\(\int \frac {1}{(d+e x)^{5/2} (a+c x^2)} \, dx\) [623]

Optimal result

Integrand size = 19, antiderivative size = 736 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {c^{3/4} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

[Out]

-2/3*e/(a*e^2+c*d^2)/(e*x+d)^(3/2)-4*c*d*e/(a*e^2+c*d^2)^2/(e*x+d)^(1/2)+1/2*c^(3/4)*e*arctanh((-c^(1/4)*2^(1/
2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))*(3*c*d^2-a*e^2-
2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/2*c^(3/4)
*e*arctanh((c^(1/4)*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2
))^(1/2))*(3*c*d^2-a*e^2-2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)
^(1/2))^(1/2)-1/4*c^(3/4)*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)-c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^(1/2)+(a
*e^2+c*d^2)^(1/2))^(1/2))*(3*c*d^2-a*e^2+2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/
2)+(a*e^2+c*d^2)^(1/2))^(1/2)+1/4*c^(3/4)*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)+c^(1/4)*2^(1/2)*(e*x+d)^(1/
2)*(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(3*c*d^2-a*e^2+2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/(a*e^2+c*d^2)^(5/2)*
2^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 736, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {724, 843, 841, 1183, 648, 632, 212, 642} \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=\frac {c^{3/4} e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}-a e^2+3 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} \left (a e^2+c d^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {c^{3/4} e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}-a e^2+3 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} \left (a e^2+c d^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {c^{3/4} e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}-a e^2+3 c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (a e^2+c d^2\right )^{5/2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {c^{3/4} e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}-a e^2+3 c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (a e^2+c d^2\right )^{5/2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {4 c d e}{\sqrt {d+e x} \left (a e^2+c d^2\right )^2}-\frac {2 e}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )} \]

[In]

Int[1/((d + e*x)^(5/2)*(a + c*x^2)),x]

[Out]

(-2*e)/(3*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) - (4*c*d*e)/((c*d^2 + a*e^2)^2*Sqrt[d + e*x]) + (c^(3/4)*e*(3*c*d^2
- a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*S
qrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d
^2 + a*e^2]]) - (c^(3/4)*e*(3*c*d^2 - a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[
c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*(c*d^2 + a*e
^2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(3/4)*e*(3*c*d^2 - a*e^2 + 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^
2])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d
+ e*x)])/(2*Sqrt[2]*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (c^(3/4)*e*(3*c*d^2 - a*e^
2 + 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a
*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^
2]])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

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

Rule 724

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

Rule 841

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[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]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 843

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

Rule 1183

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {c \int \frac {d-e x}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx}{c d^2+a e^2} \\ & = -\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {c \int \frac {c d^2-a e^2-2 c d e x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{\left (c d^2+a e^2\right )^2} \\ & = -\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {(2 c) \text {Subst}\left (\int \frac {2 c d^2 e+e \left (c d^2-a e^2\right )-2 c d e x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{\left (c d^2+a e^2\right )^2} \\ & = -\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {c^{3/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \left (2 c d^2 e+e \left (c d^2-a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\left (2 c d^2 e+e \left (c d^2-a e^2\right )+2 \sqrt {c} d e \sqrt {c d^2+a e^2}\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \left (2 c d^2 e+e \left (c d^2-a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\left (2 c d^2 e+e \left (c d^2-a e^2\right )+2 \sqrt {c} d e \sqrt {c d^2+a e^2}\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = -\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {\left (\sqrt {c} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \left (c d^2+a e^2\right )^{5/2}}+\frac {\left (\sqrt {c} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \left (c d^2+a e^2\right )^{5/2}}-\frac {\left (c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = -\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}-\frac {c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (\sqrt {c} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\left (c d^2+a e^2\right )^{5/2}}-\frac {\left (\sqrt {c} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\left (c d^2+a e^2\right )^{5/2}} \\ & = -\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {c^{3/4} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.29 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.36 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=-\frac {2 e \left (a e^2+c d (7 d+6 e x)\right )}{3 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {\sqrt {c} \sqrt {-c d-i \sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {a} \left (-i \sqrt {c} d+\sqrt {a} e\right )^3}+\frac {\sqrt {c} \sqrt {-c d+i \sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {a} \left (i \sqrt {c} d+\sqrt {a} e\right )^3} \]

