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

Optimal. Leaf size=506 \[ -\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt {d+e x} (c d-b e)}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (b^2 (-e) (4 B d-5 A e)-4 b c d (2 B d-3 A e)+16 A c^2 d^2\right )}{4 b^5 d^{7/2}}+\frac {3 c^{5/2} \left (3 b^2 c e (11 A e+8 B d)-4 b c^2 d (11 A e+2 B d)+16 A c^3 d^2-21 b^3 B e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}}+\frac {b (c d-b e) \left (b^2 e (4 B d-5 A e)-b c d (5 A e+6 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-5 A e)+b^2 c d e (2 A e+21 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{4 b^4 d^2 \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)^2}+\frac {3 e \left (b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)+b^2 c^2 d^2 e (5 A e+9 B d)-4 b c^3 d^3 (4 A e+B d)+8 A c^4 d^4\right )}{4 b^4 d^3 \sqrt {d+e x} (c d-b e)^3} \]

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Rubi [A]  time = 1.45, antiderivative size = 506, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {822, 828, 826, 1166, 208} \begin {gather*} \frac {c x \left (b^2 c d e (2 A e+21 B d)+b^3 \left (-e^2\right ) (4 B d-5 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )+b (c d-b e) \left (b^2 e (4 B d-5 A e)-b c d (5 A e+6 B d)+12 A c^2 d^2\right )}{4 b^4 d^2 \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)^2}+\frac {3 e \left (b^2 c^2 d^2 e (5 A e+9 B d)-b^3 c d e^2 (4 B d-3 A e)+b^4 e^3 (4 B d-5 A e)-4 b c^3 d^3 (4 A e+B d)+8 A c^4 d^4\right )}{4 b^4 d^3 \sqrt {d+e x} (c d-b e)^3}+\frac {3 c^{5/2} \left (3 b^2 c e (11 A e+8 B d)-4 b c^2 d (11 A e+2 B d)+16 A c^3 d^2-21 b^3 B e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (b^2 (-e) (4 B d-5 A e)-4 b c d (2 B d-3 A e)+16 A c^2 d^2\right )}{4 b^5 d^{7/2}}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt {d+e x} (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*e*(8*A*c^4*d^4 + b^4*e^3*(4*B*d - 5*A*e) - b^3*c*d*e^2*(4*B*d - 3*A*e) - 4*b*c^3*d^3*(B*d + 4*A*e) + b^2*c^
2*d^2*e*(9*B*d + 5*A*e)))/(4*b^4*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) - (A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A
*e))*x)/(2*b^2*d*(c*d - b*e)*Sqrt[d + e*x]*(b*x + c*x^2)^2) + (b*(c*d - b*e)*(12*A*c^2*d^2 + b^2*e*(4*B*d - 5*
A*e) - b*c*d*(6*B*d + 5*A*e)) + c*(24*A*c^3*d^3 - b^3*e^2*(4*B*d - 5*A*e) + b^2*c*d*e*(21*B*d + 2*A*e) - 12*b*
c^2*d^2*(B*d + 3*A*e))*x)/(4*b^4*d^2*(c*d - b*e)^2*Sqrt[d + e*x]*(b*x + c*x^2)) - (3*(16*A*c^2*d^2 - b^2*e*(4*
B*d - 5*A*e) - 4*b*c*d*(2*B*d - 3*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(7/2)) + (3*c^(5/2)*(16*A*c^3
*d^2 - 21*b^3*B*e^2 - 4*b*c^2*d*(2*B*d + 11*A*e) + 3*b^2*c*e*(8*B*d + 11*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])
/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(7/2))

Rule 208

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

Rule 822

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

Rule 826

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 828

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

Rule 1166

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*} \int \frac {A+B x}{(d+e x)^{3/2} \left (b x+c x^2\right )^3} \, dx &=-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (12 A c^2 d^2+b^2 e (4 B d-5 A e)-b c d (6 B d+5 A e)\right )-\frac {7}{2} c e (b B d-2 A c d+A b e) x}{(d+e x)^{3/2} \left (b x+c x^2\right )^2} \, dx}{2 b^2 d (c d-b e)}\\ &=-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-5 A e)-b c d (6 B d+5 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-5 A e)+b^2 c d e (21 B d+2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} (c d-b e)^2 \left (16 A c^2 d^2-b^2 e (4 B d-5 A e)-4 b c d (2 B d-3 A e)\right )+\frac {3}{4} c e \left (24 A c^3 d^3-b^3 e^2 (4 B d-5 A e)+b^2 c d e (21 B d+2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^2 (c d-b e)^2}\\ &=\frac {3 e \left (8 A c^4 d^4+b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)-4 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (9 B d+5 A e)\right )}{4 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-5 A e)-b c d (6 B d+5 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-5 A e)+b^2 c d e (21 B d+2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} (c d-b e)^3 \left (16 A c^2 d^2-b^2 e (4 B d-5 A e)-4 b c d (2 B d-3 A e)\right )+\frac {3}{4} c e \left (8 A c^4 d^4+b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)-4 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (9 B d+5 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d^3 (c d-b e)^3}\\ &=\frac {3 e \left (8 A c^4 d^4+b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)-4 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (9 B d+5 A e)\right )}{4 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-5 A e)-b c d (6 B d+5 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-5 A e)+b^2 c d e (21 B d+2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {3}{4} e (c d-b e)^3 \left (16 A c^2 d^2-b^2 e (4 B d-5 A e)-4 b c d (2 B d-3 A e)\right )-\frac {3}{4} c d e \left (8 A c^4 d^4+b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)-4 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (9 B d+5 A e)\right )+\frac {3}{4} c e \left (8 A c^4 d^4+b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)-4 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (9 B d+5 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 d^3 (c d-b e)^3}\\ &=\frac {3 e \left (8 A c^4 d^4+b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)-4 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (9 B d+5 A e)\right )}{4 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-5 A e)-b c d (6 B d+5 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-5 A e)+b^2 c d e (21 B d+2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {\left (3 c \left (16 A c^2 d^2-b^2 e (4 B d-5 A e)-4 b c d (2 B d-3 A e)\right )\right ) \operatorname {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 d^3}-\frac {\left (3 c^3 \left (16 A c^3 d^2-21 b^3 B e^2-4 b c^2 d (2 B d+11 A e)+3 b^2 c e (8 B d+11 A e)\right )\right ) \operatorname {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 d-b e)^3}\\ &=\frac {3 e \left (8 A c^4 d^4+b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)-4 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (9 B d+5 A e)\right )}{4 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-5 A e)-b c d (6 B d+5 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-5 A e)+b^2 c d e (21 B d+2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}-\frac {3 \left (16 A c^2 d^2-b^2 e (4 B d-5 A e)-4 b c d (2 B d-3 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{7/2}}+\frac {3 c^{5/2} \left (16 A c^3 d^2-21 b^3 B e^2-4 b c^2 d (2 B d+11 A e)+3 b^2 c e (8 B d+11 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}}\\ \end {align*}

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Mathematica [C]  time = 0.55, size = 387, normalized size = 0.76 \begin {gather*} \frac {2 A b^4 d^2 (b e-c d)^3+b^3 d x (b e-c d)^3 (-5 A b e-8 A c d+4 b B d)+x^2 \left (b^2 c d (c d-b e)^2 \left (b^2 e (4 B d-5 A e)-b c d (5 A e+6 B d)+12 A c^2 d^2\right )+(b+c x) \left (b c d (b e-c d) \left (b^3 e^2 (4 B d-5 A e)-b^2 c d e (2 A e+21 B d)+12 b c^2 d^2 (3 A e+B d)-24 A c^3 d^3\right )-(b+c x) \left (3 c^2 d^3 \left (3 b^2 c e (11 A e+8 B d)-4 b c^2 d (11 A e+2 B d)+16 A c^3 d^2-21 b^3 B e^2\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c (d+e x)}{c d-b e}\right )-3 (c d-b e)^3 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {e x}{d}+1\right ) \left (b^2 e (5 A e-4 B d)+4 b c d (3 A e-2 B d)+16 A c^2 d^2\right )\right )\right )\right )}{4 b^5 d^3 x^2 (b+c x)^2 \sqrt {d+e x} (c d-b e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^4*d^2*(-(c*d) + b*e)^3 + b^3*d*(-(c*d) + b*e)^3*(4*b*B*d - 8*A*c*d - 5*A*b*e)*x + x^2*(b^2*c*d*(c*d - b
*e)^2*(12*A*c^2*d^2 + b^2*e*(4*B*d - 5*A*e) - b*c*d*(6*B*d + 5*A*e)) + (b + c*x)*(b*c*d*(-(c*d) + b*e)*(-24*A*
c^3*d^3 + b^3*e^2*(4*B*d - 5*A*e) - b^2*c*d*e*(21*B*d + 2*A*e) + 12*b*c^2*d^2*(B*d + 3*A*e)) - (b + c*x)*(3*c^
2*d^3*(16*A*c^3*d^2 - 21*b^3*B*e^2 - 4*b*c^2*d*(2*B*d + 11*A*e) + 3*b^2*c*e*(8*B*d + 11*A*e))*Hypergeometric2F
1[-1/2, 1, 1/2, (c*(d + e*x))/(c*d - b*e)] - 3*(c*d - b*e)^3*(16*A*c^2*d^2 + 4*b*c*d*(-2*B*d + 3*A*e) + b^2*e*
(-4*B*d + 5*A*e))*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (e*x)/d]))))/(4*b^5*d^3*(c*d - b*e)^3*x^2*(b + c*x)^2*Sq
rt[d + e*x])

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IntegrateAlgebraic [B]  time = 2.31, size = 1208, normalized size = 2.39 \begin {gather*} -\frac {-24 A c^6 (d+e x) d^7+12 b B c^5 (d+e x) d^7+72 A c^6 (d+e x)^2 d^6-36 b B c^5 (d+e x)^2 d^6+84 A b c^5 e (d+e x) d^6-45 b^2 B c^4 e (d+e x) d^6+8 b^4 B c^2 e^3 d^5-72 A c^6 (d+e x)^3 d^5+36 b B c^5 (d+e x)^3 d^5-216 A b c^5 e (d+e x)^2 d^5+117 b^2 B c^4 e (d+e x)^2 d^5-96 A b^2 c^4 e^2 (d+e x) d^5+57 b^3 B c^3 e^2 (d+e x) d^5-8 A b^4 c^2 e^4 d^4-16 b^5 B c e^4 d^4+24 A c^6 (d+e x)^4 d^4-12 b B c^5 (d+e x)^4 d^4+180 A b c^5 e (d+e x)^3 d^4-99 b^2 B c^4 e (d+e x)^3 d^4+199 A b^2 c^4 e^2 (d+e x)^2 d^4-122 b^3 B c^3 e^2 (d+e x)^2 d^4+30 A b^3 c^3 e^3 (d+e x) d^4-72 b^4 B c^2 e^3 (d+e x) d^4+8 b^6 B e^5 d^3+16 A b^5 c e^5 d^3-48 A b c^5 e (d+e x)^4 d^3+27 b^2 B c^4 e (d+e x)^4 d^3-118 A b^2 c^4 e^2 (d+e x)^3 d^3+77 b^3 B c^3 e^2 (d+e x)^3 d^3-38 A b^3 c^3 e^3 (d+e x)^2 d^3+120 b^4 B c^2 e^3 (d+e x)^2 d^3+62 A b^4 c^2 e^4 (d+e x) d^3+68 b^5 B c e^4 (d+e x) d^3-8 A b^6 e^6 d^2+15 A b^2 c^4 e^2 (d+e x)^4 d^2-12 b^3 B c^3 e^2 (d+e x)^4 d^2-3 A b^3 c^3 e^3 (d+e x)^3 d^2-68 b^4 B c^2 e^3 (d+e x)^3 d^2-106 A b^4 c^2 e^4 (d+e x)^2 d^2-76 b^5 B c e^4 (d+e x)^2 d^2-20 b^6 B e^5 (d+e x) d^2-81 A b^5 c e^5 (d+e x) d^2+9 A b^3 c^3 e^3 (d+e x)^4 d+12 b^4 B c^2 e^3 (d+e x)^4 d+73 A b^4 c^2 e^4 (d+e x)^3 d+24 b^5 B c e^4 (d+e x)^3 d+12 b^6 B e^5 (d+e x)^2 d+89 A b^5 c e^5 (d+e x)^2 d+25 A b^6 e^6 (d+e x) d-15 A b^4 c^2 e^4 (d+e x)^4-30 A b^5 c e^5 (d+e x)^3-15 A b^6 e^6 (d+e x)^2}{4 b^4 d^3 e (b e-c d)^3 x^2 \sqrt {d+e x} (-c d+b e+c (d+e x))^2}-\frac {3 \left (16 A d^2 c^{11/2}-8 b B d^2 c^{9/2}-44 A b d e c^{9/2}+33 A b^2 e^2 c^{7/2}+24 b^2 B d e c^{7/2}-21 b^3 B e^2 c^{5/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {b e-c d} \sqrt {d+e x}}{c d-b e}\right )}{4 b^5 (b e-c d)^{7/2}}+\frac {3 \left (-5 A e^2 b^2+4 B d e b^2+8 B c d^2 b-12 A c d e b-16 A c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(8*b^4*B*c^2*d^5*e^3 - 16*b^5*B*c*d^4*e^4 - 8*A*b^4*c^2*d^4*e^4 + 8*b^6*B*d^3*e^5 + 16*A*b^5*c*d^3*e^5 -
8*A*b^6*d^2*e^6 + 12*b*B*c^5*d^7*(d + e*x) - 24*A*c^6*d^7*(d + e*x) - 45*b^2*B*c^4*d^6*e*(d + e*x) + 84*A*b*c^
5*d^6*e*(d + e*x) + 57*b^3*B*c^3*d^5*e^2*(d + e*x) - 96*A*b^2*c^4*d^5*e^2*(d + e*x) - 72*b^4*B*c^2*d^4*e^3*(d
+ e*x) + 30*A*b^3*c^3*d^4*e^3*(d + e*x) + 68*b^5*B*c*d^3*e^4*(d + e*x) + 62*A*b^4*c^2*d^3*e^4*(d + e*x) - 20*b
^6*B*d^2*e^5*(d + e*x) - 81*A*b^5*c*d^2*e^5*(d + e*x) + 25*A*b^6*d*e^6*(d + e*x) - 36*b*B*c^5*d^6*(d + e*x)^2
+ 72*A*c^6*d^6*(d + e*x)^2 + 117*b^2*B*c^4*d^5*e*(d + e*x)^2 - 216*A*b*c^5*d^5*e*(d + e*x)^2 - 122*b^3*B*c^3*d
^4*e^2*(d + e*x)^2 + 199*A*b^2*c^4*d^4*e^2*(d + e*x)^2 + 120*b^4*B*c^2*d^3*e^3*(d + e*x)^2 - 38*A*b^3*c^3*d^3*
e^3*(d + e*x)^2 - 76*b^5*B*c*d^2*e^4*(d + e*x)^2 - 106*A*b^4*c^2*d^2*e^4*(d + e*x)^2 + 12*b^6*B*d*e^5*(d + e*x
)^2 + 89*A*b^5*c*d*e^5*(d + e*x)^2 - 15*A*b^6*e^6*(d + e*x)^2 + 36*b*B*c^5*d^5*(d + e*x)^3 - 72*A*c^6*d^5*(d +
e*x)^3 - 99*b^2*B*c^4*d^4*e*(d + e*x)^3 + 180*A*b*c^5*d^4*e*(d + e*x)^3 + 77*b^3*B*c^3*d^3*e^2*(d + e*x)^3 -
118*A*b^2*c^4*d^3*e^2*(d + e*x)^3 - 68*b^4*B*c^2*d^2*e^3*(d + e*x)^3 - 3*A*b^3*c^3*d^2*e^3*(d + e*x)^3 + 24*b^
5*B*c*d*e^4*(d + e*x)^3 + 73*A*b^4*c^2*d*e^4*(d + e*x)^3 - 30*A*b^5*c*e^5*(d + e*x)^3 - 12*b*B*c^5*d^4*(d + e*
x)^4 + 24*A*c^6*d^4*(d + e*x)^4 + 27*b^2*B*c^4*d^3*e*(d + e*x)^4 - 48*A*b*c^5*d^3*e*(d + e*x)^4 - 12*b^3*B*c^3
*d^2*e^2*(d + e*x)^4 + 15*A*b^2*c^4*d^2*e^2*(d + e*x)^4 + 12*b^4*B*c^2*d*e^3*(d + e*x)^4 + 9*A*b^3*c^3*d*e^3*(
d + e*x)^4 - 15*A*b^4*c^2*e^4*(d + e*x)^4)/(b^4*d^3*e*(-(c*d) + b*e)^3*x^2*Sqrt[d + e*x]*(-(c*d) + b*e + c*(d
+ e*x))^2) - (3*(-8*b*B*c^(9/2)*d^2 + 16*A*c^(11/2)*d^2 + 24*b^2*B*c^(7/2)*d*e - 44*A*b*c^(9/2)*d*e - 21*b^3*B
*c^(5/2)*e^2 + 33*A*b^2*c^(7/2)*e^2)*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(4*b^5*(-
(c*d) + b*e)^(7/2)) + (3*(8*b*B*c*d^2 - 16*A*c^2*d^2 + 4*b^2*B*d*e - 12*A*b*c*d*e - 5*A*b^2*e^2)*ArcTanh[Sqrt[
d + e*x]/Sqrt[d]])/(4*b^5*d^(7/2))

