\(\int \frac {1}{\sqrt {d+e x} (a+c x^2)^2} \, dx\) [635]

Optimal result

Integrand size = 19, antiderivative size = 739 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^2} \, dx=\frac {(a e+c d x) \sqrt {d+e x}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e \left (c d^2+3 a e^2+\sqrt {c} d \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c d^2+3 a e^2+\sqrt {c} d \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c d^2+3 a e^2-\sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+3 a e^2-\sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

[Out]

1/2*(c*d*x+a*e)*(e*x+d)^(1/2)/a/(a*e^2+c*d^2)/(c*x^2+a)+1/8*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*d^2+3*a*e^2+d*c^(1/2)*(a*e^2+c*d^2)^(
1/2))/a/c^(1/4)/(a*e^2+c*d^2)^(3/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/8*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*d^2+3*a*e^2+
d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/c^(1/4)/(a*e^2+c*d^2)^(3/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/1
6*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*d^2+3*a*e^2-d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/c^(1/4)/(a*e^2+c*d^2)^(3/2)*2^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)
^(1/2))^(1/2)+1/16*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*d^2+3*a*e^2-d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/c^(1/4)/(a*e^2+c*d^2)^(3/2)*2^(1/2)/(d*c^(1
/2)+(a*e^2+c*d^2)^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 739, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {755, 841, 1183, 648, 632, 212, 642} \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^2} \, dx=\frac {e \left (\sqrt {c} d \sqrt {a e^2+c d^2}+3 a e^2+c d^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{4 \sqrt {2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {e \left (\sqrt {c} d \sqrt {a e^2+c d^2}+3 a e^2+c d^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{4 \sqrt {2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {\sqrt {d+e x} (a e+c d x)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac {e \left (-\sqrt {c} d \sqrt {a e^2+c d^2}+3 a e^2+c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {e \left (-\sqrt {c} d \sqrt {a e^2+c d^2}+3 a e^2+c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}} \]

[In]

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

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(2*a*(c*d^2 + a*e^2)*(a + c*x^2)) + (e*(c*d^2 + 3*a*e^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]]])/(4*Sqrt[2]*a*c^(1/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c*d
^2 + 3*a*e^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]]])/(4*Sqrt[2]*a*c^(1/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c
]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c*d^2 + 3*a*e^2 - Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] - S
qrt[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^(1/4)*
(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(c*d^2 + 3*a*e^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)])/(8*Sqrt[2]*a*c^(1/4)*(c*d^2 + a*e^2)^(3/2)*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 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) \sqrt {d+e x}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (-2 c d^2-3 a e^2\right )-\frac {1}{2} c d e x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )} \\ & = \frac {(a e+c d x) \sqrt {d+e x}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} c d^2 e+\frac {1}{2} e \left (-2 c d^2-3 a e^2\right )-\frac {1}{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 )}{a \left (c d^2+a e^2\right )} \\ & = \frac {(a e+c d x) \sqrt {d+e x}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \left (\frac {1}{2} c d^2 e+\frac {1}{2} e \left (-2 c d^2-3 a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\left (\frac {1}{2} c d^2 e+\frac {1}{2} e \left (-2 c d^2-3 a e^2\right )+\frac {1}{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 )}{2 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \left (\frac {1}{2} c d^2 e+\frac {1}{2} e \left (-2 c d^2-3 a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\left (\frac {1}{2} c d^2 e+\frac {1}{2} e \left (-2 c d^2-3 a e^2\right )+\frac {1}{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 )}{2 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = \frac {(a e+c d x) \sqrt {d+e x}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\left (e \left (c d^2+3 a e^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 )}{8 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c d^2+3 a e^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 )}{8 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c d^2+3 a e^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 )}{8 a \sqrt {c} \left (c d^2+a e^2\right )^{3/2}}+\frac {\left (e \left (c d^2+3 a e^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 )}{8 a \sqrt {c} \left (c d^2+a e^2\right )^{3/2}} \\ & = \frac {(a e+c d x) \sqrt {d+e x}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {e \left (c d^2+3 a e^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 )}{8 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+3 a e^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 )}{8 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (c d^2+3 a e^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 )}{4 a \sqrt {c} \left (c d^2+a e^2\right )^{3/2}}-\frac {\left (e \left (c d^2+3 a e^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 )}{4 a \sqrt {c} \left (c d^2+a e^2\right )^{3/2}} \\ & = \frac {(a e+c d x) \sqrt {d+e x}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e \left (c d^2+3 a e^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 )}{4 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c d^2+3 a e^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 )}{4 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c d^2+3 a e^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 )}{8 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+3 a e^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 )}{8 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.15 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.41 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {a} (a e+c d x) \sqrt {d+e x}}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {i \left (2 \sqrt {c} d+3 i \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\left (\sqrt {c} d+i \sqrt {a} e\right ) \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {i \left (2 \sqrt {c} d-3 i \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\left (\sqrt {c} d-i \sqrt {a} e\right ) \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{4 a^{3/2}} \]