[In]

Integrate[1/((d + e*x)^(5/2)*(a + c*x^2)),x]

[Out]

(-2*e*(a*e^2 + c*d*(7*d + 6*e*x)))/(3*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)) + (Sqrt[c]*Sqrt[-(c*d) - I*Sqrt[a]*Sq
rt[c]*e]*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/(Sqrt[a]*((-I)*
Sqrt[c]*d + Sqrt[a]*e)^3) + (Sqrt[c]*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c
]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/(Sqrt[a]*(I*Sqrt[c]*d + Sqrt[a]*e)^3)

Maple [A] (verified)

Time = 2.62 (sec) , antiderivative size = 793, normalized size of antiderivative = 1.08

method	result	size
pseudoelliptic	\(-\frac {-\frac {\left (e x +d \right )^{\frac {3}{2}} \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \left (\left (e^{2} a -2 d \sqrt {c}\, \sqrt {e^{2} a +c \,d^{2}}-3 c \,d^{2}\right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-d \,e^{2} a c +3 c^{2} d^{3}+2 c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d^{2}\right ) \ln \left (\left (e x +d \right ) \sqrt {c}-\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}+\frac {\left (e x +d \right )^{\frac {3}{2}} \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \left (\left (e^{2} a -2 d \sqrt {c}\, \sqrt {e^{2} a +c \,d^{2}}-3 c \,d^{2}\right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-d \,e^{2} a c +3 c^{2} d^{3}+2 c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d^{2}\right ) \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}+\left (\frac {2 \sqrt {e^{2} a +c \,d^{2}}\, \left (6 x c d e +e^{2} a +7 c \,d^{2}\right ) \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}{3}+\left (\arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )-\arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )\right ) \left (a c \,e^{2}-3 c^{2} d^{2}+2 c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d \right ) \left (e x +d \right )^{\frac {3}{2}}\right ) e^{2} a}{\left (e^{2} a +c \,d^{2}\right )^{\frac {5}{2}} \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \left (e x +d \right )^{\frac {3}{2}} a e}\)	\(793\)
derivativedivides	\(\text {Expression too large to display}\)	\(2547\)
default	\(\text {Expression too large to display}\)	\(2547\)

[In]

int(1/(e*x+d)^(5/2)/(c*x^2+a),x,method=_RETURNVERBOSE)

[Out]

-1/(a*e^2+c*d^2)^(5/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*(-1/4*(e*x+d)^(3/
2)*(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/
2)*((e^2*a-2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2)-3*c*d^2)*((a*e^2+c*d^2)*c)^(1/2)-d*e^2*a*c+3*c^2*d^3+2*c^(3/2)*(a*e
^2+c*d^2)^(1/2)*d^2)*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1
/2))+1/4*(e*x+d)^(3/2)*(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*(2*((a*e^2+c*d^2)
*c)^(1/2)+2*c*d)^(1/2)*((e^2*a-2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2)-3*c*d^2)*((a*e^2+c*d^2)*c)^(1/2)-d*e^2*a*c+3*c^
2*d^3+2*c^(3/2)*(a*e^2+c*d^2)^(1/2)*d^2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1
/2)+(a*e^2+c*d^2)^(1/2))+(2/3*(a*e^2+c*d^2)^(1/2)*(6*c*d*e*x+a*e^2+7*c*d^2)*(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(
(a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)+(arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2))/
(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2))-arctan((-2*c^(1/2)*(e*x+d)^(1/2)+(2*((a
*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)))*(a*
c*e^2-3*c^2*d^2+2*c^(3/2)*(a*e^2+c*d^2)^(1/2)*d)*(e*x+d)^(3/2))*e^2*a)/(e*x+d)^(3/2)/a/e

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5149 vs. \(2 (597) = 1194\).