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fricas [B]  time = 116.35, size = 7400, normalized size = 14.62

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*(3*((8*(B*b*c^6 - 2*A*c^7)*d^6*e - 4*(6*B*b^2*c^5 - 11*A*b*c^6)*d^5*e^2 + 3*(7*B*b^3*c^4 - 11*A*b^2*c^5)
*d^4*e^3)*x^5 + (8*(B*b*c^6 - 2*A*c^7)*d^7 - 4*(2*B*b^2*c^5 - 3*A*b*c^6)*d^6*e - (27*B*b^3*c^4 - 55*A*b^2*c^5)
*d^5*e^2 + 6*(7*B*b^4*c^3 - 11*A*b^3*c^4)*d^4*e^3)*x^4 + (16*(B*b^2*c^5 - 2*A*b*c^6)*d^7 - 8*(5*B*b^3*c^4 - 9*
A*b^2*c^5)*d^6*e + 2*(9*B*b^4*c^3 - 11*A*b^3*c^4)*d^5*e^2 + 3*(7*B*b^5*c^2 - 11*A*b^4*c^3)*d^4*e^3)*x^3 + (8*(
B*b^3*c^4 - 2*A*b^2*c^5)*d^7 - 4*(6*B*b^4*c^3 - 11*A*b^3*c^4)*d^6*e + 3*(7*B*b^5*c^2 - 11*A*b^4*c^3)*d^5*e^2)*
x^2)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)
) - 3*((5*A*b^5*c^2*e^6 + 8*(B*b*c^6 - 2*A*c^7)*d^5*e - 4*(5*B*b^2*c^5 - 9*A*b*c^6)*d^4*e^2 + (12*B*b^3*c^4 -
17*A*b^2*c^5)*d^3*e^3 + (4*B*b^4*c^3 - 5*A*b^3*c^4)*d^2*e^4 - (4*B*b^5*c^2 + 3*A*b^4*c^3)*d*e^5)*x^5 + (10*A*b
^6*c*e^6 + 8*(B*b*c^6 - 2*A*c^7)*d^6 - 4*(B*b^2*c^5 - A*b*c^6)*d^5*e - (28*B*b^3*c^4 - 55*A*b^2*c^5)*d^4*e^2 +
(28*B*b^4*c^3 - 39*A*b^3*c^4)*d^3*e^3 + (4*B*b^5*c^2 - 13*A*b^4*c^3)*d^2*e^4 - (8*B*b^6*c + A*b^5*c^2)*d*e^5)
*x^4 + (5*A*b^7*e^6 + 16*(B*b^2*c^5 - 2*A*b*c^6)*d^6 - 8*(4*B*b^3*c^4 - 7*A*b^2*c^5)*d^5*e + 2*(2*B*b^4*c^3 +
A*b^3*c^4)*d^4*e^2 + (20*B*b^5*c^2 - 27*A*b^4*c^3)*d^3*e^3 - (4*B*b^6*c + 11*A*b^5*c^2)*d^2*e^4 - (4*B*b^7 - 7
*A*b^6*c)*d*e^5)*x^3 + (5*A*b^7*d*e^5 + 8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^6 - 4*(5*B*b^4*c^3 - 9*A*b^3*c^4)*d^5*e
+ (12*B*b^5*c^2 - 17*A*b^4*c^3)*d^4*e^2 + (4*B*b^6*c - 5*A*b^5*c^2)*d^3*e^3 - (4*B*b^7 + 3*A*b^6*c)*d^2*e^4)*x
^2)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c^3*d^6 - 6*A*b^5*c^2*d^5*e + 6*A*b^6*c*
d^4*e^2 - 2*A*b^7*d^3*e^3 + 3*(5*A*b^5*c^2*d*e^5 + 4*(B*b^2*c^5 - 2*A*b*c^6)*d^5*e - (9*B*b^3*c^4 - 16*A*b^2*c
^5)*d^4*e^2 + (4*B*b^4*c^3 - 5*A*b^3*c^4)*d^3*e^3 - (4*B*b^5*c^2 + 3*A*b^4*c^3)*d^2*e^4)*x^4 + (30*A*b^6*c*d*e
^5 + 12*(B*b^2*c^5 - 2*A*b*c^6)*d^6 - 3*(3*B*b^3*c^4 - 4*A*b^2*c^5)*d^5*e - 29*(B*b^4*c^3 - 2*A*b^3*c^4)*d^4*e
^2 + (20*B*b^5*c^2 - 33*A*b^4*c^3)*d^3*e^3 - (24*B*b^6*c + 13*A*b^5*c^2)*d^2*e^4)*x^3 + (15*A*b^7*d*e^5 + 18*(
B*b^3*c^4 - 2*A*b^2*c^5)*d^6 - (37*B*b^4*c^3 - 65*A*b^3*c^4)*d^5*e + (12*B*b^5*c^2 - 7*A*b^4*c^3)*d^4*e^2 + (4
*B*b^6*c - 23*A*b^5*c^2)*d^3*e^3 - (12*B*b^7 - A*b^6*c)*d^2*e^4)*x^2 + (5*A*b^7*d^2*e^4 + 4*(B*b^4*c^3 - 2*A*b
^3*c^4)*d^6 - (12*B*b^5*c^2 - 19*A*b^4*c^3)*d^5*e + 3*(4*B*b^6*c - 3*A*b^5*c^2)*d^4*e^2 - (4*B*b^7 + 7*A*b^6*c
)*d^3*e^3)*x)*sqrt(e*x + d))/((b^5*c^5*d^7*e - 3*b^6*c^4*d^6*e^2 + 3*b^7*c^3*d^5*e^3 - b^8*c^2*d^4*e^4)*x^5 +
(b^5*c^5*d^8 - b^6*c^4*d^7*e - 3*b^7*c^3*d^6*e^2 + 5*b^8*c^2*d^5*e^3 - 2*b^9*c*d^4*e^4)*x^4 + (2*b^6*c^4*d^8 -
5*b^7*c^3*d^7*e + 3*b^8*c^2*d^6*e^2 + b^9*c*d^5*e^3 - b^10*d^4*e^4)*x^3 + (b^7*c^3*d^8 - 3*b^8*c^2*d^7*e + 3*
b^9*c*d^6*e^2 - b^10*d^5*e^3)*x^2), -1/8*(6*((8*(B*b*c^6 - 2*A*c^7)*d^6*e - 4*(6*B*b^2*c^5 - 11*A*b*c^6)*d^5*e
^2 + 3*(7*B*b^3*c^4 - 11*A*b^2*c^5)*d^4*e^3)*x^5 + (8*(B*b*c^6 - 2*A*c^7)*d^7 - 4*(2*B*b^2*c^5 - 3*A*b*c^6)*d^
6*e - (27*B*b^3*c^4 - 55*A*b^2*c^5)*d^5*e^2 + 6*(7*B*b^4*c^3 - 11*A*b^3*c^4)*d^4*e^3)*x^4 + (16*(B*b^2*c^5 - 2
*A*b*c^6)*d^7 - 8*(5*B*b^3*c^4 - 9*A*b^2*c^5)*d^6*e + 2*(9*B*b^4*c^3 - 11*A*b^3*c^4)*d^5*e^2 + 3*(7*B*b^5*c^2
- 11*A*b^4*c^3)*d^4*e^3)*x^3 + (8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^7 - 4*(6*B*b^4*c^3 - 11*A*b^3*c^4)*d^6*e + 3*(7*
B*b^5*c^2 - 11*A*b^4*c^3)*d^5*e^2)*x^2)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d -
b*e))/(c*e*x + c*d)) - 3*((5*A*b^5*c^2*e^6 + 8*(B*b*c^6 - 2*A*c^7)*d^5*e - 4*(5*B*b^2*c^5 - 9*A*b*c^6)*d^4*e^2
+ (12*B*b^3*c^4 - 17*A*b^2*c^5)*d^3*e^3 + (4*B*b^4*c^3 - 5*A*b^3*c^4)*d^2*e^4 - (4*B*b^5*c^2 + 3*A*b^4*c^3)*d
*e^5)*x^5 + (10*A*b^6*c*e^6 + 8*(B*b*c^6 - 2*A*c^7)*d^6 - 4*(B*b^2*c^5 - A*b*c^6)*d^5*e - (28*B*b^3*c^4 - 55*A
*b^2*c^5)*d^4*e^2 + (28*B*b^4*c^3 - 39*A*b^3*c^4)*d^3*e^3 + (4*B*b^5*c^2 - 13*A*b^4*c^3)*d^2*e^4 - (8*B*b^6*c
+ A*b^5*c^2)*d*e^5)*x^4 + (5*A*b^7*e^6 + 16*(B*b^2*c^5 - 2*A*b*c^6)*d^6 - 8*(4*B*b^3*c^4 - 7*A*b^2*c^5)*d^5*e
+ 2*(2*B*b^4*c^3 + A*b^3*c^4)*d^4*e^2 + (20*B*b^5*c^2 - 27*A*b^4*c^3)*d^3*e^3 - (4*B*b^6*c + 11*A*b^5*c^2)*d^2
*e^4 - (4*B*b^7 - 7*A*b^6*c)*d*e^5)*x^3 + (5*A*b^7*d*e^5 + 8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^6 - 4*(5*B*b^4*c^3 -
9*A*b^3*c^4)*d^5*e + (12*B*b^5*c^2 - 17*A*b^4*c^3)*d^4*e^2 + (4*B*b^6*c - 5*A*b^5*c^2)*d^3*e^3 - (4*B*b^7 + 3*
A*b^6*c)*d^2*e^4)*x^2)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c^3*d^6 - 6*A*b^5*c^2
*d^5*e + 6*A*b^6*c*d^4*e^2 - 2*A*b^7*d^3*e^3 + 3*(5*A*b^5*c^2*d*e^5 + 4*(B*b^2*c^5 - 2*A*b*c^6)*d^5*e - (9*B*b
^3*c^4 - 16*A*b^2*c^5)*d^4*e^2 + (4*B*b^4*c^3 - 5*A*b^3*c^4)*d^3*e^3 - (4*B*b^5*c^2 + 3*A*b^4*c^3)*d^2*e^4)*x^
4 + (30*A*b^6*c*d*e^5 + 12*(B*b^2*c^5 - 2*A*b*c^6)*d^6 - 3*(3*B*b^3*c^4 - 4*A*b^2*c^5)*d^5*e - 29*(B*b^4*c^3 -
2*A*b^3*c^4)*d^4*e^2 + (20*B*b^5*c^2 - 33*A*b^4*c^3)*d^3*e^3 - (24*B*b^6*c + 13*A*b^5*c^2)*d^2*e^4)*x^3 + (15
*A*b^7*d*e^5 + 18*(B*b^3*c^4 - 2*A*b^2*c^5)*d^6 - (37*B*b^4*c^3 - 65*A*b^3*c^4)*d^5*e + (12*B*b^5*c^2 - 7*A*b^
4*c^3)*d^4*e^2 + (4*B*b^6*c - 23*A*b^5*c^2)*d^3*e^3 - (12*B*b^7 - A*b^6*c)*d^2*e^4)*x^2 + (5*A*b^7*d^2*e^4 + 4
*(B*b^4*c^3 - 2*A*b^3*c^4)*d^6 - (12*B*b^5*c^2 - 19*A*b^4*c^3)*d^5*e + 3*(4*B*b^6*c - 3*A*b^5*c^2)*d^4*e^2 - (
4*B*b^7 + 7*A*b^6*c)*d^3*e^3)*x)*sqrt(e*x + d))/((b^5*c^5*d^7*e - 3*b^6*c^4*d^6*e^2 + 3*b^7*c^3*d^5*e^3 - b^8*
c^2*d^4*e^4)*x^5 + (b^5*c^5*d^8 - b^6*c^4*d^7*e - 3*b^7*c^3*d^6*e^2 + 5*b^8*c^2*d^5*e^3 - 2*b^9*c*d^4*e^4)*x^4
+ (2*b^6*c^4*d^8 - 5*b^7*c^3*d^7*e + 3*b^8*c^2*d^6*e^2 + b^9*c*d^5*e^3 - b^10*d^4*e^4)*x^3 + (b^7*c^3*d^8 - 3
*b^8*c^2*d^7*e + 3*b^9*c*d^6*e^2 - b^10*d^5*e^3)*x^2), -1/8*(6*((5*A*b^5*c^2*e^6 + 8*(B*b*c^6 - 2*A*c^7)*d^5*e
- 4*(5*B*b^2*c^5 - 9*A*b*c^6)*d^4*e^2 + (12*B*b^3*c^4 - 17*A*b^2*c^5)*d^3*e^3 + (4*B*b^4*c^3 - 5*A*b^3*c^4)*d
^2*e^4 - (4*B*b^5*c^2 + 3*A*b^4*c^3)*d*e^5)*x^5 + (10*A*b^6*c*e^6 + 8*(B*b*c^6 - 2*A*c^7)*d^6 - 4*(B*b^2*c^5 -
A*b*c^6)*d^5*e - (28*B*b^3*c^4 - 55*A*b^2*c^5)*d^4*e^2 + (28*B*b^4*c^3 - 39*A*b^3*c^4)*d^3*e^3 + (4*B*b^5*c^2
- 13*A*b^4*c^3)*d^2*e^4 - (8*B*b^6*c + A*b^5*c^2)*d*e^5)*x^4 + (5*A*b^7*e^6 + 16*(B*b^2*c^5 - 2*A*b*c^6)*d^6
- 8*(4*B*b^3*c^4 - 7*A*b^2*c^5)*d^5*e + 2*(2*B*b^4*c^3 + A*b^3*c^4)*d^4*e^2 + (20*B*b^5*c^2 - 27*A*b^4*c^3)*d^
3*e^3 - (4*B*b^6*c + 11*A*b^5*c^2)*d^2*e^4 - (4*B*b^7 - 7*A*b^6*c)*d*e^5)*x^3 + (5*A*b^7*d*e^5 + 8*(B*b^3*c^4
- 2*A*b^2*c^5)*d^6 - 4*(5*B*b^4*c^3 - 9*A*b^3*c^4)*d^5*e + (12*B*b^5*c^2 - 17*A*b^4*c^3)*d^4*e^2 + (4*B*b^6*c
- 5*A*b^5*c^2)*d^3*e^3 - (4*B*b^7 + 3*A*b^6*c)*d^2*e^4)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + 3*((8
*(B*b*c^6 - 2*A*c^7)*d^6*e - 4*(6*B*b^2*c^5 - 11*A*b*c^6)*d^5*e^2 + 3*(7*B*b^3*c^4 - 11*A*b^2*c^5)*d^4*e^3)*x^
5 + (8*(B*b*c^6 - 2*A*c^7)*d^7 - 4*(2*B*b^2*c^5 - 3*A*b*c^6)*d^6*e - (27*B*b^3*c^4 - 55*A*b^2*c^5)*d^5*e^2 + 6
*(7*B*b^4*c^3 - 11*A*b^3*c^4)*d^4*e^3)*x^4 + (16*(B*b^2*c^5 - 2*A*b*c^6)*d^7 - 8*(5*B*b^3*c^4 - 9*A*b^2*c^5)*d
^6*e + 2*(9*B*b^4*c^3 - 11*A*b^3*c^4)*d^5*e^2 + 3*(7*B*b^5*c^2 - 11*A*b^4*c^3)*d^4*e^3)*x^3 + (8*(B*b^3*c^4 -
2*A*b^2*c^5)*d^7 - 4*(6*B*b^4*c^3 - 11*A*b^3*c^4)*d^6*e + 3*(7*B*b^5*c^2 - 11*A*b^4*c^3)*d^5*e^2)*x^2)*sqrt(c/
(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + 2*(2*A*b
^4*c^3*d^6 - 6*A*b^5*c^2*d^5*e + 6*A*b^6*c*d^4*e^2 - 2*A*b^7*d^3*e^3 + 3*(5*A*b^5*c^2*d*e^5 + 4*(B*b^2*c^5 - 2
*A*b*c^6)*d^5*e - (9*B*b^3*c^4 - 16*A*b^2*c^5)*d^4*e^2 + (4*B*b^4*c^3 - 5*A*b^3*c^4)*d^3*e^3 - (4*B*b^5*c^2 +
3*A*b^4*c^3)*d^2*e^4)*x^4 + (30*A*b^6*c*d*e^5 + 12*(B*b^2*c^5 - 2*A*b*c^6)*d^6 - 3*(3*B*b^3*c^4 - 4*A*b^2*c^5)
*d^5*e - 29*(B*b^4*c^3 - 2*A*b^3*c^4)*d^4*e^2 + (20*B*b^5*c^2 - 33*A*b^4*c^3)*d^3*e^3 - (24*B*b^6*c + 13*A*b^5
*c^2)*d^2*e^4)*x^3 + (15*A*b^7*d*e^5 + 18*(B*b^3*c^4 - 2*A*b^2*c^5)*d^6 - (37*B*b^4*c^3 - 65*A*b^3*c^4)*d^5*e
+ (12*B*b^5*c^2 - 7*A*b^4*c^3)*d^4*e^2 + (4*B*b^6*c - 23*A*b^5*c^2)*d^3*e^3 - (12*B*b^7 - A*b^6*c)*d^2*e^4)*x^
2 + (5*A*b^7*d^2*e^4 + 4*(B*b^4*c^3 - 2*A*b^3*c^4)*d^6 - (12*B*b^5*c^2 - 19*A*b^4*c^3)*d^5*e + 3*(4*B*b^6*c -
3*A*b^5*c^2)*d^4*e^2 - (4*B*b^7 + 7*A*b^6*c)*d^3*e^3)*x)*sqrt(e*x + d))/((b^5*c^5*d^7*e - 3*b^6*c^4*d^6*e^2 +
3*b^7*c^3*d^5*e^3 - b^8*c^2*d^4*e^4)*x^5 + (b^5*c^5*d^8 - b^6*c^4*d^7*e - 3*b^7*c^3*d^6*e^2 + 5*b^8*c^2*d^5*e^
3 - 2*b^9*c*d^4*e^4)*x^4 + (2*b^6*c^4*d^8 - 5*b^7*c^3*d^7*e + 3*b^8*c^2*d^6*e^2 + b^9*c*d^5*e^3 - b^10*d^4*e^4
)*x^3 + (b^7*c^3*d^8 - 3*b^8*c^2*d^7*e + 3*b^9*c*d^6*e^2 - b^10*d^5*e^3)*x^2), -1/4*(3*((8*(B*b*c^6 - 2*A*c^7)
*d^6*e - 4*(6*B*b^2*c^5 - 11*A*b*c^6)*d^5*e^2 + 3*(7*B*b^3*c^4 - 11*A*b^2*c^5)*d^4*e^3)*x^5 + (8*(B*b*c^6 - 2*
A*c^7)*d^7 - 4*(2*B*b^2*c^5 - 3*A*b*c^6)*d^6*e - (27*B*b^3*c^4 - 55*A*b^2*c^5)*d^5*e^2 + 6*(7*B*b^4*c^3 - 11*A
*b^3*c^4)*d^4*e^3)*x^4 + (16*(B*b^2*c^5 - 2*A*b*c^6)*d^7 - 8*(5*B*b^3*c^4 - 9*A*b^2*c^5)*d^6*e + 2*(9*B*b^4*c^
3 - 11*A*b^3*c^4)*d^5*e^2 + 3*(7*B*b^5*c^2 - 11*A*b^4*c^3)*d^4*e^3)*x^3 + (8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^7 - 4
*(6*B*b^4*c^3 - 11*A*b^3*c^4)*d^6*e + 3*(7*B*b^5*c^2 - 11*A*b^4*c^3)*d^5*e^2)*x^2)*sqrt(-c/(c*d - b*e))*arctan
(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) + 3*((5*A*b^5*c^2*e^6 + 8*(B*b*c^6 - 2*A*c^7)*
d^5*e - 4*(5*B*b^2*c^5 - 9*A*b*c^6)*d^4*e^2 + (12*B*b^3*c^4 - 17*A*b^2*c^5)*d^3*e^3 + (4*B*b^4*c^3 - 5*A*b^3*c
^4)*d^2*e^4 - (4*B*b^5*c^2 + 3*A*b^4*c^3)*d*e^5)*x^5 + (10*A*b^6*c*e^6 + 8*(B*b*c^6 - 2*A*c^7)*d^6 - 4*(B*b^2*
c^5 - A*b*c^6)*d^5*e - (28*B*b^3*c^4 - 55*A*b^2*c^5)*d^4*e^2 + (28*B*b^4*c^3 - 39*A*b^3*c^4)*d^3*e^3 + (4*B*b^
5*c^2 - 13*A*b^4*c^3)*d^2*e^4 - (8*B*b^6*c + A*b^5*c^2)*d*e^5)*x^4 + (5*A*b^7*e^6 + 16*(B*b^2*c^5 - 2*A*b*c^6)
*d^6 - 8*(4*B*b^3*c^4 - 7*A*b^2*c^5)*d^5*e + 2*(2*B*b^4*c^3 + A*b^3*c^4)*d^4*e^2 + (20*B*b^5*c^2 - 27*A*b^4*c^
3)*d^3*e^3 - (4*B*b^6*c + 11*A*b^5*c^2)*d^2*e^4 - (4*B*b^7 - 7*A*b^6*c)*d*e^5)*x^3 + (5*A*b^7*d*e^5 + 8*(B*b^3
*c^4 - 2*A*b^2*c^5)*d^6 - 4*(5*B*b^4*c^3 - 9*A*b^3*c^4)*d^5*e + (12*B*b^5*c^2 - 17*A*b^4*c^3)*d^4*e^2 + (4*B*b
^6*c - 5*A*b^5*c^2)*d^3*e^3 - (4*B*b^7 + 3*A*b^6*c)*d^2*e^4)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) +
(2*A*b^4*c^3*d^6 - 6*A*b^5*c^2*d^5*e + 6*A*b^6*c*d^4*e^2 - 2*A*b^7*d^3*e^3 + 3*(5*A*b^5*c^2*d*e^5 + 4*(B*b^2*c
^5 - 2*A*b*c^6)*d^5*e - (9*B*b^3*c^4 - 16*A*b^2*c^5)*d^4*e^2 + (4*B*b^4*c^3 - 5*A*b^3*c^4)*d^3*e^3 - (4*B*b^5*
c^2 + 3*A*b^4*c^3)*d^2*e^4)*x^4 + (30*A*b^6*c*d*e^5 + 12*(B*b^2*c^5 - 2*A*b*c^6)*d^6 - 3*(3*B*b^3*c^4 - 4*A*b^
2*c^5)*d^5*e - 29*(B*b^4*c^3 - 2*A*b^3*c^4)*d^4*e^2 + (20*B*b^5*c^2 - 33*A*b^4*c^3)*d^3*e^3 - (24*B*b^6*c + 13
*A*b^5*c^2)*d^2*e^4)*x^3 + (15*A*b^7*d*e^5 + 18*(B*b^3*c^4 - 2*A*b^2*c^5)*d^6 - (37*B*b^4*c^3 - 65*A*b^3*c^4)*
d^5*e + (12*B*b^5*c^2 - 7*A*b^4*c^3)*d^4*e^2 + (4*B*b^6*c - 23*A*b^5*c^2)*d^3*e^3 - (12*B*b^7 - A*b^6*c)*d^2*e
^4)*x^2 + (5*A*b^7*d^2*e^4 + 4*(B*b^4*c^3 - 2*A*b^3*c^4)*d^6 - (12*B*b^5*c^2 - 19*A*b^4*c^3)*d^5*e + 3*(4*B*b^
6*c - 3*A*b^5*c^2)*d^4*e^2 - (4*B*b^7 + 7*A*b^6*c)*d^3*e^3)*x)*sqrt(e*x + d))/((b^5*c^5*d^7*e - 3*b^6*c^4*d^6*
e^2 + 3*b^7*c^3*d^5*e^3 - b^8*c^2*d^4*e^4)*x^5 + (b^5*c^5*d^8 - b^6*c^4*d^7*e - 3*b^7*c^3*d^6*e^2 + 5*b^8*c^2*
d^5*e^3 - 2*b^9*c*d^4*e^4)*x^4 + (2*b^6*c^4*d^8 - 5*b^7*c^3*d^7*e + 3*b^8*c^2*d^6*e^2 + b^9*c*d^5*e^3 - b^10*d
^4*e^4)*x^3 + (b^7*c^3*d^8 - 3*b^8*c^2*d^7*e + 3*b^9*c*d^6*e^2 - b^10*d^5*e^3)*x^2)]

