\(\int \frac {1}{\sqrt {d+e x} (a-c x^2)^3} \, dx\) [642]

Optimal result

Integrand size = 20, antiderivative size = 315 \[ \int \frac {1}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=-\frac {(a e-c d x) \sqrt {d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d \left (c d^2-2 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac {3 \left (4 c d^2-10 \sqrt {a} \sqrt {c} d e+7 a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {3 \left (4 c d^2+10 \sqrt {a} \sqrt {c} d e+7 a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2}} \]

[Out]

-3/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(4*c*d^2+7*a*e^2-10*d*e*a^(1/2)*c^(1/2))/a^(
5/2)/c^(1/4)/(-e*a^(1/2)+d*c^(1/2))^(5/2)+3/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(4*c
*d^2+7*a*e^2+10*d*e*a^(1/2)*c^(1/2))/a^(5/2)/c^(1/4)/(e*a^(1/2)+d*c^(1/2))^(5/2)-1/4*(-c*d*x+a*e)*(e*x+d)^(1/2
)/a/(-a*e^2+c*d^2)/(-c*x^2+a)^2-1/16*(a*e*(-7*a*e^2+c*d^2)-6*c*d*(-2*a*e^2+c*d^2)*x)*(e*x+d)^(1/2)/a^2/(-a*e^2
+c*d^2)^2/(-c*x^2+a)

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 315, 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, 837, 841, 1180, 214} \[ \int \frac {1}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=-\frac {3 \left (-10 \sqrt {a} \sqrt {c} d e+7 a e^2+4 c d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {3 \left (10 \sqrt {a} \sqrt {c} d e+7 a e^2+4 c d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2}}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d x \left (c d^2-2 a e^2\right )\right )}{16 a^2 \left (a-c x^2\right ) \left (c d^2-a e^2\right )^2}-\frac {\sqrt {d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )} \]

[In]

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

[Out]

-1/4*((a*e - c*d*x)*Sqrt[d + e*x])/(a*(c*d^2 - a*e^2)*(a - c*x^2)^2) - (Sqrt[d + e*x]*(a*e*(c*d^2 - 7*a*e^2) -
6*c*d*(c*d^2 - 2*a*e^2)*x))/(16*a^2*(c*d^2 - a*e^2)^2*(a - c*x^2)) - (3*(4*c*d^2 - 10*Sqrt[a]*Sqrt[c]*d*e + 7
*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(1/4)*(Sqrt[c]*d - Sqrt[a]
*e)^(5/2)) + (3*(4*c*d^2 + 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d +
Sqrt[a]*e]])/(32*a^(5/2)*c^(1/4)*(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 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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 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) \sqrt {d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}+\frac {\int \frac {\frac {1}{2} \left (6 c d^2-7 a e^2\right )+\frac {5}{2} c d e x}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx}{4 a \left (c d^2-a e^2\right )} \\ & = -\frac {(a e-c d x) \sqrt {d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d \left (c d^2-2 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac {\int \frac {-\frac {3}{4} c \left (4 c^2 d^4-9 a c d^2 e^2+7 a^2 e^4\right )-\frac {3}{2} c^2 d e \left (c d^2-2 a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{8 a^2 c \left (c d^2-a e^2\right )^2} \\ & = -\frac {(a e-c d x) \sqrt {d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d \left (c d^2-2 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {3}{2} c^2 d^2 e \left (c d^2-2 a e^2\right )-\frac {3}{4} c e \left (4 c^2 d^4-9 a c d^2 e^2+7 a^2 e^4\right )-\frac {3}{2} c^2 d e \left (c d^2-2 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 )}{4 a^2 c \left (c d^2-a e^2\right )^2} \\ & = -\frac {(a e-c d x) \sqrt {d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d \left (c d^2-2 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac {\left (3 \sqrt {c} \left (4 c d^2-10 \sqrt {a} \sqrt {c} d e+7 a e^2\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 )}{32 a^{5/2} \left (\sqrt {c} d-\sqrt {a} e\right )^2}+\frac {\left (3 \sqrt {c} \left (4 c d^2+10 \sqrt {a} \sqrt {c} d e+7 a e^2\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 )}{32 a^{5/2} \left (\sqrt {c} d+\sqrt {a} e\right )^2} \\ & = -\frac {(a e-c d x) \sqrt {d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d \left (c d^2-2 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac {3 \left (4 c d^2-10 \sqrt {a} \sqrt {c} d e+7 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {3 \left (4 c d^2+10 \sqrt {a} \sqrt {c} d e+7 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.66 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {d+e x} \left (-11 a^3 e^3+6 c^3 d^3 x^3+a^2 c e \left (5 d^2+16 d e x+7 e^2 x^2\right )-a c^2 d x \left (10 d^2+d e x+12 e^2 x^2\right )\right )}{\left (c d^2-a e^2\right )^2 \left (a-c x^2\right )^2}+\frac {3 \left (4 c d^2+10 \sqrt {a} \sqrt {c} d e+7 a e^2\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\left (\sqrt {c} d+\sqrt {a} e\right )^2 \sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {3 \left (4 c d^2-10 \sqrt {a} \sqrt {c} d e+7 a e^2\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\left (\sqrt {c} d-\sqrt {a} e\right )^2 \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{32 a^{5/2}} \]