Time = 0.40 (sec) , antiderivative size = 5149, normalized size of antiderivative = 7.00 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+a),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=\int \frac {1}{\left (a + c x^{2}\right ) \left (d + e x\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/(e*x+d)**(5/2)/(c*x**2+a),x)

[Out]

Integral(1/((a + c*x**2)*(d + e*x)**(5/2)), x)

Maxima [F]

\[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+a),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)*(e*x + d)^(5/2)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1215 vs. \(2 (597) = 1194\).

Time = 0.37 (sec) , antiderivative size = 1215, normalized size of antiderivative = 1.65 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=-\frac {{\left (2 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}^{2} a c d e {\left | c \right |} + {\left (3 \, \sqrt {-a c} c^{3} d^{6} e + 5 \, \sqrt {-a c} a c^{2} d^{4} e^{3} + \sqrt {-a c} a^{2} c d^{2} e^{5} - \sqrt {-a c} a^{3} e^{7}\right )} {\left | c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5} \right |} {\left | c \right |} - {\left (c^{6} d^{11} e + 3 \, a c^{5} d^{9} e^{3} + 2 \, a^{2} c^{4} d^{7} e^{5} - 2 \, a^{3} c^{3} d^{5} e^{7} - 3 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4} + \sqrt {{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}^{2} - {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}}}{c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}}}}\right )}{{\left (a c^{4} d^{8} e + 4 \, a^{2} c^{3} d^{6} e^{3} + 6 \, a^{3} c^{2} d^{4} e^{5} + 4 \, a^{4} c d^{2} e^{7} + a^{5} e^{9} + \sqrt {-a c} c^{4} d^{9} + 4 \, \sqrt {-a c} a c^{3} d^{7} e^{2} + 6 \, \sqrt {-a c} a^{2} c^{2} d^{5} e^{4} + 4 \, \sqrt {-a c} a^{3} c d^{3} e^{6} + \sqrt {-a c} a^{4} d e^{8}\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5} \right |}} - \frac {{\left (2 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}^{2} a c d e {\left | c \right |} - {\left (3 \, \sqrt {-a c} c^{3} d^{6} e + 5 \, \sqrt {-a c} a c^{2} d^{4} e^{3} + \sqrt {-a c} a^{2} c d^{2} e^{5} - \sqrt {-a c} a^{3} e^{7}\right )} {\left | c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5} \right |} {\left | c \right |} - {\left (c^{6} d^{11} e + 3 \, a c^{5} d^{9} e^{3} + 2 \, a^{2} c^{4} d^{7} e^{5} - 2 \, a^{3} c^{3} d^{5} e^{7} - 3 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4} - \sqrt {{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}^{2} - {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}}}{c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}}}}\right )}{{\left (a c^{4} d^{8} e + 4 \, a^{2} c^{3} d^{6} e^{3} + 6 \, a^{3} c^{2} d^{4} e^{5} + 4 \, a^{4} c d^{2} e^{7} + a^{5} e^{9} - \sqrt {-a c} c^{4} d^{9} - 4 \, \sqrt {-a c} a c^{3} d^{7} e^{2} - 6 \, \sqrt {-a c} a^{2} c^{2} d^{5} e^{4} - 4 \, \sqrt {-a c} a^{3} c d^{3} e^{6} - \sqrt {-a c} a^{4} d e^{8}\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5} \right |}} - \frac {2 \, {\left (6 \, {\left (e x + d\right )} c d e + c d^{2} e + a e^{3}\right )}}{3 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}}} \]

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+a),x, algorithm="giac")

[Out]