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giac [B]  time = 0.41, size = 1313, normalized size = 2.59

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(8*B*b*c^5*d^2 - 16*A*c^6*d^2 - 24*B*b^2*c^4*d*e + 44*A*b*c^5*d*e + 21*B*b^3*c^3*e^2 - 33*A*b^2*c^4*e^2)*a
rctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^3*d^3 - 3*b^6*c^2*d^2*e + 3*b^7*c*d*e^2 - b^8*e^3)*sqrt(-c
^2*d + b*c*e)) + 2*(B*d*e^4 - A*e^5)/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*sqrt(x*e + d))
- 1/4*(12*(x*e + d)^(7/2)*B*b*c^5*d^4*e - 24*(x*e + d)^(7/2)*A*c^6*d^4*e - 36*(x*e + d)^(5/2)*B*b*c^5*d^5*e +
72*(x*e + d)^(5/2)*A*c^6*d^5*e + 36*(x*e + d)^(3/2)*B*b*c^5*d^6*e - 72*(x*e + d)^(3/2)*A*c^6*d^6*e - 12*sqrt(
x*e + d)*B*b*c^5*d^7*e + 24*sqrt(x*e + d)*A*c^6*d^7*e - 27*(x*e + d)^(7/2)*B*b^2*c^4*d^3*e^2 + 48*(x*e + d)^(7
/2)*A*b*c^5*d^3*e^2 + 99*(x*e + d)^(5/2)*B*b^2*c^4*d^4*e^2 - 180*(x*e + d)^(5/2)*A*b*c^5*d^4*e^2 - 117*(x*e +
d)^(3/2)*B*b^2*c^4*d^5*e^2 + 216*(x*e + d)^(3/2)*A*b*c^5*d^5*e^2 + 45*sqrt(x*e + d)*B*b^2*c^4*d^6*e^2 - 84*sqr
t(x*e + d)*A*b*c^5*d^6*e^2 + 12*(x*e + d)^(7/2)*B*b^3*c^3*d^2*e^3 - 15*(x*e + d)^(7/2)*A*b^2*c^4*d^2*e^3 - 77*
(x*e + d)^(5/2)*B*b^3*c^3*d^3*e^3 + 118*(x*e + d)^(5/2)*A*b^2*c^4*d^3*e^3 + 122*(x*e + d)^(3/2)*B*b^3*c^3*d^4*
e^3 - 199*(x*e + d)^(3/2)*A*b^2*c^4*d^4*e^3 - 57*sqrt(x*e + d)*B*b^3*c^3*d^5*e^3 + 96*sqrt(x*e + d)*A*b^2*c^4*
d^5*e^3 - 4*(x*e + d)^(7/2)*B*b^4*c^2*d*e^4 - 9*(x*e + d)^(7/2)*A*b^3*c^3*d*e^4 + 36*(x*e + d)^(5/2)*B*b^4*c^2
*d^2*e^4 + 3*(x*e + d)^(5/2)*A*b^3*c^3*d^2*e^4 - 72*(x*e + d)^(3/2)*B*b^4*c^2*d^3*e^4 + 38*(x*e + d)^(3/2)*A*b
^3*c^3*d^3*e^4 + 40*sqrt(x*e + d)*B*b^4*c^2*d^4*e^4 - 30*sqrt(x*e + d)*A*b^3*c^3*d^4*e^4 + 7*(x*e + d)^(7/2)*A
*b^4*c^2*e^5 - 8*(x*e + d)^(5/2)*B*b^5*c*d*e^5 - 41*(x*e + d)^(5/2)*A*b^4*c^2*d*e^5 + 28*(x*e + d)^(3/2)*B*b^5
*c*d^2*e^5 + 58*(x*e + d)^(3/2)*A*b^4*c^2*d^2*e^5 - 20*sqrt(x*e + d)*B*b^5*c*d^3*e^5 - 30*sqrt(x*e + d)*A*b^4*
c^2*d^3*e^5 + 14*(x*e + d)^(5/2)*A*b^5*c*e^6 - 4*(x*e + d)^(3/2)*B*b^6*d*e^6 - 41*(x*e + d)^(3/2)*A*b^5*c*d*e^
6 + 4*sqrt(x*e + d)*B*b^6*d^2*e^6 + 33*sqrt(x*e + d)*A*b^5*c*d^2*e^6 + 7*(x*e + d)^(3/2)*A*b^6*e^7 - 9*sqrt(x*
e + d)*A*b^6*d*e^7)/((b^4*c^3*d^6 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 - b^7*d^3*e^3)*((x*e + d)^2*c - 2*(x*e +
d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2) - 3/4*(8*B*b*c*d^2 - 16*A*c^2*d^2 + 4*B*b^2*d*e - 12*A*b*c*d*e - 5
*A*b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)*d^3)

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maple [B]  time = 0.10, size = 1022, normalized size = 2.02

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18*e*c^4/(b*e-c*d)^3/b^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d+2*e*c^5/(b*e-c*d)
^3/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*d-3*e*c^6/(b*e-c*d)^3/b^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*d+3*e*c^6/(b*e-c*
d)^3/b^4/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d^2-33/4*e^2*c^5/(b*e-c*d)^3/b^3/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d+25/4*e
^2*c^4/(b*e-c*d)^3/b^2/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d-33*e*c^5/(b*e-c*d)^3/b^4/((b*e-c*d)*c)^(1/2)*arctan((e*
x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d-2*e*c^5/(b*e-c*d)^3/b^3/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d^2+2*e^5/(b*e-c*d
)^3/d^3/(e*x+d)^(1/2)*A+3*e/b^3/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B-2*e^4/(b*e-c*d)^3/d^2/(e*x+d)^(1/2)*B
-15/4*e^2/b^3/d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A-9/4/b^3/d^2/x^2*(e*x+d)^(1/2)*A-12/b^5/d^(3/2)*arctanh(
(e*x+d)^(1/2)/d^(1/2))*A*c^2+6/b^4/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B*c+7/4/b^3/d^3/x^2*A*(e*x+d)^(3/2)+
3/e/b^4/d^2/x^2*A*(e*x+d)^(3/2)*c-3/e/b^4/d/x^2*(e*x+d)^(1/2)*A*c+19/4*e^2*c^5/(b*e-c*d)^3/b^3/(c*e*x+b*e)^2*(
e*x+d)^(3/2)*A-15/4*e^2*c^4/(b*e-c*d)^3/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B+21/4*e^3*c^4/(b*e-c*d)^3/b^2/(c*e*x+
b*e)^2*A*(e*x+d)^(1/2)-17/4*e^3*c^3/(b*e-c*d)^3/b/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)+99/4*e^2*c^4/(b*e-c*d)^3/b^3/(
(b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A+12*c^6/(b*e-c*d)^3/b^5/((b*e-c*d)*c)^(1/2)*ar
ctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^2-6*c^5/(b*e-c*d)^3/b^4/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)
/((b*e-c*d)*c)^(1/2)*c)*B*d^2-63/4*e^2*c^3/(b*e-c*d)^3/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)
*c)^(1/2)*c)*B+1/e/b^3/d/x^2*(e*x+d)^(1/2)*B-9*e/b^4/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c-1/e/b^3/d^2/x^
2*B*(e*x+d)^(3/2)

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(3/2)/(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 details)Is b*e-c*d positive or negative?

