\(\int \frac {A+B x}{(d+e x)^{3/2} (a-c x^2)} \, dx\) [1451]

Optimal result

Integrand size = 25, antiderivative size = 197 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=-\frac {2 (B d-A e)}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} \sqrt [4]{c} \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {843, 841, 1180, 214} \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} \sqrt [4]{c} \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2}}-\frac {2 (B d-A e)}{\sqrt {d+e x} \left (c d^2-a e^2\right )} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.29 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=\frac {-2 B d+2 A e}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {-c d-\sqrt {a} \sqrt {c} e}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {-c d+\sqrt {a} \sqrt {c} e}} \]

[In]

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

[Out]

(-2*B*d + 2*A*e)/((c*d^2 - a*e^2)*Sqrt[d + e*x]) + ((Sqrt[a]*B + A*Sqrt[c])*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt
[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(Sqrt[a]*(Sqrt[c]*d + Sqrt[a]*e)*Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]
*e]) + ((Sqrt[a]*B - A*Sqrt[c])*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e
)])/(Sqrt[a]*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e])

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.16

method	result	size
derivativedivides	\(\frac {-2 A e +2 B d}{\left (e^{2} a -c \,d^{2}\right ) \sqrt {e x +d}}-\frac {2 c \left (-\frac {\left (-A c d e +B a \,e^{2}+A \sqrt {a c \,e^{2}}\, e -B \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (A c d e -B a \,e^{2}+A \sqrt {a c \,e^{2}}\, e -B \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{e^{2} a -c \,d^{2}}\)	\(229\)
default	\(-\frac {2 \left (A e -B d \right )}{\left (e^{2} a -c \,d^{2}\right ) \sqrt {e x +d}}+\frac {2 c \left (-\frac {\left (A c d e -B a \,e^{2}-A \sqrt {a c \,e^{2}}\, e +B \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-A c d e +B a \,e^{2}-A \sqrt {a c \,e^{2}}\, e +B \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{e^{2} a -c \,d^{2}}\)	\(229\)
pseudoelliptic	\(-\frac {\sqrt {e x +d}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, c \left (A c d e -B a \,e^{2}+A \sqrt {a c \,e^{2}}\, e -B \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\left (c \left (\left (-A e +B d \right ) \sqrt {a c \,e^{2}}+e \left (A c d -B a e \right )\right ) \sqrt {e x +d}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+2 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \left (A e -B d \right )\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}{\sqrt {e x +d}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \left (e^{2} a -c \,d^{2}\right )}\)	\(272\)

[In]

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

[Out]

2*(-A*e+B*d)/(a*e^2-c*d^2)/(e*x+d)^(1/2)-2/(a*e^2-c*d^2)*c*(-1/2*(-A*c*d*e+B*a*e^2+A*(a*c*e^2)^(1/2)*e-B*(a*c*
e^2)^(1/2)*d)/(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*(A*c*d*e-B*a*e^2+A*(a*c*e^2)^(1/2)*e-B*(a*c*e^2)^(1/2)*d)/(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)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6448 vs. \(2 (147) = 294\).

Time = 4.30 (sec) , antiderivative size = 6448, normalized size of antiderivative = 32.73 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

-Integral(A/(-a*d*sqrt(d + e*x) - a*e*x*sqrt(d + e*x) + c*d*x**2*sqrt(d + e*x) + c*e*x**3*sqrt(d + e*x)), x) -
Integral(B*x/(-a*d*sqrt(d + e*x) - a*e*x*sqrt(d + e*x) + c*d*x**2*sqrt(d + e*x) + c*e*x**3*sqrt(d + e*x)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 941 vs. \(2 (147) = 294\).

