Integrand size = 26, antiderivative size = 181 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx=-\frac {\sqrt {d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}-\frac {\sqrt {d} (2 b B d-4 A c d+3 A b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {\sqrt {c d-b e} \left (4 A c^2 d-b^2 B e-b c (2 B d+A e)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{3/2}} \]
-(3*A*b*e-4*A*c*d+2*B*b*d)*arctanh((e*x+d)^(1/2)/d^(1/2))*d^(1/2)/b^3-(4*A *c^2*d-b^2*B*e-b*c*(A*e+2*B*d))*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^( 1/2))*(-b*e+c*d)^(1/2)/b^3/c^(3/2)-(A*b*c*d+(2*A*c^2*d+b^2*B*e-b*c*(A*e+B* d))*x)*(e*x+d)^(1/2)/b^2/c/(c*x^2+b*x)
Time = 0.61 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {\frac {b \sqrt {d+e x} (b B (c d-b e) x+A c (-b d-2 c d x+b e x))}{c x (b+c x)}-\frac {\sqrt {-c d+b e} \left (4 A c^2 d-b^2 B e-b c (2 B d+A e)\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{3/2}}-\sqrt {d} (2 b B d-4 A c d+3 A b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3} \]
((b*Sqrt[d + e*x]*(b*B*(c*d - b*e)*x + A*c*(-(b*d) - 2*c*d*x + b*e*x)))/(c *x*(b + c*x)) - (Sqrt[-(c*d) + b*e]*(4*A*c^2*d - b^2*B*e - b*c*(2*B*d + A* e))*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/c^(3/2) - Sqrt[d]* (2*b*B*d - 4*A*c*d + 3*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3
Time = 0.47 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1233, 27, 1197, 27, 1480, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {\int \frac {c d (2 b B d-4 A c d+3 A b e)-e \left (-B e b^2-c (B d+A e) b+2 A c^2 d\right ) x}{2 \sqrt {d+e x} \left (c x^2+b x\right )}dx}{b^2 c}-\frac {\sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {c d (2 b B d-4 A c d+3 A b e)-e \left (-B e b^2-c (B d+A e) b+2 A c^2 d\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{2 b^2 c}-\frac {\sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {\int \frac {e \left ((b B-2 A c) d (c d-b e)-\left (-B e b^2-c (B d+A e) b+2 A c^2 d\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2 c}-\frac {\sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int \frac {(b B-2 A c) d (c d-b e)-\left (-B e b^2-c (B d+A e) b+2 A c^2 d\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2 c}-\frac {\sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {e \left (\frac {(c d-b e) \left (-b c (A e+2 B d)+4 A c^2 d+b^2 (-B) e\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}+\frac {c^2 d (3 A b e-4 A c d+2 b B d) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}\right )}{b^2 c}-\frac {\sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {e \left (-\frac {\sqrt {c d-b e} \left (-b c (A e+2 B d)+4 A c^2 d+b^2 (-B) e\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c} e}-\frac {c \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (3 A b e-4 A c d+2 b B d)}{b e}\right )}{b^2 c}-\frac {\sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
-((Sqrt[d + e*x]*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(b ^2*c*(b*x + c*x^2))) + (e*(-((c*Sqrt[d]*(2*b*B*d - 4*A*c*d + 3*A*b*e)*ArcT anh[Sqrt[d + e*x]/Sqrt[d]])/(b*e)) - (Sqrt[c*d - b*e]*(4*A*c^2*d - b^2*B*e - b*c*(2*B*d + A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b *Sqrt[c]*e)))/(b^2*c)
3.13.41.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[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] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(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]