[In]

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

[Out]

((-2*Sqrt[a]*Sqrt[d + e*x]*(-11*a^3*e^3 + 6*c^3*d^3*x^3 + a^2*c*e*(5*d^2 + 16*d*e*x + 7*e^2*x^2) - a*c^2*d*x*(
10*d^2 + d*e*x + 12*e^2*x^2)))/((c*d^2 - a*e^2)^2*(a - c*x^2)^2) + (3*(4*c*d^2 + 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*
e^2)*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
)^2*Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]) - (3*(4*c*d^2 - 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*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)^2*Sqrt[-(c*d) + Sqrt[a]*S
qrt[c]*e]))/(32*a^(5/2))

Maple [A] (verified)

Time = 2.95 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.60

method	result	size
derivativedivides	\(-2 e^{5} c^{3} \left (-\frac {\frac {-\frac {3 \sqrt {a c \,e^{2}}\, \left (2 c d +3 \sqrt {a c \,e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{3} \left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right )}+\frac {\sqrt {a c \,e^{2}}\, \left (6 c d +11 \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{4 c^{3} \left (c d +\sqrt {a c \,e^{2}}\right )}}{{\left (-e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}+\frac {3 \left (7 e^{2} a +4 c \,d^{2}+10 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 c \left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 c \,e^{4} a^{2} \sqrt {a c \,e^{2}}}+\frac {\frac {\frac {3 \sqrt {a c \,e^{2}}\, \left (2 c d -3 \sqrt {a c \,e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{3} \left (e^{2} a +c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right )}-\frac {\sqrt {a c \,e^{2}}\, \left (6 c d -11 \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{4 c^{3} \left (c d -\sqrt {a c \,e^{2}}\right )}}{{\left (-e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}-\frac {3 \left (-7 e^{2} a -4 c \,d^{2}+10 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 c \left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 c \,e^{4} a^{2} \sqrt {a c \,e^{2}}}\right )\)	\(505\)
default	\(-2 e^{5} c^{3} \left (-\frac {\frac {-\frac {3 \sqrt {a c \,e^{2}}\, \left (2 c d +3 \sqrt {a c \,e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{3} \left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right )}+\frac {\sqrt {a c \,e^{2}}\, \left (6 c d +11 \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{4 c^{3} \left (c d +\sqrt {a c \,e^{2}}\right )}}{{\left (-e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}+\frac {3 \left (7 e^{2} a +4 c \,d^{2}+10 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 c \left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 c \,e^{4} a^{2} \sqrt {a c \,e^{2}}}+\frac {\frac {\frac {3 \sqrt {a c \,e^{2}}\, \left (2 c d -3 \sqrt {a c \,e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{3} \left (e^{2} a +c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right )}-\frac {\sqrt {a c \,e^{2}}\, \left (6 c d -11 \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{4 c^{3} \left (c d -\sqrt {a c \,e^{2}}\right )}}{{\left (-e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}-\frac {3 \left (-7 e^{2} a -4 c \,d^{2}+10 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 c \left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 c \,e^{4} a^{2} \sqrt {a c \,e^{2}}}\right )\)	\(505\)