Time = 0.41 (sec) , antiderivative size = 941, normalized size of antiderivative = 4.78 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=-\frac {2 \, {\left (B d - A e\right )}}{{\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d}} + \frac {{\left ({\left (c d^{2} e - a e^{3}\right )}^{2} \sqrt {a c} B a d {\left | c \right |} - {\left (c d^{2} e - a e^{3}\right )}^{2} \sqrt {a c} A a e {\left | c \right |} + 2 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} A {\left | c d^{2} e - a e^{3} \right |} {\left | c \right |} - {\left (a c^{2} d^{4} - a^{3} e^{4}\right )} B {\left | c d^{2} e - a e^{3} \right |} {\left | c \right |} - {\left (\sqrt {a c} c^{3} d^{6} e - 2 \, \sqrt {a c} a c^{2} d^{4} e^{3} + \sqrt {a c} a^{2} c d^{2} e^{5}\right )} A {\left | c \right |} + {\left (\sqrt {a c} a c^{2} d^{5} e^{2} - 2 \, \sqrt {a c} a^{2} c d^{3} e^{4} + \sqrt {a c} a^{3} d e^{6}\right )} B {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d^{3} - a c d e^{2} + \sqrt {{\left (c^{2} d^{3} - a c d e^{2}\right )}^{2} - {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} - a c e^{2}\right )}}}{c^{2} d^{2} - a c e^{2}}}}\right )}{{\left (a c^{3} d^{5} - 2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4} - \sqrt {a c} a c^{2} d^{4} e + 2 \, \sqrt {a c} a^{2} c d^{2} e^{3} - \sqrt {a c} a^{3} e^{5}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | c d^{2} e - a e^{3} \right |}} - \frac {{\left ({\left (c d^{2} e - a e^{3}\right )}^{2} \sqrt {a c} B a d {\left | c \right |} - {\left (c d^{2} e - a e^{3}\right )}^{2} \sqrt {a c} A a e {\left | c \right |} - 2 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} A {\left | c d^{2} e - a e^{3} \right |} {\left | c \right |} + {\left (a c^{2} d^{4} - a^{3} e^{4}\right )} B {\left | c d^{2} e - a e^{3} \right |} {\left | c \right |} - {\left (\sqrt {a c} c^{3} d^{6} e - 2 \, \sqrt {a c} a c^{2} d^{4} e^{3} + \sqrt {a c} a^{2} c d^{2} e^{5}\right )} A {\left | c \right |} + {\left (\sqrt {a c} a c^{2} d^{5} e^{2} - 2 \, \sqrt {a c} a^{2} c d^{3} e^{4} + \sqrt {a c} a^{3} d e^{6}\right )} B {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d^{3} - a c d e^{2} - \sqrt {{\left (c^{2} d^{3} - a c d e^{2}\right )}^{2} - {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} - a c e^{2}\right )}}}{c^{2} d^{2} - a c e^{2}}}}\right )}{{\left (a c^{3} d^{5} - 2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4} + \sqrt {a c} a c^{2} d^{4} e - 2 \, \sqrt {a c} a^{2} c d^{2} e^{3} + \sqrt {a c} a^{3} e^{5}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | c d^{2} e - a e^{3} \right |}} \]

[In]

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

[Out]

-2*(B*d - A*e)/((c*d^2 - a*e^2)*sqrt(e*x + d)) + ((c*d^2*e - a*e^3)^2*sqrt(a*c)*B*a*d*abs(c) - (c*d^2*e - a*e^
3)^2*sqrt(a*c)*A*a*e*abs(c) + 2*(a*c^2*d^3*e - a^2*c*d*e^3)*A*abs(c*d^2*e - a*e^3)*abs(c) - (a*c^2*d^4 - a^3*e
^4)*B*abs(c*d^2*e - a*e^3)*abs(c) - (sqrt(a*c)*c^3*d^6*e - 2*sqrt(a*c)*a*c^2*d^4*e^3 + sqrt(a*c)*a^2*c*d^2*e^5
)*A*abs(c) + (sqrt(a*c)*a*c^2*d^5*e^2 - 2*sqrt(a*c)*a^2*c*d^3*e^4 + sqrt(a*c)*a^3*d*e^6)*B*abs(c))*arctan(sqrt
(e*x + d)/sqrt(-(c^2*d^3 - a*c*d*e^2 + sqrt((c^2*d^3 - a*c*d*e^2)^2 - (c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(c^2
*d^2 - a*c*e^2)))/(c^2*d^2 - a*c*e^2)))/((a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4 - sqrt(a*c)*a*c^2*d^4*e
+ 2*sqrt(a*c)*a^2*c*d^2*e^3 - sqrt(a*c)*a^3*e^5)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(c*d^2*e - a*e^3)) - ((c*d^2*
e - a*e^3)^2*sqrt(a*c)*B*a*d*abs(c) - (c*d^2*e - a*e^3)^2*sqrt(a*c)*A*a*e*abs(c) - 2*(a*c^2*d^3*e - a^2*c*d*e^
3)*A*abs(c*d^2*e - a*e^3)*abs(c) + (a*c^2*d^4 - a^3*e^4)*B*abs(c*d^2*e - a*e^3)*abs(c) - (sqrt(a*c)*c^3*d^6*e
- 2*sqrt(a*c)*a*c^2*d^4*e^3 + sqrt(a*c)*a^2*c*d^2*e^5)*A*abs(c) + (sqrt(a*c)*a*c^2*d^5*e^2 - 2*sqrt(a*c)*a^2*c
*d^3*e^4 + sqrt(a*c)*a^3*d*e^6)*B*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(c^2*d^3 - a*c*d*e^2 - sqrt((c^2*d^3 - a*
c*d*e^2)^2 - (c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(c^2*d^2 - a*c*e^2)))/(c^2*d^2 - a*c*e^2)))/((a*c^3*d^5 - 2*a
^2*c^2*d^3*e^2 + a^3*c*d*e^4 + sqrt(a*c)*a*c^2*d^4*e - 2*sqrt(a*c)*a^2*c*d^2*e^3 + sqrt(a*c)*a^3*e^5)*sqrt(-c^
2*d + sqrt(a*c)*c*e)*abs(c*d^2*e - a*e^3))