[In]

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

[Out]

((2*Sqrt[a]*(a*e + c*d*x)*Sqrt[d + e*x])/((c*d^2 + a*e^2)*(a + c*x^2)) + (I*(2*Sqrt[c]*d + (3*I)*Sqrt[a]*e)*Ar
cTan[(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)
*Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]) - (I*(2*Sqrt[c]*d - (3*I)*Sqrt[a]*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)*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e
]))/(4*a^(3/2))

Maple [F(-1)]

Timed out.

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3267 vs. \(2 (597) = 1194\).

Time = 0.90 (sec) , antiderivative size = 3267, normalized size of antiderivative = 4.42 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/8*((a^2*c*d^2 + a^3*e^2 + (a*c^2*d^2 + a^2*c*e^2)*x^2)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + (a
^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^
10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*
c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))*log((20*c^2*d^4*e^
3 + 81*a*c*d^2*e^5 + 81*a^2*e^7)*sqrt(e*x + d) + (5*a^2*c^2*d^4*e^4 + 24*a^3*c*d^2*e^6 + 27*a^4*e^8 + 2*(a^3*c
^5*d^9 + 5*a^4*c^4*d^7*e^2 + 9*a^5*c^3*d^5*e^4 + 7*a^6*c^2*d^3*e^6 + 2*a^7*c*d*e^8)*sqrt(-(25*c^2*d^4*e^6 + 90
*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*
a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + (a^3*c
^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/
(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*
d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))) - (a^2*c*d^2 + a^3*e^
2 + (a*c^2*d^2 + a^2*c*e^2)*x^2)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + (a^3*c^3*d^6 + 3*a^4*c^2*d
^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^
4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^1
2)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))*log((20*c^2*d^4*e^3 + 81*a*c*d^2*e^5 + 81*
a^2*e^7)*sqrt(e*x + d) - (5*a^2*c^2*d^4*e^4 + 24*a^3*c*d^2*e^6 + 27*a^4*e^8 + 2*(a^3*c^5*d^9 + 5*a^4*c^4*d^7*e
^2 + 9*a^5*c^3*d^5*e^4 + 7*a^6*c^2*d^3*e^6 + 2*a^7*c*d*e^8)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^
10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*
c^2*d^2*e^10 + a^9*c*e^12)))*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + (a^3*c^3*d^6 + 3*a^4*c^2*d^4*e
^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^
6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))
/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))) + (a^2*c*d^2 + a^3*e^2 + (a*c^2*d^2 + a^2*c*e
^2)*x^2)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - (a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4
+ a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*
c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^
4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))*log((20*c^2*d^4*e^3 + 81*a*c*d^2*e^5 + 81*a^2*e^7)*sqrt(e*x + d) +
(5*a^2*c^2*d^4*e^4 + 24*a^3*c*d^2*e^6 + 27*a^4*e^8 - 2*(a^3*c^5*d^9 + 5*a^4*c^4*d^7*e^2 + 9*a^5*c^3*d^5*e^4 +
7*a^6*c^2*d^3*e^6 + 2*a^7*c*d*e^8)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^
4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^1
2)))*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - (a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a
^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*
d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^
