\(\int \frac {1}{(d+e x)^{3/2} (a-c x^2)^2} \, dx\) [629]

Optimal result

Integrand size = 20, antiderivative size = 265 \[ \int \frac {1}{(d+e x)^{3/2} \left (a-c x^2\right )^2} \, dx=-\frac {e \left (c d^2+5 a e^2\right )}{2 a \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}-\frac {a e-c d x}{2 a \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a-c x^2\right )}-\frac {\sqrt [4]{c} \left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {\sqrt [4]{c} \left (2 \sqrt {c} d+5 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {755, 843, 841, 1180, 214} \[ \int \frac {1}{(d+e x)^{3/2} \left (a-c x^2\right )^2} \, dx=-\frac {\sqrt [4]{c} \left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {\sqrt [4]{c} \left (5 \sqrt {a} e+2 \sqrt {c} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2}}-\frac {a e-c d x}{2 a \left (a-c x^2\right ) \sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {e \left (5 a e^2+c d^2\right )}{2 a \sqrt {d+e x} \left (c d^2-a e^2\right )^2} \]

[In]

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

[Out]

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

Rule 214

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

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 1180

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

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

Mathematica [A] (verified)

Time = 2.52 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.17 \[ \int \frac {1}{(d+e x)^{3/2} \left (a-c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {a} \left (4 a^2 e^3-c^2 d^2 x (d+e x)+a c e \left (2 d^2+d e x-5 e^2 x^2\right )\right )}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x} \left (-a+c x^2\right )}-\frac {\left (2 \sqrt {c} d+5 \sqrt {a} e\right ) \sqrt {-c d-\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\left (\sqrt {c} d+\sqrt {a} e\right )^3}+\frac {\left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \sqrt {-c d+\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\left (\sqrt {c} d-\sqrt {a} e\right )^3}}{4 a^{3/2}} \]

[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)) - ((2*Sqrt[c]*d + 5*Sqrt[a]*e)*Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*ArcTan[(Sqrt[-(c*d) - Sqrt
[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(Sqrt[c]*d + Sqrt[a]*e)^3 + ((2*Sqrt[c]*d - 5*Sqrt[a]*
e)*Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[
a]*e)])/(Sqrt[c]*d - Sqrt[a]*e)^3)/(4*a^(3/2))

Maple [A] (verified)

Time = 2.83 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.31

method	result	size
derivativedivides	\(2 e^{3} \left (-\frac {c \left (\frac {-\frac {\left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a \,e^{2}}+\frac {d \left (3 e^{2} a +c \,d^{2}\right ) \sqrt {e x +d}}{4 a \,e^{2}}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {c \left (-\frac {\left (-8 d \,e^{2} a c +2 c^{2} d^{3}+5 \sqrt {a c \,e^{2}}\, a \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (8 d \,e^{2} a c -2 c^{2} d^{3}+5 \sqrt {a c \,e^{2}}\, a \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 a \,e^{2}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{2}}-\frac {1}{\left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {e x +d}}\right )\)	\(347\)
default	\(2 e^{3} \left (-\frac {c \left (\frac {-\frac {\left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a \,e^{2}}+\frac {d \left (3 e^{2} a +c \,d^{2}\right ) \sqrt {e x +d}}{4 a \,e^{2}}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {c \left (-\frac {\left (-8 d \,e^{2} a c +2 c^{2} d^{3}+5 \sqrt {a c \,e^{2}}\, a \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (8 d \,e^{2} a c -2 c^{2} d^{3}+5 \sqrt {a c \,e^{2}}\, a \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 a \,e^{2}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{2}}-\frac {1}{\left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {e x +d}}\right )\)	\(347\)
pseudoelliptic	\(-\frac {2 \left (\left (-c \,x^{2}+a \right ) \sqrt {e x +d}\, c \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, e \left (\frac {\left (5 e^{2} a +c \,d^{2}\right ) \sqrt {a c \,e^{2}}}{8}+c d \left (e^{2} a -\frac {c \,d^{2}}{4}\right )\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\left (\left (-c \,x^{2}+a \right ) \sqrt {e x +d}\, c \left (\frac {\left (-5 e^{2} a -c \,d^{2}\right ) \sqrt {a c \,e^{2}}}{8}+c d \left (e^{2} a -\frac {c \,d^{2}}{4}\right )\right ) e \,\operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \left (-\frac {d^{2} x \left (e x +d \right ) c^{2}}{4}+\frac {e \left (-\frac {5}{2} x^{2} e^{2}+\frac {1}{2} d e x +d^{2}\right ) a c}{2}+a^{2} e^{3}\right )\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\right )}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \sqrt {e x +d}\, \left (-c \,x^{2}+a \right ) \left (e^{2} a -c \,d^{2}\right )^{2} a}\)	\(349\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5703 vs. \(2 (206) = 412\).