pseudoelliptic	\(-\frac {e^{5} c^{3} \left (\frac {-\frac {\sqrt {a c \,e^{2}}\, \sqrt {e x +d}\, \left (-9 a c \,e^{3} x -6 c^{2} d^{2} e x -15 \sqrt {a c \,e^{2}}\, c d e x +19 d \,e^{2} a c +11 \sqrt {a c \,e^{2}}\, a \,e^{2}+8 \sqrt {a c \,e^{2}}\, c \,d^{2}\right )}{\left (c d +\sqrt {a c \,e^{2}}\right ) \left (-c e x +\sqrt {a c \,e^{2}}\right )^{2}}-\frac {3 \left (7 e^{2} a +4 c \,d^{2}+10 \sqrt {a c \,e^{2}}\, d \right ) \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}}}{4 c^{2} e^{4} a^{2} \sqrt {a c \,e^{2}}\, \left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right )}-\frac {3 \left (e x +d \right )^{\frac {3}{2}} d}{2 \left (c e x +\sqrt {a c \,e^{2}}\right )^{2} c \left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) e^{4} a^{2}}+\frac {9 \left (e x +d \right )^{\frac {3}{2}}}{4 \left (c e x +\sqrt {a c \,e^{2}}\right )^{2} c a \,e^{2} \left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {a c \,e^{2}}}+\frac {3 \sqrt {e x +d}\, d}{2 \left (c e x +\sqrt {a c \,e^{2}}\right )^{2} c \left (-c d +\sqrt {a c \,e^{2}}\right ) e^{4} a^{2}}-\frac {11 \sqrt {e x +d}}{4 \left (c e x +\sqrt {a c \,e^{2}}\right )^{2} c a \,e^{2} \left (-c d +\sqrt {a c \,e^{2}}\right ) \sqrt {a c \,e^{2}}}+\frac {21 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 c^{2} \left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, e^{2} a \sqrt {a c \,e^{2}}}+\frac {3 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) d^{2}}{\left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, c \,e^{4} a^{2} \sqrt {a c \,e^{2}}}-\frac {15 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) d}{2 c^{2} \left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, e^{4} a^{2}}\right )}{8}\)	\(733\)

[In]

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

[Out]

-2*e^5*c^3*(-1/16/c/e^4/a^2/(a*c*e^2)^(1/2)*((-3/4*(a*c*e^2)^(1/2)/c^3*(2*c*d+3*(a*c*e^2)^(1/2))/(e^2*a+c*d^2+
2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)+1/4*(a*c*e^2)^(1/2)/c^3*(6*c*d+11*(a*c*e^2)^(1/2))/(c*d+(a*c*e^2)^(1/2))*(e
*x+d)^(1/2))/(-e*x+(a*c*e^2)^(1/2)/c)^2+3/4*(7*e^2*a+4*c*d^2+10*(a*c*e^2)^(1/2)*d)/c/(e^2*a+c*d^2+2*(a*c*e^2)^
(1/2)*d)/((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)))+1/16/c/e^4/
a^2/(a*c*e^2)^(1/2)*((3/4*(a*c*e^2)^(1/2)/c^3*(2*c*d-3*(a*c*e^2)^(1/2))/(e^2*a+c*d^2-2*(a*c*e^2)^(1/2)*d)*(e*x
+d)^(3/2)-1/4*(a*c*e^2)^(1/2)/c^3*(6*c*d-11*(a*c*e^2)^(1/2))/(c*d-(a*c*e^2)^(1/2))*(e*x+d)^(1/2))/(-e*x-(a*c*e
^2)^(1/2)/c)^2-3/4*(-7*e^2*a-4*c*d^2+10*(a*c*e^2)^(1/2)*d)/c/(-e^2*a-c*d^2+2*(a*c*e^2)^(1/2)*d)/((-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))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5776 vs. \(2 (258) = 516\).