2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))) - (a^2*c*d^2 + a^3*e^2 + (a*c^2*d^2 + a^2*c*e^2)*x^2)*sqrt(-(4*c^2*d^
5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - (a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^
2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4
*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*
d^2*e^4 + a^6*e^6))*log((20*c^2*d^4*e^3 + 81*a*c*d^2*e^5 + 81*a^2*e^7)*sqrt(e*x + d) - (5*a^2*c^2*d^4*e^4 + 24
*a^3*c*d^2*e^6 + 27*a^4*e^8 - 2*(a^3*c^5*d^9 + 5*a^4*c^4*d^7*e^2 + 9*a^5*c^3*d^5*e^4 + 7*a^6*c^2*d^3*e^6 + 2*a
^7*c*d*e^8)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*
c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))*sqrt(-(4*c^2*d^5 +
15*a*c*d^3*e^2 + 15*a^2*d*e^4 - (a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^
4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6
*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*
e^4 + a^6*e^6))) + 4*(c*d*x + a*e)*sqrt(e*x + d))/(a^2*c*d^2 + a^3*e^2 + (a*c^2*d^2 + a^2*c*e^2)*x^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 875, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^2} \, dx=-\frac {{\left ({\left (a c d^{2} e + a^{2} e^{3}\right )}^{2} \sqrt {-a c} d e {\left | c \right |} + {\left (a c^{2} d^{4} e + 4 \, a^{2} c d^{2} e^{3} + 3 \, a^{3} e^{5}\right )} {\left | -a c d^{2} e - a^{2} e^{3} \right |} {\left | c \right |} + {\left (2 \, \sqrt {-a c} a c^{3} d^{7} e + 7 \, \sqrt {-a c} a^{2} c^{2} d^{5} e^{3} + 8 \, \sqrt {-a c} a^{3} c d^{3} e^{5} + 3 \, \sqrt {-a c} a^{4} d e^{7}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d^{3} + a^{2} c d e^{2} + \sqrt {{\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )}^{2} - {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )}}}{a c^{2} d^{2} + a^{2} c e^{2}}}}\right )}{4 \, {\left (a^{2} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4} + \sqrt {-a c} a^{2} c^{2} d^{4} e + 2 \, \sqrt {-a c} a^{3} c d^{2} e^{3} + \sqrt {-a c} a^{4} e^{5}\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | -a c d^{2} e - a^{2} e^{3} \right |}} - \frac {{\left ({\left (a c d^{2} e + a^{2} e^{3}\right )}^{2} c d e {\left | c \right |} + {\left (\sqrt {-a c} c^{2} d^{4} e + 4 \, \sqrt {-a c} a c d^{2} e^{3} + 3 \, \sqrt {-a c} a^{2} e^{5}\right )} {\left | -a c d^{2} e - a^{2} e^{3} \right |} {\left | c \right |} + {\left (2 \, a c^{4} d^{7} e + 7 \, a^{2} c^{3} d^{5} e^{3} + 8 \, a^{3} c^{2} d^{3} e^{5} + 3 \, a^{4} c d e^{7}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d^{3} + a^{2} c d e^{2} - \sqrt {{\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )}^{2} - {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )}}}{a c^{2} d^{2} + a^{2} c e^{2}}}}\right )}{4 \, {\left (a^{2} c^{3} d^{4} e + 2 \, a^{3} c^{2} d^{2} e^{3} + a^{4} c e^{5} + \sqrt {-a c} a c^{3} d^{5} + 2 \, \sqrt {-a c} a^{2} c^{2} d^{3} e^{2} + \sqrt {-a c} a^{3} c d e^{4}\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | -a c d^{2} e - a^{2} e^{3} \right |}} + \frac {{\left (e x + d\right )}^{\frac {3}{2}} c d e - \sqrt {e x + d} c d^{2} e + \sqrt {e x + d} a e^{3}}{2 \, {\left (a c d^{2} + a^{2} e^{2}\right )} {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + a e^{2}\right )}} \]

[In]

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

[Out]

