\(\int \frac {A+B x}{(d+e x)^{7/2} (b x+c x^2)^2} \, dx\) [1246]
Optimal result
Integrand size = 26, antiderivative size = 457 \[
\int \frac {A+B x}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=-\frac {e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac {e \left (6 A c^3 d^3-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 b c^2 d^2 (B d+3 A e)\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )}{b^2 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac {(2 b B d-4 A c d-7 A b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{9/2}}+\frac {c^{7/2} \left (2 b B c d-4 A c^2 d-9 b^2 B e+11 A b c e\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}
\]
[Out]
-1/5*e*(10*A*c^2*d^2-b^2*e*(-7*A*e+2*B*d)-5*b*c*d*(2*A*e+B*d))/b^2/d^2/(-b*e+c*d)^2/(e*x+d)^(5/2)-1/3*e*(6*A*c
^3*d^3-b^2*c*d*e*(-17*A*e+6*B*d)+b^3*e^2*(-7*A*e+2*B*d)-3*b*c^2*d^2*(3*A*e+B*d))/b^2/d^3/(-b*e+c*d)^3/(e*x+d)^
(3/2)+(-A*b*(-b*e+c*d)-c*(2*A*c*d-b*(A*e+B*d))*x)/b^2/d/(-b*e+c*d)/(e*x+d)^(5/2)/(c*x^2+b*x)-(-7*A*b*e-4*A*c*d
+2*B*b*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3/d^(9/2)+c^(7/2)*(11*A*b*c*e-4*A*c^2*d-9*B*b^2*e+2*B*b*c*d)*arctan
h(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3/(-b*e+c*d)^(9/2)-e*(2*A*c^4*d^4-2*b^2*c^2*d^2*e*(-13*A*e+6*B*d)-
b^4*e^3*(-7*A*e+2*B*d)+8*b^3*c*d*e^2*(-3*A*e+B*d)-b*c^3*d^3*(4*A*e+B*d))/b^2/d^4/(-b*e+c*d)^4/(e*x+d)^(1/2)
Rubi [A] (verified)
Time = 0.92 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.00, number of
steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {836, 842, 840, 1180, 214}
\[
\int \frac {A+B x}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (-7 A b e-4 A c d+2 b B d)}{b^3 d^{9/2}}+\frac {c^{7/2} \left (11 A b c e-4 A c^2 d-9 b^2 B e+2 b B c d\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}-\frac {e \left (b^2 (-e) (2 B d-7 A e)-5 b c d (2 A e+B d)+10 A c^2 d^2\right )}{5 b^2 d^2 (d+e x)^{5/2} (c d-b e)^2}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)}-\frac {e \left (b^3 e^2 (2 B d-7 A e)-b^2 c d e (6 B d-17 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{3 b^2 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac {e \left (b^4 \left (-e^3\right ) (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-2 b^2 c^2 d^2 e (6 B d-13 A e)-b c^3 d^3 (4 A e+B d)+2 A c^4 d^4\right )}{b^2 d^4 \sqrt {d+e x} (c d-b e)^4}
\]
[In]
Int[(A + B*x)/((d + e*x)^(7/2)*(b*x + c*x^2)^2),x]
[Out]
-1/5*(e*(10*A*c^2*d^2 - b^2*e*(2*B*d - 7*A*e) - 5*b*c*d*(B*d + 2*A*e)))/(b^2*d^2*(c*d - b*e)^2*(d + e*x)^(5/2)
) - (e*(6*A*c^3*d^3 - b^2*c*d*e*(6*B*d - 17*A*e) + b^3*e^2*(2*B*d - 7*A*e) - 3*b*c^2*d^2*(B*d + 3*A*e)))/(3*b^
2*d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) - (e*(2*A*c^4*d^4 - 2*b^2*c^2*d^2*e*(6*B*d - 13*A*e) - b^4*e^3*(2*B*d - 7
*A*e) + 8*b^3*c*d*e^2*(B*d - 3*A*e) - b*c^3*d^3*(B*d + 4*A*e)))/(b^2*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - (A*b*(
c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x)/(b^2*d*(c*d - b*e)*(d + e*x)^(5/2)*(b*x + c*x^2)) - ((2*b*B*d - 4*
A*c*d - 7*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(9/2)) + (c^(7/2)*(2*b*B*c*d - 4*A*c^2*d - 9*b^2*B*e +
11*A*b*c*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(c*d - b*e)^(9/2))
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 836
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 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 842
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 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 {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac {\int \frac {-\frac {1}{2} (c d-b e) (2 b B d-4 A c d-7 A b e)-\frac {7}{2} c e (b B d-2 A c d+A b e) x}{(d+e x)^{7/2} \left (b x+c x^2\right )} \, dx}{b^2 d (c d-b e)} \\ & = -\frac {e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac {\int \frac {-\frac {1}{2} (c d-b e)^2 (2 b B d-4 A c d-7 A b e)+\frac {1}{2} c e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right ) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{b^2 d^2 (c d-b e)^2} \\ & = -\frac {e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac {e \left (6 A c^3 d^3-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 b c^2 d^2 (B d+3 A e)\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac {\int \frac {-\frac {1}{2} (c d-b e)^3 (2 b B d-4 A c d-7 A b e)+\frac {1}{2} c e \left (6 A c^3 d^3-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 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}{b^2 d^3 (c d-b e)^3} \\ & = -\frac {e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac {e \left (6 A c^3 d^3-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 b c^2 d^2 (B d+3 A e)\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )}{b^2 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac {\int \frac {-\frac {1}{2} (c d-b e)^4 (2 b B d-4 A c d-7 A b e)+\frac {1}{2} c e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2 d^4 (c d-b e)^4} \\ & = -\frac {e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac {e \left (6 A c^3 d^3-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 b c^2 d^2 (B d+3 A e)\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )}{b^2 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac {2 \text {Subst}\left (\int \frac {-\frac {1}{2} e (c d-b e)^4 (2 b B d-4 A c d-7 A b e)-\frac {1}{2} c d e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )+\frac {1}{2} c e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 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^2 d^4 (c d-b e)^4} \\ & = -\frac {e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac {e \left (6 A c^3 d^3-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 b c^2 d^2 (B d+3 A e)\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )}{b^2 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}+\frac {(c (2 b B d-4 A c d-7 A b e)) \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 )}{b^3 d^4}-\frac {\left (2 \left (\frac {1}{4} c e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )-\frac {-\frac {1}{2} c e (-2 c d+b e) \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )+2 c \left (-\frac {1}{2} e (c d-b e)^4 (2 b B d-4 A c d-7 A b e)-\frac {1}{2} c d e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )\right )}{2 b 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 )}{b^2 d^4 (c d-b e)^4} \\ & = -\frac {e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac {e \left (6 A c^3 d^3-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 b c^2 d^2 (B d+3 A e)\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )}{b^2 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac {(2 b B d-4 A c d-7 A b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{9/2}}+\frac {c^{7/2} \left (2 b B c d-4 A c^2 d-9 b^2 B e+11 A b c e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{9/2}} \\
\end{align*}
Mathematica [A] (verified)
Time = 2.04 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.21
\[
\int \frac {A+B x}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=\frac {\frac {b \left (b B d x \left (15 c^4 d^3 (d+e x)^3+2 b^4 e^4 \left (23 d^2+35 d e x+15 e^2 x^2\right )+6 b c^3 d^2 e^2 x \left (36 d^2+65 d e x+30 e^2 x^2\right )+2 b^2 c^2 d e^2 \left (108 d^3+109 d^2 e x-50 d e^2 x^2-60 e^3 x^3\right )-2 b^3 c e^3 \left (86 d^3+117 d^2 e x+25 d e^2 x^2-15 e^3 x^3\right )\right )-A \left (30 c^5 d^4 x (d+e x)^3+15 b c^4 d^3 (d-4 e x) (d+e x)^3+b^5 e^4 \left (15 d^3+161 d^2 e x+245 d e^2 x^2+105 e^3 x^3\right )+2 b^3 c^2 d e^2 \left (45 d^4+278 d^3 e x+179 d^2 e^2 x^2-225 d e^3 x^3-180 e^4 x^4\right )-b^4 c e^3 \left (60 d^4+537 d^3 e x+679 d^2 e^2 x^2+115 d e^3 x^3-105 e^4 x^4\right )+2 b^2 c^3 d^2 e \left (-30 d^4-45 d^3 e x+218 d^2 e^2 x^2+425 d e^3 x^3+195 e^4 x^4\right )\right )\right )}{d^4 (c d-b e)^4 x (b+c x) (d+e x)^{5/2}}+\frac {15 c^{7/2} \left (4 A c^2 d+9 b^2 B e-b c (2 B d+11 A e)\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{9/2}}+\frac {15 (-2 b B d+4 A c d+7 A b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{9/2}}}{15 b^3}
\]
[In]
Integrate[(A + B*x)/((d + e*x)^(7/2)*(b*x + c*x^2)^2),x]
[Out]
((b*(b*B*d*x*(15*c^4*d^3*(d + e*x)^3 + 2*b^4*e^4*(23*d^2 + 35*d*e*x + 15*e^2*x^2) + 6*b*c^3*d^2*e^2*x*(36*d^2
+ 65*d*e*x + 30*e^2*x^2) + 2*b^2*c^2*d*e^2*(108*d^3 + 109*d^2*e*x - 50*d*e^2*x^2 - 60*e^3*x^3) - 2*b^3*c*e^3*(
86*d^3 + 117*d^2*e*x + 25*d*e^2*x^2 - 15*e^3*x^3)) - A*(30*c^5*d^4*x*(d + e*x)^3 + 15*b*c^4*d^3*(d - 4*e*x)*(d
+ e*x)^3 + b^5*e^4*(15*d^3 + 161*d^2*e*x + 245*d*e^2*x^2 + 105*e^3*x^3) + 2*b^3*c^2*d*e^2*(45*d^4 + 278*d^3*e
*x + 179*d^2*e^2*x^2 - 225*d*e^3*x^3 - 180*e^4*x^4) - b^4*c*e^3*(60*d^4 + 537*d^3*e*x + 679*d^2*e^2*x^2 + 115*
d*e^3*x^3 - 105*e^4*x^4) + 2*b^2*c^3*d^2*e*(-30*d^4 - 45*d^3*e*x + 218*d^2*e^2*x^2 + 425*d*e^3*x^3 + 195*e^4*x
^4))))/(d^4*(c*d - b*e)^4*x*(b + c*x)*(d + e*x)^(5/2)) + (15*c^(7/2)*(4*A*c^2*d + 9*b^2*B*e - b*c*(2*B*d + 11*
A*e))*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(-(c*d) + b*e)^(9/2) + (15*(-2*b*B*d + 4*A*c*d + 7*A
*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(9/2))/(15*b^3)
Maple [A] (verified)
Time = 1.30 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.76
| | |
method | result | size |
| | |
derivativedivides |
\(2 e^{2} \left (-\frac {A e -B d}{5 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 A b \,e^{2}-4 A c d e -B b d e +3 B c \,d^{2}}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {3 A \,b^{2} e^{3}-10 A b c d \,e^{2}+10 A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e -6 B \,c^{2} d^{3}}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}-\frac {c^{4} \left (\frac {\left (\frac {1}{2} A b c e -\frac {1}{2} b^{2} B e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (11 A b c e -4 A \,c^{2} d -9 b^{2} B e +2 B b c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{4} e^{2} b^{3}}+\frac {-\frac {A b \sqrt {e x +d}}{2 x}+\frac {\left (7 A b e +4 A c d -2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{d^{4} e^{2} b^{3}}\right )\) |
\(347\) |
default |
\(2 e^{2} \left (-\frac {A e -B d}{5 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 A b \,e^{2}-4 A c d e -B b d e +3 B c \,d^{2}}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {3 A \,b^{2} e^{3}-10 A b c d \,e^{2}+10 A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e -6 B \,c^{2} d^{3}}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}-\frac {c^{4} \left (\frac {\left (\frac {1}{2} A b c e -\frac {1}{2} b^{2} B e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (11 A b c e -4 A \,c^{2} d -9 b^{2} B e +2 B b c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{4} e^{2} b^{3}}+\frac {-\frac {A b \sqrt {e x +d}}{2 x}+\frac {\left (7 A b e +4 A c d -2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{d^{4} e^{2} b^{3}}\right )\) |
\(347\) |
risch |
\(-\frac {A \sqrt {e x +d}}{d^{4} b^{2} x}-\frac {e \left (\frac {2 c^{4} d^{4} \left (\frac {\left (\frac {1}{2} A b c e -\frac {1}{2} b^{2} B e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (11 A b c e -4 A \,c^{2} d -9 b^{2} B e +2 B b c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{4} b e}-\frac {\left (7 A b e +4 A c d -2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}+\frac {2 b^{2} e \left (3 A \,b^{2} e^{3}-10 A b c d \,e^{2}+10 A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e -6 B \,c^{2} d^{3}\right )}{\left (b e -c d \right )^{4} \sqrt {e x +d}}+\frac {2 b^{2} d e \left (2 A b \,e^{2}-4 A c d e -B b d e +3 B c \,d^{2}\right )}{3 \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 b^{2} d^{2} e \left (A e -B d \right )}{5 \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}\right )}{d^{4} b^{2}}\) |
\(362\) |
pseudoelliptic |
\(e^{2} \left (-\frac {\sqrt {e x +d}\, A \,c^{5}}{e^{2} \left (c x +b \right ) b^{2} \left (b e -c d \right )^{4}}+\frac {\sqrt {e x +d}\, B \,c^{4}}{e^{2} \left (c x +b \right ) b \left (b e -c d \right )^{4}}-\frac {11 A \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) c^{5}}{\sqrt {\left (b e -c d \right ) c}\, b^{2} e \left (b e -c d \right )^{4}}+\frac {4 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) A \,c^{6} d}{\sqrt {\left (b e -c d \right ) c}\, \left (b e -c d \right )^{4} e^{2} b^{3}}+\frac {9 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) B \,c^{4}}{\sqrt {\left (b e -c d \right ) c}\, b e \left (b e -c d \right )^{4}}-\frac {2 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) B \,c^{5} d}{\sqrt {\left (b e -c d \right ) c}\, b^{2} \left (b e -c d \right )^{4} e^{2}}+\frac {-\frac {A b \sqrt {e x +d}}{x}+\frac {\left (7 A b e +4 A c d -2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{\sqrt {d}}}{b^{3} e^{2} d^{4}}-\frac {2 \left (A e -B d \right )}{5 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (2 A b \,e^{2}-4 A c d e -B b d e +3 B c \,d^{2}\right )}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (3 A \,b^{2} e^{3}-10 A b c d \,e^{2}+10 A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e -6 B \,c^{2} d^{3}\right )}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}\right )\) |
\(518\) |
| | |
|
|
|
[In]
int((B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)
[Out]
2*e^2*(-1/5*(A*e-B*d)/d^2/(b*e-c*d)^2/(e*x+d)^(5/2)-1/3*(2*A*b*e^2-4*A*c*d*e-B*b*d*e+3*B*c*d^2)/d^3/(b*e-c*d)^
3/(e*x+d)^(3/2)-(3*A*b^2*e^3-10*A*b*c*d*e^2+10*A*c^2*d^2*e-B*b^2*d*e^2+4*B*b*c*d^2*e-6*B*c^2*d^3)/d^4/(b*e-c*d
)^4/(e*x+d)^(1/2)-c^4/(b*e-c*d)^4/e^2/b^3*((1/2*A*b*c*e-1/2*b^2*B*e)*(e*x+d)^(1/2)/(c*(e*x+d)+b*e-c*d)+1/2*(11
*A*b*c*e-4*A*c^2*d-9*B*b^2*e+2*B*b*c*d)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)))+1/d^4
/e^2/b^3*(-1/2*A*b*(e*x+d)^(1/2)/x+1/2*(7*A*b*e+4*A*c*d-2*B*b*d)/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. 2127 vs. \(2 (429) = 858\).