Time = 2.45 (sec) , antiderivative size = 5776, normalized size of antiderivative = 18.34 \[ \int \frac {1}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1642 vs. \(2 (258) = 516\).

Time = 0.43 (sec) , antiderivative size = 1642, normalized size of antiderivative = 5.21 \[ \int \frac {1}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-3/32*(2*(a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)^2*(c^2*d^3*e - 2*a*c*d*e^3)*abs(c) + (2*sqrt(a*c)*a*c^4*d
^8*e - 9*sqrt(a*c)*a^2*c^3*d^6*e^3 + 19*sqrt(a*c)*a^3*c^2*d^4*e^5 - 19*sqrt(a*c)*a^4*c*d^2*e^7 + 7*sqrt(a*c)*a
^5*e^9)*abs(a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)*abs(c) - (4*a^3*c^7*d^13*e - 25*a^4*c^6*d^11*e^3 + 67*a
^5*c^5*d^9*e^5 - 98*a^6*c^4*d^7*e^7 + 82*a^7*c^3*d^5*e^9 - 37*a^8*c^2*d^3*e^11 + 7*a^9*c*d*e^13)*abs(c))*arcta
n(sqrt(e*x + d)/sqrt(-(a^2*c^3*d^5 - 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4 + sqrt((a^2*c^3*d^5 - 2*a^3*c^2*d^3*e^2 +
a^4*c*d*e^4)^2 - (a^2*c^3*d^6 - 3*a^3*c^2*d^4*e^2 + 3*a^4*c*d^2*e^4 - a^5*e^6)*(a^2*c^3*d^4 - 2*a^3*c^2*d^2*e
^2 + a^4*c*e^4)))/(a^2*c^3*d^4 - 2*a^3*c^2*d^2*e^2 + a^4*c*e^4)))/((a^4*c^5*d^8*e - 4*a^5*c^4*d^6*e^3 + 6*a^6*
c^3*d^4*e^5 - 4*a^7*c^2*d^2*e^7 + a^8*c*e^9 - sqrt(a*c)*a^3*c^5*d^9 + 4*sqrt(a*c)*a^4*c^4*d^7*e^2 - 6*sqrt(a*c
)*a^5*c^3*d^5*e^4 + 4*sqrt(a*c)*a^6*c^2*d^3*e^6 - sqrt(a*c)*a^7*c*d*e^8)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(a^2*
c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)) - 3/32*(2*(a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)^2*(sqrt(a*c)*c*d
^3*e - 2*sqrt(a*c)*a*d*e^3)*abs(c) - (2*a^2*c^4*d^8*e - 9*a^3*c^3*d^6*e^3 + 19*a^4*c^2*d^4*e^5 - 19*a^5*c*d^2*
e^7 + 7*a^6*e^9)*abs(a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)*abs(c) - (4*sqrt(a*c)*a^3*c^6*d^13*e - 25*sqrt
(a*c)*a^4*c^5*d^11*e^3 + 67*sqrt(a*c)*a^5*c^4*d^9*e^5 - 98*sqrt(a*c)*a^6*c^3*d^7*e^7 + 82*sqrt(a*c)*a^7*c^2*d^
5*e^9 - 37*sqrt(a*c)*a^8*c*d^3*e^11 + 7*sqrt(a*c)*a^9*d*e^13)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a^2*c^3*d^5
- 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4 - sqrt((a^2*c^3*d^5 - 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4)^2 - (a^2*c^3*d^6 - 3*
a^3*c^2*d^4*e^2 + 3*a^4*c*d^2*e^4 - a^5*e^6)*(a^2*c^3*d^4 - 2*a^3*c^2*d^2*e^2 + a^4*c*e^4)))/(a^2*c^3*d^4 - 2*
a^3*c^2*d^2*e^2 + a^4*c*e^4)))/((a^4*c^5*d^9 - 4*a^5*c^4*d^7*e^2 + 6*a^6*c^3*d^5*e^4 - 4*a^7*c^2*d^3*e^6 + a^8
*c*d*e^8 + sqrt(a*c)*a^4*c^4*d^8*e - 4*sqrt(a*c)*a^5*c^3*d^6*e^3 + 6*sqrt(a*c)*a^6*c^2*d^4*e^5 - 4*sqrt(a*c)*a
^7*c*d^2*e^7 + sqrt(a*c)*a^8*e^9)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5))
- 1/16*(6*(e*x + d)^(7/2)*c^3*d^3*e - 18*(e*x + d)^(5/2)*c^3*d^4*e + 18*(e*x + d)^(3/2)*c^3*d^5*e - 6*sqrt(e*
x + d)*c^3*d^6*e - 12*(e*x + d)^(7/2)*a*c^2*d*e^3 + 35*(e*x + d)^(5/2)*a*c^2*d^2*e^3 - 44*(e*x + d)^(3/2)*a*c^
2*d^3*e^3 + 21*sqrt(e*x + d)*a*c^2*d^4*e^3 + 7*(e*x + d)^(5/2)*a^2*c*e^5 + 2*(e*x + d)^(3/2)*a^2*c*d*e^5 - 4*s
qrt(e*x + d)*a^2*c*d^2*e^5 - 11*sqrt(e*x + d)*a^3*e^7)/((a^2*c^2*d^4 - 2*a^3*c*d^2*e^2 + a^4*e^4)*((e*x + d)^2
*c - 2*(e*x + d)*c*d + c*d^2 - a*e^2)^2)