________________________________________________________________________________________

mupad [B]  time = 9.53, size = 23541, normalized size = 46.52

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*(A*e^5 - B*d*e^4))/(c*d^2 - b*d*e) + ((d + e*x)^2*(15*A*b^6*e^7 - 72*A*c^6*d^6*e - 12*B*b^6*d*e^6 + 216*
A*b*c^5*d^5*e^2 + 76*B*b^5*c*d^2*e^5 - 199*A*b^2*c^4*d^4*e^3 + 38*A*b^3*c^3*d^3*e^4 + 106*A*b^4*c^2*d^2*e^5 -
117*B*b^2*c^4*d^5*e^2 + 122*B*b^3*c^3*d^4*e^3 - 120*B*b^4*c^2*d^3*e^4 - 89*A*b^5*c*d*e^6 + 36*B*b*c^5*d^6*e))/
(4*b^4*(c*d^2 - b*d*e)^3) + ((d + e*x)^3*(30*A*b^5*c*e^6 + 72*A*c^6*d^5*e - 180*A*b*c^5*d^4*e^2 - 73*A*b^4*c^2
*d*e^5 + 118*A*b^2*c^4*d^3*e^3 + 3*A*b^3*c^3*d^2*e^4 + 99*B*b^2*c^4*d^4*e^2 - 77*B*b^3*c^3*d^3*e^3 + 68*B*b^4*
c^2*d^2*e^4 - 36*B*b*c^5*d^5*e - 24*B*b^5*c*d*e^5))/(4*b^4*(c*d^2 - b*d*e)^3) + ((d + e*x)*(25*A*b^5*e^6 + 24*
A*c^5*d^5*e - 20*B*b^5*d*e^5 - 60*A*b*c^4*d^4*e^2 + 48*B*b^4*c*d^2*e^4 + 36*A*b^2*c^3*d^3*e^3 + 6*A*b^3*c^2*d^
2*e^4 + 33*B*b^2*c^3*d^4*e^2 - 24*B*b^3*c^2*d^3*e^3 - 56*A*b^4*c*d*e^5 - 12*B*b*c^4*d^5*e))/(4*b^4*(c*d^2 - b*
d*e)^2) - (3*(d + e*x)^4*(8*A*c^6*d^4*e - 5*A*b^4*c^2*e^5 - 16*A*b*c^5*d^3*e^2 + 3*A*b^3*c^3*d*e^4 + 4*B*b^4*c
^2*d*e^4 + 5*A*b^2*c^4*d^2*e^3 + 9*B*b^2*c^4*d^3*e^2 - 4*B*b^3*c^3*d^2*e^3 - 4*B*b*c^5*d^4*e))/(4*b^4*(c*d^2 -
b*d*e)^3))/(c^2*(d + e*x)^(9/2) - (4*c^2*d - 2*b*c*e)*(d + e*x)^(7/2) - (d + e*x)^(3/2)*(4*c^2*d^3 + 2*b^2*d*
e^2 - 6*b*c*d^2*e) + (d + e*x)^(5/2)*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e) + (d + e*x)^(1/2)*(c^2*d^4 + b^2*d^2*e^
2 - 2*b*c*d^3*e)) - atan(-(((d + e*x)^(1/2)*(589824*A^2*b^12*c^22*d^28*e^2 - 8257536*A^2*b^13*c^21*d^27*e^3 +
53342208*A^2*b^14*c^20*d^26*e^4 - 210382848*A^2*b^15*c^19*d^25*e^5 + 564860160*A^2*b^16*c^18*d^24*e^6 - 108983
8080*A^2*b^17*c^17*d^23*e^7 + 1555380864*A^2*b^18*c^16*d^22*e^8 - 1667850624*A^2*b^19*c^15*d^21*e^9 + 13582575
36*A^2*b^20*c^14*d^20*e^10 - 855642240*A^2*b^21*c^13*d^19*e^11 + 438185088*A^2*b^22*c^12*d^18*e^12 - 201386880
*A^2*b^23*c^11*d^17*e^13 + 90100224*A^2*b^24*c^10*d^16*e^14 - 37986048*A^2*b^25*c^9*d^15*e^15 + 15108480*A^2*b
^26*c^8*d^14*e^16 - 6844032*A^2*b^27*c^7*d^13*e^17 + 3399552*A^2*b^28*c^6*d^12*e^18 - 1300608*A^2*b^29*c^5*d^1
1*e^19 + 293760*A^2*b^30*c^4*d^10*e^20 - 28800*A^2*b^31*c^3*d^9*e^21 + 147456*B^2*b^14*c^20*d^28*e^2 - 2138112
*B^2*b^15*c^19*d^27*e^3 + 14340096*B^2*b^16*c^18*d^26*e^4 - 58816512*B^2*b^17*c^17*d^25*e^5 + 164257920*B^2*b^
18*c^16*d^24*e^6 - 328809600*B^2*b^19*c^15*d^23*e^7 + 482904576*B^2*b^20*c^14*d^22*e^8 - 521961984*B^2*b^21*c^
13*d^21*e^9 + 407418624*B^2*b^22*c^12*d^20*e^10 - 216610560*B^2*b^23*c^11*d^19*e^11 + 65382912*B^2*b^24*c^10*d
^18*e^12 - 1276416*B^2*b^25*c^9*d^17*e^13 - 6007680*B^2*b^26*c^8*d^16*e^14 + 137088*B^2*b^27*c^7*d^15*e^15 + 1
751040*B^2*b^28*c^6*d^14*e^16 - 903168*B^2*b^29*c^5*d^13*e^17 + 202752*B^2*b^30*c^4*d^12*e^18 - 18432*B^2*b^31
*c^3*d^11*e^19 - 589824*A*B*b^13*c^21*d^28*e^2 + 8404992*A*B*b^14*c^20*d^27*e^3 - 55332864*A*B*b^15*c^19*d^26*
e^4 + 222584832*A*B*b^16*c^18*d^25*e^5 - 609557760*A*B*b^17*c^17*d^24*e^6 + 1197861120*A*B*b^18*c^16*d^23*e^7
- 1733566464*A*B*b^19*c^15*d^22*e^8 + 1864765440*A*B*b^20*c^14*d^21*e^9 - 1485494784*A*B*b^21*c^13*d^20*e^10 +
864115200*A*B*b^22*c^12*d^19*e^11 - 361248768*A*B*b^23*c^11*d^18*e^12 + 110656512*A*B*b^24*c^10*d^17*e^13 - 2
7548928*A*B*b^25*c^9*d^16*e^14 + 3209472*A*B*b^26*c^8*d^15*e^15 + 4930560*A*B*b^27*c^7*d^14*e^16 - 4912128*A*B
*b^28*c^6*d^13*e^17 + 2165760*A*B*b^29*c^5*d^12*e^18 - 488448*A*B*b^30*c^4*d^11*e^19 + 46080*A*B*b^31*c^3*d^10
*e^20) - ((9*(25*A^2*b^4*e^4 + 256*A^2*c^4*d^4 + 64*B^2*b^2*c^2*d^4 + 16*B^2*b^4*d^2*e^2 + 304*A^2*b^2*c^2*d^2
*e^2 + 384*A^2*b*c^3*d^3*e + 120*A^2*b^3*c*d*e^3 + 64*B^2*b^3*c*d^3*e - 256*A*B*b*c^3*d^4 - 40*A*B*b^4*d*e^3 -
320*A*B*b^2*c^2*d^3*e - 176*A*B*b^3*c*d^2*e^2))/(64*b^10*d^7))^(1/2)*((d + e*x)^(1/2)*((9*(25*A^2*b^4*e^4 + 2
56*A^2*c^4*d^4 + 64*B^2*b^2*c^2*d^4 + 16*B^2*b^4*d^2*e^2 + 304*A^2*b^2*c^2*d^2*e^2 + 384*A^2*b*c^3*d^3*e + 120
*A^2*b^3*c*d*e^3 + 64*B^2*b^3*c*d^3*e - 256*A*B*b*c^3*d^4 - 40*A*B*b^4*d*e^3 - 320*A*B*b^2*c^2*d^3*e - 176*A*B
*b^3*c*d^2*e^2))/(64*b^10*d^7))^(1/2)*(16384*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d^30*e^3 + 1843200*b^24*c^1
6*d^29*e^4 - 8314880*b^25*c^15*d^28*e^5 + 26091520*b^26*c^14*d^27*e^6 - 60383232*b^27*c^13*d^26*e^7 + 10660249
6*b^28*c^12*d^25*e^8 - 146432000*b^29*c^11*d^24*e^9 + 158146560*b^30*c^10*d^23*e^10 - 134717440*b^31*c^9*d^22*
e^11 + 90202112*b^32*c^8*d^21*e^12 - 46964736*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6*d^19*e^14 - 5447680*b^35*
c^5*d^18*e^15 + 1105920*b^36*c^4*d^17*e^16 - 139264*b^37*c^3*d^16*e^17 + 8192*b^38*c^2*d^15*e^18) - 24576*A*b^
18*c^19*d^29*e^3 + 356352*A*b^19*c^18*d^28*e^4 - 2396160*A*b^20*c^17*d^27*e^5 + 9897984*A*b^21*c^16*d^26*e^6 -
28065792*A*b^22*c^15*d^25*e^7 + 57891840*A*b^23*c^14*d^24*e^8 - 90071040*A*b^24*c^13*d^23*e^9 + 108810240*A*b
^25*c^12*d^22*e^10 - 105566208*A*b^26*c^11*d^21*e^11 + 86406144*A*b^27*c^10*d^20*e^12 - 63393792*A*b^28*c^9*d^
19*e^13 + 43075584*A*b^29*c^8*d^18*e^14 - 26173440*A*b^30*c^7*d^17*e^15 + 13108224*A*b^31*c^6*d^16*e^16 - 4964
352*A*b^32*c^5*d^15*e^17 + 1302528*A*b^33*c^4*d^14*e^18 - 208896*A*b^34*c^3*d^13*e^19 + 15360*A*b^35*c^2*d^12*
e^20 + 12288*B*b^19*c^18*d^29*e^3 - 181248*B*b^20*c^17*d^28*e^4 + 1241088*B*b^21*c^16*d^27*e^5 - 5203968*B*b^2
2*c^15*d^26*e^6 + 14831616*B*b^23*c^14*d^25*e^7 - 30096384*B*b^24*c^13*d^24*e^8 + 44064768*B*b^25*c^12*d^23*e^
9 - 45551616*B*b^26*c^11*d^22*e^10 + 30007296*B*b^27*c^10*d^21*e^11 - 6454272*B*b^28*c^9*d^20*e^12 - 10407936*
B*b^29*c^8*d^19*e^13 + 14112768*B*b^30*c^7*d^18*e^14 - 9449472*B*b^31*c^6*d^17*e^15 + 3996672*B*b^32*c^5*d^16*
e^16 - 1081344*B*b^33*c^4*d^15*e^17 + 172032*B*b^34*c^3*d^14*e^18 - 12288*B*b^35*c^2*d^13*e^19))*((9*(25*A^2*b
^4*e^4 + 256*A^2*c^4*d^4 + 64*B^2*b^2*c^2*d^4 + 16*B^2*b^4*d^2*e^2 + 304*A^2*b^2*c^2*d^2*e^2 + 384*A^2*b*c^3*d
^3*e + 120*A^2*b^3*c*d*e^3 + 64*B^2*b^3*c*d^3*e - 256*A*B*b*c^3*d^4 - 40*A*B*b^4*d*e^3 - 320*A*B*b^2*c^2*d^3*e
- 176*A*B*b^3*c*d^2*e^2))/(64*b^10*d^7))^(1/2)*1i + ((d + e*x)^(1/2)*(589824*A^2*b^12*c^22*d^28*e^2 - 8257536
*A^2*b^13*c^21*d^27*e^3 + 53342208*A^2*b^14*c^20*d^26*e^4 - 210382848*A^2*b^15*c^19*d^25*e^5 + 564860160*A^2*b
^16*c^18*d^24*e^6 - 1089838080*A^2*b^17*c^17*d^23*e^7 + 1555380864*A^2*b^18*c^16*d^22*e^8 - 1667850624*A^2*b^1
9*c^15*d^21*e^9 + 1358257536*A^2*b^20*c^14*d^20*e^10 - 855642240*A^2*b^21*c^13*d^19*e^11 + 438185088*A^2*b^22*
c^12*d^18*e^12 - 201386880*A^2*b^23*c^11*d^17*e^13 + 90100224*A^2*b^24*c^10*d^16*e^14 - 37986048*A^2*b^25*c^9*
d^15*e^15 + 15108480*A^2*b^26*c^8*d^14*e^16 - 6844032*A^2*b^27*c^7*d^13*e^17 + 3399552*A^2*b^28*c^6*d^12*e^18
- 1300608*A^2*b^29*c^5*d^11*e^19 + 293760*A^2*b^30*c^4*d^10*e^20 - 28800*A^2*b^31*c^3*d^9*e^21 + 147456*B^2*b^
14*c^20*d^28*e^2 - 2138112*B^2*b^15*c^19*d^27*e^3 + 14340096*B^2*b^16*c^18*d^26*e^4 - 58816512*B^2*b^17*c^17*d
^25*e^5 + 164257920*B^2*b^18*c^16*d^24*e^6 - 328809600*B^2*b^19*c^15*d^23*e^7 + 482904576*B^2*b^20*c^14*d^22*e
^8 - 521961984*B^2*b^21*c^13*d^21*e^9 + 407418624*B^2*b^22*c^12*d^20*e^10 - 216610560*B^2*b^23*c^11*d^19*e^11
+ 65382912*B^2*b^24*c^10*d^18*e^12 - 1276416*B^2*b^25*c^9*d^17*e^13 - 6007680*B^2*b^26*c^8*d^16*e^14 + 137088*
B^2*b^27*c^7*d^15*e^15 + 1751040*B^2*b^28*c^6*d^14*e^16 - 903168*B^2*b^29*c^5*d^13*e^17 + 202752*B^2*b^30*c^4*
d^12*e^18 - 18432*B^2*b^31*c^3*d^11*e^19 - 589824*A*B*b^13*c^21*d^28*e^2 + 8404992*A*B*b^14*c^20*d^27*e^3 - 55
332864*A*B*b^15*c^19*d^26*e^4 + 222584832*A*B*b^16*c^18*d^25*e^5 - 609557760*A*B*b^17*c^17*d^24*e^6 + 11978611
20*A*B*b^18*c^16*d^23*e^7 - 1733566464*A*B*b^19*c^15*d^22*e^8 + 1864765440*A*B*b^20*c^14*d^21*e^9 - 1485494784
*A*B*b^21*c^13*d^20*e^10 + 864115200*A*B*b^22*c^12*d^19*e^11 - 361248768*A*B*b^23*c^11*d^18*e^12 + 110656512*A
*B*b^24*c^10*d^17*e^13 - 27548928*A*B*b^25*c^9*d^16*e^14 + 3209472*A*B*b^26*c^8*d^15*e^15 + 4930560*A*B*b^27*c
^7*d^14*e^16 - 4912128*A*B*b^28*c^6*d^13*e^17 + 2165760*A*B*b^29*c^5*d^12*e^18 - 488448*A*B*b^30*c^4*d^11*e^19
+ 46080*A*B*b^31*c^3*d^10*e^20) - ((9*(25*A^2*b^4*e^4 + 256*A^2*c^4*d^4 + 64*B^2*b^2*c^2*d^4 + 16*B^2*b^4*d^2
*e^2 + 304*A^2*b^2*c^2*d^2*e^2 + 384*A^2*b*c^3*d^3*e + 120*A^2*b^3*c*d*e^3 + 64*B^2*b^3*c*d^3*e - 256*A*B*b*c^
3*d^4 - 40*A*B*b^4*d*e^3 - 320*A*B*b^2*c^2*d^3*e - 176*A*B*b^3*c*d^2*e^2))/(64*b^10*d^7))^(1/2)*((d + e*x)^(1/
2)*((9*(25*A^2*b^4*e^4 + 256*A^2*c^4*d^4 + 64*B^2*b^2*c^2*d^4 + 16*B^2*b^4*d^2*e^2 + 304*A^2*b^2*c^2*d^2*e^2 +
384*A^2*b*c^3*d^3*e + 120*A^2*b^3*c*d*e^3 + 64*B^2*b^3*c*d^3*e - 256*A*B*b*c^3*d^4 - 40*A*B*b^4*d*e^3 - 320*A
*B*b^2*c^2*d^3*e - 176*A*B*b^3*c*d^2*e^2))/(64*b^10*d^7))^(1/2)*(16384*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d
^30*e^3 + 1843200*b^24*c^16*d^29*e^4 - 8314880*b^25*c^15*d^28*e^5 + 26091520*b^26*c^14*d^27*e^6 - 60383232*b^2
7*c^13*d^26*e^7 + 106602496*b^28*c^12*d^25*e^8 - 146432000*b^29*c^11*d^24*e^9 + 158146560*b^30*c^10*d^23*e^10
- 134717440*b^31*c^9*d^22*e^11 + 90202112*b^32*c^8*d^21*e^12 - 46964736*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6
*d^19*e^14 - 5447680*b^35*c^5*d^18*e^15 + 1105920*b^36*c^4*d^17*e^16 - 139264*b^37*c^3*d^16*e^17 + 8192*b^38*c
^2*d^15*e^18) + 24576*A*b^18*c^19*d^29*e^3 - 356352*A*b^19*c^18*d^28*e^4 + 2396160*A*b^20*c^17*d^27*e^5 - 9897