Time = 93.47 (sec) , antiderivative size = 8537, normalized size of antiderivative = 18.68
\[
\int \frac {A+B x}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=\text {Too large to display}
\]
[In]
integrate((B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x)^2,x, algorithm="fricas")
[Out]
Too large to include
Sympy [F(-1)]
Timed out. \[
\int \frac {A+B x}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=\text {Timed out}
\]
[In]
integrate((B*x+A)/(e*x+d)**(7/2)/(c*x**2+b*x)**2,x)
[Out]
Timed out
Maxima [F(-2)]
Exception generated. \[
\int \frac {A+B x}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError}
\]
[In]
integrate((B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more detail
Giac [A] (verification not implemented)
none
Time = 0.32 (sec) , antiderivative size = 858, normalized size of antiderivative = 1.88
\[
\int \frac {A+B x}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=-\frac {{\left (2 \, B b c^{5} d - 4 \, A c^{6} d - 9 \, B b^{2} c^{4} e + 11 \, A b c^{5} e\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{4} d^{4} - 4 \, b^{4} c^{3} d^{3} e + 6 \, b^{5} c^{2} d^{2} e^{2} - 4 \, b^{6} c d e^{3} + b^{7} e^{4}\right )} \sqrt {-c^{2} d + b c e}} + \frac {{\left (e x + d\right )}^{\frac {3}{2}} B b c^{4} d^{4} e - 2 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{5} d^{4} e - \sqrt {e x + d} B b c^{4} d^{5} e + 2 \, \sqrt {e x + d} A c^{5} d^{5} e + 4 \, {\left (e x + d\right )}^{\frac {3}{2}} A b c^{4} d^{3} e^{2} - 5 \, \sqrt {e x + d} A b c^{4} d^{4} e^{2} - 6 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{2} c^{3} d^{2} e^{3} + 10 \, \sqrt {e x + d} A b^{2} c^{3} d^{3} e^{3} + 4 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{3} c^{2} d e^{4} - 10 \, \sqrt {e x + d} A b^{3} c^{2} d^{2} e^{4} - {\left (e x + d\right )}^{\frac {3}{2}} A b^{4} c e^{5} + 5 \, \sqrt {e x + d} A b^{4} c d e^{5} - \sqrt {e x + d} A b^{5} e^{6}}{{\left (b^{2} c^{4} d^{8} - 4 \, b^{3} c^{3} d^{7} e + 6 \, b^{4} c^{2} d^{6} e^{2} - 4 \, b^{5} c d^{5} e^{3} + b^{6} d^{4} e^{4}\right )} {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )}} + \frac {2 \, {\left (90 \, {\left (e x + d\right )}^{2} B c^{2} d^{3} e^{2} + 15 \, {\left (e x + d\right )} B c^{2} d^{4} e^{2} + 3 \, B c^{2} d^{5} e^{2} - 60 \, {\left (e x + d\right )}^{2} B b c d^{2} e^{3} - 150 \, {\left (e x + d\right )}^{2} A c^{2} d^{2} e^{3} - 20 \, {\left (e x + d\right )} B b c d^{3} e^{3} - 20 \, {\left (e x + d\right )} A c^{2} d^{3} e^{3} - 6 \, B b c d^{4} e^{3} - 3 \, A c^{2} d^{4} e^{3} + 15 \, {\left (e x + d\right )}^{2} B b^{2} d e^{4} + 150 \, {\left (e x + d\right )}^{2} A b c d e^{4} + 5 \, {\left (e x + d\right )} B b^{2} d^{2} e^{4} + 30 \, {\left (e x + d\right )} A b c d^{2} e^{4} + 3 \, B b^{2} d^{3} e^{4} + 6 \, A b c d^{3} e^{4} - 45 \, {\left (e x + d\right )}^{2} A b^{2} e^{5} - 10 \, {\left (e x + d\right )} A b^{2} d e^{5} - 3 \, A b^{2} d^{2} e^{5}\right )}}{15 \, {\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}}} + \frac {{\left (2 \, B b d - 4 \, A c d - 7 \, A b e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d} d^{4}}
\]
[In]
integrate((B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x)^2,x, algorithm="giac")
[Out]
-(2*B*b*c^5*d - 4*A*c^6*d - 9*B*b^2*c^4*e + 11*A*b*c^5*e)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c
^4*d^4 - 4*b^4*c^3*d^3*e + 6*b^5*c^2*d^2*e^2 - 4*b^6*c*d*e^3 + b^7*e^4)*sqrt(-c^2*d + b*c*e)) + ((e*x + d)^(3/
2)*B*b*c^4*d^4*e - 2*(e*x + d)^(3/2)*A*c^5*d^4*e - sqrt(e*x + d)*B*b*c^4*d^5*e + 2*sqrt(e*x + d)*A*c^5*d^5*e +
4*(e*x + d)^(3/2)*A*b*c^4*d^3*e^2 - 5*sqrt(e*x + d)*A*b*c^4*d^4*e^2 - 6*(e*x + d)^(3/2)*A*b^2*c^3*d^2*e^3 + 1
0*sqrt(e*x + d)*A*b^2*c^3*d^3*e^3 + 4*(e*x + d)^(3/2)*A*b^3*c^2*d*e^4 - 10*sqrt(e*x + d)*A*b^3*c^2*d^2*e^4 - (
e*x + d)^(3/2)*A*b^4*c*e^5 + 5*sqrt(e*x + d)*A*b^4*c*d*e^5 - sqrt(e*x + d)*A*b^5*e^6)/((b^2*c^4*d^8 - 4*b^3*c^
3*d^7*e + 6*b^4*c^2*d^6*e^2 - 4*b^5*c*d^5*e^3 + b^6*d^4*e^4)*((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x +
d)*b*e - b*d*e)) + 2/15*(90*(e*x + d)^2*B*c^2*d^3*e^2 + 15*(e*x + d)*B*c^2*d^4*e^2 + 3*B*c^2*d^5*e^2 - 60*(e*
x + d)^2*B*b*c*d^2*e^3 - 150*(e*x + d)^2*A*c^2*d^2*e^3 - 20*(e*x + d)*B*b*c*d^3*e^3 - 20*(e*x + d)*A*c^2*d^3*e
^3 - 6*B*b*c*d^4*e^3 - 3*A*c^2*d^4*e^3 + 15*(e*x + d)^2*B*b^2*d*e^4 + 150*(e*x + d)^2*A*b*c*d*e^4 + 5*(e*x + d
)*B*b^2*d^2*e^4 + 30*(e*x + d)*A*b*c*d^2*e^4 + 3*B*b^2*d^3*e^4 + 6*A*b*c*d^3*e^4 - 45*(e*x + d)^2*A*b^2*e^5 -
10*(e*x + d)*A*b^2*d*e^5 - 3*A*b^2*d^2*e^5)/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 +
b^4*d^4*e^4)*(e*x + d)^(5/2)) + (2*B*b*d - 4*A*c*d - 7*A*b*e)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^3*sqrt(-d)*d^4
)
Mupad [B] (verification not implemented)
Time = 15.71 (sec) , antiderivative size = 20597, normalized size of antiderivative = 45.07
\[
\int \frac {A+B x}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=\text {Too large to display}
\]
[In]
int((A + B*x)/((b*x + c*x^2)^2*(d + e*x)^(7/2)),x)
[Out]
log((((49*A^2*b^2*e^2 + 16*A^2*c^2*d^2 + 4*B^2*b^2*d^2 - 16*A*B*b*c*d^2 - 28*A*B*b^2*d*e + 56*A^2*b*c*d*e)/(4*
b^6*d^9))^(1/2)*((d + e*x)^(1/2)*((49*A^2*b^2*e^2 + 16*A^2*c^2*d^2 + 4*B^2*b^2*d^2 - 16*A*B*b*c*d^2 - 28*A*B*b
^2*d*e + 56*A^2*b*c*d*e)/(4*b^6*d^9))^(1/2)*(16*b^12*c^23*d^41*e^2 - 328*b^13*c^22*d^40*e^3 + 3200*b^14*c^21*d
^39*e^4 - 19760*b^15*c^20*d^38*e^5 + 86640*b^16*c^19*d^37*e^6 - 286824*b^17*c^18*d^36*e^7 + 744192*b^18*c^17*d
^35*e^8 - 1550400*b^19*c^16*d^34*e^9 + 2635680*b^20*c^15*d^33*e^10 - 3695120*b^21*c^14*d^32*e^11 + 4299776*b^2
2*c^13*d^31*e^12 - 4165408*b^23*c^12*d^30*e^13 + 3359200*b^24*c^11*d^29*e^14 - 2248080*b^25*c^10*d^28*e^15 + 1
240320*b^26*c^9*d^27*e^16 - 558144*b^27*c^8*d^26*e^17 + 201552*b^28*c^7*d^25*e^18 - 57000*b^29*c^6*d^24*e^19 +
12160*b^30*c^5*d^23*e^20 - 1840*b^31*c^4*d^22*e^21 + 176*b^32*c^3*d^21*e^22 - 8*b^33*c^2*d^20*e^23) - 8*A*b^1
0*c^23*d^37*e^3 + 148*A*b^11*c^22*d^36*e^4 - 1160*A*b^12*c^21*d^35*e^5 + 4760*A*b^13*c^20*d^34*e^6 - 8036*A*b^
14*c^19*d^33*e^7 - 21868*A*b^15*c^18*d^32*e^8 + 194304*A*b^16*c^17*d^31*e^9 - 709280*A*b^17*c^16*d^30*e^10 + 1
744160*A*b^18*c^15*d^29*e^11 - 3218072*A*b^19*c^14*d^28*e^12 + 4654832*A*b^20*c^13*d^27*e^13 - 5394480*A*b^21*