-(2*(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)^2*a*c*d*e*abs(c) + (3*sqrt(-a*c)*c^3*d^6*e + 5*sqrt(-a*c)*a*c^2*d^4*
e^3 + sqrt(-a*c)*a^2*c*d^2*e^5 - sqrt(-a*c)*a^3*e^7)*abs(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)*abs(c) - (c^6*d^
11*e + 3*a*c^5*d^9*e^3 + 2*a^2*c^4*d^7*e^5 - 2*a^3*c^3*d^5*e^7 - 3*a^4*c^2*d^3*e^9 - a^5*c*d*e^11)*abs(c))*arc
tan(sqrt(e*x + d)/sqrt(-(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4 + sqrt((c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e
^4)^2 - (c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*(c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/(c^3
*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/((a*c^4*d^8*e + 4*a^2*c^3*d^6*e^3 + 6*a^3*c^2*d^4*e^5 + 4*a^4*c*d^2*e^7
+ a^5*e^9 + sqrt(-a*c)*c^4*d^9 + 4*sqrt(-a*c)*a*c^3*d^7*e^2 + 6*sqrt(-a*c)*a^2*c^2*d^5*e^4 + 4*sqrt(-a*c)*a^3*
c*d^3*e^6 + sqrt(-a*c)*a^4*d*e^8)*sqrt(-c^2*d - sqrt(-a*c)*c*e)*abs(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)) - (2
*(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)^2*a*c*d*e*abs(c) - (3*sqrt(-a*c)*c^3*d^6*e + 5*sqrt(-a*c)*a*c^2*d^4*e^3
+ sqrt(-a*c)*a^2*c*d^2*e^5 - sqrt(-a*c)*a^3*e^7)*abs(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)*abs(c) - (c^6*d^11*
e + 3*a*c^5*d^9*e^3 + 2*a^2*c^4*d^7*e^5 - 2*a^3*c^3*d^5*e^7 - 3*a^4*c^2*d^3*e^9 - a^5*c*d*e^11)*abs(c))*arctan
(sqrt(e*x + d)/sqrt(-(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4 - sqrt((c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)
^2 - (c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*(c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/(c^3*d^
4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/((a*c^4*d^8*e + 4*a^2*c^3*d^6*e^3 + 6*a^3*c^2*d^4*e^5 + 4*a^4*c*d^2*e^7 + a
^5*e^9 - sqrt(-a*c)*c^4*d^9 - 4*sqrt(-a*c)*a*c^3*d^7*e^2 - 6*sqrt(-a*c)*a^2*c^2*d^5*e^4 - 4*sqrt(-a*c)*a^3*c*d
^3*e^6 - sqrt(-a*c)*a^4*d*e^8)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)) - 2/3*(
6*(e*x + d)*c*d*e + c*d^2*e + a*e^3)/((c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*(e*x + d)^(3/2))

Mupad [B] (verification not implemented)

Time = 11.62 (sec) , antiderivative size = 7908, normalized size of antiderivative = 10.74 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=\text {Too large to display} \]

[In]

int(1/((a + c*x^2)*(d + e*x)^(5/2)),x)

[Out]

