Integrand size = 25, antiderivative size = 225 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx=\frac {(a B+A c x) \sqrt {d+e x}}{2 a c \left (a-c x^2\right )}-\frac {\left (2 A c d-a B e-\sqrt {a} A \sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{5/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (2 A c d-a B e+\sqrt {a} A \sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{5/4} \sqrt {\sqrt {c} d+\sqrt {a} e}} \]
1/2*(A*c*x+B*a)*(e*x+d)^(1/2)/a/c/(-c*x^2+a)-1/4*arctanh(c^(1/4)*(e*x+d)^( 1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(2*A*c*d-B*a*e-A*e*a^(1/2)*c^(1/2))/a^( 3/2)/c^(5/4)/(-e*a^(1/2)+d*c^(1/2))^(1/2)+1/4*arctanh(c^(1/4)*(e*x+d)^(1/2 )/(e*a^(1/2)+d*c^(1/2))^(1/2))*(2*A*c*d-B*a*e+A*e*a^(1/2)*c^(1/2))/a^(3/2) /c^(5/4)/(e*a^(1/2)+d*c^(1/2))^(1/2)
Time = 0.98 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.27 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {c} (a B+A c x) \sqrt {d+e x}}{-a+c x^2}-\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \left (2 A c d-a B e+\sqrt {a} A \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {c} d+\sqrt {a} e}+\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \left (2 A c d-a B e-\sqrt {a} A \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {c} d-\sqrt {a} e}}{4 a^{3/2} c^{3/2}} \]
((-2*Sqrt[a]*Sqrt[c]*(a*B + A*c*x)*Sqrt[d + e*x])/(-a + c*x^2) - (Sqrt[-(c *d) - Sqrt[a]*Sqrt[c]*e]*(2*A*c*d - a*B*e + Sqrt[a]*A*Sqrt[c]*e)*ArcTan[(S qrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/( Sqrt[c]*d + Sqrt[a]*e) + (Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*(2*A*c*d - a*B* e - Sqrt[a]*A*Sqrt[c]*e)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(Sqrt[c]*d - Sqrt[a]*e))/(4*a^(3/2)*c^(3/ 2))
Time = 0.42 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {685, 27, 654, 25, 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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 685 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{2 a c \left (a-c x^2\right )}-\frac {\int -\frac {2 A c d-a B e+A c e x}{2 \sqrt {d+e x} \left (a-c x^2\right )}dx}{2 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 A c d-a B e+A c e x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{4 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\int -\frac {e (A c d-a B e+A c (d+e x))}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {e (A c d-a B e+A c (d+e x))}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{2 a c \left (a-c x^2\right )}-\frac {e \int \frac {A c d-a B e+A c (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{2 a c \left (a-c x^2\right )}-\frac {e \left (\frac {1}{2} \sqrt {c} \left (A \sqrt {c}-\frac {2 A c d-a B e}{\sqrt {a} e}\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}+\frac {1}{2} \sqrt {c} \left (\frac {2 A c d-a B e}{\sqrt {a} e}+A \sqrt {c}\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}\right )}{2 a c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{2 a c \left (a-c x^2\right )}-\frac {e \left (-\frac {\left (A \sqrt {c}-\frac {2 A c d-a B e}{\sqrt {a} e}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {a} e}}-\frac {\left (\frac {2 A c d-a B e}{\sqrt {a} e}+A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt [4]{c} \sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 a c}\) |
((a*B + A*c*x)*Sqrt[d + e*x])/(2*a*c*(a - c*x^2)) - (e*(-1/2*((A*Sqrt[c] - (2*A*c*d - a*B*e)/(Sqrt[a]*e))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[ c]*d - Sqrt[a]*e]])/(c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) - ((A*Sqrt[c] + (2*A*c*d - a*B*e)/(Sqrt[a]*e))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c ]*d + Sqrt[a]*e]])/(2*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*e])))/(2*a*c)