984*A*b^21*c^16*d^26*e^6 + 28065792*A*b^22*c^15*d^25*e^7 - 57891840*A*b^23*c^14*d^24*e^8 + 90071040*A*b^24*c^1
3*d^23*e^9 - 108810240*A*b^25*c^12*d^22*e^10 + 105566208*A*b^26*c^11*d^21*e^11 - 86406144*A*b^27*c^10*d^20*e^1
2 + 63393792*A*b^28*c^9*d^19*e^13 - 43075584*A*b^29*c^8*d^18*e^14 + 26173440*A*b^30*c^7*d^17*e^15 - 13108224*A
*b^31*c^6*d^16*e^16 + 4964352*A*b^32*c^5*d^15*e^17 - 1302528*A*b^33*c^4*d^14*e^18 + 208896*A*b^34*c^3*d^13*e^1
9 - 15360*A*b^35*c^2*d^12*e^20 - 12288*B*b^19*c^18*d^29*e^3 + 181248*B*b^20*c^17*d^28*e^4 - 1241088*B*b^21*c^1
6*d^27*e^5 + 5203968*B*b^22*c^15*d^26*e^6 - 14831616*B*b^23*c^14*d^25*e^7 + 30096384*B*b^24*c^13*d^24*e^8 - 44
064768*B*b^25*c^12*d^23*e^9 + 45551616*B*b^26*c^11*d^22*e^10 - 30007296*B*b^27*c^10*d^21*e^11 + 6454272*B*b^28
*c^9*d^20*e^12 + 10407936*B*b^29*c^8*d^19*e^13 - 14112768*B*b^30*c^7*d^18*e^14 + 9449472*B*b^31*c^6*d^17*e^15
- 3996672*B*b^32*c^5*d^16*e^16 + 1081344*B*b^33*c^4*d^15*e^17 - 172032*B*b^34*c^3*d^14*e^18 + 12288*B*b^35*c^2
*d^13*e^19))*((9*(25*A^2*b^4*e^4 + 256*A^2*c^4*d^4 + 64*B^2*b^2*c^2*d^4 + 16*B^2*b^4*d^2*e^2 + 304*A^2*b^2*c^2
*d^2*e^2 + 384*A^2*b*c^3*d^3*e + 120*A^2*b^3*c*d*e^3 + 64*B^2*b^3*c*d^3*e - 256*A*B*b*c^3*d^4 - 40*A*B*b^4*d*e
^3 - 320*A*B*b^2*c^2*d^3*e - 176*A*B*b^3*c*d^2*e^2))/(64*b^10*d^7))^(1/2)*1i)/(((d + e*x)^(1/2)*(589824*A^2*b^
12*c^22*d^28*e^2 - 8257536*A^2*b^13*c^21*d^27*e^3 + 53342208*A^2*b^14*c^20*d^26*e^4 - 210382848*A^2*b^15*c^19*
d^25*e^5 + 564860160*A^2*b^16*c^18*d^24*e^6 - 1089838080*A^2*b^17*c^17*d^23*e^7 + 1555380864*A^2*b^18*c^16*d^2
2*e^8 - 1667850624*A^2*b^19*c^15*d^21*e^9 + 1358257536*A^2*b^20*c^14*d^20*e^10 - 855642240*A^2*b^21*c^13*d^19*
e^11 + 438185088*A^2*b^22*c^12*d^18*e^12 - 201386880*A^2*b^23*c^11*d^17*e^13 + 90100224*A^2*b^24*c^10*d^16*e^1
4 - 37986048*A^2*b^25*c^9*d^15*e^15 + 15108480*A^2*b^26*c^8*d^14*e^16 - 6844032*A^2*b^27*c^7*d^13*e^17 + 33995
52*A^2*b^28*c^6*d^12*e^18 - 1300608*A^2*b^29*c^5*d^11*e^19 + 293760*A^2*b^30*c^4*d^10*e^20 - 28800*A^2*b^31*c^
3*d^9*e^21 + 147456*B^2*b^14*c^20*d^28*e^2 - 2138112*B^2*b^15*c^19*d^27*e^3 + 14340096*B^2*b^16*c^18*d^26*e^4
- 58816512*B^2*b^17*c^17*d^25*e^5 + 164257920*B^2*b^18*c^16*d^24*e^6 - 328809600*B^2*b^19*c^15*d^23*e^7 + 4829
04576*B^2*b^20*c^14*d^22*e^8 - 521961984*B^2*b^21*c^13*d^21*e^9 + 407418624*B^2*b^22*c^12*d^20*e^10 - 21661056
0*B^2*b^23*c^11*d^19*e^11 + 65382912*B^2*b^24*c^10*d^18*e^12 - 1276416*B^2*b^25*c^9*d^17*e^13 - 6007680*B^2*b^
26*c^8*d^16*e^14 + 137088*B^2*b^27*c^7*d^15*e^15 + 1751040*B^2*b^28*c^6*d^14*e^16 - 903168*B^2*b^29*c^5*d^13*e
^17 + 202752*B^2*b^30*c^4*d^12*e^18 - 18432*B^2*b^31*c^3*d^11*e^19 - 589824*A*B*b^13*c^21*d^28*e^2 + 8404992*A
*B*b^14*c^20*d^27*e^3 - 55332864*A*B*b^15*c^19*d^26*e^4 + 222584832*A*B*b^16*c^18*d^25*e^5 - 609557760*A*B*b^1
7*c^17*d^24*e^6 + 1197861120*A*B*b^18*c^16*d^23*e^7 - 1733566464*A*B*b^19*c^15*d^22*e^8 + 1864765440*A*B*b^20*
c^14*d^21*e^9 - 1485494784*A*B*b^21*c^13*d^20*e^10 + 864115200*A*B*b^22*c^12*d^19*e^11 - 361248768*A*B*b^23*c^
11*d^18*e^12 + 110656512*A*B*b^24*c^10*d^17*e^13 - 27548928*A*B*b^25*c^9*d^16*e^14 + 3209472*A*B*b^26*c^8*d^15
*e^15 + 4930560*A*B*b^27*c^7*d^14*e^16 - 4912128*A*B*b^28*c^6*d^13*e^17 + 2165760*A*B*b^29*c^5*d^12*e^18 - 488
448*A*B*b^30*c^4*d^11*e^19 + 46080*A*B*b^31*c^3*d^10*e^20) - ((9*(25*A^2*b^4*e^4 + 256*A^2*c^4*d^4 + 64*B^2*b^
2*c^2*d^4 + 16*B^2*b^4*d^2*e^2 + 304*A^2*b^2*c^2*d^2*e^2 + 384*A^2*b*c^3*d^3*e + 120*A^2*b^3*c*d*e^3 + 64*B^2*
b^3*c*d^3*e - 256*A*B*b*c^3*d^4 - 40*A*B*b^4*d*e^3 - 320*A*B*b^2*c^2*d^3*e - 176*A*B*b^3*c*d^2*e^2))/(64*b^10*
d^7))^(1/2)*((d + e*x)^(1/2)*((9*(25*A^2*b^4*e^4 + 256*A^2*c^4*d^4 + 64*B^2*b^2*c^2*d^4 + 16*B^2*b^4*d^2*e^2 +
304*A^2*b^2*c^2*d^2*e^2 + 384*A^2*b*c^3*d^3*e + 120*A^2*b^3*c*d*e^3 + 64*B^2*b^3*c*d^3*e - 256*A*B*b*c^3*d^4
- 40*A*B*b^4*d*e^3 - 320*A*B*b^2*c^2*d^3*e - 176*A*B*b^3*c*d^2*e^2))/(64*b^10*d^7))^(1/2)*(16384*b^22*c^18*d^3
1*e^2 - 253952*b^23*c^17*d^30*e^3 + 1843200*b^24*c^16*d^29*e^4 - 8314880*b^25*c^15*d^28*e^5 + 26091520*b^26*c^
14*d^27*e^6 - 60383232*b^27*c^13*d^26*e^7 + 106602496*b^28*c^12*d^25*e^8 - 146432000*b^29*c^11*d^24*e^9 + 1581
46560*b^30*c^10*d^23*e^10 - 134717440*b^31*c^9*d^22*e^11 + 90202112*b^32*c^8*d^21*e^12 - 46964736*b^33*c^7*d^2
0*e^13 + 18636800*b^34*c^6*d^19*e^14 - 5447680*b^35*c^5*d^18*e^15 + 1105920*b^36*c^4*d^17*e^16 - 139264*b^37*c
^3*d^16*e^17 + 8192*b^38*c^2*d^15*e^18) - 24576*A*b^18*c^19*d^29*e^3 + 356352*A*b^19*c^18*d^28*e^4 - 2396160*A
*b^20*c^17*d^27*e^5 + 9897984*A*b^21*c^16*d^26*e^6 - 28065792*A*b^22*c^15*d^25*e^7 + 57891840*A*b^23*c^14*d^24
*e^8 - 90071040*A*b^24*c^13*d^23*e^9 + 108810240*A*b^25*c^12*d^22*e^10 - 105566208*A*b^26*c^11*d^21*e^11 + 864
06144*A*b^27*c^10*d^20*e^12 - 63393792*A*b^28*c^9*d^19*e^13 + 43075584*A*b^29*c^8*d^18*e^14 - 26173440*A*b^30*
c^7*d^17*e^15 + 13108224*A*b^31*c^6*d^16*e^16 - 4964352*A*b^32*c^5*d^15*e^17 + 1302528*A*b^33*c^4*d^14*e^18 -
208896*A*b^34*c^3*d^13*e^19 + 15360*A*b^35*c^2*d^12*e^20 + 12288*B*b^19*c^18*d^29*e^3 - 181248*B*b^20*c^17*d^2
8*e^4 + 1241088*B*b^21*c^16*d^27*e^5 - 5203968*B*b^22*c^15*d^26*e^6 + 14831616*B*b^23*c^14*d^25*e^7 - 30096384
*B*b^24*c^13*d^24*e^8 + 44064768*B*b^25*c^12*d^23*e^9 - 45551616*B*b^26*c^11*d^22*e^10 + 30007296*B*b^27*c^10*
d^21*e^11 - 6454272*B*b^28*c^9*d^20*e^12 - 10407936*B*b^29*c^8*d^19*e^13 + 14112768*B*b^30*c^7*d^18*e^14 - 944
9472*B*b^31*c^6*d^17*e^15 + 3996672*B*b^32*c^5*d^16*e^16 - 1081344*B*b^33*c^4*d^15*e^17 + 172032*B*b^34*c^3*d^
14*e^18 - 12288*B*b^35*c^2*d^13*e^19))*((9*(25*A^2*b^4*e^4 + 256*A^2*c^4*d^4 + 64*B^2*b^2*c^2*d^4 + 16*B^2*b^4
*d^2*e^2 + 304*A^2*b^2*c^2*d^2*e^2 + 384*A^2*b*c^3*d^3*e + 120*A^2*b^3*c*d*e^3 + 64*B^2*b^3*c*d^3*e - 256*A*B*
b*c^3*d^4 - 40*A*B*b^4*d*e^3 - 320*A*B*b^2*c^2*d^3*e - 176*A*B*b^3*c*d^2*e^2))/(64*b^10*d^7))^(1/2) - ((d + e*
x)^(1/2)*(589824*A^2*b^12*c^22*d^28*e^2 - 8257536*A^2*b^13*c^21*d^27*e^3 + 53342208*A^2*b^14*c^20*d^26*e^4 - 2
10382848*A^2*b^15*c^19*d^25*e^5 + 564860160*A^2*b^16*c^18*d^24*e^6 - 1089838080*A^2*b^17*c^17*d^23*e^7 + 15553
80864*A^2*b^18*c^16*d^22*e^8 - 1667850624*A^2*b^19*c^15*d^21*e^9 + 1358257536*A^2*b^20*c^14*d^20*e^10 - 855642
240*A^2*b^21*c^13*d^19*e^11 + 438185088*A^2*b^22*c^12*d^18*e^12 - 201386880*A^2*b^23*c^11*d^17*e^13 + 90100224
*A^2*b^24*c^10*d^16*e^14 - 37986048*A^2*b^25*c^9*d^15*e^15 + 15108480*A^2*b^26*c^8*d^14*e^16 - 6844032*A^2*b^2
7*c^7*d^13*e^17 + 3399552*A^2*b^28*c^6*d^12*e^18 - 1300608*A^2*b^29*c^5*d^11*e^19 + 293760*A^2*b^30*c^4*d^10*e
^20 - 28800*A^2*b^31*c^3*d^9*e^21 + 147456*B^2*b^14*c^20*d^28*e^2 - 2138112*B^2*b^15*c^19*d^27*e^3 + 14340096*
B^2*b^16*c^18*d^26*e^4 - 58816512*B^2*b^17*c^17*d^25*e^5 + 164257920*B^2*b^18*c^16*d^24*e^6 - 328809600*B^2*b^
19*c^15*d^23*e^7 + 482904576*B^2*b^20*c^14*d^22*e^8 - 521961984*B^2*b^21*c^13*d^21*e^9 + 407418624*B^2*b^22*c^
12*d^20*e^10 - 216610560*B^2*b^23*c^11*d^19*e^11 + 65382912*B^2*b^24*c^10*d^18*e^12 - 1276416*B^2*b^25*c^9*d^1
7*e^13 - 6007680*B^2*b^26*c^8*d^16*e^14 + 137088*B^2*b^27*c^7*d^15*e^15 + 1751040*B^2*b^28*c^6*d^14*e^16 - 903
168*B^2*b^29*c^5*d^13*e^17 + 202752*B^2*b^30*c^4*d^12*e^18 - 18432*B^2*b^31*c^3*d^11*e^19 - 589824*A*B*b^13*c^
21*d^28*e^2 + 8404992*A*B*b^14*c^20*d^27*e^3 - 55332864*A*B*b^15*c^19*d^26*e^4 + 222584832*A*B*b^16*c^18*d^25*
e^5 - 609557760*A*B*b^17*c^17*d^24*e^6 + 1197861120*A*B*b^18*c^16*d^23*e^7 - 1733566464*A*B*b^19*c^15*d^22*e^8
+ 1864765440*A*B*b^20*c^14*d^21*e^9 - 1485494784*A*B*b^21*c^13*d^20*e^10 + 864115200*A*B*b^22*c^12*d^19*e^11
- 361248768*A*B*b^23*c^11*d^18*e^12 + 110656512*A*B*b^24*c^10*d^17*e^13 - 27548928*A*B*b^25*c^9*d^16*e^14 + 32
09472*A*B*b^26*c^8*d^15*e^15 + 4930560*A*B*b^27*c^7*d^14*e^16 - 4912128*A*B*b^28*c^6*d^13*e^17 + 2165760*A*B*b
^29*c^5*d^12*e^18 - 488448*A*B*b^30*c^4*d^11*e^19 + 46080*A*B*b^31*c^3*d^10*e^20) - ((9*(25*A^2*b^4*e^4 + 256*
A^2*c^4*d^4 + 64*B^2*b^2*c^2*d^4 + 16*B^2*b^4*d^2*e^2 + 304*A^2*b^2*c^2*d^2*e^2 + 384*A^2*b*c^3*d^3*e + 120*A^
2*b^3*c*d*e^3 + 64*B^2*b^3*c*d^3*e - 256*A*B*b*c^3*d^4 - 40*A*B*b^4*d*e^3 - 320*A*B*b^2*c^2*d^3*e - 176*A*B*b^
3*c*d^2*e^2))/(64*b^10*d^7))^(1/2)*((d + e*x)^(1/2)*((9*(25*A^2*b^4*e^4 + 256*A^2*c^4*d^4 + 64*B^2*b^2*c^2*d^4
+ 16*B^2*b^4*d^2*e^2 + 304*A^2*b^2*c^2*d^2*e^2 + 384*A^2*b*c^3*d^3*e + 120*A^2*b^3*c*d*e^3 + 64*B^2*b^3*c*d^3
*e - 256*A*B*b*c^3*d^4 - 40*A*B*b^4*d*e^3 - 320*A*B*b^2*c^2*d^3*e - 176*A*B*b^3*c*d^2*e^2))/(64*b^10*d^7))^(1/
2)*(16384*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d^30*e^3 + 1843200*b^24*c^16*d^29*e^4 - 8314880*b^25*c^15*d^28
*e^5 + 26091520*b^26*c^14*d^27*e^6 - 60383232*b^27*c^13*d^26*e^7 + 106602496*b^28*c^12*d^25*e^8 - 146432000*b^
29*c^11*d^24*e^9 + 158146560*b^30*c^10*d^23*e^10 - 134717440*b^31*c^9*d^22*e^11 + 90202112*b^32*c^8*d^21*e^12
- 46964736*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6*d^19*e^14 - 5447680*b^35*c^5*d^18*e^15 + 1105920*b^36*c^4*d^
17*e^16 - 139264*b^37*c^3*d^16*e^17 + 8192*b^38*c^2*d^15*e^18) + 24576*A*b^18*c^19*d^29*e^3 - 356352*A*b^19*c^
18*d^28*e^4 + 2396160*A*b^20*c^17*d^27*e^5 - 9897984*A*b^21*c^16*d^26*e^6 + 28065792*A*b^22*c^15*d^25*e^7 - 57
891840*A*b^23*c^14*d^24*e^8 + 90071040*A*b^24*c^13*d^23*e^9 - 108810240*A*b^25*c^12*d^22*e^10 + 105566208*A*b^
26*c^11*d^21*e^11 - 86406144*A*b^27*c^10*d^20*e^12 + 63393792*A*b^28*c^9*d^19*e^13 - 43075584*A*b^29*c^8*d^18*