- ((2*e)/(3*(a*e^2 + c*d^2)) + (4*c*d*e*(d + e*x))/(a*e^2 + c*d^2)^2)/(d + e*x)^(3/2) - atan((((d + e*x)^(1/2)
*(320*a^2*c^11*d^12*e^6 - 16*c^13*d^16*e^2 - 16*a^8*c^5*e^18 + 1024*a^3*c^10*d^10*e^8 + 1440*a^4*c^9*d^8*e^10
+ 1024*a^5*c^8*d^6*e^12 + 320*a^6*c^7*d^4*e^14) + (-(a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 -
10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^
10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(96*a*c^13*d^18*e^
3 - (d + e*x)^(1/2)*(-(a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4
*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c
^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^22 + 640*a
^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 + 7680*a^4*c^11*d^15*e^8 + 13440*a^5*c^10*d^13*e^10 + 16128*a^6*c^9*
d^11*e^12 + 13440*a^7*c^8*d^9*e^14 + 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5*e^18 + 640*a^10*c^5*d^3*e^20) -
32*a^10*c^4*e^21 + 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 + 4480*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10
*e^11 + 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 - 128*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*(-(a^2*e^
5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*
d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e
^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*1i + ((d + e*x)^(1/2)*(320*a^2*c^11*d^12*e^6 - 16*c^13*d^16*e^2 - 16*a^8*c^5*
e^18 + 1024*a^3*c^10*d^10*e^8 + 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 + 320*a^6*c^7*d^4*e^14) - (-(a^2
*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a
*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^
6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4
- 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5
*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d^21
*e^2 + 64*a^11*c^4*d*e^22 + 640*a^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 + 7680*a^4*c^11*d^15*e^8 + 13440*a^
5*c^10*d^13*e^10 + 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^14 + 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5
*e^18 + 640*a^10*c^5*d^3*e^20) - 32*a^10*c^4*e^21 + 96*a*c^13*d^18*e^3 + 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11
*d^14*e^7 + 4480*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10*e^11 + 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 - 1
28*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*(-(a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^
2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 +
5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*1i)/(((d + e*x)^(1/2)*(
320*a^2*c^11*d^12*e^6 - 16*c^13*d^16*e^2 - 16*a^8*c^5*e^18 + 1024*a^3*c^10*d^10*e^8 + 1440*a^4*c^9*d^8*e^10 +
1024*a^5*c^8*d^6*e^12 + 320*a^6*c^7*d^4*e^14) - (-(a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10
*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10
+ 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(
a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 1
0*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3
*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^22 + 640*a^2*c^13*d^19*e^4 + 2880
*a^3*c^12*d^17*e^6 + 7680*a^4*c^11*d^15*e^8 + 13440*a^5*c^10*d^13*e^10 + 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c
^8*d^9*e^14 + 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5*e^18 + 640*a^10*c^5*d^3*e^20) - 32*a^10*c^4*e^21 + 96*a
*c^13*d^18*e^3 + 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 + 4480*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10*e
^11 + 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 - 128*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*(-(a^2*e^5*
(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^
2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4
+ 10*a^5*c^2*d^4*e^6)))^(1/2) - ((d + e*x)^(1/2)*(320*a^2*c^11*d^12*e^6 - 16*c^13*d^16*e^2 - 16*a^8*c^5*e^18
+ 1024*a^3*c^10*d^10*e^8 + 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 + 320*a^6*c^7*d^4*e^14) + (-(a^2*e^5*
(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^
2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4
+ 10*a^5*c^2*d^4*e^6)))^(1/2)*(96*a*c^13*d^18*e^3 - (d + e*x)^(1/2)*(-(a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 +
5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^
7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)
*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^22 + 640*a^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 + 7680*a^4*c^11*d^1
5*e^8 + 13440*a^5*c^10*d^13*e^10 + 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^14 + 7680*a^8*c^7*d^7*e^16 +
2880*a^9*c^6*d^5*e^18 + 640*a^10*c^5*d^3*e^20) - 32*a^10*c^4*e^21 + 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^14
*e^7 + 4480*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10*e^11 + 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 - 128*a^
8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*(-(a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3
*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6
*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2) + 32*c^12*d^13*e^3 + 192*a*c
^11*d^11*e^5 + 32*a^6*c^6*d*e^15 + 480*a^2*c^10*d^9*e^7 + 640*a^3*c^9*d^7*e^9 + 480*a^4*c^8*d^5*e^11 + 192*a^5
*c^7*d^3*e^13))*(-(a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(
-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d
