\(\int \frac {1}{(d+e x)^{3/2} (a+c x^2)^2} \, dx\) [636]
Optimal result
Integrand size = 19, antiderivative size = 845 \[
\int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}
\]
[Out]
1/2*e*(-5*a*e^2+c*d^2)/a/(a*e^2+c*d^2)^2/(e*x+d)^(1/2)+1/2*(c*d*x+a*e)/a/(a*e^2+c*d^2)/(c*x^2+a)/(e*x+d)^(1/2)
+1/8*c^(1/4)*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))*(c^(3/2)*d^3+13*a*d*e^2*c^(1/2)+(-5*a*e^2+c*d^2)*(a*e^2+c*d^2)^(1/2))/a/(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)-1/8*c^(1/4)*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))*(c^(3/2)*d^3+13*a*d*e^2*c^(1/2)+(-5*a*
e^2+c*d^2)*(a*e^2+c*d^2)^(1/2))/a/(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)-1/16*c^(1/
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
))*(c^(3/2)*d^3+13*a*d*e^2*c^(1/2)-(-5*a*e^2+c*d^2)*(a*e^2+c*d^2)^(1/2))/a/(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)+1/16*c^(1/4)*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))*(c^(3/2)*d^3+13*a*d*e^2*c^(1/2)-(-5*a*e^2+c*d^2)*(a*e^2+c*d^2)^(1/
2))/a/(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 = 2.26 (sec) , antiderivative size = 845, 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 = {755, 843, 841, 1183, 648, 632,
212, 642} \[
\int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (c x^2+a\right )}
\]
[In]
Int[1/((d + e*x)^(3/2)*(a + c*x^2)^2),x]
[Out]
(e*(c*d^2 - 5*a*e^2))/(2*a*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]) + (a*e + c*d*x)/(2*a*(c*d^2 + a*e^2)*Sqrt[d + e*x]
*(a + c*x^2)) + (c^(1/4)*e*(c^(3/2)*d^3 + 13*a*Sqrt[c]*d*e^2 + (c*d^2 - 5*a*e^2)*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]]]
)/(4*Sqrt[2]*a*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(1/4)*e*(c^(3/2)*d^3 + 13*a*S
qrt[c]*d*e^2 + (c*d^2 - 5*a*e^2)*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]]])/(4*Sqrt[2]*a*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c
]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(1/4)*e*(c^(3/2)*d^3 + 13*a*Sqrt[c]*d*e^2 - (c*d^2 - 5*a*e^2)*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)])/(8*Sqrt[2]*a*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (c^(1/4)*e*(c^(3/2)*d^
3 + 13*a*Sqrt[c]*d*e^2 - (c*d^2 - 5*a*e^2)*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)])/(8*Sqrt[2]*a*(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 755
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*
((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^
m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[
{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
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 {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (-2 c d^2-5 a e^2\right )-\frac {3}{2} c d e x}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )} \\ & = \frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\int \frac {-c d \left (c d^2+4 a e^2\right )-\frac {1}{2} c e \left (c d^2-5 a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )^2} \\ & = \frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} c d e \left (c d^2-5 a e^2\right )-c d e \left (c d^2+4 a e^2\right )-\frac {1}{2} c e \left (c d^2-5 a e^2\right ) x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{a \left (c d^2+a e^2\right )^2} \\ & = \frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\frac {1}{2} c d e \left (c d^2-5 a e^2\right )-c d e \left (c d^2+4 a e^2\right )\right )}{\sqrt [4]{c}}-\left (\frac {1}{2} c d e \left (c d^2-5 a e^2\right )+\frac {1}{2} \sqrt {c} e \left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-c d e \left (c d^2+4 a e^2\right )\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 )}{2 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\frac {1}{2} c d e \left (c d^2-5 a e^2\right )-c d e \left (c d^2+4 a e^2\right )\right )}{\sqrt [4]{c}}+\left (\frac {1}{2} c d e \left (c d^2-5 a e^2\right )+\frac {1}{2} \sqrt {c} e \left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-c d e \left (c d^2+4 a e^2\right )\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 )}{2 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = \frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\left (\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \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 )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \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 )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \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 )}{8 a \left (c d^2+a e^2\right )^{5/2}}+\frac {\left (e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \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 )}{8 a \left (c d^2+a e^2\right )^{5/2}} \\ & = \frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \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 )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \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 )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \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 )}{4 a \left (c d^2+a e^2\right )^{5/2}}-\frac {\left (e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \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 )}{4 a \left (c d^2+a e^2\right )^{5/2}} \\ & = \frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \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 )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \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 )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \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 )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \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 )}{8 \sqrt {2} a \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 = 1.83 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.40
\[
\int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {a} \left (-4 a^2 e^3+c^2 d^2 x (d+e x)+a c e \left (2 d^2+d e x-5 e^2 x^2\right )\right )}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x} \left (a+c x^2\right )}+\frac {i \left (2 c d+5 i \sqrt {a} \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\left (\sqrt {c} d+i \sqrt {a} e\right )^2 \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {i \left (2 c d-5 i \sqrt {a} \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\left (\sqrt {c} d-i \sqrt {a} e\right )^2 \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{4 a^{3/2}}
\]
[In]
Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)^2),x]
[Out]
((2*Sqrt[a]*(-4*a^2*e^3 + c^2*d^2*x*(d + e*x) + a*c*e*(2*d^2 + d*e*x - 5*e^2*x^2)))/((c*d^2 + a*e^2)^2*Sqrt[d
+ e*x]*(a + c*x^2)) + (I*(2*c*d + (5*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[c]*d + I*Sqrt[a]*e)^2*Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]) - (I*(2*c*
d - (5*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[c]*d - I*Sqrt[a]*e)^2*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]))/(4*a^(3/2))
Maple [A] (verified)
Time = 3.74 (sec) , antiderivative size = 1059, normalized size of antiderivative =
1.25
| | |
method | result | size |
| | |
pseudoelliptic |
\(\text {Expression too large to display}\) |
\(1059\) |
derivativedivides |
\(\text {Expression too large to display}\) |
\(3080\) |
default |
\(\text {Expression too large to display}\) |
\(3080\) |
| | |
|
|
|
[In]
int(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x,method=_RETURNVERBOSE)
[Out]
-5/4/(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)/(e*x+d)^(1/2)/c^(1/2)*(1/4*(e*x+d)^
(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)*(((e^2*a-1/5*c*d^2)*(c*x^2+a)*(a*e^2+c*d^2)^(1/2)+13/5*(1/13*a*(
13*e^2*x^2+d^2)*c^(3/2)+c^(1/2)*a^2*e^2+1/13*x^2*c^(5/2)*d^2)*d)*((a*e^2+c*d^2)*c)^(1/2)-(c*(e^2*a-1/5*c*d^2)*
(c*x^2+a)*(a*e^2+c*d^2)^(1/2)+13/5*d*(1/13*a*(13*e^2*x^2+d^2)*c^(5/2)+a^2*e^2*c^(3/2)+1/13*c^(7/2)*d^2*x^2))*d
)*(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)*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)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1
/2)*(((e^2*a-1/5*c*d^2)*(c*x^2+a)*(a*e^2+c*d^2)^(1/2)+13/5*(1/13*a*(13*e^2*x^2+d^2)*c^(3/2)+c^(1/2)*a^2*e^2+1/
13*x^2*c^(5/2)*d^2)*d)*((a*e^2+c*d^2)*c)^(1/2)-(c*(e^2*a-1/5*c*d^2)*(c*x^2+a)*(a*e^2+c*d^2)^(1/2)+13/5*d*(1/13
*a*(13*e^2*x^2+d^2)*c^(5/2)+a^2*e^2*c^(3/2)+1/13*c^(7/2)*d^2*x^2))*d)*(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)*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))+(8/5*(-1/2*e*(-5/2*x^2*e^2+1/2*d*e*x+d^2)*a*c^(3/2)-1/4*d^2*x*(e*x+d)*c^(5/2)+a^2*e^3*c^(1/2))*(
a*e^2+c*d^2)^(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)+(e*x+d)^(1/2)*(c*(e^2
*a-1/5*c*d^2)*(c*x^2+a)*(a*e^2+c*d^2)^(1/2)-13/5*d*(1/13*a*(13*e^2*x^2+d^2)*c^(5/2)+a^2*e^2*c^(3/2)+1/13*c^(7/
2)*d^2*x^2))*e*(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))))*e*a)/(a*e^2+c*d^2)^(5/2
)/a^2/e/(c*x^2+a)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 5698 vs. \(2 (691) = 1382\).
Time = 1.15 (sec) , antiderivative size = 5698, normalized size of antiderivative = 6.74
\[
\int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display}
\]
[In]
integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x, algorithm="fricas")
[Out]
Too large to include
Sympy [F]
\[
\int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\int \frac {1}{\left (a + c x^{2}\right )^{2} \left (d + e x\right )^{\frac {3}{2}}}\, dx
\]
[In]
integrate(1/(e*x+d)**(3/2)/(c*x**2+a)**2,x)
[Out]
Integral(1/((a + c*x**2)**2*(d + e*x)**(3/2)), x)
Maxima [F]
\[
\int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{2} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x }
\]
[In]
integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x, algorithm="maxima")
[Out]
integrate(1/((c*x^2 + a)^2*(e*x + d)^(3/2)), x)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1397 vs. \(2 (691) = 1382\).