Time = 2.09 (sec) , antiderivative size = 5703, normalized size of antiderivative = 21.52 \[ \int \frac {1}{(d+e x)^{3/2} \left (a-c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

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. 1411 vs. \(2 (206) = 412\).

Time = 0.43 (sec) , antiderivative size = 1411, normalized size of antiderivative = 5.32 \[ \int \frac {1}{(d+e x)^{3/2} \left (a-c x^2\right )^2} \, dx=\text {Too large to display} \]

[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 - sq
rt(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*
(sqrt(a*c)*c*d^2*e + 5*sqrt(a*c)*a*e^3)*abs(c) - (a*c^4*d^7*e - 15*a^2*c^3*d^5*e^3 + 27*a^3*c^2*d^3*e^5 - 13*a
^4*c*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*(sqrt(a*c)*a*c^6*d^12*e - 8*sqrt(a*c)*a^2*
c^5*d^10*e^3 + 22*sqrt(a*c)*a^3*c^4*d^8*e^5 - 28*sqrt(a*c)*a^4*c^3*d^6*e^7 + 17*sqrt(a*c)*a^5*c^2*d^4*e^9 - 4*
sqrt(a*c)*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 - sq
rt((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^5*d^9 -
4*a^3*c^4*d^7*e^2 + 6*a^4*c^3*d^5*e^4 - 4*a^5*c^2*d^3*e^6 + a^6*c*d*e^8 + sqrt(a*c)*a^2*c^4*d^8*e - 4*sqrt(a*
c)*a^3*c^3*d^6*e^3 + 6*sqrt(a*c)*a^4*c^2*d^4*e^5 - 4*sqrt(a*c)*a^5*c*d^2*e^7 + sqrt(a*c)*a^6*e^9)*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 = 12.35 (sec) , antiderivative size = 8700, normalized size of antiderivative = 32.83 \[ \int \frac {1}{(d+e x)^{3/2} \left (a-c x^2\right )^2} \, dx=\text {Too large to display} \]

[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^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 - 95
232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19) - (d + e*x)^(1/2)*(800*a^12*c^4*e^20 + 128*a^3*c^13*d^18*e^2
- 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)*((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 + 2472
96*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)*(800*a^12*c^4*e^20 + 128*a^3*c^13*d^18*e^2 - 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)*((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^1
1*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)*(800*a^12*c^4*e^20 + 128*a^3*c^13*d^18*e^2 - 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) - ((-(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))/(6
4*(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)*
(800*a^12*c^4*e^20 + 128*a^3*c^13*d^18*e^2 - 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) + 1000*a^10*c^
4*e^19 - 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))/((a*e^2 - c*d
^2)*(d + e*x)^(1/2) - c*(d + e*x)^(5/2) + 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 - 24
5760*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 - 2
45760*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)*(800*a^12*c^4*e^20 + 128*a^3*c^13*d^18*e^2 - 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)*((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 - 20
480*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 - 5160
96*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)*(800*a^12*c^4*e^20 + 128*a^3*c^13*
d^18*e^2 - 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^1
0 - 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)*((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)*(800*a^12*c^4*e^20 + 128*a^3*c^13*d^18*e^2 - 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) - ((-(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 + 95
232*a^12*c^6*d^5*e^17 - 26880*a^13*c^5*d^3*e^19) - (d + e*x)^(1/2)*(800*a^12*c^4*e^20 + 128*a^3*c^13*d^18*e^2
- 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) + 1000*a^10*c^4*e^19 - 32*a^2*c^12*d^16*e^3 + 232*a^3*c^1
1*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))/(6
4*(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