e^14 + 26173440*A*b^30*c^7*d^17*e^15 - 13108224*A*b^31*c^6*d^16*e^16 + 4964352*A*b^32*c^5*d^15*e^17 - 1302528*
A*b^33*c^4*d^14*e^18 + 208896*A*b^34*c^3*d^13*e^19 - 15360*A*b^35*c^2*d^12*e^20 - 12288*B*b^19*c^18*d^29*e^3 +
181248*B*b^20*c^17*d^28*e^4 - 1241088*B*b^21*c^16*d^27*e^5 + 5203968*B*b^22*c^15*d^26*e^6 - 14831616*B*b^23*c
^14*d^25*e^7 + 30096384*B*b^24*c^13*d^24*e^8 - 44064768*B*b^25*c^12*d^23*e^9 + 45551616*B*b^26*c^11*d^22*e^10
- 30007296*B*b^27*c^10*d^21*e^11 + 6454272*B*b^28*c^9*d^20*e^12 + 10407936*B*b^29*c^8*d^19*e^13 - 14112768*B*b
^30*c^7*d^18*e^14 + 9449472*B*b^31*c^6*d^17*e^15 - 3996672*B*b^32*c^5*d^16*e^16 + 1081344*B*b^33*c^4*d^15*e^17
- 172032*B*b^34*c^3*d^14*e^18 + 12288*B*b^35*c^2*d^13*e^19))*((9*(25*A^2*b^4*e^4 + 256*A^2*c^4*d^4 + 64*B^2*b
^2*c^2*d^4 + 16*B^2*b^4*d^2*e^2 + 304*A^2*b^2*c^2*d^2*e^2 + 384*A^2*b*c^3*d^3*e + 120*A^2*b^3*c*d*e^3 + 64*B^2
*b^3*c*d^3*e - 256*A*B*b*c^3*d^4 - 40*A*B*b^4*d*e^3 - 320*A*B*b^2*c^2*d^3*e - 176*A*B*b^3*c*d^2*e^2))/(64*b^10
*d^7))^(1/2) - 1769472*A^3*b^8*c^23*d^26*e^3 + 23003136*A^3*b^9*c^22*d^25*e^4 - 136138752*A^3*b^10*c^21*d^24*e
^5 + 483508224*A^3*b^11*c^20*d^23*e^6 - 1141579008*A^3*b^12*c^19*d^22*e^7 + 1869094656*A^3*b^13*c^18*d^21*e^8
- 2133106272*A^3*b^14*c^17*d^20*e^9 + 1631703744*A^3*b^15*c^16*d^19*e^10 - 716335488*A^3*b^16*c^15*d^18*e^11 +
36390816*A^3*b^17*c^14*d^17*e^12 + 153641664*A^3*b^18*c^13*d^16*e^13 - 89697024*A^3*b^19*c^12*d^15*e^14 + 400
65408*A^3*b^20*c^11*d^14*e^15 - 43695936*A^3*b^21*c^10*d^13*e^16 + 41388192*A^3*b^22*c^9*d^12*e^17 - 21843648*
A^3*b^23*c^8*d^11*e^18 + 6082560*A^3*b^24*c^7*d^10*e^19 - 712800*A^3*b^25*c^6*d^9*e^20 + 221184*B^3*b^11*c^20*
d^26*e^3 - 3041280*B^3*b^12*c^19*d^25*e^4 + 19132416*B^3*b^13*c^18*d^24*e^5 - 72873216*B^3*b^14*c^17*d^23*e^6
+ 187373952*B^3*b^15*c^16*d^22*e^7 - 343108224*B^3*b^16*c^15*d^21*e^8 + 459302400*B^3*b^17*c^14*d^20*e^9 - 452
086272*B^3*b^18*c^13*d^19*e^10 + 320101632*B^3*b^19*c^12*d^18*e^11 - 148172544*B^3*b^20*c^11*d^17*e^12 + 24731
136*B^3*b^21*c^10*d^16*e^13 + 23604480*B^3*b^22*c^9*d^15*e^14 - 23497344*B^3*b^23*c^8*d^14*e^15 + 10675584*B^3
*b^24*c^7*d^13*e^16 - 2654208*B^3*b^25*c^6*d^12*e^17 + 290304*B^3*b^26*c^5*d^11*e^18 - 1327104*A*B^2*b^10*c^21
*d^26*e^3 + 17915904*A*B^2*b^11*c^20*d^25*e^4 - 110481408*A*B^2*b^12*c^19*d^24*e^5 + 411360768*A*B^2*b^13*c^18
*d^23*e^6 - 1029158784*A*B^2*b^14*c^17*d^22*e^7 + 1819508832*A*B^2*b^15*c^16*d^21*e^8 - 2321496288*A*B^2*b^16*
c^15*d^20*e^9 + 2131940736*A*B^2*b^17*c^14*d^19*e^10 - 1360146816*A*B^2*b^18*c^13*d^18*e^11 + 537046848*A*B^2*
b^19*c^12*d^17*e^12 - 75442752*A*B^2*b^20*c^11*d^16*e^13 - 26096256*A*B^2*b^21*c^10*d^15*e^14 - 9808128*A*B^2*
b^22*c^9*d^14*e^15 + 30634848*A*B^2*b^23*c^8*d^13*e^16 - 19613664*A*B^2*b^24*c^7*d^12*e^17 + 5889024*A*B^2*b^2
5*c^6*d^11*e^18 - 725760*A*B^2*b^26*c^5*d^10*e^19 + 2654208*A^2*B*b^9*c^22*d^26*e^3 - 35168256*A^2*B*b^10*c^21
*d^25*e^4 + 212502528*A^2*B*b^11*c^20*d^24*e^5 - 772996608*A^2*B*b^12*c^19*d^23*e^6 + 1879770240*A^2*B*b^13*c^
18*d^22*e^7 - 3201998688*A^2*B*b^14*c^17*d^21*e^8 + 3875314752*A^2*B*b^15*c^16*d^20*e^9 - 3278408256*A^2*B*b^1
6*c^15*d^19*e^10 + 1809184896*A^2*B*b^17*c^14*d^18*e^11 - 509470560*A^2*B*b^18*c^13*d^17*e^12 - 26137728*A^2*B
*b^19*c^12*d^16*e^13 + 20559744*A^2*B*b^20*c^11*d^15*e^14 + 65536128*A^2*B*b^21*c^10*d^14*e^15 - 57254688*A^2*
B*b^22*c^9*d^13*e^16 + 15059520*A^2*B*b^23*c^8*d^12*e^17 + 3043008*A^2*B*b^24*c^7*d^11*e^18 - 2643840*A^2*B*b^
25*c^6*d^10*e^19 + 453600*A^2*B*b^26*c^5*d^9*e^20))*((9*(25*A^2*b^4*e^4 + 256*A^2*c^4*d^4 + 64*B^2*b^2*c^2*d^4
+ 16*B^2*b^4*d^2*e^2 + 304*A^2*b^2*c^2*d^2*e^2 + 384*A^2*b*c^3*d^3*e + 120*A^2*b^3*c*d*e^3 + 64*B^2*b^3*c*d^3
*e - 256*A*B*b*c^3*d^4 - 40*A*B*b^4*d*e^3 - 320*A*B*b^2*c^2*d^3*e - 176*A*B*b^3*c*d^2*e^2))/(64*b^10*d^7))^(1/
2)*2i - atan(-(((d + e*x)^(1/2)*(589824*A^2*b^12*c^22*d^28*e^2 - 8257536*A^2*b^13*c^21*d^27*e^3 + 53342208*A^2
*b^14*c^20*d^26*e^4 - 210382848*A^2*b^15*c^19*d^25*e^5 + 564860160*A^2*b^16*c^18*d^24*e^6 - 1089838080*A^2*b^1
7*c^17*d^23*e^7 + 1555380864*A^2*b^18*c^16*d^22*e^8 - 1667850624*A^2*b^19*c^15*d^21*e^9 + 1358257536*A^2*b^20*
c^14*d^20*e^10 - 855642240*A^2*b^21*c^13*d^19*e^11 + 438185088*A^2*b^22*c^12*d^18*e^12 - 201386880*A^2*b^23*c^
11*d^17*e^13 + 90100224*A^2*b^24*c^10*d^16*e^14 - 37986048*A^2*b^25*c^9*d^15*e^15 + 15108480*A^2*b^26*c^8*d^14
*e^16 - 6844032*A^2*b^27*c^7*d^13*e^17 + 3399552*A^2*b^28*c^6*d^12*e^18 - 1300608*A^2*b^29*c^5*d^11*e^19 + 293
760*A^2*b^30*c^4*d^10*e^20 - 28800*A^2*b^31*c^3*d^9*e^21 + 147456*B^2*b^14*c^20*d^28*e^2 - 2138112*B^2*b^15*c^
19*d^27*e^3 + 14340096*B^2*b^16*c^18*d^26*e^4 - 58816512*B^2*b^17*c^17*d^25*e^5 + 164257920*B^2*b^18*c^16*d^24
*e^6 - 328809600*B^2*b^19*c^15*d^23*e^7 + 482904576*B^2*b^20*c^14*d^22*e^8 - 521961984*B^2*b^21*c^13*d^21*e^9
+ 407418624*B^2*b^22*c^12*d^20*e^10 - 216610560*B^2*b^23*c^11*d^19*e^11 + 65382912*B^2*b^24*c^10*d^18*e^12 - 1
276416*B^2*b^25*c^9*d^17*e^13 - 6007680*B^2*b^26*c^8*d^16*e^14 + 137088*B^2*b^27*c^7*d^15*e^15 + 1751040*B^2*b
^28*c^6*d^14*e^16 - 903168*B^2*b^29*c^5*d^13*e^17 + 202752*B^2*b^30*c^4*d^12*e^18 - 18432*B^2*b^31*c^3*d^11*e^
19 - 589824*A*B*b^13*c^21*d^28*e^2 + 8404992*A*B*b^14*c^20*d^27*e^3 - 55332864*A*B*b^15*c^19*d^26*e^4 + 222584
832*A*B*b^16*c^18*d^25*e^5 - 609557760*A*B*b^17*c^17*d^24*e^6 + 1197861120*A*B*b^18*c^16*d^23*e^7 - 1733566464
*A*B*b^19*c^15*d^22*e^8 + 1864765440*A*B*b^20*c^14*d^21*e^9 - 1485494784*A*B*b^21*c^13*d^20*e^10 + 864115200*A
*B*b^22*c^12*d^19*e^11 - 361248768*A*B*b^23*c^11*d^18*e^12 + 110656512*A*B*b^24*c^10*d^17*e^13 - 27548928*A*B*
b^25*c^9*d^16*e^14 + 3209472*A*B*b^26*c^8*d^15*e^15 + 4930560*A*B*b^27*c^7*d^14*e^16 - 4912128*A*B*b^28*c^6*d^
13*e^17 + 2165760*A*B*b^29*c^5*d^12*e^18 - 488448*A*B*b^30*c^4*d^11*e^19 + 46080*A*B*b^31*c^3*d^10*e^20) - (-(
9*(256*A^2*c^11*d^4 + 1089*A^2*b^4*c^7*e^4 + 64*B^2*b^2*c^9*d^4 + 441*B^2*b^6*c^5*e^4 + 2992*A^2*b^2*c^9*d^2*e
^2 + 912*B^2*b^4*c^7*d^2*e^2 - 1386*A*B*b^5*c^6*e^4 - 1408*A^2*b*c^10*d^3*e - 2904*A^2*b^3*c^8*d*e^3 - 384*B^2
*b^3*c^8*d^3*e - 1008*B^2*b^5*c^6*d*e^3 - 256*A*B*b*c^10*d^4 + 1472*A*B*b^2*c^9*d^3*e + 3432*A*B*b^4*c^7*d*e^3
- 3312*A*B*b^3*c^8*d^2*e^2))/(64*(b^17*e^7 - b^10*c^7*d^7 + 7*b^11*c^6*d^6*e - 21*b^12*c^5*d^5*e^2 + 35*b^13*
c^4*d^4*e^3 - 35*b^14*c^3*d^3*e^4 + 21*b^15*c^2*d^2*e^5 - 7*b^16*c*d*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(9*(256*A
^2*c^11*d^4 + 1089*A^2*b^4*c^7*e^4 + 64*B^2*b^2*c^9*d^4 + 441*B^2*b^6*c^5*e^4 + 2992*A^2*b^2*c^9*d^2*e^2 + 912
*B^2*b^4*c^7*d^2*e^2 - 1386*A*B*b^5*c^6*e^4 - 1408*A^2*b*c^10*d^3*e - 2904*A^2*b^3*c^8*d*e^3 - 384*B^2*b^3*c^8
*d^3*e - 1008*B^2*b^5*c^6*d*e^3 - 256*A*B*b*c^10*d^4 + 1472*A*B*b^2*c^9*d^3*e + 3432*A*B*b^4*c^7*d*e^3 - 3312*
A*B*b^3*c^8*d^2*e^2))/(64*(b^17*e^7 - b^10*c^7*d^7 + 7*b^11*c^6*d^6*e - 21*b^12*c^5*d^5*e^2 + 35*b^13*c^4*d^4*
e^3 - 35*b^14*c^3*d^3*e^4 + 21*b^15*c^2*d^2*e^5 - 7*b^16*c*d*e^6)))^(1/2)*(16384*b^22*c^18*d^31*e^2 - 253952*b
^23*c^17*d^30*e^3 + 1843200*b^24*c^16*d^29*e^4 - 8314880*b^25*c^15*d^28*e^5 + 26091520*b^26*c^14*d^27*e^6 - 60
383232*b^27*c^13*d^26*e^7 + 106602496*b^28*c^12*d^25*e^8 - 146432000*b^29*c^11*d^24*e^9 + 158146560*b^30*c^10*
d^23*e^10 - 134717440*b^31*c^9*d^22*e^11 + 90202112*b^32*c^8*d^21*e^12 - 46964736*b^33*c^7*d^20*e^13 + 1863680
0*b^34*c^6*d^19*e^14 - 5447680*b^35*c^5*d^18*e^15 + 1105920*b^36*c^4*d^17*e^16 - 139264*b^37*c^3*d^16*e^17 + 8
192*b^38*c^2*d^15*e^18) - 24576*A*b^18*c^19*d^29*e^3 + 356352*A*b^19*c^18*d^28*e^4 - 2396160*A*b^20*c^17*d^27*
e^5 + 9897984*A*b^21*c^16*d^26*e^6 - 28065792*A*b^22*c^15*d^25*e^7 + 57891840*A*b^23*c^14*d^24*e^8 - 90071040*
A*b^24*c^13*d^23*e^9 + 108810240*A*b^25*c^12*d^22*e^10 - 105566208*A*b^26*c^11*d^21*e^11 + 86406144*A*b^27*c^1
0*d^20*e^12 - 63393792*A*b^28*c^9*d^19*e^13 + 43075584*A*b^29*c^8*d^18*e^14 - 26173440*A*b^30*c^7*d^17*e^15 +
13108224*A*b^31*c^6*d^16*e^16 - 4964352*A*b^32*c^5*d^15*e^17 + 1302528*A*b^33*c^4*d^14*e^18 - 208896*A*b^34*c^
3*d^13*e^19 + 15360*A*b^35*c^2*d^12*e^20 + 12288*B*b^19*c^18*d^29*e^3 - 181248*B*b^20*c^17*d^28*e^4 + 1241088*
B*b^21*c^16*d^27*e^5 - 5203968*B*b^22*c^15*d^26*e^6 + 14831616*B*b^23*c^14*d^25*e^7 - 30096384*B*b^24*c^13*d^2
4*e^8 + 44064768*B*b^25*c^12*d^23*e^9 - 45551616*B*b^26*c^11*d^22*e^10 + 30007296*B*b^27*c^10*d^21*e^11 - 6454
272*B*b^28*c^9*d^20*e^12 - 10407936*B*b^29*c^8*d^19*e^13 + 14112768*B*b^30*c^7*d^18*e^14 - 9449472*B*b^31*c^6*
d^17*e^15 + 3996672*B*b^32*c^5*d^16*e^16 - 1081344*B*b^33*c^4*d^15*e^17 + 172032*B*b^34*c^3*d^14*e^18 - 12288*
B*b^35*c^2*d^13*e^19))*(-(9*(256*A^2*c^11*d^4 + 1089*A^2*b^4*c^7*e^4 + 64*B^2*b^2*c^9*d^4 + 441*B^2*b^6*c^5*e^
4 + 2992*A^2*b^2*c^9*d^2*e^2 + 912*B^2*b^4*c^7*d^2*e^2 - 1386*A*B*b^5*c^6*e^4 - 1408*A^2*b*c^10*d^3*e - 2904*A
^2*b^3*c^8*d*e^3 - 384*B^2*b^3*c^8*d^3*e - 1008*B^2*b^5*c^6*d*e^3 - 256*A*B*b*c^10*d^4 + 1472*A*B*b^2*c^9*d^3*
e + 3432*A*B*b^4*c^7*d*e^3 - 3312*A*B*b^3*c^8*d^2*e^2))/(64*(b^17*e^7 - b^10*c^7*d^7 + 7*b^11*c^6*d^6*e - 21*b
^12*c^5*d^5*e^2 + 35*b^13*c^4*d^4*e^3 - 35*b^14*c^3*d^3*e^4 + 21*b^15*c^2*d^2*e^5 - 7*b^16*c*d*e^6)))^(1/2)*1i
+ ((d + e*x)^(1/2)*(589824*A^2*b^12*c^22*d^28*e^2 - 8257536*A^2*b^13*c^21*d^27*e^3 + 53342208*A^2*b^14*c^20*d
^26*e^4 - 210382848*A^2*b^15*c^19*d^25*e^5 + 564860160*A^2*b^16*c^18*d^24*e^6 - 1089838080*A^2*b^17*c^17*d^23*
e^7 + 1555380864*A^2*b^18*c^16*d^22*e^8 - 1667850624*A^2*b^19*c^15*d^21*e^9 + 1358257536*A^2*b^20*c^14*d^20*e^
