399*A*B*b^3*c*d^
5*e^2)/4)/b^10)^(1/2) + log((d^2*e^3*(b*e - c*d)^2*(35*A^3*b^7*c^2*e^7 - 55296*A^3*c^9*d^7 + 6912*B^3*b^3*c^6*
d^7 + 315*A*B^2*b^9*e^7 + 252*B^3*b^9*d*e^6 - 252288*A^3*b^2*c^7*d^5*e^2 + 146880*A^3*b^3*c^6*d^4*e^3 - 33048*
A^3*b^4*c^5*d^3*e^4 - 540*A^3*b^5*c^4*d^2*e^5 + 2304*B^3*b^5*c^4*d^5*e^2 - 104*B^3*b^6*c^3*d^4*e^3 + 2668*B^3*
b^7*c^2*d^3*e^4 + 82944*A^2*B*b*c^8*d^7 + 210*A^2*B*b^8*c*e^7 + 193536*A^3*b*c^8*d^6*e - 41472*A*B^2*b^2*c^7*d
^7 + 686*A^3*b^6*c^3*d*e^6 - 12096*B^3*b^4*c^5*d^6*e + 624*B^3*b^8*c*d^2*e^5 - 58176*A*B^2*b^4*c^5*d^5*e^2 + 7
344*A*B^2*b^5*c^4*d^4*e^3 - 10212*A*B^2*b^6*c^3*d^3*e^4 + 4131*A*B^2*b^7*c^2*d^2*e^5 + 233280*A^2*B*b^3*c^6*d^
5*e^2 - 82224*A^2*B*b^4*c^5*d^4*e^3 + 15300*A^2*B*b^5*c^4*d^3*e^4 - 9459*A^2*B*b^6*c^3*d^2*e^5 + 462*A*B^2*b^8
*c*d*e^6 + 96768*A*B^2*b^3*c^6*d^6*e - 241920*A^2*B*b^2*c^7*d^6*e + 2184*A^2*B*b^7*c^2*d*e^6))/(64*b^12*c^3) -
(((((d*e^3*(b*e - c*d)*(24*A*c^3*d^2 + 3*B*b^3*e^2 + A*b^2*c*e^2 - 12*B*b*c^2*d^2 - 24*A*b*c^2*d*e + 5*B*b^2*
c*d*e))/b^2 + b^2*c^2*e^2*(b*e - 2*c*d)*(d + e*x)^(1/2)*((d^3*(35*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 28
*B*b^2*d*e - 84*A*b*c*d*e)^2)/b^10)^(1/2))*((d^3*(35*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 28*B*b^2*d*e -
84*A*b*c*d*e)^2)/b^10)^(1/2))/8 + ((d + e*x)^(1/2)*(9*B^2*b^10*e^10 + A^2*b^8*c^2*e^10 + 4608*A^2*c^10*d^8*e^2
+ 28896*A^2*b^2*c^8*d^6*e^4 - 22176*A^2*b^3*c^7*d^5*e^5 + 8330*A^2*b^4*c^6*d^4*e^6 - 1204*A^2*b^5*c^5*d^3*e^7
- 42*A^2*b^6*c^4*d^2*e^8 + 1152*B^2*b^2*c^8*d^8*e^2 - 3264*B^2*b^3*c^7*d^7*e^3 + 2912*B^2*b^4*c^6*d^6*e^4 - 7
84*B^2*b^5*c^5*d^5*e^5 + 105*B^2*b^6*c^4*d^4*e^6 - 196*B^2*b^7*c^3*d^3*e^7 + 70*B^2*b^8*c^2*d^2*e^8 + 12*B^2*b
^9*c*d*e^9 - 18432*A^2*b*c^9*d^7*e^3 + 20*A^2*b^7*c^3*d*e^9 + 6*A*B*b^9*c*e^10 - 4608*A*B*b*c^9*d^8*e^2 + 64*A
*B*b^8*c^2*d*e^9 + 15744*A*B*b^2*c^8*d^7*e^3 - 19488*A*B*b^3*c^7*d^6*e^4 + 10304*A*B*b^4*c^6*d^5*e^5 - 2170*A*
B*b^5*c^5*d^4*e^6 + 504*A*B*b^6*c^4*d^3*e^7 - 364*A*B*b^7*c^3*d^2*e^8))/(8*b^8*c^3))*((d^3*(35*A*b^2*e^2 + 48*
A*c^2*d^2 - 24*B*b*c*d^2 + 28*B*b^2*d*e - 84*A*b*c*d*e)^2)/b^10)^(1/2))/8)*((2304*A^2*c^4*d^7 + 576*B^2*b^2*c^
2*d^7 + 1225*A^2*b^4*d^3*e^4 + 784*B^2*b^4*d^5*e^2 + 10416*A^2*b^2*c^2*d^5*e^2 + 1960*A*B*b^4*d^4*e^3 - 8064*A
^2*b*c^3*d^6*e - 1344*B^2*b^3*c*d^6*e - 5880*A^2*b^3*c*d^4*e^3 - 2304*A*B*b*c^3*d^7 + 6720*A*B*b^2*c^2*d^6*e -
6384*A*B*b^3*c*d^5*e^2)/(64*b^10))^(1/2) + (((d + e*x)^(7/2)*(A*b^3*c*e^4 - 5*B*b^4*e^4 + 24*A*c^4*d^3*e - 36
*A*b*c^3*d^2*e^2 + 10*A*b^2*c^2*d*e^3 + 11*B*b^2*c^2*d^2*e^2 - 12*B*b*c^3*d^3*e + 2*B*b^3*c*d*e^3))/(4*b^4*c)
- ((d + e*x)^(5/2)*(3*B*b^5*e^5 + A*b^4*c*e^5 + 72*A*c^5*d^4*e - 144*A*b*c^4*d^3*e^2 - 13*A*b^3*c^2*d*e^4 + 85
*A*b^2*c^3*d^2*e^3 + 51*B*b^2*c^3*d^3*e^2 - 11*B*b^3*c^2*d^2*e^3 - 36*B*b*c^4*d^4*e - 11*B*b^4*c*d*e^4))/(4*b^
4*c^2) - ((d + e*x)^(1/2)*(24*A*c^5*d^6*e + 3*B*b^5*d^2*e^5 - 72*A*b*c^4*d^5*e^2 + A*b^4*c*d^2*e^5 - B*b^4*c*d
^3*e^4 + 73*A*b^2*c^3*d^4*e^3 - 26*A*b^3*c^2*d^3*e^4 + 29*B*b^2*c^3*d^5*e^2 - 19*B*b^3*c^2*d^4*e^3 - 12*B*b*c^
4*d^6*e))/(4*b^4*c^2) + ((d + e*x)^(3/2)*(72*A*c^5*d^5*e + 6*B*b^5*d*e^5 - 180*A*b*c^4*d^4*e^2 - 7*B*b^4*c*d^2
*e^4 + 148*A*b^2*c^3*d^3*e^3 - 42*A*b^3*c^2*d^2*e^4 + 69*B*b^2*c^3*d^4*e^2 - 32*B*b^3*c^2*d^3*e^3 + 2*A*b^4*c*
d*e^5 - 36*B*b*c^4*d^5*e))/(4*b^4*c^2))/(c^2*(d + e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) -
(4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e) + c^2*d^4 + b^2*d^2*e^2 - 2*b
*c*d^3*e)