Time = 0.40 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(2 e^{2} \left (\frac {\left (b e -c d \right ) \left (\frac {b e \left (A c -B b \right ) \sqrt {e x +d}}{2 c \left (c \left (e x +d \right )+b e -c d \right )}+\frac {\left (A b c e -4 A \,c^{2} d +b^{2} B e +2 B b c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 c \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{2}}-\frac {d \left (\frac {A b \sqrt {e x +d}}{2 x}+\frac {\left (3 A b e -4 A c d +2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{e^{2} b^{3}}\right )\) | \(183\) |
| default | \(2 e^{2} \left (\frac {\left (b e -c d \right ) \left (\frac {b e \left (A c -B b \right ) \sqrt {e x +d}}{2 c \left (c \left (e x +d \right )+b e -c d \right )}+\frac {\left (A b c e -4 A \,c^{2} d +b^{2} B e +2 B b c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 c \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{2}}-\frac {d \left (\frac {A b \sqrt {e x +d}}{2 x}+\frac {\left (3 A b e -4 A c d +2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{e^{2} b^{3}}\right )\) | \(183\) |
| pseudoelliptic | \(\frac {4 \left (A \,c^{2} d -\frac {b c \left (A e +2 B d \right )}{4}-\frac {b^{2} B e}{4}\right ) \left (-b e +c d \right ) \sqrt {d}\, x \left (c x +b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )-\left (3 c x \left (c x +b \right ) d \left (-\frac {4 A c d}{3}+b \left (A e +\frac {2 B d}{3}\right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )+\sqrt {e x +d}\, \sqrt {d}\, \left (2 A \,c^{2} d x +\left (\left (-B x +A \right ) d -A e x \right ) b c +B \,b^{2} e x \right ) b \right ) \sqrt {\left (b e -c d \right ) c}}{\left (c x +b \right ) b^{3} c \sqrt {\left (b e -c d \right ) c}\, x \sqrt {d}}\) | \(199\) |
| risch | \(-\frac {d A \sqrt {e x +d}}{b^{2} x}-\frac {e \left (\frac {-\frac {b e \left (A b c e -A \,c^{2} d -b^{2} B e +B b c d \right ) \sqrt {e x +d}}{c \left (c \left (e x +d \right )+b e -c d \right )}-\frac {\left (A \,b^{2} c \,e^{2}-5 A b \,c^{2} d e +4 A \,c^{3} d^{2}+b^{3} B \,e^{2}+B \,b^{2} c d e -2 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{c \sqrt {\left (b e -c d \right ) c}}}{b e}+\frac {\sqrt {d}\, \left (3 A b e -4 A c d +2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e}\right )}{b^{2}}\) | \(222\) |
2*e^2*((b*e-c*d)/b^3/e^2*(1/2*b*e*(A*c-B*b)/c*(e*x+d)^(1/2)/(c*(e*x+d)+b*e -c*d)+1/2*(A*b*c*e-4*A*c^2*d+B*b^2*e+2*B*b*c*d)/c/((b*e-c*d)*c)^(1/2)*arct an(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)))-d/e^2/b^3*(1/2*A*b*(e*x+d)^(1/2)/ x+1/2*(3*A*b*e-4*A*c*d+2*B*b*d)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))))
Time = 0.71 (sec) , antiderivative size = 1146, normalized size of antiderivative = 6.33 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
[1/2*(((2*(B*b*c^2 - 2*A*c^3)*d + (B*b^2*c + A*b*c^2)*e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d + (B*b^3 + A*b^2*c)*e)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + ((3*A*b *c^2*e + 2*(B*b*c^2 - 2*A*c^3)*d)*x^2 + (3*A*b^2*c*e + 2*(B*b^2*c - 2*A*b* c^2)*d)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(A*b^2 *c*d - ((B*b^2*c - 2*A*b*c^2)*d - (B*b^3 - A*b^2*c)*e)*x)*sqrt(e*x + d))/( b^3*c^2*x^2 + b^4*c*x), 1/2*(2*((2*(B*b*c^2 - 2*A*c^3)*d + (B*b^2*c + A*b* c^2)*e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d + (B*b^3 + A*b^2*c)*e)*x)*sqrt(-( c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + ((3*A*b*c^2*e + 2*(B*b*c^2 - 2*A*c^3)*d)*x^2 + (3*A*b^2*c*e + 2*(B*b^2*c - 2*A*b*c^2)*d)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2 *(A*b^2*c*d - ((B*b^2*c - 2*A*b*c^2)*d - (B*b^3 - A*b^2*c)*e)*x)*sqrt(e*x + d))/(b^3*c^2*x^2 + b^4*c*x), 1/2*(2*((3*A*b*c^2*e + 2*(B*b*c^2 - 2*A*c^3 )*d)*x^2 + (3*A*b^2*c*e + 2*(B*b^2*c - 2*A*b*c^2)*d)*x)*sqrt(-d)*arctan(sq rt(e*x + d)*sqrt(-d)/d) + ((2*(B*b*c^2 - 2*A*c^3)*d + (B*b^2*c + A*b*c^2)* e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d + (B*b^3 + A*b^2*c)*e)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/ (c*x + b)) - 2*(A*b^2*c*d - ((B*b^2*c - 2*A*b*c^2)*d - (B*b^3 - A*b^2*c)*e )*x)*sqrt(e*x + d))/(b^3*c^2*x^2 + b^4*c*x), (((2*(B*b*c^2 - 2*A*c^3)*d + (B*b^2*c + A*b*c^2)*e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d + (B*b^3 + A*b^...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m ore detail