Mupad [B] (verification not implemented)

Time = 12.81 (sec) , antiderivative size = 8961, normalized size of antiderivative = 28.45 \[ \int \frac {1}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- atan(((((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*d^4*e^7 - 3891
2*a^8*c^4*d^2*e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4))
- ((d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4
- 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(40
96*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*
d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 -
16384*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)
))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d
^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10
- a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)
+ ((d + e*x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e^6 - 990*a^3*c
^4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*
a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*
c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d
^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)*1i - (((3*(
14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*d^4*e^7 - 38912*a^8*c^4*d^2*
e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) + ((d + e*x)^(
1/2)*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2
*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^1
0 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/
2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 - 16384*a^8*c^5
*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^
5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*c^
2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^1
0 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2) - ((d + e*x)^
(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e^6 - 990*a^3*c^4*d^2*e^8))/(
64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 -
49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a
^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c
^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)*1i)/((((3*(14336*a^9*c^3*
e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*d^4*e^7 - 38912*a^8*c^4*d^2*e^9))/(2048*(a
^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) - ((d + e*x)^(1/2)*(-(9*(16*
a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*
c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d
^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)*(4096*a^9*c
^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 - 16384*a^8*c^5*d^3*e^8))/(64
*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 - 49
*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^1
5*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5
*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2) + ((d + e*x)^(1/2)*(441*a^4
*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e^6 - 990*a^3*c^4*d^2*e^8))/(64*(a^8*e^8 +
a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^
15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) +
105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 1
0*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2) - (3*(144*c^6*d^7*e^3 - 684*a*c^5*d^5*e
^5 - 882*a^3*c^3*d*e^9 + 1233*a^2*c^4*d^3*e^7))/(1024*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^
6*e^2 + 6*a^8*c^2*d^4*e^4)) + (((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^6*e^5 + 38912*
a^7*c^5*d^4*e^7 - 38912*a^8*c^4*d^2*e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2
+ 6*a^8*c^2*d^4*e^4)) + ((d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2
+ 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^
7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3
*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 2
4576*a^7*c^6*d^5*e^6 - 16384*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^
2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d
^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)
))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^1
4*c^2*d^2*e^8)))^(1/2) - ((d + e*x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a^2*c
^5*d^4*e^6 - 990*a^3*c^4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^
2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210
*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^
15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^
8)))^(1/2)))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210
*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^
15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^
8)))^(1/2)*2i - atan(((((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*
d^4*e^7 - 38912*a^8*c^4*d^2*e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2 + 6*a^8*
c^2*d^4*e^4)) - ((d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a