Time = 0.43 (sec) , antiderivative size = 1397, normalized size of antiderivative = 1.65
\[
\int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display}
\]
[In]
integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x, algorithm="giac")
[Out]
1/4*((a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)^2*(c*d^2*e - 5*a*e^3)*abs(c) - (sqrt(-a*c)*c^3*d^7*e + 15*sqrt(
-a*c)*a*c^2*d^5*e^3 + 27*sqrt(-a*c)*a^2*c*d^3*e^5 + 13*sqrt(-a*c)*a^3*d*e^7)*abs(a*c^2*d^4*e + 2*a^2*c*d^2*e^3
+ a^3*e^5)*abs(c) + 2*(a*c^6*d^12*e + 8*a^2*c^5*d^10*e^3 + 22*a^3*c^4*d^8*e^5 + 28*a^4*c^3*d^6*e^7 + 17*a^5*c
^2*d^4*e^9 + 4*a^6*c*d^2*e^11)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4
+ sqrt((a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)^2 - (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a
^4*e^6)*(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/((a^2*c^4*
d^8*e + 4*a^3*c^3*d^6*e^3 + 6*a^4*c^2*d^4*e^5 + 4*a^5*c*d^2*e^7 + a^6*e^9 + sqrt(-a*c)*a*c^4*d^9 + 4*sqrt(-a*c
)*a^2*c^3*d^7*e^2 + 6*sqrt(-a*c)*a^3*c^2*d^5*e^4 + 4*sqrt(-a*c)*a^4*c*d^3*e^6 + sqrt(-a*c)*a^5*d*e^8)*sqrt(-c^
2*d - sqrt(-a*c)*c*e)*abs(a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)) + 1/4*((a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^
3*e^5)^2*(c*d^2*e - 5*a*e^3)*abs(c) + (sqrt(-a*c)*c^3*d^7*e + 15*sqrt(-a*c)*a*c^2*d^5*e^3 + 27*sqrt(-a*c)*a^2*
c*d^3*e^5 + 13*sqrt(-a*c)*a^3*d*e^7)*abs(a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)*abs(c) + 2*(a*c^6*d^12*e + 8
*a^2*c^5*d^10*e^3 + 22*a^3*c^4*d^8*e^5 + 28*a^4*c^3*d^6*e^7 + 17*a^5*c^2*d^4*e^9 + 4*a^6*c*d^2*e^11)*abs(c))*a
rctan(sqrt(e*x + d)/sqrt(-(a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4 - sqrt((a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 +
a^3*c*d*e^4)^2 - (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 +
a^3*c*e^4)))/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/((a^2*c^4*d^8*e + 4*a^3*c^3*d^6*e^3 + 6*a^4*c^2*d^
4*e^5 + 4*a^5*c*d^2*e^7 + a^6*e^9 - sqrt(-a*c)*a*c^4*d^9 - 4*sqrt(-a*c)*a^2*c^3*d^7*e^2 - 6*sqrt(-a*c)*a^3*c^2
*d^5*e^4 - 4*sqrt(-a*c)*a^4*c*d^3*e^6 - sqrt(-a*c)*a^5*d*e^8)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(a*c^2*d^4*e +
2*a^2*c*d^2*e^3 + a^3*e^5)) + 1/2*((e*x + d)^2*c^2*d^2*e - (e*x + d)*c^2*d^3*e - 5*(e*x + d)^2*a*c*e^3 + 11*(e
*x + d)*a*c*d*e^3 - 4*a*c*d^2*e^3 - 4*a^2*e^5)/((a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4)*((e*x + d)^(5/2)*c - 2
*(e*x + d)^(3/2)*c*d + sqrt(e*x + d)*c*d^2 + sqrt(e*x + d)*a*e^2))
Mupad [B] (verification not implemented)
Time = 12.38 (sec) , antiderivative size = 8777, normalized size of antiderivative = 10.39
\[
\int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display}
\]
[In]
int(1/((a + c*x^2)^2*(d + e*x)^(3/2)),x)
[Out]
atan((((-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3
*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d
^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^
7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*
a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8
*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480
*a^7*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*
a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^1
5*c^5*d^3*e^20) + 3328*a^14*c^4*d*e^21 + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*
e^7 + 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^1
5 + 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19) + (d + e*x)^(1/2)*(128*a^3*c^13*d^18*e^2 - 800*a^12*c^4
*e^20 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 +
51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*
c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2)
- 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c
^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*1i - ((-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/
2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e
^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 +
10*a^9*c^2*d^4*e^6)))^(1/2)*(3328*a^14*c^4*d*e^21 - (d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1
/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*
e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4
+ 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c^13*d^19*e^4 + 92160
*a^8*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12 + 43008
0*a^12*c^8*d^9*e^14 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^3*e^20) + 256*a^5*
c^13*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d
^11*e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*
e^19) - (d + e*x)^(1/2)*(128*a^3*c^13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d
^14*e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14
+ 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*
d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2
))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*
e^6)))^(1/2)*1i)/(1000*a^10*c^4*e^19 - ((-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70
*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11
*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)
*(3328*a^14*c^4*d*e^21 - (d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 7
0*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^1
1*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2
)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 + 245760*
a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 + 245760
*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^3*e^20) + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^1
2*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d
^9*e^13 + 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19) - (d + e*x)^(1/2)*(128
*a^3*c^13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 + 30848*a^6*c^10*d^1
2*e^8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2
432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4
+ 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*
d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2) - ((-(4*a^3*c^4
*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 1
05*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*
d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^
9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a