-1/4*((a*c*d^2*e + a^2*e^3)^2*sqrt(-a*c)*d*e*abs(c) + (a*c^2*d^4*e + 4*a^2*c*d^2*e^3 + 3*a^3*e^5)*abs(-a*c*d^2
*e - a^2*e^3)*abs(c) + (2*sqrt(-a*c)*a*c^3*d^7*e + 7*sqrt(-a*c)*a^2*c^2*d^5*e^3 + 8*sqrt(-a*c)*a^3*c*d^3*e^5 +
3*sqrt(-a*c)*a^4*d*e^7)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a*c^2*d^3 + a^2*c*d*e^2 + sqrt((a*c^2*d^3 + a^2*c
*d*e^2)^2 - (a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4)*(a*c^2*d^2 + a^2*c*e^2)))/(a*c^2*d^2 + a^2*c*e^2)))/((a^2*
c^3*d^5 + 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4 + sqrt(-a*c)*a^2*c^2*d^4*e + 2*sqrt(-a*c)*a^3*c*d^2*e^3 + sqrt(-a*c)
*a^4*e^5)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(-a*c*d^2*e - a^2*e^3)) - 1/4*((a*c*d^2*e + a^2*e^3)^2*c*d*e*abs(c)
+ (sqrt(-a*c)*c^2*d^4*e + 4*sqrt(-a*c)*a*c*d^2*e^3 + 3*sqrt(-a*c)*a^2*e^5)*abs(-a*c*d^2*e - a^2*e^3)*abs(c) +
(2*a*c^4*d^7*e + 7*a^2*c^3*d^5*e^3 + 8*a^3*c^2*d^3*e^5 + 3*a^4*c*d*e^7)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a
*c^2*d^3 + a^2*c*d*e^2 - sqrt((a*c^2*d^3 + a^2*c*d*e^2)^2 - (a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4)*(a*c^2*d^2
+ a^2*c*e^2)))/(a*c^2*d^2 + a^2*c*e^2)))/((a^2*c^3*d^4*e + 2*a^3*c^2*d^2*e^3 + a^4*c*e^5 + sqrt(-a*c)*a*c^3*d
^5 + 2*sqrt(-a*c)*a^2*c^2*d^3*e^2 + sqrt(-a*c)*a^3*c*d*e^4)*sqrt(-c^2*d - sqrt(-a*c)*c*e)*abs(-a*c*d^2*e - a^2
*e^3)) + 1/2*((e*x + d)^(3/2)*c*d*e - sqrt(e*x + d)*c*d^2*e + sqrt(e*x + d)*a*e^3)/((a*c*d^2 + a^2*e^2)*((e*x
+ d)^2*c - 2*(e*x + d)*c*d + c*d^2 + a*e^2))

Mupad [B] (verification not implemented)