c^12*d^26*e^14 + 5063240*A*b^22*c^11*d^25*e^15 - 3863800*A*b^23*c^10*d^24*e^16 + 2393152*A*b^24*c^9*d^23*e^17
- 1194528*A*b^25*c^8*d^22*e^18 + 474056*A*b^26*c^7*d^21*e^19 - 146300*A*b^27*c^6*d^20*e^20 + 33880*A*b^28*c^5*
d^19*e^21 - 5544*A*b^29*c^4*d^18*e^22 + 572*A*b^30*c^3*d^17*e^23 - 28*A*b^31*c^2*d^16*e^24 + 4*B*b^11*c^22*d^3
7*e^3 - 144*B*b^12*c^21*d^36*e^4 + 1840*B*b^13*c^20*d^35*e^5 - 13160*B*b^14*c^19*d^34*e^6 + 62328*B*b^15*c^18*
d^33*e^7 - 212800*B*b^16*c^17*d^32*e^8 + 550432*B*b^17*c^16*d^31*e^9 - 1113120*B*b^18*c^15*d^30*e^10 + 1796600
*B*b^19*c^14*d^29*e^11 - 2345824*B*b^20*c^13*d^28*e^12 + 2498496*B*b^21*c^12*d^27*e^13 - 2179632*B*b^22*c^11*d
^26*e^14 + 1557920*B*b^23*c^10*d^25*e^15 - 909120*B*b^24*c^9*d^24*e^16 + 429664*B*b^25*c^8*d^23*e^17 - 162208*
B*b^26*c^7*d^22*e^18 + 47844*B*b^27*c^6*d^21*e^19 - 10640*B*b^28*c^5*d^20*e^20 + 1680*B*b^29*c^4*d^19*e^21 - 1
68*B*b^30*c^3*d^18*e^22 + 8*B*b^31*c^2*d^17*e^23) + (d + e*x)^(1/2)*(1088*A^2*b^7*c^24*d^33*e^3 - 64*A^2*b^6*c
^25*d^34*e^2 - 8404*A^2*b^8*c^23*d^32*e^4 + 38720*A^2*b^9*c^22*d^31*e^5 - 116512*A^2*b^10*c^21*d^30*e^6 + 2309
12*A^2*b^11*c^20*d^29*e^7 - 267432*A^2*b^12*c^19*d^28*e^8 + 38544*A^2*b^13*c^18*d^27*e^9 + 473880*A^2*b^14*c^1
7*d^26*e^10 - 851136*A^2*b^15*c^16*d^25*e^11 + 393646*A^2*b^16*c^15*d^24*e^12 + 1207368*A^2*b^17*c^14*d^23*e^1
3 - 3343724*A^2*b^18*c^13*d^22*e^14 + 4835160*A^2*b^19*c^12*d^21*e^15 - 4903382*A^2*b^20*c^11*d^20*e^16 + 3751
968*A^2*b^21*c^10*d^19*e^17 - 2217072*A^2*b^22*c^9*d^18*e^18 + 1013232*A^2*b^23*c^8*d^17*e^19 - 353210*A^2*b^2
4*c^7*d^16*e^20 + 91080*A^2*b^25*c^6*d^15*e^21 - 16412*A^2*b^26*c^5*d^14*e^22 + 1848*A^2*b^27*c^4*d^13*e^23 -
98*A^2*b^28*c^3*d^12*e^24 - 16*B^2*b^8*c^23*d^34*e^2 + 328*B^2*b^9*c^22*d^33*e^3 - 3074*B^2*b^10*c^21*d^32*e^4
+ 17576*B^2*b^11*c^20*d^31*e^5 - 69252*B^2*b^12*c^19*d^30*e^6 + 201648*B^2*b^13*c^18*d^29*e^7 - 454686*B^2*b^
14*c^17*d^28*e^8 + 821328*B^2*b^15*c^16*d^27*e^9 - 1218432*B^2*b^16*c^15*d^26*e^10 + 1509384*B^2*b^17*c^14*d^2
5*e^11 - 1574606*B^2*b^18*c^13*d^24*e^12 + 1384168*B^2*b^19*c^12*d^23*e^13 - 1019324*B^2*b^20*c^11*d^22*e^14 +
622176*B^2*b^21*c^10*d^21*e^15 - 310242*B^2*b^22*c^9*d^20*e^16 + 124032*B^2*b^23*c^8*d^19*e^17 - 38760*B^2*b^
24*c^7*d^18*e^18 + 9120*B^2*b^25*c^6*d^17*e^19 - 1520*B^2*b^26*c^5*d^16*e^20 + 160*B^2*b^27*c^4*d^15*e^21 - 8*
B^2*b^28*c^3*d^14*e^22 + 64*A*B*b^7*c^24*d^34*e^2 - 1200*A*B*b^8*c^23*d^33*e^3 + 10252*A*B*b^9*c^22*d^32*e^4 -
52944*A*B*b^10*c^21*d^31*e^5 + 184216*A*B*b^11*c^20*d^30*e^6 - 452112*A*B*b^12*c^19*d^29*e^7 + 781428*A*B*b^1
3*c^18*d^28*e^8 - 863424*A*B*b^14*c^17*d^27*e^9 + 255408*A*B*b^15*c^16*d^26*e^10 + 1244088*A*B*b^16*c^15*d^25*
e^11 - 3244396*A*B*b^17*c^14*d^24*e^12 + 4868800*A*B*b^18*c^13*d^23*e^13 - 5345768*A*B*b^19*c^12*d^22*e^14 + 4
568696*A*B*b^20*c^11*d^21*e^15 - 3100404*A*B*b^21*c^10*d^20*e^16 + 1674432*A*B*b^22*c^9*d^19*e^17 - 713184*A*B
*b^23*c^8*d^18*e^18 + 234840*A*B*b^24*c^7*d^17*e^19 - 57760*A*B*b^25*c^6*d^16*e^20 + 10000*A*B*b^26*c^5*d^15*e
^21 - 1088*A*B*b^27*c^4*d^14*e^22 + 56*A*B*b^28*c^3*d^13*e^23))*((49*A^2*b^2*e^2 + 16*A^2*c^2*d^2 + 4*B^2*b^2*
d^2 - 16*A*B*b*c*d^2 - 28*A*B*b^2*d*e + 56*A^2*b*c*d*e)/(4*b^6*d^9))^(1/2) + 32*A^3*b^4*c^25*d^30*e^3 - 480*A^
3*b^5*c^24*d^29*e^4 + 3590*A^3*b^6*c^23*d^28*e^5 - 17780*A^3*b^7*c^22*d^27*e^6 + 62874*A^3*b^8*c^21*d^26*e^7 -
157248*A^3*b^9*c^20*d^25*e^8 + 254443*A^3*b^10*c^19*d^24*e^9 - 163416*A^3*b^11*c^18*d^23*e^10 - 380204*A^3*b^
12*c^17*d^22*e^11 + 1403292*A^3*b^13*c^16*d^21*e^12 - 2458995*A^3*b^14*c^15*d^20*e^13 + 2901724*A^3*b^15*c^14*
d^19*e^14 - 2487478*A^3*b^16*c^13*d^18*e^15 + 1581048*A^3*b^17*c^12*d^17*e^16 - 741891*A^3*b^18*c^11*d^16*e^17
+ 250736*A^3*b^19*c^10*d^15*e^18 - 57912*A^3*b^20*c^9*d^14*e^19 + 8204*A^3*b^21*c^8*d^13*e^20 - 539*A^3*b^22*
c^7*d^12*e^21 - 4*B^3*b^7*c^22*d^30*e^3 + 18*B^3*b^8*c^21*d^29*e^4 + 344*B^3*b^9*c^20*d^28*e^5 - 4228*B^3*b^10
*c^19*d^27*e^6 + 22848*B^3*b^11*c^18*d^26*e^7 - 76706*B^3*b^12*c^17*d^25*e^8 + 178640*B^3*b^13*c^16*d^24*e^9 -
304128*B^3*b^14*c^15*d^23*e^10 + 389136*B^3*b^15*c^14*d^22*e^11 - 379346*B^3*b^16*c^13*d^21*e^12 + 282744*B^3
*b^17*c^12*d^20*e^13 - 160244*B^3*b^18*c^11*d^19*e^14 + 67984*B^3*b^19*c^10*d^18*e^15 - 20958*B^3*b^20*c^9*d^1
7*e^16 + 4448*B^3*b^21*c^8*d^16*e^17 - 584*B^3*b^22*c^7*d^15*e^18 + 36*B^3*b^23*c^6*d^14*e^19 + 24*A*B^2*b^6*c
^23*d^30*e^3 - 192*A*B^2*b^7*c^22*d^29*e^4 - 111*A*B^2*b^8*c^21*d^28*e^5 + 6636*A*B^2*b^9*c^20*d^27*e^6 - 3297
0*A*B^2*b^10*c^19*d^26*e^7 + 75432*A*B^2*b^11*c^18*d^25*e^8 - 55881*A*B^2*b^12*c^17*d^24*e^9 - 172920*A*B^2*b^
13*c^16*d^23*e^10 + 664488*A*B^2*b^14*c^15*d^22*e^11 - 1211100*A*B^2*b^15*c^14*d^21*e^12 + 1461999*A*B^2*b^16*
c^13*d^20*e^13 - 1264452*A*B^2*b^17*c^12*d^19*e^14 + 802158*A*B^2*b^18*c^11*d^18*e^15 - 372624*A*B^2*b^19*c^10
*d^17*e^16 + 123945*A*B^2*b^20*c^9*d^16*e^17 - 28080*A*B^2*b^21*c^8*d^15*e^18 + 3900*A*B^2*b^22*c^7*d^14*e^19
- 252*A*B^2*b^23*c^6*d^13*e^20 - 48*A^2*B*b^5*c^24*d^30*e^3 + 552*A^2*B*b^6*c^23*d^29*e^4 - 2949*A^2*B*b^7*c^2
2*d^28*e^5 + 11844*A^2*B*b^8*c^21*d^27*e^6 - 47628*A^2*B*b^9*c^20*d^26*e^7 + 176274*A^2*B*b^10*c^19*d^25*e^8 -
502782*A^2*B*b^11*c^18*d^24*e^9 + 1030776*A^2*B*b^12*c^17*d^23*e^10 - 1480512*A^2*B*b^13*c^16*d^22*e^11 + 141
1806*A^2*B*b^14*c^15*d^21*e^12 - 703164*A^2*B*b^15*c^14*d^20*e^13 - 205212*A^2*B*b^16*c^13*d^19*e^14 + 729540*
A^2*B*b^17*c^12*d^18*e^15 - 708498*A^2*B*b^18*c^11*d^17*e^16 + 417222*A^2*B*b^19*c^10*d^16*e^17 - 162912*A^2*B
*b^20*c^9*d^15*e^18 + 41592*A^2*B*b^21*c^8*d^14*e^19 - 6342*A^2*B*b^22*c^7*d^13*e^20 + 441*A^2*B*b^23*c^6*d^12
*e^21)*((49*A^2*b^2*e^2 + 16*A^2*c^2*d^2 + 4*B^2*b^2*d^2 - 16*A*B*b*c*d^2 - 28*A*B*b^2*d*e + 56*A^2*b*c*d*e)/(
4*b^6*d^9))^(1/2) - log(32*A^3*b^4*c^25*d^30*e^3 - ((((49*A^2*b^2*e^2)/4 + 4*A^2*c^2*d^2 + B^2*b^2*d^2 - 4*A*B
*b*c*d^2 - 7*A*B*b^2*d*e + 14*A^2*b*c*d*e)/(b^6*d^9))^(1/2)*((d + e*x)^(1/2)*(((49*A^2*b^2*e^2)/4 + 4*A^2*c^2*
d^2 + B^2*b^2*d^2 - 4*A*B*b*c*d^2 - 7*A*B*b^2*d*e + 14*A^2*b*c*d*e)/(b^6*d^9))^(1/2)*(16*b^12*c^23*d^41*e^2 -
328*b^13*c^22*d^40*e^3 + 3200*b^14*c^21*d^39*e^4 - 19760*b^15*c^20*d^38*e^5 + 86640*b^16*c^19*d^37*e^6 - 28682
4*b^17*c^18*d^36*e^7 + 744192*b^18*c^17*d^35*e^8 - 1550400*b^19*c^16*d^34*e^9 + 2635680*b^20*c^15*d^33*e^10 -
