3.7.0 ∫ 1 ____________ (d+ex)3∕2(a+cx2)2 dx [636]

Integrand size = 19, antiderivative size = 845 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

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} \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (c x^2+a\right )} \]

(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[

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])

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]

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]

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]

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]

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]

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] &&

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

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)

Time = 1.83 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.40 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {a} \left (-4 a^2 e^3+c^2 d^2 x (d+e x)+a c e \left (2 d^2+d e x-5 e^2 x^2\right )\right )}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x} \left (a+c x^2\right )}+\frac {i \left (2 c d+5 i \sqrt {a} \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\left (\sqrt {c} d+i \sqrt {a} e\right )^2 \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {i \left (2 c d-5 i \sqrt {a} \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\left (\sqrt {c} d-i \sqrt {a} e\right )^2 \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{4 a^{3/2}} \]

Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)^2),x]

((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	\(\text {Expression too large to display}\)	\(3080\)

int(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x,method=_RETURNVERBOSE)

-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

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 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x, algorithm="fricas")

Sympy [F]

\[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\int \frac {1}{\left (a + c x^{2}\right )^{2} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]

integrate(1/(e*x+d)**(3/2)/(c*x**2+a)**2,x)

Integral(1/((a + c*x**2)**2*(d + e*x)**(3/2)), x)

Maxima [F]

\[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{2} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x, algorithm="maxima")

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 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x, algorithm="giac")

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 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]

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

\(\int \frac {1}{(d+e x)^{3/2} (a+c x^2)^2} \, dx\) [636]

Optimal result