Time = 11.71 (sec) , antiderivative size = 5300, normalized size of antiderivative = 7.17 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(((a*e^3 - c*d^2*e)*(d + e*x)^(1/2))/(2*a*(a*e^2 + c*d^2)) + (c*d*e*(d + e*x)^(3/2))/(2*a*(a*e^2 + c*d^2)))/(c
*(d + e*x)^2 + a*e^2 + c*d^2 - 2*c*d*(d + e*x)) - atan(((((192*a^5*c^3*e^7 + 64*a^3*c^5*d^4*e^3 + 256*a^4*c^4*
d^2*e^5)/(8*(a^5*e^4 + a^3*c^2*d^4 + 2*a^4*c*d^2*e^2)) + ((d + e*x)^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5*e
^2 + 128*a^4*c^5*d^3*e^4)*(-(9*a*e^5*(-a^9*c)^(1/2) + 4*a^3*c^3*d^5 + 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*
d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2))/(a^4*
e^4 + a^2*c^2*d^4 + 2*a^3*c*d^2*e^2))*(-(9*a*e^5*(-a^9*c)^(1/2) + 4*a^3*c^3*d^5 + 5*c*d^2*e^3*(-a^9*c)^(1/2) +
15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^
(1/2) + ((d + e*x)^(1/2)*(9*a^2*c^3*e^6 + 4*c^5*d^4*e^2 + 11*a*c^4*d^2*e^4))/(a^4*e^4 + a^2*c^2*d^4 + 2*a^3*c*
d^2*e^2))*(-(9*a*e^5*(-a^9*c)^(1/2) + 4*a^3*c^3*d^5 + 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5
*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2)*1i - (((192*a^5*c^3*e^
7 + 64*a^3*c^5*d^4*e^3 + 256*a^4*c^4*d^2*e^5)/(8*(a^5*e^4 + a^3*c^2*d^4 + 2*a^4*c*d^2*e^2)) - ((d + e*x)^(1/2)
*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5*e^2 + 128*a^4*c^5*d^3*e^4)*(-(9*a*e^5*(-a^9*c)^(1/2) + 4*a^3*c^3*d^5 + 5*c
*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^
2 + 3*a^8*c^2*d^2*e^4)))^(1/2))/(a^4*e^4 + a^2*c^2*d^4 + 2*a^3*c*d^2*e^2))*(-(9*a*e^5*(-a^9*c)^(1/2) + 4*a^3*c
^3*d^5 + 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^
7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c^3*e^6 + 4*c^5*d^4*e^2 + 11*a*c^4*d^2*e^
4))/(a^4*e^4 + a^2*c^2*d^4 + 2*a^3*c*d^2*e^2))*(-(9*a*e^5*(-a^9*c)^(1/2) + 4*a^3*c^3*d^5 + 5*c*d^2*e^3*(-a^9*c
)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^
2*e^4)))^(1/2)*1i)/((4*c^4*d^3*e^3 + 9*a*c^3*d*e^5)/(4*(a^5*e^4 + a^3*c^2*d^4 + 2*a^4*c*d^2*e^2)) + (((192*a^5
*c^3*e^7 + 64*a^3*c^5*d^4*e^3 + 256*a^4*c^4*d^2*e^5)/(8*(a^5*e^4 + a^3*c^2*d^4 + 2*a^4*c*d^2*e^2)) + ((d + e*x
)^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5*e^2 + 128*a^4*c^5*d^3*e^4)*(-(9*a*e^5*(-a^9*c)^(1/2) + 4*a^3*c^3*d^
5 + 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3
*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2))/(a^4*e^4 + a^2*c^2*d^4 + 2*a^3*c*d^2*e^2))*(-(9*a*e^5*(-a^9*c)^(1/2) +
4*a^3*c^3*d^5 + 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6
+ 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2)*(9*a^2*c^3*e^6 + 4*c^5*d^4*e^2 + 11*a*c^4
*d^2*e^4))/(a^4*e^4 + a^2*c^2*d^4 + 2*a^3*c*d^2*e^2))*(-(9*a*e^5*(-a^9*c)^(1/2) + 4*a^3*c^3*d^5 + 5*c*d^2*e^3*
(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8
*c^2*d^2*e^4)))^(1/2) + (((192*a^5*c^3*e^7 + 64*a^3*c^5*d^4*e^3 + 256*a^4*c^4*d^2*e^5)/(8*(a^5*e^4 + a^3*c^2*d
^4 + 2*a^4*c*d^2*e^2)) - ((d + e*x)^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5*e^2 + 128*a^4*c^5*d^3*e^4)*(-(9*a
*e^5*(-a^9*c)^(1/2) + 4*a^3*c^3*d^5 + 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a
^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2))/(a^4*e^4 + a^2*c^2*d^4 + 2*a^3*c*d^2*
e^2))*(-(9*a*e^5*(-a^9*c)^(1/2) + 4*a^3*c^3*d^5 + 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d
*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*
c^3*e^6 + 4*c^5*d^4*e^2 + 11*a*c^4*d^2*e^4))/(a^4*e^4 + a^2*c^2*d^4 + 2*a^3*c*d^2*e^2))*(-(9*a*e^5*(-a^9*c)^(1
/2) + 4*a^3*c^3*d^5 + 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c
^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2)))*(-(9*a*e^5*(-a^9*c)^(1/2) + 4*a^3*c^3*d^5 + 5*c*d^2*