10 - 855642240*A^2*b^21*c^13*d^19*e^11 + 438185088*A^2*b^22*c^12*d^18*e^12 - 201386880*A^2*b^23*c^11*d^17*e^13
+ 90100224*A^2*b^24*c^10*d^16*e^14 - 37986048*A^2*b^25*c^9*d^15*e^15 + 15108480*A^2*b^26*c^8*d^14*e^16 - 6844
032*A^2*b^27*c^7*d^13*e^17 + 3399552*A^2*b^28*c^6*d^12*e^18 - 1300608*A^2*b^29*c^5*d^11*e^19 + 293760*A^2*b^30
*c^4*d^10*e^20 - 28800*A^2*b^31*c^3*d^9*e^21 + 147456*B^2*b^14*c^20*d^28*e^2 - 2138112*B^2*b^15*c^19*d^27*e^3
+ 14340096*B^2*b^16*c^18*d^26*e^4 - 58816512*B^2*b^17*c^17*d^25*e^5 + 164257920*B^2*b^18*c^16*d^24*e^6 - 32880
9600*B^2*b^19*c^15*d^23*e^7 + 482904576*B^2*b^20*c^14*d^22*e^8 - 521961984*B^2*b^21*c^13*d^21*e^9 + 407418624*
B^2*b^22*c^12*d^20*e^10 - 216610560*B^2*b^23*c^11*d^19*e^11 + 65382912*B^2*b^24*c^10*d^18*e^12 - 1276416*B^2*b
^25*c^9*d^17*e^13 - 6007680*B^2*b^26*c^8*d^16*e^14 + 137088*B^2*b^27*c^7*d^15*e^15 + 1751040*B^2*b^28*c^6*d^14
*e^16 - 903168*B^2*b^29*c^5*d^13*e^17 + 202752*B^2*b^30*c^4*d^12*e^18 - 18432*B^2*b^31*c^3*d^11*e^19 - 589824*
A*B*b^13*c^21*d^28*e^2 + 8404992*A*B*b^14*c^20*d^27*e^3 - 55332864*A*B*b^15*c^19*d^26*e^4 + 222584832*A*B*b^16
*c^18*d^25*e^5 - 609557760*A*B*b^17*c^17*d^24*e^6 + 1197861120*A*B*b^18*c^16*d^23*e^7 - 1733566464*A*B*b^19*c^
15*d^22*e^8 + 1864765440*A*B*b^20*c^14*d^21*e^9 - 1485494784*A*B*b^21*c^13*d^20*e^10 + 864115200*A*B*b^22*c^12
*d^19*e^11 - 361248768*A*B*b^23*c^11*d^18*e^12 + 110656512*A*B*b^24*c^10*d^17*e^13 - 27548928*A*B*b^25*c^9*d^1
6*e^14 + 3209472*A*B*b^26*c^8*d^15*e^15 + 4930560*A*B*b^27*c^7*d^14*e^16 - 4912128*A*B*b^28*c^6*d^13*e^17 + 21
65760*A*B*b^29*c^5*d^12*e^18 - 488448*A*B*b^30*c^4*d^11*e^19 + 46080*A*B*b^31*c^3*d^10*e^20) - (-(9*(256*A^2*c
^11*d^4 + 1089*A^2*b^4*c^7*e^4 + 64*B^2*b^2*c^9*d^4 + 441*B^2*b^6*c^5*e^4 + 2992*A^2*b^2*c^9*d^2*e^2 + 912*B^2
*b^4*c^7*d^2*e^2 - 1386*A*B*b^5*c^6*e^4 - 1408*A^2*b*c^10*d^3*e - 2904*A^2*b^3*c^8*d*e^3 - 384*B^2*b^3*c^8*d^3
*e - 1008*B^2*b^5*c^6*d*e^3 - 256*A*B*b*c^10*d^4 + 1472*A*B*b^2*c^9*d^3*e + 3432*A*B*b^4*c^7*d*e^3 - 3312*A*B*
b^3*c^8*d^2*e^2))/(64*(b^17*e^7 - b^10*c^7*d^7 + 7*b^11*c^6*d^6*e - 21*b^12*c^5*d^5*e^2 + 35*b^13*c^4*d^4*e^3
- 35*b^14*c^3*d^3*e^4 + 21*b^15*c^2*d^2*e^5 - 7*b^16*c*d*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(9*(256*A^2*c^11*d^4
+ 1089*A^2*b^4*c^7*e^4 + 64*B^2*b^2*c^9*d^4 + 441*B^2*b^6*c^5*e^4 + 2992*A^2*b^2*c^9*d^2*e^2 + 912*B^2*b^4*c^7
*d^2*e^2 - 1386*A*B*b^5*c^6*e^4 - 1408*A^2*b*c^10*d^3*e - 2904*A^2*b^3*c^8*d*e^3 - 384*B^2*b^3*c^8*d^3*e - 100
8*B^2*b^5*c^6*d*e^3 - 256*A*B*b*c^10*d^4 + 1472*A*B*b^2*c^9*d^3*e + 3432*A*B*b^4*c^7*d*e^3 - 3312*A*B*b^3*c^8*
d^2*e^2))/(64*(b^17*e^7 - b^10*c^7*d^7 + 7*b^11*c^6*d^6*e - 21*b^12*c^5*d^5*e^2 + 35*b^13*c^4*d^4*e^3 - 35*b^1
4*c^3*d^3*e^4 + 21*b^15*c^2*d^2*e^5 - 7*b^16*c*d*e^6)))^(1/2)*(16384*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d^3
0*e^3 + 1843200*b^24*c^16*d^29*e^4 - 8314880*b^25*c^15*d^28*e^5 + 26091520*b^26*c^14*d^27*e^6 - 60383232*b^27*
c^13*d^26*e^7 + 106602496*b^28*c^12*d^25*e^8 - 146432000*b^29*c^11*d^24*e^9 + 158146560*b^30*c^10*d^23*e^10 -
134717440*b^31*c^9*d^22*e^11 + 90202112*b^32*c^8*d^21*e^12 - 46964736*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6*d
^19*e^14 - 5447680*b^35*c^5*d^18*e^15 + 1105920*b^36*c^4*d^17*e^16 - 139264*b^37*c^3*d^16*e^17 + 8192*b^38*c^2
*d^15*e^18) + 24576*A*b^18*c^19*d^29*e^3 - 356352*A*b^19*c^18*d^28*e^4 + 2396160*A*b^20*c^17*d^27*e^5 - 989798
4*A*b^21*c^16*d^26*e^6 + 28065792*A*b^22*c^15*d^25*e^7 - 57891840*A*b^23*c^14*d^24*e^8 + 90071040*A*b^24*c^13*
d^23*e^9 - 108810240*A*b^25*c^12*d^22*e^10 + 105566208*A*b^26*c^11*d^21*e^11 - 86406144*A*b^27*c^10*d^20*e^12
+ 63393792*A*b^28*c^9*d^19*e^13 - 43075584*A*b^29*c^8*d^18*e^14 + 26173440*A*b^30*c^7*d^17*e^15 - 13108224*A*b
^31*c^6*d^16*e^16 + 4964352*A*b^32*c^5*d^15*e^17 - 1302528*A*b^33*c^4*d^14*e^18 + 208896*A*b^34*c^3*d^13*e^19
- 15360*A*b^35*c^2*d^12*e^20 - 12288*B*b^19*c^18*d^29*e^3 + 181248*B*b^20*c^17*d^28*e^4 - 1241088*B*b^21*c^16*
d^27*e^5 + 5203968*B*b^22*c^15*d^26*e^6 - 14831616*B*b^23*c^14*d^25*e^7 + 30096384*B*b^24*c^13*d^24*e^8 - 4406
4768*B*b^25*c^12*d^23*e^9 + 45551616*B*b^26*c^11*d^22*e^10 - 30007296*B*b^27*c^10*d^21*e^11 + 6454272*B*b^28*c
^9*d^20*e^12 + 10407936*B*b^29*c^8*d^19*e^13 - 14112768*B*b^30*c^7*d^18*e^14 + 9449472*B*b^31*c^6*d^17*e^15 -
3996672*B*b^32*c^5*d^16*e^16 + 1081344*B*b^33*c^4*d^15*e^17 - 172032*B*b^34*c^3*d^14*e^18 + 12288*B*b^35*c^2*d
^13*e^19))*(-(9*(256*A^2*c^11*d^4 + 1089*A^2*b^4*c^7*e^4 + 64*B^2*b^2*c^9*d^4 + 441*B^2*b^6*c^5*e^4 + 2992*A^2
*b^2*c^9*d^2*e^2 + 912*B^2*b^4*c^7*d^2*e^2 - 1386*A*B*b^5*c^6*e^4 - 1408*A^2*b*c^10*d^3*e - 2904*A^2*b^3*c^8*d
*e^3 - 384*B^2*b^3*c^8*d^3*e - 1008*B^2*b^5*c^6*d*e^3 - 256*A*B*b*c^10*d^4 + 1472*A*B*b^2*c^9*d^3*e + 3432*A*B
*b^4*c^7*d*e^3 - 3312*A*B*b^3*c^8*d^2*e^2))/(64*(b^17*e^7 - b^10*c^7*d^7 + 7*b^11*c^6*d^6*e - 21*b^12*c^5*d^5*
e^2 + 35*b^13*c^4*d^4*e^3 - 35*b^14*c^3*d^3*e^4 + 21*b^15*c^2*d^2*e^5 - 7*b^16*c*d*e^6)))^(1/2)*1i)/(((d + e*x
)^(1/2)*(589824*A^2*b^12*c^22*d^28*e^2 - 8257536*A^2*b^13*c^21*d^27*e^3 + 53342208*A^2*b^14*c^20*d^26*e^4 - 21
0382848*A^2*b^15*c^19*d^25*e^5 + 564860160*A^2*b^16*c^18*d^24*e^6 - 1089838080*A^2*b^17*c^17*d^23*e^7 + 155538
0864*A^2*b^18*c^16*d^22*e^8 - 1667850624*A^2*b^19*c^15*d^21*e^9 + 1358257536*A^2*b^20*c^14*d^20*e^10 - 8556422
40*A^2*b^21*c^13*d^19*e^11 + 438185088*A^2*b^22*c^12*d^18*e^12 - 201386880*A^2*b^23*c^11*d^17*e^13 + 90100224*
A^2*b^24*c^10*d^16*e^14 - 37986048*A^2*b^25*c^9*d^15*e^15 + 15108480*A^2*b^26*c^8*d^14*e^16 - 6844032*A^2*b^27
*c^7*d^13*e^17 + 3399552*A^2*b^28*c^6*d^12*e^18 - 1300608*A^2*b^29*c^5*d^11*e^19 + 293760*A^2*b^30*c^4*d^10*e^
20 - 28800*A^2*b^31*c^3*d^9*e^21 + 147456*B^2*b^14*c^20*d^28*e^2 - 2138112*B^2*b^15*c^19*d^27*e^3 + 14340096*B
^2*b^16*c^18*d^26*e^4 - 58816512*B^2*b^17*c^17*d^25*e^5 + 164257920*B^2*b^18*c^16*d^24*e^6 - 328809600*B^2*b^1
9*c^15*d^23*e^7 + 482904576*B^2*b^20*c^14*d^22*e^8 - 521961984*B^2*b^21*c^13*d^21*e^9 + 407418624*B^2*b^22*c^1
2*d^20*e^10 - 216610560*B^2*b^23*c^11*d^19*e^11 + 65382912*B^2*b^24*c^10*d^18*e^12 - 1276416*B^2*b^25*c^9*d^17
*e^13 - 6007680*B^2*b^26*c^8*d^16*e^14 + 137088*B^2*b^27*c^7*d^15*e^15 + 1751040*B^2*b^28*c^6*d^14*e^16 - 9031
68*B^2*b^29*c^5*d^13*e^17 + 202752*B^2*b^30*c^4*d^12*e^18 - 18432*B^2*b^31*c^3*d^11*e^19 - 589824*A*B*b^13*c^2
1*d^28*e^2 + 8404992*A*B*b^14*c^20*d^27*e^3 - 55332864*A*B*b^15*c^19*d^26*e^4 + 222584832*A*B*b^16*c^18*d^25*e
^5 - 609557760*A*B*b^17*c^17*d^24*e^6 + 1197861120*A*B*b^18*c^16*d^23*e^7 - 1733566464*A*B*b^19*c^15*d^22*e^8
+ 1864765440*A*B*b^20*c^14*d^21*e^9 - 1485494784*A*B*b^21*c^13*d^20*e^10 + 864115200*A*B*b^22*c^12*d^19*e^11 -
361248768*A*B*b^23*c^11*d^18*e^12 + 110656512*A*B*b^24*c^10*d^17*e^13 - 27548928*A*B*b^25*c^9*d^16*e^14 + 320
9472*A*B*b^26*c^8*d^15*e^15 + 4930560*A*B*b^27*c^7*d^14*e^16 - 4912128*A*B*b^28*c^6*d^13*e^17 + 2165760*A*B*b^
29*c^5*d^12*e^18 - 488448*A*B*b^30*c^4*d^11*e^19 + 46080*A*B*b^31*c^3*d^10*e^20) - (-(9*(256*A^2*c^11*d^4 + 10
89*A^2*b^4*c^7*e^4 + 64*B^2*b^2*c^9*d^4 + 441*B^2*b^6*c^5*e^4 + 2992*A^2*b^2*c^9*d^2*e^2 + 912*B^2*b^4*c^7*d^2
*e^2 - 1386*A*B*b^5*c^6*e^4 - 1408*A^2*b*c^10*d^3*e - 2904*A^2*b^3*c^8*d*e^3 - 384*B^2*b^3*c^8*d^3*e - 1008*B^
2*b^5*c^6*d*e^3 - 256*A*B*b*c^10*d^4 + 1472*A*B*b^2*c^9*d^3*e + 3432*A*B*b^4*c^7*d*e^3 - 3312*A*B*b^3*c^8*d^2*
e^2))/(64*(b^17*e^7 - b^10*c^7*d^7 + 7*b^11*c^6*d^6*e - 21*b^12*c^5*d^5*e^2 + 35*b^13*c^4*d^4*e^3 - 35*b^14*c^
3*d^3*e^4 + 21*b^15*c^2*d^2*e^5 - 7*b^16*c*d*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(9*(256*A^2*c^11*d^4 + 1089*A^2*b
^4*c^7*e^4 + 64*B^2*b^2*c^9*d^4 + 441*B^2*b^6*c^5*e^4 + 2992*A^2*b^2*c^9*d^2*e^2 + 912*B^2*b^4*c^7*d^2*e^2 - 1
386*A*B*b^5*c^6*e^4 - 1408*A^2*b*c^10*d^3*e - 2904*A^2*b^3*c^8*d*e^3 - 384*B^2*b^3*c^8*d^3*e - 1008*B^2*b^5*c^
6*d*e^3 - 256*A*B*b*c^10*d^4 + 1472*A*B*b^2*c^9*d^3*e + 3432*A*B*b^4*c^7*d*e^3 - 3312*A*B*b^3*c^8*d^2*e^2))/(6
4*(b^17*e^7 - b^10*c^7*d^7 + 7*b^11*c^6*d^6*e - 21*b^12*c^5*d^5*e^2 + 35*b^13*c^4*d^4*e^3 - 35*b^14*c^3*d^3*e^
4 + 21*b^15*c^2*d^2*e^5 - 7*b^16*c*d*e^6)))^(1/2)*(16384*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d^30*e^3 + 1843
200*b^24*c^16*d^29*e^4 - 8314880*b^25*c^15*d^28*e^5 + 26091520*b^26*c^14*d^27*e^6 - 60383232*b^27*c^13*d^26*e^
7 + 106602496*b^28*c^12*d^25*e^8 - 146432000*b^29*c^11*d^24*e^9 + 158146560*b^30*c^10*d^23*e^10 - 134717440*b^
31*c^9*d^22*e^11 + 90202112*b^32*c^8*d^21*e^12 - 46964736*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6*d^19*e^14 - 5
447680*b^35*c^5*d^18*e^15 + 1105920*b^36*c^4*d^17*e^16 - 139264*b^37*c^3*d^16*e^17 + 8192*b^38*c^2*d^15*e^18)
- 24576*A*b^18*c^19*d^29*e^3 + 356352*A*b^19*c^18*d^28*e^4 - 2396160*A*b^20*c^17*d^27*e^5 + 9897984*A*b^21*c^1
6*d^26*e^6 - 28065792*A*b^22*c^15*d^25*e^7 + 57891840*A*b^23*c^14*d^24*e^8 - 90071040*A*b^24*c^13*d^23*e^9 + 1
08810240*A*b^25*c^12*d^22*e^10 - 105566208*A*b^26*c^11*d^21*e^11 + 86406144*A*b^27*c^10*d^20*e^12 - 63393792*A
*b^28*c^9*d^19*e^13 + 43075584*A*b^29*c^8*d^18*e^14 - 26173440*A*b^30*c^7*d^17*e^15 + 13108224*A*b^31*c^6*d^16
*e^16 - 4964352*A*b^32*c^5*d^15*e^17 + 1302528*A*b^33*c^4*d^14*e^18 - 208896*A*b^34*c^3*d^13*e^19 + 15360*A*b^
35*c^2*d^12*e^20 + 12288*B*b^19*c^18*d^29*e^3 - 181248*B*b^20*c^17*d^28*e^4 + 1241088*B*b^21*c^16*d^27*e^5 - 5
203968*B*b^22*c^15*d^26*e^6 + 14831616*B*b^23*c^14*d^25*e^7 - 30096384*B*b^24*c^13*d^24*e^8 + 44064768*B*b^25*