3695120*b^21*c^14*d^32*e^11 + 4299776*b^22*c^13*d^31*e^12 - 4165408*b^23*c^12*d^30*e^13 + 3359200*b^24*c^11*d^
29*e^14 - 2248080*b^25*c^10*d^28*e^15 + 1240320*b^26*c^9*d^27*e^16 - 558144*b^27*c^8*d^26*e^17 + 201552*b^28*c
^7*d^25*e^18 - 57000*b^29*c^6*d^24*e^19 + 12160*b^30*c^5*d^23*e^20 - 1840*b^31*c^4*d^22*e^21 + 176*b^32*c^3*d^
21*e^22 - 8*b^33*c^2*d^20*e^23) + 8*A*b^10*c^23*d^37*e^3 - 148*A*b^11*c^22*d^36*e^4 + 1160*A*b^12*c^21*d^35*e^
5 - 4760*A*b^13*c^20*d^34*e^6 + 8036*A*b^14*c^19*d^33*e^7 + 21868*A*b^15*c^18*d^32*e^8 - 194304*A*b^16*c^17*d^
31*e^9 + 709280*A*b^17*c^16*d^30*e^10 - 1744160*A*b^18*c^15*d^29*e^11 + 3218072*A*b^19*c^14*d^28*e^12 - 465483
2*A*b^20*c^13*d^27*e^13 + 5394480*A*b^21*c^12*d^26*e^14 - 5063240*A*b^22*c^11*d^25*e^15 + 3863800*A*b^23*c^10*
d^24*e^16 - 2393152*A*b^24*c^9*d^23*e^17 + 1194528*A*b^25*c^8*d^22*e^18 - 474056*A*b^26*c^7*d^21*e^19 + 146300
*A*b^27*c^6*d^20*e^20 - 33880*A*b^28*c^5*d^19*e^21 + 5544*A*b^29*c^4*d^18*e^22 - 572*A*b^30*c^3*d^17*e^23 + 28
*A*b^31*c^2*d^16*e^24 - 4*B*b^11*c^22*d^37*e^3 + 144*B*b^12*c^21*d^36*e^4 - 1840*B*b^13*c^20*d^35*e^5 + 13160*
B*b^14*c^19*d^34*e^6 - 62328*B*b^15*c^18*d^33*e^7 + 212800*B*b^16*c^17*d^32*e^8 - 550432*B*b^17*c^16*d^31*e^9
+ 1113120*B*b^18*c^15*d^30*e^10 - 1796600*B*b^19*c^14*d^29*e^11 + 2345824*B*b^20*c^13*d^28*e^12 - 2498496*B*b^
21*c^12*d^27*e^13 + 2179632*B*b^22*c^11*d^26*e^14 - 1557920*B*b^23*c^10*d^25*e^15 + 909120*B*b^24*c^9*d^24*e^1
6 - 429664*B*b^25*c^8*d^23*e^17 + 162208*B*b^26*c^7*d^22*e^18 - 47844*B*b^27*c^6*d^21*e^19 + 10640*B*b^28*c^5*
d^20*e^20 - 1680*B*b^29*c^4*d^19*e^21 + 168*B*b^30*c^3*d^18*e^22 - 8*B*b^31*c^2*d^17*e^23) + (d + e*x)^(1/2)*(
1088*A^2*b^7*c^24*d^33*e^3 - 64*A^2*b^6*c^25*d^34*e^2 - 8404*A^2*b^8*c^23*d^32*e^4 + 38720*A^2*b^9*c^22*d^31*e
^5 - 116512*A^2*b^10*c^21*d^30*e^6 + 230912*A^2*b^11*c^20*d^29*e^7 - 267432*A^2*b^12*c^19*d^28*e^8 + 38544*A^2
*b^13*c^18*d^27*e^9 + 473880*A^2*b^14*c^17*d^26*e^10 - 851136*A^2*b^15*c^16*d^25*e^11 + 393646*A^2*b^16*c^15*d
^24*e^12 + 1207368*A^2*b^17*c^14*d^23*e^13 - 3343724*A^2*b^18*c^13*d^22*e^14 + 4835160*A^2*b^19*c^12*d^21*e^15
- 4903382*A^2*b^20*c^11*d^20*e^16 + 3751968*A^2*b^21*c^10*d^19*e^17 - 2217072*A^2*b^22*c^9*d^18*e^18 + 101323
2*A^2*b^23*c^8*d^17*e^19 - 353210*A^2*b^24*c^7*d^16*e^20 + 91080*A^2*b^25*c^6*d^15*e^21 - 16412*A^2*b^26*c^5*d
^14*e^22 + 1848*A^2*b^27*c^4*d^13*e^23 - 98*A^2*b^28*c^3*d^12*e^24 - 16*B^2*b^8*c^23*d^34*e^2 + 328*B^2*b^9*c^
22*d^33*e^3 - 3074*B^2*b^10*c^21*d^32*e^4 + 17576*B^2*b^11*c^20*d^31*e^5 - 69252*B^2*b^12*c^19*d^30*e^6 + 2016
48*B^2*b^13*c^18*d^29*e^7 - 454686*B^2*b^14*c^17*d^28*e^8 + 821328*B^2*b^15*c^16*d^27*e^9 - 1218432*B^2*b^16*c
^15*d^26*e^10 + 1509384*B^2*b^17*c^14*d^25*e^11 - 1574606*B^2*b^18*c^13*d^24*e^12 + 1384168*B^2*b^19*c^12*d^23
*e^13 - 1019324*B^2*b^20*c^11*d^22*e^14 + 622176*B^2*b^21*c^10*d^21*e^15 - 310242*B^2*b^22*c^9*d^20*e^16 + 124
032*B^2*b^23*c^8*d^19*e^17 - 38760*B^2*b^24*c^7*d^18*e^18 + 9120*B^2*b^25*c^6*d^17*e^19 - 1520*B^2*b^26*c^5*d^
16*e^20 + 160*B^2*b^27*c^4*d^15*e^21 - 8*B^2*b^28*c^3*d^14*e^22 + 64*A*B*b^7*c^24*d^34*e^2 - 1200*A*B*b^8*c^23
*d^33*e^3 + 10252*A*B*b^9*c^22*d^32*e^4 - 52944*A*B*b^10*c^21*d^31*e^5 + 184216*A*B*b^11*c^20*d^30*e^6 - 45211
2*A*B*b^12*c^19*d^29*e^7 + 781428*A*B*b^13*c^18*d^28*e^8 - 863424*A*B*b^14*c^17*d^27*e^9 + 255408*A*B*b^15*c^1
6*d^26*e^10 + 1244088*A*B*b^16*c^15*d^25*e^11 - 3244396*A*B*b^17*c^14*d^24*e^12 + 4868800*A*B*b^18*c^13*d^23*e
^13 - 5345768*A*B*b^19*c^12*d^22*e^14 + 4568696*A*B*b^20*c^11*d^21*e^15 - 3100404*A*B*b^21*c^10*d^20*e^16 + 16
74432*A*B*b^22*c^9*d^19*e^17 - 713184*A*B*b^23*c^8*d^18*e^18 + 234840*A*B*b^24*c^7*d^17*e^19 - 57760*A*B*b^25*
c^6*d^16*e^20 + 10000*A*B*b^26*c^5*d^15*e^21 - 1088*A*B*b^27*c^4*d^14*e^22 + 56*A*B*b^28*c^3*d^13*e^23))*(((49
*A^2*b^2*e^2)/4 + 4*A^2*c^2*d^2 + B^2*b^2*d^2 - 4*A*B*b*c*d^2 - 7*A*B*b^2*d*e + 14*A^2*b*c*d*e)/(b^6*d^9))^(1/
2) - 480*A^3*b^5*c^24*d^29*e^4 + 3590*A^3*b^6*c^23*d^28*e^5 - 17780*A^3*b^7*c^22*d^27*e^6 + 62874*A^3*b^8*c^21
*d^26*e^7 - 157248*A^3*b^9*c^20*d^25*e^8 + 254443*A^3*b^10*c^19*d^24*e^9 - 163416*A^3*b^11*c^18*d^23*e^10 - 38
0204*A^3*b^12*c^17*d^22*e^11 + 1403292*A^3*b^13*c^16*d^21*e^12 - 2458995*A^3*b^14*c^15*d^20*e^13 + 2901724*A^3
*b^15*c^14*d^19*e^14 - 2487478*A^3*b^16*c^13*d^18*e^15 + 1581048*A^3*b^17*c^12*d^17*e^16 - 741891*A^3*b^18*c^1
1*d^16*e^17 + 250736*A^3*b^19*c^10*d^15*e^18 - 57912*A^3*b^20*c^9*d^14*e^19 + 8204*A^3*b^21*c^8*d^13*e^20 - 53
9*A^3*b^22*c^7*d^12*e^21 - 4*B^3*b^7*c^22*d^30*e^3 + 18*B^3*b^8*c^21*d^29*e^4 + 344*B^3*b^9*c^20*d^28*e^5 - 42
28*B^3*b^10*c^19*d^27*e^6 + 22848*B^3*b^11*c^18*d^26*e^7 - 76706*B^3*b^12*c^17*d^25*e^8 + 178640*B^3*b^13*c^16
*d^24*e^9 - 304128*B^3*b^14*c^15*d^23*e^10 + 389136*B^3*b^15*c^14*d^22*e^11 - 379346*B^3*b^16*c^13*d^21*e^12 +
282744*B^3*b^17*c^12*d^20*e^13 - 160244*B^3*b^18*c^11*d^19*e^14 + 67984*B^3*b^19*c^10*d^18*e^15 - 20958*B^3*b
^20*c^9*d^17*e^16 + 4448*B^3*b^21*c^8*d^16*e^17 - 584*B^3*b^22*c^7*d^15*e^18 + 36*B^3*b^23*c^6*d^14*e^19 + 24*
A*B^2*b^6*c^23*d^30*e^3 - 192*A*B^2*b^7*c^22*d^29*e^4 - 111*A*B^2*b^8*c^21*d^28*e^5 + 6636*A*B^2*b^9*c^20*d^27
*e^6 - 32970*A*B^2*b^10*c^19*d^26*e^7 + 75432*A*B^2*b^11*c^18*d^25*e^8 - 55881*A*B^2*b^12*c^17*d^24*e^9 - 1729
20*A*B^2*b^13*c^16*d^23*e^10 + 664488*A*B^2*b^14*c^15*d^22*e^11 - 1211100*A*B^2*b^15*c^14*d^21*e^12 + 1461999*
A*B^2*b^16*c^13*d^20*e^13 - 1264452*A*B^2*b^17*c^12*d^19*e^14 + 802158*A*B^2*b^18*c^11*d^18*e^15 - 372624*A*B^
2*b^19*c^10*d^17*e^16 + 123945*A*B^2*b^20*c^9*d^16*e^17 - 28080*A*B^2*b^21*c^8*d^15*e^18 + 3900*A*B^2*b^22*c^7
*d^14*e^19 - 252*A*B^2*b^23*c^6*d^13*e^20 - 48*A^2*B*b^5*c^24*d^30*e^3 + 552*A^2*B*b^6*c^23*d^29*e^4 - 2949*A^
2*B*b^7*c^22*d^28*e^5 + 11844*A^2*B*b^8*c^21*d^27*e^6 - 47628*A^2*B*b^9*c^20*d^26*e^7 + 176274*A^2*B*b^10*c^19
*d^25*e^8 - 502782*A^2*B*b^11*c^18*d^24*e^9 + 1030776*A^2*B*b^12*c^17*d^23*e^10 - 1480512*A^2*B*b^13*c^16*d^22
*e^11 + 1411806*A^2*B*b^14*c^15*d^21*e^12 - 703164*A^2*B*b^15*c^14*d^20*e^13 - 205212*A^2*B*b^16*c^13*d^19*e^1
4 + 729540*A^2*B*b^17*c^12*d^18*e^15 - 708498*A^2*B*b^18*c^11*d^17*e^16 + 417222*A^2*B*b^19*c^10*d^16*e^17 - 1