*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d
^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c^13*d^19*e^4
+ 92160*a^8*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12
+ 430080*a^12*c^8*d^9*e^14 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^3*e^20) + 3
328*a^14*c^4*d*e^21 + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^
10*d^13*e^9 + 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*
d^5*e^17 + 26880*a^13*c^5*d^3*e^19) + (d + e*x)^(1/2)*(128*a^3*c^13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^
12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c^8*d^8*
e^12 + 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 - 25*a^2*e^
7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 +
154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8
*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2) - 32*a^2*c^12*d^16*e^3 - 232*a^3*c^11*d^14*e^5 + 280*a^4*c^10*d^12*
e^7 + 4760*a^5*c^9*d^10*e^9 + 13720*a^6*c^8*d^8*e^11 + 19208*a^7*c^7*d^6*e^13 + 14728*a^8*c^6*d^4*e^15 + 5960*
a^9*c^5*d^2*e^17))*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35
*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10
+ 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*2i - ((2*e^3)/(a*e^2
+ c*d^2) + (c*e*(5*a*e^2 - c*d^2)*(d + e*x)^2)/(2*a*(a*e^2 + c*d^2)^2) - (c*d*e*(11*a*e^2 - c*d^2)*(d + e*x))
/(2*a*(a*e^2 + c*d^2)^2))/(c*(d + e*x)^(5/2) + (a*e^2 + c*d^2)*(d + e*x)^(1/2) - 2*c*d*(d + e*x)^(3/2)) + atan
((((-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a
^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e
^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 +
25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*
c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2
+ 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7
*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11
*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^
5*d^3*e^20) + 3328*a^14*c^4*d*e^21 + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7
+ 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 +
95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19) + (d + e*x)^(1/2)*(128*a^3*c^13*d^18*e^2 - 800*a^12*c^4*e^2
0 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 + 510
08*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*
d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 10
5*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d
^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*1i - ((-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) +
35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(
-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*
a^9*c^2*d^4*e^6)))^(1/2)*(3328*a^14*c^4*d*e^21 - (d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2)
+ 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*
(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10
*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c^13*d^19*e^4 + 92160*a^8
*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12 + 430080*a^
12*c^8*d^9*e^14 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^3*e^20) + 256*a^5*c^13
*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*
e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19
) - (d + e*x)^(1/2)*(128*a^3*c^13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*
e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 + 32
00*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*
e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(
64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)
))^(1/2)*1i)/(1000*a^10*c^4*e^19 - ((-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5
*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^1
0 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(33
28*a^14*c^4*d*e^21 - (d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^
5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^
10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2
048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 + 245760*a^9*
c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 + 245760*a^1
3*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^3*e^20) + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^12*d^
17*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d^9*e
^13 + 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19) - (d + e*x)^(1/2)*(128*a^3
*c^13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 + 30848*a^6*c^10*d^12*e^
8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2432*
a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 3
5*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10
+ 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2) - ((-(4*a^3*c^4*d^7
+ 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a
^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*
e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)
^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d
^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e
^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c^13*d^19*e^4 + 92
160*a^8*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12 + 43
0080*a^12*c^8*d^9*e^14 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^3*e^20) + 3328*
a^14*c^4*d*e^21 + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d
^13*e^9 + 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*d^5*
e^17 + 26880*a^13*c^5*d^3*e^19) + (d + e*x)^(1/2)*(128*a^3*c^13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^12*d
^16*e^4 + 10240*a^5*c^11*d^14*e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c^8*d^8*e^12
+ 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-
a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154
*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3
*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2) - 32*a^2*c^12*d^16*e^3 - 232*a^3*c^11*d^14*e^5 + 280*a^4*c^10*d^12*e^7
+ 4760*a^5*c^9*d^10*e^9 + 13720*a^6*c^8*d^8*e^11 + 19208*a^7*c^7*d^6*e^13 + 14728*a^8*c^6*d^4*e^15 + 5960*a^9*
c^5*d^2*e^17))*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2
*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*
a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*2i