c^12*d^23*e^9 - 45551616*B*b^26*c^11*d^22*e^10 + 30007296*B*b^27*c^10*d^21*e^11 - 6454272*B*b^28*c^9*d^20*e^12
- 10407936*B*b^29*c^8*d^19*e^13 + 14112768*B*b^30*c^7*d^18*e^14 - 9449472*B*b^31*c^6*d^17*e^15 + 3996672*B*b^
32*c^5*d^16*e^16 - 1081344*B*b^33*c^4*d^15*e^17 + 172032*B*b^34*c^3*d^14*e^18 - 12288*B*b^35*c^2*d^13*e^19))*(
-(9*(256*A^2*c^11*d^4 + 1089*A^2*b^4*c^7*e^4 + 64*B^2*b^2*c^9*d^4 + 441*B^2*b^6*c^5*e^4 + 2992*A^2*b^2*c^9*d^2
*e^2 + 912*B^2*b^4*c^7*d^2*e^2 - 1386*A*B*b^5*c^6*e^4 - 1408*A^2*b*c^10*d^3*e - 2904*A^2*b^3*c^8*d*e^3 - 384*B
^2*b^3*c^8*d^3*e - 1008*B^2*b^5*c^6*d*e^3 - 256*A*B*b*c^10*d^4 + 1472*A*B*b^2*c^9*d^3*e + 3432*A*B*b^4*c^7*d*e
^3 - 3312*A*B*b^3*c^8*d^2*e^2))/(64*(b^17*e^7 - b^10*c^7*d^7 + 7*b^11*c^6*d^6*e - 21*b^12*c^5*d^5*e^2 + 35*b^1
3*c^4*d^4*e^3 - 35*b^14*c^3*d^3*e^4 + 21*b^15*c^2*d^2*e^5 - 7*b^16*c*d*e^6)))^(1/2) - ((d + e*x)^(1/2)*(589824
*A^2*b^12*c^22*d^28*e^2 - 8257536*A^2*b^13*c^21*d^27*e^3 + 53342208*A^2*b^14*c^20*d^26*e^4 - 210382848*A^2*b^1
5*c^19*d^25*e^5 + 564860160*A^2*b^16*c^18*d^24*e^6 - 1089838080*A^2*b^17*c^17*d^23*e^7 + 1555380864*A^2*b^18*c
^16*d^22*e^8 - 1667850624*A^2*b^19*c^15*d^21*e^9 + 1358257536*A^2*b^20*c^14*d^20*e^10 - 855642240*A^2*b^21*c^1
3*d^19*e^11 + 438185088*A^2*b^22*c^12*d^18*e^12 - 201386880*A^2*b^23*c^11*d^17*e^13 + 90100224*A^2*b^24*c^10*d
^16*e^14 - 37986048*A^2*b^25*c^9*d^15*e^15 + 15108480*A^2*b^26*c^8*d^14*e^16 - 6844032*A^2*b^27*c^7*d^13*e^17
+ 3399552*A^2*b^28*c^6*d^12*e^18 - 1300608*A^2*b^29*c^5*d^11*e^19 + 293760*A^2*b^30*c^4*d^10*e^20 - 28800*A^2*
b^31*c^3*d^9*e^21 + 147456*B^2*b^14*c^20*d^28*e^2 - 2138112*B^2*b^15*c^19*d^27*e^3 + 14340096*B^2*b^16*c^18*d^
26*e^4 - 58816512*B^2*b^17*c^17*d^25*e^5 + 164257920*B^2*b^18*c^16*d^24*e^6 - 328809600*B^2*b^19*c^15*d^23*e^7
+ 482904576*B^2*b^20*c^14*d^22*e^8 - 521961984*B^2*b^21*c^13*d^21*e^9 + 407418624*B^2*b^22*c^12*d^20*e^10 - 2
16610560*B^2*b^23*c^11*d^19*e^11 + 65382912*B^2*b^24*c^10*d^18*e^12 - 1276416*B^2*b^25*c^9*d^17*e^13 - 6007680
*B^2*b^26*c^8*d^16*e^14 + 137088*B^2*b^27*c^7*d^15*e^15 + 1751040*B^2*b^28*c^6*d^14*e^16 - 903168*B^2*b^29*c^5
*d^13*e^17 + 202752*B^2*b^30*c^4*d^12*e^18 - 18432*B^2*b^31*c^3*d^11*e^19 - 589824*A*B*b^13*c^21*d^28*e^2 + 84
04992*A*B*b^14*c^20*d^27*e^3 - 55332864*A*B*b^15*c^19*d^26*e^4 + 222584832*A*B*b^16*c^18*d^25*e^5 - 609557760*
A*B*b^17*c^17*d^24*e^6 + 1197861120*A*B*b^18*c^16*d^23*e^7 - 1733566464*A*B*b^19*c^15*d^22*e^8 + 1864765440*A*
B*b^20*c^14*d^21*e^9 - 1485494784*A*B*b^21*c^13*d^20*e^10 + 864115200*A*B*b^22*c^12*d^19*e^11 - 361248768*A*B*
b^23*c^11*d^18*e^12 + 110656512*A*B*b^24*c^10*d^17*e^13 - 27548928*A*B*b^25*c^9*d^16*e^14 + 3209472*A*B*b^26*c
^8*d^15*e^15 + 4930560*A*B*b^27*c^7*d^14*e^16 - 4912128*A*B*b^28*c^6*d^13*e^17 + 2165760*A*B*b^29*c^5*d^12*e^1
8 - 488448*A*B*b^30*c^4*d^11*e^19 + 46080*A*B*b^31*c^3*d^10*e^20) - (-(9*(256*A^2*c^11*d^4 + 1089*A^2*b^4*c^7*
e^4 + 64*B^2*b^2*c^9*d^4 + 441*B^2*b^6*c^5*e^4 + 2992*A^2*b^2*c^9*d^2*e^2 + 912*B^2*b^4*c^7*d^2*e^2 - 1386*A*B
*b^5*c^6*e^4 - 1408*A^2*b*c^10*d^3*e - 2904*A^2*b^3*c^8*d*e^3 - 384*B^2*b^3*c^8*d^3*e - 1008*B^2*b^5*c^6*d*e^3
- 256*A*B*b*c^10*d^4 + 1472*A*B*b^2*c^9*d^3*e + 3432*A*B*b^4*c^7*d*e^3 - 3312*A*B*b^3*c^8*d^2*e^2))/(64*(b^17
*e^7 - b^10*c^7*d^7 + 7*b^11*c^6*d^6*e - 21*b^12*c^5*d^5*e^2 + 35*b^13*c^4*d^4*e^3 - 35*b^14*c^3*d^3*e^4 + 21*
b^15*c^2*d^2*e^5 - 7*b^16*c*d*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(9*(256*A^2*c^11*d^4 + 1089*A^2*b^4*c^7*e^4 + 64
*B^2*b^2*c^9*d^4 + 441*B^2*b^6*c^5*e^4 + 2992*A^2*b^2*c^9*d^2*e^2 + 912*B^2*b^4*c^7*d^2*e^2 - 1386*A*B*b^5*c^6
*e^4 - 1408*A^2*b*c^10*d^3*e - 2904*A^2*b^3*c^8*d*e^3 - 384*B^2*b^3*c^8*d^3*e - 1008*B^2*b^5*c^6*d*e^3 - 256*A
*B*b*c^10*d^4 + 1472*A*B*b^2*c^9*d^3*e + 3432*A*B*b^4*c^7*d*e^3 - 3312*A*B*b^3*c^8*d^2*e^2))/(64*(b^17*e^7 - b
^10*c^7*d^7 + 7*b^11*c^6*d^6*e - 21*b^12*c^5*d^5*e^2 + 35*b^13*c^4*d^4*e^3 - 35*b^14*c^3*d^3*e^4 + 21*b^15*c^2
*d^2*e^5 - 7*b^16*c*d*e^6)))^(1/2)*(16384*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d^30*e^3 + 1843200*b^24*c^16*d
^29*e^4 - 8314880*b^25*c^15*d^28*e^5 + 26091520*b^26*c^14*d^27*e^6 - 60383232*b^27*c^13*d^26*e^7 + 106602496*b
^28*c^12*d^25*e^8 - 146432000*b^29*c^11*d^24*e^9 + 158146560*b^30*c^10*d^23*e^10 - 134717440*b^31*c^9*d^22*e^1
1 + 90202112*b^32*c^8*d^21*e^12 - 46964736*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6*d^19*e^14 - 5447680*b^35*c^5
*d^18*e^15 + 1105920*b^36*c^4*d^17*e^16 - 139264*b^37*c^3*d^16*e^17 + 8192*b^38*c^2*d^15*e^18) + 24576*A*b^18*
c^19*d^29*e^3 - 356352*A*b^19*c^18*d^28*e^4 + 2396160*A*b^20*c^17*d^27*e^5 - 9897984*A*b^21*c^16*d^26*e^6 + 28
065792*A*b^22*c^15*d^25*e^7 - 57891840*A*b^23*c^14*d^24*e^8 + 90071040*A*b^24*c^13*d^23*e^9 - 108810240*A*b^25
*c^12*d^22*e^10 + 105566208*A*b^26*c^11*d^21*e^11 - 86406144*A*b^27*c^10*d^20*e^12 + 63393792*A*b^28*c^9*d^19*
e^13 - 43075584*A*b^29*c^8*d^18*e^14 + 26173440*A*b^30*c^7*d^17*e^15 - 13108224*A*b^31*c^6*d^16*e^16 + 4964352
*A*b^32*c^5*d^15*e^17 - 1302528*A*b^33*c^4*d^14*e^18 + 208896*A*b^34*c^3*d^13*e^19 - 15360*A*b^35*c^2*d^12*e^2
0 - 12288*B*b^19*c^18*d^29*e^3 + 181248*B*b^20*c^17*d^28*e^4 - 1241088*B*b^21*c^16*d^27*e^5 + 5203968*B*b^22*c
^15*d^26*e^6 - 14831616*B*b^23*c^14*d^25*e^7 + 30096384*B*b^24*c^13*d^24*e^8 - 44064768*B*b^25*c^12*d^23*e^9 +
45551616*B*b^26*c^11*d^22*e^10 - 30007296*B*b^27*c^10*d^21*e^11 + 6454272*B*b^28*c^9*d^20*e^12 + 10407936*B*b
^29*c^8*d^19*e^13 - 14112768*B*b^30*c^7*d^18*e^14 + 9449472*B*b^31*c^6*d^17*e^15 - 3996672*B*b^32*c^5*d^16*e^1
6 + 1081344*B*b^33*c^4*d^15*e^17 - 172032*B*b^34*c^3*d^14*e^18 + 12288*B*b^35*c^2*d^13*e^19))*(-(9*(256*A^2*c^
11*d^4 + 1089*A^2*b^4*c^7*e^4 + 64*B^2*b^2*c^9*d^4 + 441*B^2*b^6*c^5*e^4 + 2992*A^2*b^2*c^9*d^2*e^2 + 912*B^2*
b^4*c^7*d^2*e^2 - 1386*A*B*b^5*c^6*e^4 - 1408*A^2*b*c^10*d^3*e - 2904*A^2*b^3*c^8*d*e^3 - 384*B^2*b^3*c^8*d^3*
e - 1008*B^2*b^5*c^6*d*e^3 - 256*A*B*b*c^10*d^4 + 1472*A*B*b^2*c^9*d^3*e + 3432*A*B*b^4*c^7*d*e^3 - 3312*A*B*b
^3*c^8*d^2*e^2))/(64*(b^17*e^7 - b^10*c^7*d^7 + 7*b^11*c^6*d^6*e - 21*b^12*c^5*d^5*e^2 + 35*b^13*c^4*d^4*e^3 -
35*b^14*c^3*d^3*e^4 + 21*b^15*c^2*d^2*e^5 - 7*b^16*c*d*e^6)))^(1/2) - 1769472*A^3*b^8*c^23*d^26*e^3 + 2300313
6*A^3*b^9*c^22*d^25*e^4 - 136138752*A^3*b^10*c^21*d^24*e^5 + 483508224*A^3*b^11*c^20*d^23*e^6 - 1141579008*A^3
*b^12*c^19*d^22*e^7 + 1869094656*A^3*b^13*c^18*d^21*e^8 - 2133106272*A^3*b^14*c^17*d^20*e^9 + 1631703744*A^3*b
^15*c^16*d^19*e^10 - 716335488*A^3*b^16*c^15*d^18*e^11 + 36390816*A^3*b^17*c^14*d^17*e^12 + 153641664*A^3*b^18
*c^13*d^16*e^13 - 89697024*A^3*b^19*c^12*d^15*e^14 + 40065408*A^3*b^20*c^11*d^14*e^15 - 43695936*A^3*b^21*c^10
*d^13*e^16 + 41388192*A^3*b^22*c^9*d^12*e^17 - 21843648*A^3*b^23*c^8*d^11*e^18 + 6082560*A^3*b^24*c^7*d^10*e^1
9 - 712800*A^3*b^25*c^6*d^9*e^20 + 221184*B^3*b^11*c^20*d^26*e^3 - 3041280*B^3*b^12*c^19*d^25*e^4 + 19132416*B
^3*b^13*c^18*d^24*e^5 - 72873216*B^3*b^14*c^17*d^23*e^6 + 187373952*B^3*b^15*c^16*d^22*e^7 - 343108224*B^3*b^1
6*c^15*d^21*e^8 + 459302400*B^3*b^17*c^14*d^20*e^9 - 452086272*B^3*b^18*c^13*d^19*e^10 + 320101632*B^3*b^19*c^
12*d^18*e^11 - 148172544*B^3*b^20*c^11*d^17*e^12 + 24731136*B^3*b^21*c^10*d^16*e^13 + 23604480*B^3*b^22*c^9*d^
15*e^14 - 23497344*B^3*b^23*c^8*d^14*e^15 + 10675584*B^3*b^24*c^7*d^13*e^16 - 2654208*B^3*b^25*c^6*d^12*e^17 +
290304*B^3*b^26*c^5*d^11*e^18 - 1327104*A*B^2*b^10*c^21*d^26*e^3 + 17915904*A*B^2*b^11*c^20*d^25*e^4 - 110481
408*A*B^2*b^12*c^19*d^24*e^5 + 411360768*A*B^2*b^13*c^18*d^23*e^6 - 1029158784*A*B^2*b^14*c^17*d^22*e^7 + 1819
508832*A*B^2*b^15*c^16*d^21*e^8 - 2321496288*A*B^2*b^16*c^15*d^20*e^9 + 2131940736*A*B^2*b^17*c^14*d^19*e^10 -
1360146816*A*B^2*b^18*c^13*d^18*e^11 + 537046848*A*B^2*b^19*c^12*d^17*e^12 - 75442752*A*B^2*b^20*c^11*d^16*e^
13 - 26096256*A*B^2*b^21*c^10*d^15*e^14 - 9808128*A*B^2*b^22*c^9*d^14*e^15 + 30634848*A*B^2*b^23*c^8*d^13*e^16
- 19613664*A*B^2*b^24*c^7*d^12*e^17 + 5889024*A*B^2*b^25*c^6*d^11*e^18 - 725760*A*B^2*b^26*c^5*d^10*e^19 + 26
54208*A^2*B*b^9*c^22*d^26*e^3 - 35168256*A^2*B*b^10*c^21*d^25*e^4 + 212502528*A^2*B*b^11*c^20*d^24*e^5 - 77299
6608*A^2*B*b^12*c^19*d^23*e^6 + 1879770240*A^2*B*b^13*c^18*d^22*e^7 - 3201998688*A^2*B*b^14*c^17*d^21*e^8 + 38
75314752*A^2*B*b^15*c^16*d^20*e^9 - 3278408256*A^2*B*b^16*c^15*d^19*e^10 + 1809184896*A^2*B*b^17*c^14*d^18*e^1
1 - 509470560*A^2*B*b^18*c^13*d^17*e^12 - 26137728*A^2*B*b^19*c^12*d^16*e^13 + 20559744*A^2*B*b^20*c^11*d^15*e
^14 + 65536128*A^2*B*b^21*c^10*d^14*e^15 - 57254688*A^2*B*b^22*c^9*d^13*e^16 + 15059520*A^2*B*b^23*c^8*d^12*e^
17 + 3043008*A^2*B*b^24*c^7*d^11*e^18 - 2643840*A^2*B*b^25*c^6*d^10*e^19 + 453600*A^2*B*b^26*c^5*d^9*e^20))*(-
(9*(256*A^2*c^11*d^4 + 1089*A^2*b^4*c^7*e^4 + 64*B^2*b^2*c^9*d^4 + 441*B^2*b^6*c^5*e^4 + 2992*A^2*b^2*c^9*d^2*
e^2 + 912*B^2*b^4*c^7*d^2*e^2 - 1386*A*B*b^5*c^6*e^4 - 1408*A^2*b*c^10*d^3*e - 2904*A^2*b^3*c^8*d*e^3 - 384*B^
2*b^3*c^8*d^3*e - 1008*B^2*b^5*c^6*d*e^3 - 256*A*B*b*c^10*d^4 + 1472*A*B*b^2*c^9*d^3*e + 3432*A*B*b^4*c^7*d*e^
3 - 3312*A*B*b^3*c^8*d^2*e^2))/(64*(b^17*e^7 - b^10*c^7*d^7 + 7*b^11*c^6*d^6*e - 21*b^12*c^5*d^5*e^2 + 35*b^13
*c^4*d^4*e^3 - 35*b^14*c^3*d^3*e^4 + 21*b^15*c^2*d^2*e^5 - 7*b^16*c*d*e^6)))^(1/2)*2i

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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