62912*A^2*B*b^20*c^9*d^15*e^18 + 41592*A^2*B*b^21*c^8*d^14*e^19 - 6342*A^2*B*b^22*c^7*d^13*e^20 + 441*A^2*B*b^
23*c^6*d^12*e^21)*(((49*A^2*b^2*e^2)/4 + 4*A^2*c^2*d^2 + B^2*b^2*d^2 - 4*A*B*b*c*d^2 - 7*A*B*b^2*d*e + 14*A^2*
b*c*d*e)/(b^6*d^9))^(1/2) + atan((((d + e*x)^(1/2)*(1088*A^2*b^7*c^24*d^33*e^3 - 64*A^2*b^6*c^25*d^34*e^2 - 84
04*A^2*b^8*c^23*d^32*e^4 + 38720*A^2*b^9*c^22*d^31*e^5 - 116512*A^2*b^10*c^21*d^30*e^6 + 230912*A^2*b^11*c^20*
d^29*e^7 - 267432*A^2*b^12*c^19*d^28*e^8 + 38544*A^2*b^13*c^18*d^27*e^9 + 473880*A^2*b^14*c^17*d^26*e^10 - 851
136*A^2*b^15*c^16*d^25*e^11 + 393646*A^2*b^16*c^15*d^24*e^12 + 1207368*A^2*b^17*c^14*d^23*e^13 - 3343724*A^2*b
^18*c^13*d^22*e^14 + 4835160*A^2*b^19*c^12*d^21*e^15 - 4903382*A^2*b^20*c^11*d^20*e^16 + 3751968*A^2*b^21*c^10
*d^19*e^17 - 2217072*A^2*b^22*c^9*d^18*e^18 + 1013232*A^2*b^23*c^8*d^17*e^19 - 353210*A^2*b^24*c^7*d^16*e^20 +
91080*A^2*b^25*c^6*d^15*e^21 - 16412*A^2*b^26*c^5*d^14*e^22 + 1848*A^2*b^27*c^4*d^13*e^23 - 98*A^2*b^28*c^3*d
^12*e^24 - 16*B^2*b^8*c^23*d^34*e^2 + 328*B^2*b^9*c^22*d^33*e^3 - 3074*B^2*b^10*c^21*d^32*e^4 + 17576*B^2*b^11
*c^20*d^31*e^5 - 69252*B^2*b^12*c^19*d^30*e^6 + 201648*B^2*b^13*c^18*d^29*e^7 - 454686*B^2*b^14*c^17*d^28*e^8
+ 821328*B^2*b^15*c^16*d^27*e^9 - 1218432*B^2*b^16*c^15*d^26*e^10 + 1509384*B^2*b^17*c^14*d^25*e^11 - 1574606*
B^2*b^18*c^13*d^24*e^12 + 1384168*B^2*b^19*c^12*d^23*e^13 - 1019324*B^2*b^20*c^11*d^22*e^14 + 622176*B^2*b^21*
c^10*d^21*e^15 - 310242*B^2*b^22*c^9*d^20*e^16 + 124032*B^2*b^23*c^8*d^19*e^17 - 38760*B^2*b^24*c^7*d^18*e^18
+ 9120*B^2*b^25*c^6*d^17*e^19 - 1520*B^2*b^26*c^5*d^16*e^20 + 160*B^2*b^27*c^4*d^15*e^21 - 8*B^2*b^28*c^3*d^14
*e^22 + 64*A*B*b^7*c^24*d^34*e^2 - 1200*A*B*b^8*c^23*d^33*e^3 + 10252*A*B*b^9*c^22*d^32*e^4 - 52944*A*B*b^10*c
^21*d^31*e^5 + 184216*A*B*b^11*c^20*d^30*e^6 - 452112*A*B*b^12*c^19*d^29*e^7 + 781428*A*B*b^13*c^18*d^28*e^8 -
863424*A*B*b^14*c^17*d^27*e^9 + 255408*A*B*b^15*c^16*d^26*e^10 + 1244088*A*B*b^16*c^15*d^25*e^11 - 3244396*A*
B*b^17*c^14*d^24*e^12 + 4868800*A*B*b^18*c^13*d^23*e^13 - 5345768*A*B*b^19*c^12*d^22*e^14 + 4568696*A*B*b^20*c
^11*d^21*e^15 - 3100404*A*B*b^21*c^10*d^20*e^16 + 1674432*A*B*b^22*c^9*d^19*e^17 - 713184*A*B*b^23*c^8*d^18*e^
18 + 234840*A*B*b^24*c^7*d^17*e^19 - 57760*A*B*b^25*c^6*d^16*e^20 + 10000*A*B*b^26*c^5*d^15*e^21 - 1088*A*B*b^
27*c^4*d^14*e^22 + 56*A*B*b^28*c^3*d^13*e^23) + (-(16*A^2*c^11*d^2 + 121*A^2*b^2*c^9*e^2 + 4*B^2*b^2*c^9*d^2 +
81*B^2*b^4*c^7*e^2 - 198*A*B*b^3*c^8*e^2 - 36*B^2*b^3*c^8*d*e - 16*A*B*b*c^10*d^2 - 88*A^2*b*c^10*d*e + 116*A
*B*b^2*c^9*d*e)/(4*(b^15*e^9 - b^6*c^9*d^9 + 9*b^7*c^8*d^8*e - 36*b^8*c^7*d^7*e^2 + 84*b^9*c^6*d^6*e^3 - 126*b
^10*c^5*d^5*e^4 + 126*b^11*c^4*d^4*e^5 - 84*b^12*c^3*d^3*e^6 + 36*b^13*c^2*d^2*e^7 - 9*b^14*c*d*e^8)))^(1/2)*(
(d + e*x)^(1/2)*(-(16*A^2*c^11*d^2 + 121*A^2*b^2*c^9*e^2 + 4*B^2*b^2*c^9*d^2 + 81*B^2*b^4*c^7*e^2 - 198*A*B*b^
3*c^8*e^2 - 36*B^2*b^3*c^8*d*e - 16*A*B*b*c^10*d^2 - 88*A^2*b*c^10*d*e + 116*A*B*b^2*c^9*d*e)/(4*(b^15*e^9 - b
^6*c^9*d^9 + 9*b^7*c^8*d^8*e - 36*b^8*c^7*d^7*e^2 + 84*b^9*c^6*d^6*e^3 - 126*b^10*c^5*d^5*e^4 + 126*b^11*c^4*d
^4*e^5 - 84*b^12*c^3*d^3*e^6 + 36*b^13*c^2*d^2*e^7 - 9*b^14*c*d*e^8)))^(1/2)*(16*b^12*c^23*d^41*e^2 - 328*b^13
*c^22*d^40*e^3 + 3200*b^14*c^21*d^39*e^4 - 19760*b^15*c^20*d^38*e^5 + 86640*b^16*c^19*d^37*e^6 - 286824*b^17*c
^18*d^36*e^7 + 744192*b^18*c^17*d^35*e^8 - 1550400*b^19*c^16*d^34*e^9 + 2635680*b^20*c^15*d^33*e^10 - 3695120*
b^21*c^14*d^32*e^11 + 4299776*b^22*c^13*d^31*e^12 - 4165408*b^23*c^12*d^30*e^13 + 3359200*b^24*c^11*d^29*e^14
- 2248080*b^25*c^10*d^28*e^15 + 1240320*b^26*c^9*d^27*e^16 - 558144*b^27*c^8*d^26*e^17 + 201552*b^28*c^7*d^25*
e^18 - 57000*b^29*c^6*d^24*e^19 + 12160*b^30*c^5*d^23*e^20 - 1840*b^31*c^4*d^22*e^21 + 176*b^32*c^3*d^21*e^22
- 8*b^33*c^2*d^20*e^23) - 8*A*b^10*c^23*d^37*e^3 + 148*A*b^11*c^22*d^36*e^4 - 1160*A*b^12*c^21*d^35*e^5 + 4760
*A*b^13*c^20*d^34*e^6 - 8036*A*b^14*c^19*d^33*e^7 - 21868*A*b^15*c^18*d^32*e^8 + 194304*A*b^16*c^17*d^31*e^9 -
709280*A*b^17*c^16*d^30*e^10 + 1744160*A*b^18*c^15*d^29*e^11 - 3218072*A*b^19*c^14*d^28*e^12 + 4654832*A*b^20
*c^13*d^27*e^13 - 5394480*A*b^21*c^12*d^26*e^14 + 5063240*A*b^22*c^11*d^25*e^15 - 3863800*A*b^23*c^10*d^24*e^1
6 + 2393152*A*b^24*c^9*d^23*e^17 - 1194528*A*b^25*c^8*d^22*e^18 + 474056*A*b^26*c^7*d^21*e^19 - 146300*A*b^27*
c^6*d^20*e^20 + 33880*A*b^28*c^5*d^19*e^21 - 5544*A*b^29*c^4*d^18*e^22 + 572*A*b^30*c^3*d^17*e^23 - 28*A*b^31*
c^2*d^16*e^24 + 4*B*b^11*c^22*d^37*e^3 - 144*B*b^12*c^21*d^36*e^4 + 1840*B*b^13*c^20*d^35*e^5 - 13160*B*b^14*c
^19*d^34*e^6 + 62328*B*b^15*c^18*d^33*e^7 - 212800*B*b^16*c^17*d^32*e^8 + 550432*B*b^17*c^16*d^31*e^9 - 111312
0*B*b^18*c^15*d^30*e^10 + 1796600*B*b^19*c^14*d^29*e^11 - 2345824*B*b^20*c^13*d^28*e^12 + 2498496*B*b^21*c^12*
d^27*e^13 - 2179632*B*b^22*c^11*d^26*e^14 + 1557920*B*b^23*c^10*d^25*e^15 - 909120*B*b^24*c^9*d^24*e^16 + 4296
64*B*b^25*c^8*d^23*e^17 - 162208*B*b^26*c^7*d^22*e^18 + 47844*B*b^27*c^6*d^21*e^19 - 10640*B*b^28*c^5*d^20*e^2
0 + 1680*B*b^29*c^4*d^19*e^21 - 168*B*b^30*c^3*d^18*e^22 + 8*B*b^31*c^2*d^17*e^23))*(-(16*A^2*c^11*d^2 + 121*A
^2*b^2*c^9*e^2 + 4*B^2*b^2*c^9*d^2 + 81*B^2*b^4*c^7*e^2 - 198*A*B*b^3*c^8*e^2 - 36*B^2*b^3*c^8*d*e - 16*A*B*b*
c^10*d^2 - 88*A^2*b*c^10*d*e + 116*A*B*b^2*c^9*d*e)/(4*(b^15*e^9 - b^6*c^9*d^9 + 9*b^7*c^8*d^8*e - 36*b^8*c^7*
d^7*e^2 + 84*b^9*c^6*d^6*e^3 - 126*b^10*c^5*d^5*e^4 + 126*b^11*c^4*d^4*e^5 - 84*b^12*c^3*d^3*e^6 + 36*b^13*c^2
*d^2*e^7 - 9*b^14*c*d*e^8)))^(1/2)*1i + ((d + e*x)^(1/2)*(1088*A^2*b^7*c^24*d^33*e^3 - 64*A^2*b^6*c^25*d^34*e^
2 - 8404*A^2*b^8*c^23*d^32*e^4 + 38720*A^2*b^9*c^22*d^31*e^5 - 116512*A^2*b^10*c^21*d^30*e^6 + 230912*A^2*b^11
*c^20*d^29*e^7 - 267432*A^2*b^12*c^19*d^28*e^8 + 38544*A^2*b^13*c^18*d^27*e^9 + 473880*A^2*b^14*c^17*d^26*e^10
- 851136*A^2*b^15*c^16*d^25*e^11 + 393646*A^2*b^16*c^15*d^24*e^12 + 1207368*A^2*b^17*c^14*d^23*e^13 - 3343724
*A^2*b^18*c^13*d^22*e^14 + 4835160*A^2*b^19*c^12*d^21*e^15 - 4903382*A^2*b^20*c^11*d^20*e^16 + 3751968*A^2*b^2
1*c^10*d^19*e^17 - 2217072*A^2*b^22*c^9*d^18*e^18 + 1013232*A^2*b^23*c^8*d^17*e^19 - 353210*A^2*b^24*c^7*d^16*
e^20 + 91080*A^2*b^25*c^6*d^15*e^21 - 16412*A^2*b^26*c^5*d^14*e^22 + 1848*A^2*b^27*c^4*d^13*e^23 - 98*A^2*b^28