Mupad [B] (verification not implemented)

Time = 14.35 (sec) , antiderivative size = 10288, normalized size of antiderivative = 52.22 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan((((-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^
3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2)
- 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4
*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*((d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c
)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*
a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c
)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4))
)^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e
^8 + 320*a^5*c^5*d^3*e^10) - 32*B*a^6*c^3*e^12 + 64*A*a*c^8*d^9*e^3 + 64*A*a^5*c^4*d*e^11 - 32*B*a*c^8*d^10*e^
2 - 256*A*a^2*c^7*d^7*e^5 + 384*A*a^3*c^6*d^5*e^7 - 256*A*a^4*c^5*d^3*e^9 + 96*B*a^2*c^7*d^8*e^4 - 64*B*a^3*c^
6*d^6*e^6 - 64*B*a^4*c^5*d^4*e^8 + 96*B*a^5*c^4*d^2*e^10) + (d + e*x)^(1/2)*(16*A^2*a^4*c^4*e^10 + 16*B^2*a^5*
c^3*e^10 - 16*A^2*c^8*d^8*e^2 - 32*A^2*a^3*c^5*d^2*e^8 + 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e^8 + 32*
A^2*a*c^7*d^6*e^4 - 16*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c^6*d^5*e
^5 + 192*A*B*a^3*c^5*d^3*e^7))*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*
(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*
c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))
/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*1i + ((-(B^2*a^2*c^2*d^3 + B^2*a
^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(
1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c
*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*
c^2*d^2*e^4)))^(1/2)*((d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c
^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 +
3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)
^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a
^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 32*B
*a^6*c^3*e^12 - 64*A*a*c^8*d^9*e^3 - 64*A*a^5*c^4*d*e^11 + 32*B*a*c^8*d^10*e^2 + 256*A*a^2*c^7*d^7*e^5 - 384*A
*a^3*c^6*d^5*e^7 + 256*A*a^4*c^5*d^3*e^9 - 96*B*a^2*c^7*d^8*e^4 + 64*B*a^3*c^6*d^6*e^6 + 64*B*a^4*c^5*d^4*e^8
- 96*B*a^5*c^4*d^2*e^10) + (d + e*x)^(1/2)*(16*A^2*a^4*c^4*e^10 + 16*B^2*a^5*c^3*e^10 - 16*A^2*c^8*d^8*e^2 - 3
2*A^2*a^3*c^5*d^2*e^8 + 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e^8 + 32*A^2*a*c^7*d^6*e^4 - 16*B^2*a*c^7*
d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c^6*d^5*e^5 + 192*A*B*a^3*c^5*d^3*e^7))*(-
(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2
+ A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a
^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*
a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*1i)/(((-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*
d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*
B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*
d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*((d + e*x)^(
1/2)*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*
c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) -
6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d
^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e
^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) - 32*B*a^6*c^3*e^12 + 64*A*a*c^8*d^9*e^
3 + 64*A*a^5*c^4*d*e^11 - 32*B*a*c^8*d^10*e^2 - 256*A*a^2*c^7*d^7*e^5 + 384*A*a^3*c^6*d^5*e^7 - 256*A*a^4*c^5*
d^3*e^9 + 96*B*a^2*c^7*d^8*e^4 - 64*B*a^3*c^6*d^6*e^6 - 64*B*a^4*c^5*d^4*e^8 + 96*B*a^5*c^4*d^2*e^10) + (d + e
*x)^(1/2)*(16*A^2*a^4*c^4*e^10 + 16*B^2*a^5*c^3*e^10 - 16*A^2*c^8*d^8*e^2 - 32*A^2*a^3*c^5*d^2*e^8 + 32*B^2*a^
2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e^8 + 32*A^2*a*c^7*d^6*e^4 - 16*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 -
64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c^6*d^5*e^5 + 192*A*B*a^3*c^5*d^3*e^7))*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a
^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*