^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*
c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6
- 5*a^14*c^2*d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 24576*a^7
*c^6*d^5*e^6 - 16384*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^
6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 -
210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096
*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^
2*e^8)))^(1/2) + ((d + e*x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e
^6 - 990*a^3*c^4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^
4)))*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2
*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^1
0 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/
2)*1i - (((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*d^4*e^7 - 3891
2*a^8*c^4*d^2*e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4))
+ ((d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4
- 210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(40
96*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*
d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 -
16384*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)
))*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d
^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10
- a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)
- ((d + e*x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e^6 - 990*a^3*c
^4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*
a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*
c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d
^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)*1i)/((((3*(
14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*d^4*e^7 - 38912*a^8*c^4*d^2*
e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) - ((d + e*x)^(
1/2)*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2
*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^1
0 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/
2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 - 16384*a^8*c^5
*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^
5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*c^
2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^1
0 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2) + ((d + e*x)^
(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e^6 - 990*a^3*c^4*d^2*e^8))/(
64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 +
49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a
^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c
^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2) - (3*(144*c^6*d^7*e^3 - 6
84*a*c^5*d^5*e^5 - 882*a^3*c^3*d*e^9 + 1233*a^2*c^4*d^3*e^7))/(1024*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6
- 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) + (((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^
6*e^5 + 38912*a^7*c^5*d^4*e^7 - 38912*a^8*c^4*d^2*e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^
7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) + ((d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a
^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 -
54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4
+ 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c
^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 - 16384*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*
a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 +
189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*
(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d
^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2) - ((d + e*x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^
4 + 1089*a^2*c^5*d^4*e^6 - 990*a^3*c^4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*
e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3
*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/
2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a
^14*c^2*d^2*e^8)))^(1/2)))*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3
*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/
2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a
^14*c^2*d^2*e^8)))^(1/2)*2i - ((e*(d + e*x)^(3/2)*(9*c^3*d^5 - 22*a*c^2*d^3*e^2 + a^2*c*d*e^4))/(8*a^2*(a*e^2
- c*d^2)^2) - (e*(d + e*x)^(1/2)*(11*a^2*e^4 - 6*c^2*d^4 + 15*a*c*d^2*e^2))/(16*a^2*(a*e^2 - c*d^2)) + (3*c*e*
(c^2*d^3 - 2*a*c*d*e^2)*(d + e*x)^(7/2))/(8*a^2*(a*e^2 - c*d^2)^2) + (c*e*(d + e*x)^(5/2)*(7*a^2*e^4 - 18*c^2*
d^4 + 35*a*c*d^2*e^2))/(16*a^2*(a*e^2 - c*d^2)^2))/(c^2*(d + e*x)^4 + a^2*e^4 + c^2*d^4 + (6*c^2*d^2 - 2*a*c*e
^2)*(d + e*x)^2 - (4*c^2*d^3 - 4*a*c*d*e^2)*(d + e*x) - 4*c^2*d*(d + e*x)^3 - 2*a*c*d^2*e^2)