*c^3*d^12*e^24 - 16*B^2*b^8*c^23*d^34*e^2 + 328*B^2*b^9*c^22*d^33*e^3 - 3074*B^2*b^10*c^21*d^32*e^4 + 17576*B^
2*b^11*c^20*d^31*e^5 - 69252*B^2*b^12*c^19*d^30*e^6 + 201648*B^2*b^13*c^18*d^29*e^7 - 454686*B^2*b^14*c^17*d^2
8*e^8 + 821328*B^2*b^15*c^16*d^27*e^9 - 1218432*B^2*b^16*c^15*d^26*e^10 + 1509384*B^2*b^17*c^14*d^25*e^11 - 15
74606*B^2*b^18*c^13*d^24*e^12 + 1384168*B^2*b^19*c^12*d^23*e^13 - 1019324*B^2*b^20*c^11*d^22*e^14 + 622176*B^2
*b^21*c^10*d^21*e^15 - 310242*B^2*b^22*c^9*d^20*e^16 + 124032*B^2*b^23*c^8*d^19*e^17 - 38760*B^2*b^24*c^7*d^18
*e^18 + 9120*B^2*b^25*c^6*d^17*e^19 - 1520*B^2*b^26*c^5*d^16*e^20 + 160*B^2*b^27*c^4*d^15*e^21 - 8*B^2*b^28*c^
3*d^14*e^22 + 64*A*B*b^7*c^24*d^34*e^2 - 1200*A*B*b^8*c^23*d^33*e^3 + 10252*A*B*b^9*c^22*d^32*e^4 - 52944*A*B*
b^10*c^21*d^31*e^5 + 184216*A*B*b^11*c^20*d^30*e^6 - 452112*A*B*b^12*c^19*d^29*e^7 + 781428*A*B*b^13*c^18*d^28
*e^8 - 863424*A*B*b^14*c^17*d^27*e^9 + 255408*A*B*b^15*c^16*d^26*e^10 + 1244088*A*B*b^16*c^15*d^25*e^11 - 3244
396*A*B*b^17*c^14*d^24*e^12 + 4868800*A*B*b^18*c^13*d^23*e^13 - 5345768*A*B*b^19*c^12*d^22*e^14 + 4568696*A*B*
b^20*c^11*d^21*e^15 - 3100404*A*B*b^21*c^10*d^20*e^16 + 1674432*A*B*b^22*c^9*d^19*e^17 - 713184*A*B*b^23*c^8*d
^18*e^18 + 234840*A*B*b^24*c^7*d^17*e^19 - 57760*A*B*b^25*c^6*d^16*e^20 + 10000*A*B*b^26*c^5*d^15*e^21 - 1088*
A*B*b^27*c^4*d^14*e^22 + 56*A*B*b^28*c^3*d^13*e^23) + (-(16*A^2*c^11*d^2 + 121*A^2*b^2*c^9*e^2 + 4*B^2*b^2*c^9
*d^2 + 81*B^2*b^4*c^7*e^2 - 198*A*B*b^3*c^8*e^2 - 36*B^2*b^3*c^8*d*e - 16*A*B*b*c^10*d^2 - 88*A^2*b*c^10*d*e +
116*A*B*b^2*c^9*d*e)/(4*(b^15*e^9 - b^6*c^9*d^9 + 9*b^7*c^8*d^8*e - 36*b^8*c^7*d^7*e^2 + 84*b^9*c^6*d^6*e^3 -
126*b^10*c^5*d^5*e^4 + 126*b^11*c^4*d^4*e^5 - 84*b^12*c^3*d^3*e^6 + 36*b^13*c^2*d^2*e^7 - 9*b^14*c*d*e^8)))^(
1/2)*((d + e*x)^(1/2)*(-(16*A^2*c^11*d^2 + 121*A^2*b^2*c^9*e^2 + 4*B^2*b^2*c^9*d^2 + 81*B^2*b^4*c^7*e^2 - 198*
A*B*b^3*c^8*e^2 - 36*B^2*b^3*c^8*d*e - 16*A*B*b*c^10*d^2 - 88*A^2*b*c^10*d*e + 116*A*B*b^2*c^9*d*e)/(4*(b^15*e
^9 - b^6*c^9*d^9 + 9*b^7*c^8*d^8*e - 36*b^8*c^7*d^7*e^2 + 84*b^9*c^6*d^6*e^3 - 126*b^10*c^5*d^5*e^4 + 126*b^11
*c^4*d^4*e^5 - 84*b^12*c^3*d^3*e^6 + 36*b^13*c^2*d^2*e^7 - 9*b^14*c*d*e^8)))^(1/2)*(16*b^12*c^23*d^41*e^2 - 32
8*b^13*c^22*d^40*e^3 + 3200*b^14*c^21*d^39*e^4 - 19760*b^15*c^20*d^38*e^5 + 86640*b^16*c^19*d^37*e^6 - 286824*
b^17*c^18*d^36*e^7 + 744192*b^18*c^17*d^35*e^8 - 1550400*b^19*c^16*d^34*e^9 + 2635680*b^20*c^15*d^33*e^10 - 36
95120*b^21*c^14*d^32*e^11 + 4299776*b^22*c^13*d^31*e^12 - 4165408*b^23*c^12*d^30*e^13 + 3359200*b^24*c^11*d^29
*e^14 - 2248080*b^25*c^10*d^28*e^15 + 1240320*b^26*c^9*d^27*e^16 - 558144*b^27*c^8*d^26*e^17 + 201552*b^28*c^7
*d^25*e^18 - 57000*b^29*c^6*d^24*e^19 + 12160*b^30*c^5*d^23*e^20 - 1840*b^31*c^4*d^22*e^21 + 176*b^32*c^3*d^21
*e^22 - 8*b^33*c^2*d^20*e^23) + 8*A*b^10*c^23*d^37*e^3 - 148*A*b^11*c^22*d^36*e^4 + 1160*A*b^12*c^21*d^35*e^5
- 4760*A*b^13*c^20*d^34*e^6 + 8036*A*b^14*c^19*d^33*e^7 + 21868*A*b^15*c^18*d^32*e^8 - 194304*A*b^16*c^17*d^31
*e^9 + 709280*A*b^17*c^16*d^30*e^10 - 1744160*A*b^18*c^15*d^29*e^11 + 3218072*A*b^19*c^14*d^28*e^12 - 4654832*
A*b^20*c^13*d^27*e^13 + 5394480*A*b^21*c^12*d^26*e^14 - 5063240*A*b^22*c^11*d^25*e^15 + 3863800*A*b^23*c^10*d^
24*e^16 - 2393152*A*b^24*c^9*d^23*e^17 + 1194528*A*b^25*c^8*d^22*e^18 - 474056*A*b^26*c^7*d^21*e^19 + 146300*A
*b^27*c^6*d^20*e^20 - 33880*A*b^28*c^5*d^19*e^21 + 5544*A*b^29*c^4*d^18*e^22 - 572*A*b^30*c^3*d^17*e^23 + 28*A
*b^31*c^2*d^16*e^24 - 4*B*b^11*c^22*d^37*e^3 + 144*B*b^12*c^21*d^36*e^4 - 1840*B*b^13*c^20*d^35*e^5 + 13160*B*
b^14*c^19*d^34*e^6 - 62328*B*b^15*c^18*d^33*e^7 + 212800*B*b^16*c^17*d^32*e^8 - 550432*B*b^17*c^16*d^31*e^9 +
1113120*B*b^18*c^15*d^30*e^10 - 1796600*B*b^19*c^14*d^29*e^11 + 2345824*B*b^20*c^13*d^28*e^12 - 2498496*B*b^21
*c^12*d^27*e^13 + 2179632*B*b^22*c^11*d^26*e^14 - 1557920*B*b^23*c^10*d^25*e^15 + 909120*B*b^24*c^9*d^24*e^16
- 429664*B*b^25*c^8*d^23*e^17 + 162208*B*b^26*c^7*d^22*e^18 - 47844*B*b^27*c^6*d^21*e^19 + 10640*B*b^28*c^5*d^
20*e^20 - 1680*B*b^29*c^4*d^19*e^21 + 168*B*b^30*c^3*d^18*e^22 - 8*B*b^31*c^2*d^17*e^23))*(-(16*A^2*c^11*d^2 +
121*A^2*b^2*c^9*e^2 + 4*B^2*b^2*c^9*d^2 + 81*B^2*b^4*c^7*e^2 - 198*A*B*b^3*c^8*e^2 - 36*B^2*b^3*c^8*d*e - 16*
A*B*b*c^10*d^2 - 88*A^2*b*c^10*d*e + 116*A*B*b^2*c^9*d*e)/(4*(b^15*e^9 - b^6*c^9*d^9 + 9*b^7*c^8*d^8*e - 36*b^
8*c^7*d^7*e^2 + 84*b^9*c^6*d^6*e^3 - 126*b^10*c^5*d^5*e^4 + 126*b^11*c^4*d^4*e^5 - 84*b^12*c^3*d^3*e^6 + 36*b^
13*c^2*d^2*e^7 - 9*b^14*c*d*e^8)))^(1/2)*1i)/(((d + e*x)^(1/2)*(1088*A^2*b^7*c^24*d^33*e^3 - 64*A^2*b^6*c^25*d
^34*e^2 - 8404*A^2*b^8*c^23*d^32*e^4 + 38720*A^2*b^9*c^22*d^31*e^5 - 116512*A^2*b^10*c^21*d^30*e^6 + 230912*A^
2*b^11*c^20*d^29*e^7 - 267432*A^2*b^12*c^19*d^28*e^8 + 38544*A^2*b^13*c^18*d^27*e^9 + 473880*A^2*b^14*c^17*d^2
6*e^10 - 851136*A^2*b^15*c^16*d^25*e^11 + 393646*A^2*b^16*c^15*d^24*e^12 + 1207368*A^2*b^17*c^14*d^23*e^13 - 3
343724*A^2*b^18*c^13*d^22*e^14 + 4835160*A^2*b^19*c^12*d^21*e^15 - 4903382*A^2*b^20*c^11*d^20*e^16 + 3751968*A
^2*b^21*c^10*d^19*e^17 - 2217072*A^2*b^22*c^9*d^18*e^18 + 1013232*A^2*b^23*c^8*d^17*e^19 - 353210*A^2*b^24*c^7
*d^16*e^20 + 91080*A^2*b^25*c^6*d^15*e^21 - 16412*A^2*b^26*c^5*d^14*e^22 + 1848*A^2*b^27*c^4*d^13*e^23 - 98*A^
2*b^28*c^3*d^12*e^24 - 16*B^2*b^8*c^23*d^34*e^2 + 328*B^2*b^9*c^22*d^33*e^3 - 3074*B^2*b^10*c^21*d^32*e^4 + 17
576*B^2*b^11*c^20*d^31*e^5 - 69252*B^2*b^12*c^19*d^30*e^6 + 201648*B^2*b^13*c^18*d^29*e^7 - 454686*B^2*b^14*c^
17*d^28*e^8 + 821328*B^2*b^15*c^16*d^27*e^9 - 1218432*B^2*b^16*c^15*d^26*e^10 + 1509384*B^2*b^17*c^14*d^25*e^1
1 - 1574606*B^2*b^18*c^13*d^24*e^12 + 1384168*B^2*b^19*c^12*d^23*e^13 - 1019324*B^2*b^20*c^11*d^22*e^14 + 6221
76*B^2*b^21*c^10*d^21*e^15 - 310242*B^2*b^22*c^9*d^20*e^16 + 124032*B^2*b^23*c^8*d^19*e^17 - 38760*B^2*b^24*c^
7*d^18*e^18 + 9120*B^2*b^25*c^6*d^17*e^19 - 1520*B^2*b^26*c^5*d^16*e^20 + 160*B^2*b^27*c^4*d^15*e^21 - 8*B^2*b
^28*c^3*d^14*e^22 + 64*A*B*b^7*c^24*d^34*e^2 - 1200*A*B*b^8*c^23*d^33*e^3 + 10252*A*B*b^9*c^22*d^32*e^4 - 5294
4*A*B*b^10*c^21*d^31*e^5 + 184216*A*B*b^11*c^20*d^30*e^6 - 452112*A*B*b^12*c^19*d^29*e^7 + 781428*A*B*b^13*c^1
8*d^28*e^8 - 863424*A*B*b^14*c^17*d^27*e^9 + 255408*A*B*b^15*c^16*d^26*e^10 + 1244088*A*B*b^16*c^15*d^25*e^11