A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a
^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e
^4)))^(1/2) - ((-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) +
3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c
)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 -
a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*((d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^
3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2)
+ 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*
e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d
^2*e^4)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c
^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 32*B*a^6*c^3*e^12 - 64*A*a*c^8*d^9*e^3 - 64*A*a^5*c^4*d*e^11 + 32*B*a*c^8
*d^10*e^2 + 256*A*a^2*c^7*d^7*e^5 - 384*A*a^3*c^6*d^5*e^7 + 256*A*a^4*c^5*d^3*e^9 - 96*B*a^2*c^7*d^8*e^4 + 64*
B*a^3*c^6*d^6*e^6 + 64*B*a^4*c^5*d^4*e^8 - 96*B*a^5*c^4*d^2*e^10) + (d + e*x)^(1/2)*(16*A^2*a^4*c^4*e^10 + 16*
B^2*a^5*c^3*e^10 - 16*A^2*c^8*d^8*e^2 - 32*A^2*a^3*c^5*d^2*e^8 + 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e
^8 + 32*A^2*a*c^7*d^6*e^4 - 16*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c
^6*d^5*e^5 + 192*A*B*a^3*c^5*d^3*e^7))*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*
c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3
+ 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c
)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2) - 16*A^3*a^3*c^4*e^9 + 1
6*A^3*c^7*d^6*e^3 + 48*A^3*a^2*c^5*d^2*e^7 - 48*B^3*a^2*c^5*d^5*e^4 + 48*B^3*a^3*c^4*d^3*e^6 + 16*A*B^2*a^4*c^
3*e^9 - 16*A^2*B*c^7*d^7*e^2 - 48*A^3*a*c^6*d^4*e^5 + 16*B^3*a*c^6*d^7*e^2 - 16*B^3*a^4*c^3*d*e^8 + 48*A*B^2*a
^2*c^5*d^4*e^5 - 48*A*B^2*a^3*c^4*d^2*e^7 - 48*A^2*B*a^2*c^5*d^3*e^6 - 16*A*B^2*a*c^6*d^6*e^3 + 48*A^2*B*a*c^6
*d^5*e^4 + 16*A^2*B*a^3*c^4*d*e^8))*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2
*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3
*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(
1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*2i + atan((((-(B^2*a^2*c^2*
d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^
3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e
- 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e
^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*((d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^
3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*
a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*
e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*(64*a*c^9*d^11
*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e
^10) - 32*B*a^6*c^3*e^12 + 64*A*a*c^8*d^9*e^3 + 64*A*a^5*c^4*d*e^11 - 32*B*a*c^8*d^10*e^2 - 256*A*a^2*c^7*d^7*
e^5 + 384*A*a^3*c^6*d^5*e^7 - 256*A*a^4*c^5*d^3*e^9 + 96*B*a^2*c^7*d^8*e^4 - 64*B*a^3*c^6*d^6*e^6 - 64*B*a^4*c
^5*d^4*e^8 + 96*B*a^5*c^4*d^2*e^10) + (d + e*x)^(1/2)*(16*A^2*a^4*c^4*e^10 + 16*B^2*a^5*c^3*e^10 - 16*A^2*c^8*
d^8*e^2 - 32*A^2*a^3*c^5*d^2*e^8 + 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e^8 + 32*A^2*a*c^7*d^6*e^4 - 16
*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c^6*d^5*e^5 + 192*A*B*a^3*c^5*d
^3*e^7))*(-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*
a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2
) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c
^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*1i + ((-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) +
A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d
*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) +
6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*(
(d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) +
3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*
c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6
- a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^
2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 32*B*a^6*c^3*e^12 - 64*A*a
*c^8*d^9*e^3 - 64*A*a^5*c^4*d*e^11 + 32*B*a*c^8*d^10*e^2 + 256*A*a^2*c^7*d^7*e^5 - 384*A*a^3*c^6*d^5*e^7 + 256