^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*2i - atan((((d + e*x)^(1/2)*(320*a^2*c^11*d^12*e^6 -
16*c^13*d^16*e^2 - 16*a^8*c^5*e^18 + 1024*a^3*c^10*d^10*e^8 + 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 +
320*a^6*c^7*d^4*e^14) + ((a^2*e^5*(-a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2
*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a
^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(96*a*c^13*d^18*e^3 - (d + e*x)^(1/2)*((a^2*
e^5*(-a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*
c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6
*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^22 + 640*a^2*c^13*d^19*e^4 + 2880*a^3
*c^12*d^17*e^6 + 7680*a^4*c^11*d^15*e^8 + 13440*a^5*c^10*d^13*e^10 + 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d
^9*e^14 + 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5*e^18 + 640*a^10*c^5*d^3*e^20) - 32*a^10*c^4*e^21 + 736*a^2*
c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 + 4480*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10*e^11 + 3136*a^6*c^8*d^8*e^
13 + 896*a^7*c^7*d^6*e^15 - 128*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*((a^2*e^5*(-a^3*c^3)^(1/2) - a*c^4*d
^5 - 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4
*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(
1/2)*1i + ((d + e*x)^(1/2)*(320*a^2*c^11*d^12*e^6 - 16*c^13*d^16*e^2 - 16*a^8*c^5*e^18 + 1024*a^3*c^10*d^10*e^
8 + 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 + 320*a^6*c^7*d^4*e^14) - ((a^2*e^5*(-a^3*c^3)^(1/2) - a*c^4
*d^5 - 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/
(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))
^(1/2)*((d + e*x)^(1/2)*((a^2*e^5*(-a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*
d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^
3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^22 + 64
0*a^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 + 7680*a^4*c^11*d^15*e^8 + 13440*a^5*c^10*d^13*e^10 + 16128*a^6*c
^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^14 + 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5*e^18 + 640*a^10*c^5*d^3*e^20)
- 32*a^10*c^4*e^21 + 96*a*c^13*d^18*e^3 + 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 + 4480*a^4*c^10*d^12
*e^9 + 4928*a^5*c^9*d^10*e^11 + 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 - 128*a^8*c^6*d^4*e^17 - 160*a^9*
c^5*d^2*e^19))*((a^2*e^5*(-a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a
^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8
*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*1i)/(((d + e*x)^(1/2)*(320*a^2*c^11*d^12*e^6 - 16*c^13
*d^16*e^2 - 16*a^8*c^5*e^18 + 1024*a^3*c^10*d^10*e^8 + 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 + 320*a^6
*c^7*d^4*e^14) - ((a^2*e^5*(-a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(
-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d
^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*((a^2*e^5*(-a^3*c^3)^(1/2) - a*c^4*
d^5 - 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(
4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^
(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^22 + 640*a^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 + 7680*a^4*c^1
1*d^15*e^8 + 13440*a^5*c^10*d^13*e^10 + 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^14 + 7680*a^8*c^7*d^7*e^
16 + 2880*a^9*c^6*d^5*e^18 + 640*a^10*c^5*d^3*e^20) - 32*a^10*c^4*e^21 + 96*a*c^13*d^18*e^3 + 736*a^2*c^12*d^1
6*e^5 + 2432*a^3*c^11*d^14*e^7 + 4480*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10*e^11 + 3136*a^6*c^8*d^8*e^13 + 896
*a^7*c^7*d^6*e^15 - 128*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*((a^2*e^5*(-a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a
^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^
10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2) - (
(d + e*x)^(1/2)*(320*a^2*c^11*d^12*e^6 - 16*c^13*d^16*e^2 - 16*a^8*c^5*e^18 + 1024*a^3*c^10*d^10*e^8 + 1440*a^
4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 + 320*a^6*c^7*d^4*e^14) + ((a^2*e^5*(-a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^
3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^1
0 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(96*
a*c^13*d^18*e^3 - (d + e*x)^(1/2)*((a^2*e^5*(-a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^
2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*
e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d
*e^22 + 640*a^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 + 7680*a^4*c^11*d^15*e^8 + 13440*a^5*c^10*d^13*e^10 + 1
6128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^14 + 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5*e^18 + 640*a^10*c^5
*d^3*e^20) - 32*a^10*c^4*e^21 + 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 + 4480*a^4*c^10*d^12*e^9 + 4928
*a^5*c^9*d^10*e^11 + 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 - 128*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^1
9))*((a^2*e^5*(-a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/
2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a
^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2) + 32*c^12*d^13*e^3 + 192*a*c^11*d^11*e^5 + 32*a^6*c^6*d*e^15 + 48
0*a^2*c^10*d^9*e^7 + 640*a^3*c^9*d^7*e^9 + 480*a^4*c^8*d^5*e^11 + 192*a^5*c^7*d^3*e^13))*((a^2*e^5*(-a^3*c^3)^
(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3
*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c
^2*d^4*e^6)))^(1/2)*2i