- 3244396*A*B*b^17*c^14*d^24*e^12 + 4868800*A*B*b^18*c^13*d^23*e^13 - 5345768*A*B*b^19*c^12*d^22*e^14 + 456869
6*A*B*b^20*c^11*d^21*e^15 - 3100404*A*B*b^21*c^10*d^20*e^16 + 1674432*A*B*b^22*c^9*d^19*e^17 - 713184*A*B*b^23
*c^8*d^18*e^18 + 234840*A*B*b^24*c^7*d^17*e^19 - 57760*A*B*b^25*c^6*d^16*e^20 + 10000*A*B*b^26*c^5*d^15*e^21 -
1088*A*B*b^27*c^4*d^14*e^22 + 56*A*B*b^28*c^3*d^13*e^23) + (-(16*A^2*c^11*d^2 + 121*A^2*b^2*c^9*e^2 + 4*B^2*b
^2*c^9*d^2 + 81*B^2*b^4*c^7*e^2 - 198*A*B*b^3*c^8*e^2 - 36*B^2*b^3*c^8*d*e - 16*A*B*b*c^10*d^2 - 88*A^2*b*c^10
*d*e + 116*A*B*b^2*c^9*d*e)/(4*(b^15*e^9 - b^6*c^9*d^9 + 9*b^7*c^8*d^8*e - 36*b^8*c^7*d^7*e^2 + 84*b^9*c^6*d^6
*e^3 - 126*b^10*c^5*d^5*e^4 + 126*b^11*c^4*d^4*e^5 - 84*b^12*c^3*d^3*e^6 + 36*b^13*c^2*d^2*e^7 - 9*b^14*c*d*e^
8)))^(1/2)*((d + e*x)^(1/2)*(-(16*A^2*c^11*d^2 + 121*A^2*b^2*c^9*e^2 + 4*B^2*b^2*c^9*d^2 + 81*B^2*b^4*c^7*e^2
- 198*A*B*b^3*c^8*e^2 - 36*B^2*b^3*c^8*d*e - 16*A*B*b*c^10*d^2 - 88*A^2*b*c^10*d*e + 116*A*B*b^2*c^9*d*e)/(4*(
b^15*e^9 - b^6*c^9*d^9 + 9*b^7*c^8*d^8*e - 36*b^8*c^7*d^7*e^2 + 84*b^9*c^6*d^6*e^3 - 126*b^10*c^5*d^5*e^4 + 12
6*b^11*c^4*d^4*e^5 - 84*b^12*c^3*d^3*e^6 + 36*b^13*c^2*d^2*e^7 - 9*b^14*c*d*e^8)))^(1/2)*(16*b^12*c^23*d^41*e^
2 - 328*b^13*c^22*d^40*e^3 + 3200*b^14*c^21*d^39*e^4 - 19760*b^15*c^20*d^38*e^5 + 86640*b^16*c^19*d^37*e^6 - 2
86824*b^17*c^18*d^36*e^7 + 744192*b^18*c^17*d^35*e^8 - 1550400*b^19*c^16*d^34*e^9 + 2635680*b^20*c^15*d^33*e^1
0 - 3695120*b^21*c^14*d^32*e^11 + 4299776*b^22*c^13*d^31*e^12 - 4165408*b^23*c^12*d^30*e^13 + 3359200*b^24*c^1
1*d^29*e^14 - 2248080*b^25*c^10*d^28*e^15 + 1240320*b^26*c^9*d^27*e^16 - 558144*b^27*c^8*d^26*e^17 + 201552*b^
28*c^7*d^25*e^18 - 57000*b^29*c^6*d^24*e^19 + 12160*b^30*c^5*d^23*e^20 - 1840*b^31*c^4*d^22*e^21 + 176*b^32*c^
3*d^21*e^22 - 8*b^33*c^2*d^20*e^23) + 8*A*b^10*c^23*d^37*e^3 - 148*A*b^11*c^22*d^36*e^4 + 1160*A*b^12*c^21*d^3
5*e^5 - 4760*A*b^13*c^20*d^34*e^6 + 8036*A*b^14*c^19*d^33*e^7 + 21868*A*b^15*c^18*d^32*e^8 - 194304*A*b^16*c^1
7*d^31*e^9 + 709280*A*b^17*c^16*d^30*e^10 - 1744160*A*b^18*c^15*d^29*e^11 + 3218072*A*b^19*c^14*d^28*e^12 - 46
54832*A*b^20*c^13*d^27*e^13 + 5394480*A*b^21*c^12*d^26*e^14 - 5063240*A*b^22*c^11*d^25*e^15 + 3863800*A*b^23*c
^10*d^24*e^16 - 2393152*A*b^24*c^9*d^23*e^17 + 1194528*A*b^25*c^8*d^22*e^18 - 474056*A*b^26*c^7*d^21*e^19 + 14
6300*A*b^27*c^6*d^20*e^20 - 33880*A*b^28*c^5*d^19*e^21 + 5544*A*b^29*c^4*d^18*e^22 - 572*A*b^30*c^3*d^17*e^23
+ 28*A*b^31*c^2*d^16*e^24 - 4*B*b^11*c^22*d^37*e^3 + 144*B*b^12*c^21*d^36*e^4 - 1840*B*b^13*c^20*d^35*e^5 + 13
160*B*b^14*c^19*d^34*e^6 - 62328*B*b^15*c^18*d^33*e^7 + 212800*B*b^16*c^17*d^32*e^8 - 550432*B*b^17*c^16*d^31*
e^9 + 1113120*B*b^18*c^15*d^30*e^10 - 1796600*B*b^19*c^14*d^29*e^11 + 2345824*B*b^20*c^13*d^28*e^12 - 2498496*
B*b^21*c^12*d^27*e^13 + 2179632*B*b^22*c^11*d^26*e^14 - 1557920*B*b^23*c^10*d^25*e^15 + 909120*B*b^24*c^9*d^24
*e^16 - 429664*B*b^25*c^8*d^23*e^17 + 162208*B*b^26*c^7*d^22*e^18 - 47844*B*b^27*c^6*d^21*e^19 + 10640*B*b^28*
c^5*d^20*e^20 - 1680*B*b^29*c^4*d^19*e^21 + 168*B*b^30*c^3*d^18*e^22 - 8*B*b^31*c^2*d^17*e^23))*(-(16*A^2*c^11
*d^2 + 121*A^2*b^2*c^9*e^2 + 4*B^2*b^2*c^9*d^2 + 81*B^2*b^4*c^7*e^2 - 198*A*B*b^3*c^8*e^2 - 36*B^2*b^3*c^8*d*e
- 16*A*B*b*c^10*d^2 - 88*A^2*b*c^10*d*e + 116*A*B*b^2*c^9*d*e)/(4*(b^15*e^9 - b^6*c^9*d^9 + 9*b^7*c^8*d^8*e -
36*b^8*c^7*d^7*e^2 + 84*b^9*c^6*d^6*e^3 - 126*b^10*c^5*d^5*e^4 + 126*b^11*c^4*d^4*e^5 - 84*b^12*c^3*d^3*e^6 +
36*b^13*c^2*d^2*e^7 - 9*b^14*c*d*e^8)))^(1/2) - ((d + e*x)^(1/2)*(1088*A^2*b^7*c^24*d^33*e^3 - 64*A^2*b^6*c^2
5*d^34*e^2 - 8404*A^2*b^8*c^23*d^32*e^4 + 38720*A^2*b^9*c^22*d^31*e^5 - 116512*A^2*b^10*c^21*d^30*e^6 + 230912
*A^2*b^11*c^20*d^29*e^7 - 267432*A^2*b^12*c^19*d^28*e^8 + 38544*A^2*b^13*c^18*d^27*e^9 + 473880*A^2*b^14*c^17*
d^26*e^10 - 851136*A^2*b^15*c^16*d^25*e^11 + 393646*A^2*b^16*c^15*d^24*e^12 + 1207368*A^2*b^17*c^14*d^23*e^13
- 3343724*A^2*b^18*c^13*d^22*e^14 + 4835160*A^2*b^19*c^12*d^21*e^15 - 4903382*A^2*b^20*c^11*d^20*e^16 + 375196
8*A^2*b^21*c^10*d^19*e^17 - 2217072*A^2*b^22*c^9*d^18*e^18 + 1013232*A^2*b^23*c^8*d^17*e^19 - 353210*A^2*b^24*
c^7*d^16*e^20 + 91080*A^2*b^25*c^6*d^15*e^21 - 16412*A^2*b^26*c^5*d^14*e^22 + 1848*A^2*b^27*c^4*d^13*e^23 - 98
*A^2*b^28*c^3*d^12*e^24 - 16*B^2*b^8*c^23*d^34*e^2 + 328*B^2*b^9*c^22*d^33*e^3 - 3074*B^2*b^10*c^21*d^32*e^4 +
17576*B^2*b^11*c^20*d^31*e^5 - 69252*B^2*b^12*c^19*d^30*e^6 + 201648*B^2*b^13*c^18*d^29*e^7 - 454686*B^2*b^14
*c^17*d^28*e^8 + 821328*B^2*b^15*c^16*d^27*e^9 - 1218432*B^2*b^16*c^15*d^26*e^10 + 1509384*B^2*b^17*c^14*d^25*
e^11 - 1574606*B^2*b^18*c^13*d^24*e^12 + 1384168*B^2*b^19*c^12*d^23*e^13 - 1019324*B^2*b^20*c^11*d^22*e^14 + 6
22176*B^2*b^21*c^10*d^21*e^15 - 310242*B^2*b^22*c^9*d^20*e^16 + 124032*B^2*b^23*c^8*d^19*e^17 - 38760*B^2*b^24
*c^7*d^18*e^18 + 9120*B^2*b^25*c^6*d^17*e^19 - 1520*B^2*b^26*c^5*d^16*e^20 + 160*B^2*b^27*c^4*d^15*e^21 - 8*B^
2*b^28*c^3*d^14*e^22 + 64*A*B*b^7*c^24*d^34*e^2 - 1200*A*B*b^8*c^23*d^33*e^3 + 10252*A*B*b^9*c^22*d^32*e^4 - 5
2944*A*B*b^10*c^21*d^31*e^5 + 184216*A*B*b^11*c^20*d^30*e^6 - 452112*A*B*b^12*c^19*d^29*e^7 + 781428*A*B*b^13*
c^18*d^28*e^8 - 863424*A*B*b^14*c^17*d^27*e^9 + 255408*A*B*b^15*c^16*d^26*e^10 + 1244088*A*B*b^16*c^15*d^25*e^
11 - 3244396*A*B*b^17*c^14*d^24*e^12 + 4868800*A*B*b^18*c^13*d^23*e^13 - 5345768*A*B*b^19*c^12*d^22*e^14 + 456
8696*A*B*b^20*c^11*d^21*e^15 - 3100404*A*B*b^21*c^10*d^20*e^16 + 1674432*A*B*b^22*c^9*d^19*e^17 - 713184*A*B*b
^23*c^8*d^18*e^18 + 234840*A*B*b^24*c^7*d^17*e^19 - 57760*A*B*b^25*c^6*d^16*e^20 + 10000*A*B*b^26*c^5*d^15*e^2
1 - 1088*A*B*b^27*c^4*d^14*e^22 + 56*A*B*b^28*c^3*d^13*e^23) + (-(16*A^2*c^11*d^2 + 121*A^2*b^2*c^9*e^2 + 4*B^
2*b^2*c^9*d^2 + 81*B^2*b^4*c^7*e^2 - 198*A*B*b^3*c^8*e^2 - 36*B^2*b^3*c^8*d*e - 16*A*B*b*c^10*d^2 - 88*A^2*b*c
^10*d*e + 116*A*B*b^2*c^9*d*e)/(4*(b^15*e^9 - b^6*c^9*d^9 + 9*b^7*c^8*d^8*e - 36*b^8*c^7*d^7*e^2 + 84*b^9*c^6*
d^6*e^3 - 126*b^10*c^5*d^5*e^4 + 126*b^11*c^4*d^4*e^5 - 84*b^12*c^3*d^3*e^6 + 36*b^13*c^2*d^2*e^7 - 9*b^14*c*d
*e^8)))^(1/2)*((d + e*x)^(1/2)*(-(16*A^2*c^11*d^2 + 121*A^2*b^2*c^9*e^2 + 4*B^2*b^2*c^9*d^2 + 81*B^2*b^4*c^7*e
^2 - 198*A*B*b^3*c^8*e^2 - 36*B^2*b^3*c^8*d*e - 16*A*B*b*c^10*d^2 - 88*A^2*b*c^10*d*e + 116*A*B*b^2*c^9*d*e)/(