e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3
*a^8*c^2*d^2*e^4)))^(1/2)*2i - atan(((((192*a^5*c^3*e^7 + 64*a^3*c^5*d^4*e^3 + 256*a^4*c^4*d^2*e^5)/(8*(a^5*e^
4 + a^3*c^2*d^4 + 2*a^4*c*d^2*e^2)) + ((d + e*x)^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5*e^2 + 128*a^4*c^5*d^
3*e^4)*(-(4*a^3*c^3*d^5 - 9*a*e^5*(-a^9*c)^(1/2) - 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*
d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2))/(a^4*e^4 + a^2*c^2*d^4 +
2*a^3*c*d^2*e^2))*(-(4*a^3*c^3*d^5 - 9*a*e^5*(-a^9*c)^(1/2) - 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2
+ 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2) + ((d + e*x)^
(1/2)*(9*a^2*c^3*e^6 + 4*c^5*d^4*e^2 + 11*a*c^4*d^2*e^4))/(a^4*e^4 + a^2*c^2*d^4 + 2*a^3*c*d^2*e^2))*(-(4*a^3*
c^3*d^5 - 9*a*e^5*(-a^9*c)^(1/2) - 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*
c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2)*1i - (((192*a^5*c^3*e^7 + 64*a^3*c^5*d^4*
e^3 + 256*a^4*c^4*d^2*e^5)/(8*(a^5*e^4 + a^3*c^2*d^4 + 2*a^4*c*d^2*e^2)) - ((d + e*x)^(1/2)*(64*a^5*c^4*d*e^6
+ 64*a^3*c^6*d^5*e^2 + 128*a^4*c^5*d^3*e^4)*(-(4*a^3*c^3*d^5 - 9*a*e^5*(-a^9*c)^(1/2) - 5*c*d^2*e^3*(-a^9*c)^(
1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e
^4)))^(1/2))/(a^4*e^4 + a^2*c^2*d^4 + 2*a^3*c*d^2*e^2))*(-(4*a^3*c^3*d^5 - 9*a*e^5*(-a^9*c)^(1/2) - 5*c*d^2*e^
3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a
^8*c^2*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c^3*e^6 + 4*c^5*d^4*e^2 + 11*a*c^4*d^2*e^4))/(a^4*e^4 + a^2*
c^2*d^4 + 2*a^3*c*d^2*e^2))*(-(4*a^3*c^3*d^5 - 9*a*e^5*(-a^9*c)^(1/2) - 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^
2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2)*1i)/
((4*c^4*d^3*e^3 + 9*a*c^3*d*e^5)/(4*(a^5*e^4 + a^3*c^2*d^4 + 2*a^4*c*d^2*e^2)) + (((192*a^5*c^3*e^7 + 64*a^3*c
^5*d^4*e^3 + 256*a^4*c^4*d^2*e^5)/(8*(a^5*e^4 + a^3*c^2*d^4 + 2*a^4*c*d^2*e^2)) + ((d + e*x)^(1/2)*(64*a^5*c^4
*d*e^6 + 64*a^3*c^6*d^5*e^2 + 128*a^4*c^5*d^3*e^4)*(-(4*a^3*c^3*d^5 - 9*a*e^5*(-a^9*c)^(1/2) - 5*c*d^2*e^3*(-a
^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^
2*d^2*e^4)))^(1/2))/(a^4*e^4 + a^2*c^2*d^4 + 2*a^3*c*d^2*e^2))*(-(4*a^3*c^3*d^5 - 9*a*e^5*(-a^9*c)^(1/2) - 5*c
*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^
2 + 3*a^8*c^2*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2)*(9*a^2*c^3*e^6 + 4*c^5*d^4*e^2 + 11*a*c^4*d^2*e^4))/(a^4*e^4
+ a^2*c^2*d^4 + 2*a^3*c*d^2*e^2))*(-(4*a^3*c^3*d^5 - 9*a*e^5*(-a^9*c)^(1/2) - 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15
*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/
2) + (((192*a^5*c^3*e^7 + 64*a^3*c^5*d^4*e^3 + 256*a^4*c^4*d^2*e^5)/(8*(a^5*e^4 + a^3*c^2*d^4 + 2*a^4*c*d^2*e^
2)) - ((d + e*x)^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5*e^2 + 128*a^4*c^5*d^3*e^4)*(-(4*a^3*c^3*d^5 - 9*a*e^
5*(-a^9*c)^(1/2) - 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*
d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2))/(a^4*e^4 + a^2*c^2*d^4 + 2*a^3*c*d^2*e^2))*(-(4*a^3*c^3*
d^5 - 9*a*e^5*(-a^9*c)^(1/2) - 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^
6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c^3*e^6 + 4*c^5*d^4
*e^2 + 11*a*c^4*d^2*e^4))/(a^4*e^4 + a^2*c^2*d^4 + 2*a^3*c*d^2*e^2))*(-(4*a^3*c^3*d^5 - 9*a*e^5*(-a^9*c)^(1/2)
- 5*c*d^2*e^3*(-a^9*c)^(1/2) + 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*
d^4*e^2 + 3*a^8*c^2*d^2*e^4)))^(1/2)))*(-(4*a^3*c^3*d^5 - 9*a*e^5*(-a^9*c)^(1/2) - 5*c*d^2*e^3*(-a^9*c)^(1/2)
+ 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 + a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 + 3*a^8*c^2*d^2*e^4)))
^(1/2)*2i