3.15.55.3.1 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_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(d + e*x)^m*(a + c*x^2)^(p + 1)*((a*g - c*f*x)/(2*a*c *(p + 1))), x] - Simp[1/(2*a*c*(p + 1)) Int[(d + e*x)^(m - 1)*(a + c*x^2) ^(p + 1)*Simp[a*e*g*m - c*d*f*(2*p + 3) - c*e*f*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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 = 0.52 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.08
| method | result | size |
| derivativedivides | \(2 e^{2} \left (\frac {\frac {A \left (e x +d \right )^{\frac {3}{2}}}{4 a e}-\frac {\left (A c d -B a e \right ) \sqrt {e x +d}}{4 a e c}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {-\frac {\left (-2 A c d +B a e -A \sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (2 A c d -B a e -A \sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 a e}\right )\) | \(244\) |
| default | \(2 e^{2} \left (\frac {\frac {A \left (e x +d \right )^{\frac {3}{2}}}{4 a e}-\frac {\left (A c d -B a e \right ) \sqrt {e x +d}}{4 a e c}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {-\frac {\left (-2 A c d +B a e -A \sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (2 A c d -B a e -A \sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 a e}\right )\) | \(244\) |
| pseudoelliptic | \(\frac {c \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-c \,x^{2}+a \right ) e \left (A c d -\frac {B a e}{2}-\frac {A \sqrt {a c \,e^{2}}}{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (c \left (A c d -\frac {B a e}{2}+\frac {A \sqrt {a c \,e^{2}}}{2}\right ) \left (-c \,x^{2}+a \right ) e \,\operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\sqrt {e x +d}\, \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (A c x +B a \right )\right )}{2 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, a c \left (-c \,x^{2}+a \right )}\) | \(253\) |
2*e^2*((1/4*A/a/e*(e*x+d)^(3/2)-1/4*(A*c*d-B*a*e)/a/e/c*(e*x+d)^(1/2))/(-c *(e*x+d)^2+2*c*d*(e*x+d)+e^2*a-c*d^2)+1/4/a/e*(-1/2*(-2*A*c*d+B*a*e-A*(a*c *e^2)^(1/2))/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e* x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+1/2*(2*A*c*d-B*a*e-A*(a*c*e^2) ^(1/2))/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^ (1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 3195 vs. \(2 (170) = 340\).
Time = 4.45 (sec) , antiderivative size = 3195, normalized size of antiderivative = 14.20 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display} \]
1/8*((a*c^2*x^2 - a^2*c)*sqrt((4*A^2*c^2*d^3 - 4*A*B*a*c*d^2*e + 2*A*B*a^2 *e^3 + (B^2*a^2 - 3*A^2*a*c)*d*e^2 + (a^3*c^3*d^2 - a^4*c^2*e^2)*sqrt((4*A ^2*B^2*c^2*d^2*e^4 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^5 + (B^4*a^2 + 2*A^2*B^ 2*a*c + A^4*c^2)*e^6)/(a^3*c^7*d^4 - 2*a^4*c^6*d^2*e^2 + a^5*c^5*e^4)))/(a ^3*c^3*d^2 - a^4*c^2*e^2))*log((8*A^3*B*c^3*d^3*e^2 - 4*(3*A^2*B^2*a*c^2 + A^4*c^3)*d^2*e^3 + 2*(3*A*B^3*a^2*c + A^3*B*a*c^2)*d*e^4 - (B^4*a^3 - A^4 *a*c^2)*e^5)*sqrt(e*x + d) + (2*A^2*B*a^2*c^3*d^2*e^3 - (3*A*B^2*a^3*c^2 + A^3*a^2*c^3)*d*e^4 + (B^3*a^4*c + A^2*B*a^3*c^2)*e^5 + (2*A*a^3*c^6*d^4 - B*a^4*c^5*d^3*e - 3*A*a^4*c^5*d^2*e^2 + B*a^5*c^4*d*e^3 + A*a^5*c^4*e^4)* sqrt((4*A^2*B^2*c^2*d^2*e^4 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^5 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7*d^4 - 2*a^4*c^6*d^2*e^2 + a^5*c^5* e^4)))*sqrt((4*A^2*c^2*d^3 - 4*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 + (B^2*a^2 - 3*A^2*a*c)*d*e^2 + (a^3*c^3*d^2 - a^4*c^2*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^4 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^5 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e ^6)/(a^3*c^7*d^4 - 2*a^4*c^6*d^2*e^2 + a^5*c^5*e^4)))/(a^3*c^3*d^2 - a^4*c ^2*e^2))) - (a*c^2*x^2 - a^2*c)*sqrt((4*A^2*c^2*d^3 - 4*A*B*a*c*d^2*e + 2* A*B*a^2*e^3 + (B^2*a^2 - 3*A^2*a*c)*d*e^2 + (a^3*c^3*d^2 - a^4*c^2*e^2)*sq rt((4*A^2*B^2*c^2*d^2*e^4 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^5 + (B^4*a^2 + 2 *A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7*d^4 - 2*a^4*c^6*d^2*e^2 + a^5*c^5*e^ 4)))/(a^3*c^3*d^2 - a^4*c^2*e^2))*log((8*A^3*B*c^3*d^3*e^2 - 4*(3*A^2*B...
Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (c x^{2} - a\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (170) = 340\).
Time = 0.39 (sec) , antiderivative size = 449, normalized size of antiderivative = 2.00 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx=\frac {{\left (2 \, A a c^{3} d^{2} e - B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3} - \sqrt {a c} A c d e {\left | a \right |} {\left | c \right |} {\left | e \right |} + \sqrt {a c} B a e^{2} {\left | a \right |} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d + \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{2} e - \sqrt {a c} a c^{2} d\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} + \frac {{\left (2 \, A a c^{3} d^{2} e - B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3} + \sqrt {a c} A c d e {\left | a \right |} {\left | c \right |} {\left | e \right |} - \sqrt {a c} B a e^{2} {\left | a \right |} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d - \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{2} e + \sqrt {a c} a c^{2} d\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} - \frac {{\left (e x + d\right )}^{\frac {3}{2}} A c e - \sqrt {e x + d} A c d e + \sqrt {e x + d} B a e^{2}}{2 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} - a e^{2}\right )} a c} \]
1/4*(2*A*a*c^3*d^2*e - B*a^2*c^2*d*e^2 - A*a^2*c^2*e^3 - sqrt(a*c)*A*c*d*e *abs(a)*abs(c)*abs(e) + sqrt(a*c)*B*a*e^2*abs(a)*abs(c)*abs(e))*arctan(sqr t(e*x + d)/sqrt(-(a*c^2*d + sqrt(a^2*c^4*d^2 - (a*c^2*d^2 - a^2*c*e^2)*a*c ^2))/(a*c^2)))/((a^2*c^2*e - sqrt(a*c)*a*c^2*d)*sqrt(-c^2*d - sqrt(a*c)*c* e)*abs(a)*abs(e)) + 1/4*(2*A*a*c^3*d^2*e - B*a^2*c^2*d*e^2 - A*a^2*c^2*e^3 + sqrt(a*c)*A*c*d*e*abs(a)*abs(c)*abs(e) - sqrt(a*c)*B*a*e^2*abs(a)*abs(c )*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(a*c^2*d - sqrt(a^2*c^4*d^2 - (a*c^2* d^2 - a^2*c*e^2)*a*c^2))/(a*c^2)))/((a^2*c^2*e + sqrt(a*c)*a*c^2*d)*sqrt(- c^2*d + sqrt(a*c)*c*e)*abs(a)*abs(e)) - 1/2*((e*x + d)^(3/2)*A*c*e - sqrt( e*x + d)*A*c*d*e + sqrt(e*x + d)*B*a*e^2)/(((e*x + d)^2*c - 2*(e*x + d)*c* d + c*d^2 - a*e^2)*a*c)
Time = 12.82 (sec) , antiderivative size = 5062, normalized size of antiderivative = 22.50 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display} \]
atan(((((64*B*a^4*c^2*e^4 - 64*A*a^3*c^3*d*e^3)/(8*a^3) - 64*a*c^4*d*e^2*( d + e*x)^(1/2)*((4*A^2*a^3*c^5*d^3 + B^2*a*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3 *(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5*c^3*d *e^2 - 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d ^2 - a^7*c^5*e^2)))^(1/2))*((4*A^2*a^3*c^5*d^3 + B^2*a*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B ^2*a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(6 4*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2) + ((d + e*x)^(1/2)*(4*A^2*c^3*d^2*e^ 2 + A^2*a*c^2*e^4 + B^2*a^2*c*e^4 - 4*A*B*a*c^2*d*e^3))/a^2)*((4*A^2*a^3*c ^5*d^3 + B^2*a*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5 *c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2)*1i - (((64*B*a^4*c^2*e^4 - 64*A*a^3*c^3*d*e^3)/(8*a^3) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^5*d^3 + B^2*a*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a ^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5*c^3*d*e^ 2 - 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2))*((4*A^2*a^3*c^5*d^3 + B^2*a*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2* a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*( a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2) - ((d + e*x)^(1/2)*(4*A^2*c^3*d^2*e^...