4*(b^15*e^9 - b^6*c^9*d^9 + 9*b^7*c^8*d^8*e - 36*b^8*c^7*d^7*e^2 + 84*b^9*c^6*d^6*e^3 - 126*b^10*c^5*d^5*e^4 +
126*b^11*c^4*d^4*e^5 - 84*b^12*c^3*d^3*e^6 + 36*b^13*c^2*d^2*e^7 - 9*b^14*c*d*e^8)))^(1/2)*(16*b^12*c^23*d^41
*e^2 - 328*b^13*c^22*d^40*e^3 + 3200*b^14*c^21*d^39*e^4 - 19760*b^15*c^20*d^38*e^5 + 86640*b^16*c^19*d^37*e^6
- 286824*b^17*c^18*d^36*e^7 + 744192*b^18*c^17*d^35*e^8 - 1550400*b^19*c^16*d^34*e^9 + 2635680*b^20*c^15*d^33*
e^10 - 3695120*b^21*c^14*d^32*e^11 + 4299776*b^22*c^13*d^31*e^12 - 4165408*b^23*c^12*d^30*e^13 + 3359200*b^24*
c^11*d^29*e^14 - 2248080*b^25*c^10*d^28*e^15 + 1240320*b^26*c^9*d^27*e^16 - 558144*b^27*c^8*d^26*e^17 + 201552
*b^28*c^7*d^25*e^18 - 57000*b^29*c^6*d^24*e^19 + 12160*b^30*c^5*d^23*e^20 - 1840*b^31*c^4*d^22*e^21 + 176*b^32
*c^3*d^21*e^22 - 8*b^33*c^2*d^20*e^23) - 8*A*b^10*c^23*d^37*e^3 + 148*A*b^11*c^22*d^36*e^4 - 1160*A*b^12*c^21*
d^35*e^5 + 4760*A*b^13*c^20*d^34*e^6 - 8036*A*b^14*c^19*d^33*e^7 - 21868*A*b^15*c^18*d^32*e^8 + 194304*A*b^16*
c^17*d^31*e^9 - 709280*A*b^17*c^16*d^30*e^10 + 1744160*A*b^18*c^15*d^29*e^11 - 3218072*A*b^19*c^14*d^28*e^12 +
4654832*A*b^20*c^13*d^27*e^13 - 5394480*A*b^21*c^12*d^26*e^14 + 5063240*A*b^22*c^11*d^25*e^15 - 3863800*A*b^2
3*c^10*d^24*e^16 + 2393152*A*b^24*c^9*d^23*e^17 - 1194528*A*b^25*c^8*d^22*e^18 + 474056*A*b^26*c^7*d^21*e^19 -
146300*A*b^27*c^6*d^20*e^20 + 33880*A*b^28*c^5*d^19*e^21 - 5544*A*b^29*c^4*d^18*e^22 + 572*A*b^30*c^3*d^17*e^
23 - 28*A*b^31*c^2*d^16*e^24 + 4*B*b^11*c^22*d^37*e^3 - 144*B*b^12*c^21*d^36*e^4 + 1840*B*b^13*c^20*d^35*e^5 -
13160*B*b^14*c^19*d^34*e^6 + 62328*B*b^15*c^18*d^33*e^7 - 212800*B*b^16*c^17*d^32*e^8 + 550432*B*b^17*c^16*d^
31*e^9 - 1113120*B*b^18*c^15*d^30*e^10 + 1796600*B*b^19*c^14*d^29*e^11 - 2345824*B*b^20*c^13*d^28*e^12 + 24984
96*B*b^21*c^12*d^27*e^13 - 2179632*B*b^22*c^11*d^26*e^14 + 1557920*B*b^23*c^10*d^25*e^15 - 909120*B*b^24*c^9*d
^24*e^16 + 429664*B*b^25*c^8*d^23*e^17 - 162208*B*b^26*c^7*d^22*e^18 + 47844*B*b^27*c^6*d^21*e^19 - 10640*B*b^
28*c^5*d^20*e^20 + 1680*B*b^29*c^4*d^19*e^21 - 168*B*b^30*c^3*d^18*e^22 + 8*B*b^31*c^2*d^17*e^23))*(-(16*A^2*c
^11*d^2 + 121*A^2*b^2*c^9*e^2 + 4*B^2*b^2*c^9*d^2 + 81*B^2*b^4*c^7*e^2 - 198*A*B*b^3*c^8*e^2 - 36*B^2*b^3*c^8*
d*e - 16*A*B*b*c^10*d^2 - 88*A^2*b*c^10*d*e + 116*A*B*b^2*c^9*d*e)/(4*(b^15*e^9 - b^6*c^9*d^9 + 9*b^7*c^8*d^8*
e - 36*b^8*c^7*d^7*e^2 + 84*b^9*c^6*d^6*e^3 - 126*b^10*c^5*d^5*e^4 + 126*b^11*c^4*d^4*e^5 - 84*b^12*c^3*d^3*e^
6 + 36*b^13*c^2*d^2*e^7 - 9*b^14*c*d*e^8)))^(1/2) - 64*A^3*b^4*c^25*d^30*e^3 + 960*A^3*b^5*c^24*d^29*e^4 - 718
0*A^3*b^6*c^23*d^28*e^5 + 35560*A^3*b^7*c^22*d^27*e^6 - 125748*A^3*b^8*c^21*d^26*e^7 + 314496*A^3*b^9*c^20*d^2
5*e^8 - 508886*A^3*b^10*c^19*d^24*e^9 + 326832*A^3*b^11*c^18*d^23*e^10 + 760408*A^3*b^12*c^17*d^22*e^11 - 2806
584*A^3*b^13*c^16*d^21*e^12 + 4917990*A^3*b^14*c^15*d^20*e^13 - 5803448*A^3*b^15*c^14*d^19*e^14 + 4974956*A^3*
b^16*c^13*d^18*e^15 - 3162096*A^3*b^17*c^12*d^17*e^16 + 1483782*A^3*b^18*c^11*d^16*e^17 - 501472*A^3*b^19*c^10
*d^15*e^18 + 115824*A^3*b^20*c^9*d^14*e^19 - 16408*A^3*b^21*c^8*d^13*e^20 + 1078*A^3*b^22*c^7*d^12*e^21 + 8*B^
3*b^7*c^22*d^30*e^3 - 36*B^3*b^8*c^21*d^29*e^4 - 688*B^3*b^9*c^20*d^28*e^5 + 8456*B^3*b^10*c^19*d^27*e^6 - 456
96*B^3*b^11*c^18*d^26*e^7 + 153412*B^3*b^12*c^17*d^25*e^8 - 357280*B^3*b^13*c^16*d^24*e^9 + 608256*B^3*b^14*c^
15*d^23*e^10 - 778272*B^3*b^15*c^14*d^22*e^11 + 758692*B^3*b^16*c^13*d^21*e^12 - 565488*B^3*b^17*c^12*d^20*e^1
3 + 320488*B^3*b^18*c^11*d^19*e^14 - 135968*B^3*b^19*c^10*d^18*e^15 + 41916*B^3*b^20*c^9*d^17*e^16 - 8896*B^3*
b^21*c^8*d^16*e^17 + 1168*B^3*b^22*c^7*d^15*e^18 - 72*B^3*b^23*c^6*d^14*e^19 - 48*A*B^2*b^6*c^23*d^30*e^3 + 38
4*A*B^2*b^7*c^22*d^29*e^4 + 222*A*B^2*b^8*c^21*d^28*e^5 - 13272*A*B^2*b^9*c^20*d^27*e^6 + 65940*A*B^2*b^10*c^1
9*d^26*e^7 - 150864*A*B^2*b^11*c^18*d^25*e^8 + 111762*A*B^2*b^12*c^17*d^24*e^9 + 345840*A*B^2*b^13*c^16*d^23*e
^10 - 1328976*A*B^2*b^14*c^15*d^22*e^11 + 2422200*A*B^2*b^15*c^14*d^21*e^12 - 2923998*A*B^2*b^16*c^13*d^20*e^1
3 + 2528904*A*B^2*b^17*c^12*d^19*e^14 - 1604316*A*B^2*b^18*c^11*d^18*e^15 + 745248*A*B^2*b^19*c^10*d^17*e^16 -
247890*A*B^2*b^20*c^9*d^16*e^17 + 56160*A*B^2*b^21*c^8*d^15*e^18 - 7800*A*B^2*b^22*c^7*d^14*e^19 + 504*A*B^2*
b^23*c^6*d^13*e^20 + 96*A^2*B*b^5*c^24*d^30*e^3 - 1104*A^2*B*b^6*c^23*d^29*e^4 + 5898*A^2*B*b^7*c^22*d^28*e^5
- 23688*A^2*B*b^8*c^21*d^27*e^6 + 95256*A^2*B*b^9*c^20*d^26*e^7 - 352548*A^2*B*b^10*c^19*d^25*e^8 + 1005564*A^
2*B*b^11*c^18*d^24*e^9 - 2061552*A^2*B*b^12*c^17*d^23*e^10 + 2961024*A^2*B*b^13*c^16*d^22*e^11 - 2823612*A^2*B
*b^14*c^15*d^21*e^12 + 1406328*A^2*B*b^15*c^14*d^20*e^13 + 410424*A^2*B*b^16*c^13*d^19*e^14 - 1459080*A^2*B*b^
17*c^12*d^18*e^15 + 1416996*A^2*B*b^18*c^11*d^17*e^16 - 834444*A^2*B*b^19*c^10*d^16*e^17 + 325824*A^2*B*b^20*c
^9*d^15*e^18 - 83184*A^2*B*b^21*c^8*d^14*e^19 + 12684*A^2*B*b^22*c^7*d^13*e^20 - 882*A^2*B*b^23*c^6*d^12*e^21)
)*(-(16*A^2*c^11*d^2 + 121*A^2*b^2*c^9*e^2 + 4*B^2*b^2*c^9*d^2 + 81*B^2*b^4*c^7*e^2 - 198*A*B*b^3*c^8*e^2 - 36
*B^2*b^3*c^8*d*e - 16*A*B*b*c^10*d^2 - 88*A^2*b*c^10*d*e + 116*A*B*b^2*c^9*d*e)/(4*(b^15*e^9 - b^6*c^9*d^9 + 9
*b^7*c^8*d^8*e - 36*b^8*c^7*d^7*e^2 + 84*b^9*c^6*d^6*e^3 - 126*b^10*c^5*d^5*e^4 + 126*b^11*c^4*d^4*e^5 - 84*b^
12*c^3*d^3*e^6 + 36*b^13*c^2*d^2*e^7 - 9*b^14*c*d*e^8)))^(1/2)*2i - ((2*(A*e^3 - B*d*e^2))/(5*(c*d^2 - b*d*e))
- (2*(d + e*x)*(7*A*b*e^4 - 14*A*c*d*e^3 - 2*B*b*d*e^3 + 9*B*c*d^2*e^2))/(15*(c*d^2 - b*d*e)^2) + (2*(d + e*x
)^2*(35*A*b^2*e^5 - 10*B*b^2*d*e^4 + 113*A*c^2*d^2*e^3 - 63*B*c^2*d^3*e^2 - 113*A*b*c*d*e^4 + 38*B*b*c*d^2*e^3
))/(15*(c*d^2 - b*d*e)^3) + ((d + e*x)^3*(21*A*b^5*e^6 - 6*A*c^5*d^5*e - 6*B*b^5*d*e^5 + 15*A*b*c^4*d^4*e^2 +
34*B*b^4*c*d^2*e^4 - 142*A*b^2*c^3*d^3*e^3 + 198*A*b^3*c^2*d^2*e^4 + 66*B*b^2*c^3*d^4*e^2 - 76*B*b^3*c^2*d^3*e
^3 - 107*A*b^4*c*d*e^5 + 3*B*b*c^4*d^5*e))/(3*b^2*(c*d^2 - b*d*e)^4) - ((d + e*x)^4*(4*A*b*c^4*d^3*e^2 - 2*A*c
^5*d^4*e - 7*A*b^4*c*e^5 + 24*A*b^3*c^2*d*e^4 - 26*A*b^2*c^3*d^2*e^3 + 12*B*b^2*c^3*d^3*e^2 - 8*B*b^3*c^2*d^2*
e^3 + B*b*c^4*d^4*e + 2*B*b^4*c*d*e^4))/(b^2*(c*d^2 - b*d*e)^4))/(c*(d + e*x)^(9/2) + (c*d^2 - b*d*e)*(d + e*x
)^(5/2) + (b*e - 2*c*d)*(d + e*x)^(7/2))