*A*a^4*c^5*d^3*e^9 - 96*B*a^2*c^7*d^8*e^4 + 64*B*a^3*c^6*d^6*e^6 + 64*B*a^4*c^5*d^4*e^8 - 96*B*a^5*c^4*d^2*e^1
0) + (d + e*x)^(1/2)*(16*A^2*a^4*c^4*e^10 + 16*B^2*a^5*c^3*e^10 - 16*A^2*c^8*d^8*e^2 - 32*A^2*a^3*c^5*d^2*e^8
+ 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e^8 + 32*A^2*a*c^7*d^6*e^4 - 16*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7
*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c^6*d^5*e^5 + 192*A*B*a^3*c^5*d^3*e^7))*(-(B^2*a^2*c^2*d^3 - B^2
*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)
^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a
*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^
4*c^2*d^2*e^4)))^(1/2)*1i)/(((-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a
^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^
2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(
4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*((d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^
3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*
(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e -
3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2
- 3*a^4*c^2*d^2*e^4)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e
^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) - 32*B*a^6*c^3*e^12 + 64*A*a*c^8*d^9*e^3 + 64*A*a^5*c^4*d*e^1
1 - 32*B*a*c^8*d^10*e^2 - 256*A*a^2*c^7*d^7*e^5 + 384*A*a^3*c^6*d^5*e^7 - 256*A*a^4*c^5*d^3*e^9 + 96*B*a^2*c^7
*d^8*e^4 - 64*B*a^3*c^6*d^6*e^6 - 64*B*a^4*c^5*d^4*e^8 + 96*B*a^5*c^4*d^2*e^10) + (d + e*x)^(1/2)*(16*A^2*a^4*
c^4*e^10 + 16*B^2*a^5*c^3*e^10 - 16*A^2*c^8*d^8*e^2 - 32*A^2*a^3*c^5*d^2*e^8 + 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2
*a^4*c^4*d^2*e^8 + 32*A^2*a*c^7*d^6*e^4 - 16*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 -
192*A*B*a^2*c^6*d^5*e^5 + 192*A*B*a^3*c^5*d^3*e^7))*(-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^
3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*
A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*
c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2) - ((-(B^2*
a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^
2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^
2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c
^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*((d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*
a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2
- 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*
B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*(64*a*
c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c
^5*d^3*e^10) + 32*B*a^6*c^3*e^12 - 64*A*a*c^8*d^9*e^3 - 64*A*a^5*c^4*d*e^11 + 32*B*a*c^8*d^10*e^2 + 256*A*a^2*
c^7*d^7*e^5 - 384*A*a^3*c^6*d^5*e^7 + 256*A*a^4*c^5*d^3*e^9 - 96*B*a^2*c^7*d^8*e^4 + 64*B*a^3*c^6*d^6*e^6 + 64
*B*a^4*c^5*d^4*e^8 - 96*B*a^5*c^4*d^2*e^10) + (d + e*x)^(1/2)*(16*A^2*a^4*c^4*e^10 + 16*B^2*a^5*c^3*e^10 - 16*
A^2*c^8*d^8*e^2 - 32*A^2*a^3*c^5*d^2*e^8 + 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e^8 + 32*A^2*a*c^7*d^6*
e^4 - 16*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c^6*d^5*e^5 + 192*A*B*a
^3*c^5*d^3*e^7))*(-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2)
+ 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3
*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6
- a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2) - 16*A^3*a^3*c^4*e^9 + 16*A^3*c^7*d^6*e^3 + 48
*A^3*a^2*c^5*d^2*e^7 - 48*B^3*a^2*c^5*d^5*e^4 + 48*B^3*a^3*c^4*d^3*e^6 + 16*A*B^2*a^4*c^3*e^9 - 16*A^2*B*c^7*d
^7*e^2 - 48*A^3*a*c^6*d^4*e^5 + 16*B^3*a*c^6*d^7*e^2 - 16*B^3*a^4*c^3*d*e^8 + 48*A*B^2*a^2*c^5*d^4*e^5 - 48*A*
B^2*a^3*c^4*d^2*e^7 - 48*A^2*B*a^2*c^5*d^3*e^6 - 16*A*B^2*a*c^6*d^6*e^3 + 48*A^2*B*a*c^6*d^5*e^4 + 16*A^2*B*a^
3*c^4*d*e^8))*(-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3
*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)
^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 -
a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*2i - (2*(A*e - B*d))/((a*e^2 - c*d^2)*(d + e*x)^(
1/2))