Time = 0.28 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.75 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {{\left (2 \, B b d^{2} - 4 \, A c d^{2} + 3 \, A b d e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} - \frac {{\left (2 \, B b c^{2} d^{2} - 4 \, A c^{3} d^{2} - B b^{2} c d e + 5 \, A b c^{2} d e - B b^{3} e^{2} - A b^{2} c e^{2}\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3} c} + \frac {{\left (e x + d\right )}^{\frac {3}{2}} B b c d e - 2 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{2} d e - \sqrt {e x + d} B b c d^{2} e + 2 \, \sqrt {e x + d} A c^{2} d^{2} e - {\left (e x + d\right )}^{\frac {3}{2}} B b^{2} e^{2} + {\left (e x + d\right )}^{\frac {3}{2}} A b c e^{2} + \sqrt {e x + d} B b^{2} d e^{2} - 2 \, \sqrt {e x + d} A b c d e^{2}}{{\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )} b^{2} c} \]
(2*B*b*d^2 - 4*A*c*d^2 + 3*A*b*d*e)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^3*sq rt(-d)) - (2*B*b*c^2*d^2 - 4*A*c^3*d^2 - B*b^2*c*d*e + 5*A*b*c^2*d*e - B*b ^3*e^2 - A*b^2*c*e^2)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(- c^2*d + b*c*e)*b^3*c) + ((e*x + d)^(3/2)*B*b*c*d*e - 2*(e*x + d)^(3/2)*A*c ^2*d*e - sqrt(e*x + d)*B*b*c*d^2*e + 2*sqrt(e*x + d)*A*c^2*d^2*e - (e*x + d)^(3/2)*B*b^2*e^2 + (e*x + d)^(3/2)*A*b*c*e^2 + sqrt(e*x + d)*B*b^2*d*e^2 - 2*sqrt(e*x + d)*A*b*c*d*e^2)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e)*b^2*c)
Time = 11.31 (sec) , antiderivative size = 4391, normalized size of antiderivative = 24.26 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
(d^(1/2)*atan(((d^(1/2)*((2*(d + e*x)^(1/2)*(B^2*b^6*e^6 + A^2*b^4*c^2*e^6 + 32*A^2*c^6*d^4*e^2 + 42*A^2*b^2*c^4*d^2*e^4 + 8*B^2*b^2*c^4*d^4*e^2 - 4 *B^2*b^3*c^3*d^3*e^3 - 3*B^2*b^4*c^2*d^2*e^4 + 2*B^2*b^5*c*d*e^5 - 64*A^2* b*c^5*d^3*e^3 - 10*A^2*b^3*c^3*d*e^5 + 2*A*B*b^5*c*e^6 - 32*A*B*b*c^5*d^4* e^2 - 8*A*B*b^4*c^2*d*e^5 + 40*A*B*b^2*c^4*d^3*e^3 - 6*A*B*b^3*c^3*d^2*e^4 ))/(b^4*c) + (d^(1/2)*((8*A*b^7*c^3*d*e^4 - 4*B*b^8*c^2*d*e^4 - 8*A*b^6*c^ 4*d^2*e^3 + 4*B*b^7*c^3*d^2*e^3)/(b^6*c) + (d^(1/2)*(4*b^7*c^3*e^3 - 8*b^6 *c^4*d*e^2)*(d + e*x)^(1/2)*(3*A*b*e - 4*A*c*d + 2*B*b*d))/(b^7*c))*(3*A*b *e - 4*A*c*d + 2*B*b*d))/(2*b^3))*(3*A*b*e - 4*A*c*d + 2*B*b*d)*1i)/(2*b^3 ) + (d^(1/2)*((2*(d + e*x)^(1/2)*(B^2*b^6*e^6 + A^2*b^4*c^2*e^6 + 32*A^2*c ^6*d^4*e^2 + 42*A^2*b^2*c^4*d^2*e^4 + 8*B^2*b^2*c^4*d^4*e^2 - 4*B^2*b^3*c^ 3*d^3*e^3 - 3*B^2*b^4*c^2*d^2*e^4 + 2*B^2*b^5*c*d*e^5 - 64*A^2*b*c^5*d^3*e ^3 - 10*A^2*b^3*c^3*d*e^5 + 2*A*B*b^5*c*e^6 - 32*A*B*b*c^5*d^4*e^2 - 8*A*B *b^4*c^2*d*e^5 + 40*A*B*b^2*c^4*d^3*e^3 - 6*A*B*b^3*c^3*d^2*e^4))/(b^4*c) - (d^(1/2)*((8*A*b^7*c^3*d*e^4 - 4*B*b^8*c^2*d*e^4 - 8*A*b^6*c^4*d^2*e^3 + 4*B*b^7*c^3*d^2*e^3)/(b^6*c) - (d^(1/2)*(4*b^7*c^3*e^3 - 8*b^6*c^4*d*e^2) *(d + e*x)^(1/2)*(3*A*b*e - 4*A*c*d + 2*B*b*d))/(b^7*c))*(3*A*b*e - 4*A*c* d + 2*B*b*d))/(2*b^3))*(3*A*b*e - 4*A*c*d + 2*B*b*d)*1i)/(2*b^3))/((2*(32* A^3*c^6*d^5*e^3 + 2*B^3*b^6*d^2*e^6 + 70*A^3*b^2*c^4*d^3*e^5 - 25*A^3*b^3* c^3*d^2*e^6 - 4*B^3*b^3*c^3*d^5*e^3 - 2*B^3*b^4*c^2*d^4*e^4 + 3*A*B^2*b...