3.1457 \(\int \frac {A+B x}{(d+e x)^{3/2} (a-c x^2)^2} \, dx\)

Optimal. Leaf size=303 \[ -\frac {\left (-5 \sqrt {a} A \sqrt {c} e+3 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {\left (5 \sqrt {a} A \sqrt {c} e+3 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2}}+\frac {x (A c d-a B e)+a (B d-A e)}{2 a \left (a-c x^2\right ) \sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {e \left (5 a A e^2-6 a B d e+A c d^2\right )}{2 a \sqrt {d+e x} \left (c d^2-a e^2\right )^2} \]

[Out]

-1/4*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(2*A*c*d+3*a*B*e-5*A*e*a^(1/2)*c^(1/2))/a^(3/
2)/c^(1/4)/(-e*a^(1/2)+d*c^(1/2))^(5/2)+1/4*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(2*A*c*
d+3*a*B*e+5*A*e*a^(1/2)*c^(1/2))/a^(3/2)/c^(1/4)/(e*a^(1/2)+d*c^(1/2))^(5/2)-1/2*e*(5*A*a*e^2+A*c*d^2-6*B*a*d*
e)/a/(-a*e^2+c*d^2)^2/(e*x+d)^(1/2)+1/2*(a*(-A*e+B*d)+(A*c*d-B*a*e)*x)/a/(-a*e^2+c*d^2)/(-c*x^2+a)/(e*x+d)^(1/
2)

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Rubi [A]  time = 0.69, antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {823, 829, 827, 1166, 208} \[ -\frac {\left (-5 \sqrt {a} A \sqrt {c} e+3 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {\left (5 \sqrt {a} A \sqrt {c} e+3 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2}}+\frac {x (A c d-a B e)+a (B d-A e)}{2 a \left (a-c x^2\right ) \sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {e \left (5 a A e^2-6 a B d e+A c d^2\right )}{2 a \sqrt {d+e x} \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*(A*c*d^2 - 6*a*B*d*e + 5*a*A*e^2))/(2*a*(c*d^2 - a*e^2)^2*Sqrt[d + e*x]) + (a*(B*d - A*e) + (A*c*d - a*B*e
)*x)/(2*a*(c*d^2 - a*e^2)*Sqrt[d + e*x]*(a - c*x^2)) - ((2*A*c*d + 3*a*B*e - 5*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c
^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)) + ((2*A*
c*d + 3*a*B*e + 5*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2
)*c^(1/4)*(Sqrt[c]*d + Sqrt[a]*e)^(5/2))

Rule 208

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

Rule 823

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] + Di
st[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 827

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 829

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 1166

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*} \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^2} \, dx &=\frac {a (B d-A e)+(A c d-a B e) 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} c \left (2 A c d^2+3 a B d e-5 a A e^2\right )-\frac {3}{2} c e (A c d-a B e) x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx}{2 a c \left (c d^2-a e^2\right )}\\ &=-\frac {e \left (A c d^2-6 a B d e+5 a A e^2\right )}{2 a \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}+\frac {a (B d-A e)+(A c d-a B e) 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} c \left (2 A c d \left (c d^2-4 a e^2\right )+3 a B e \left (c d^2+a e^2\right )\right )+\frac {1}{2} c^2 e \left (A c d^2-6 a B d e+5 a A e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{2 a c \left (c d^2-a e^2\right )^2}\\ &=-\frac {e \left (A c d^2-6 a B d e+5 a A e^2\right )}{2 a \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}+\frac {a (B d-A e)+(A c d-a B e) x}{2 a \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a-c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2} c^2 d e \left (A c d^2-6 a B d e+5 a A e^2\right )+\frac {1}{2} c e \left (2 A c d \left (c d^2-4 a e^2\right )+3 a B e \left (c d^2+a e^2\right )\right )+\frac {1}{2} c^2 e \left (A c d^2-6 a B d e+5 a 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 c \left (c d^2-a e^2\right )^2}\\ &=-\frac {e \left (A c d^2-6 a B d e+5 a A e^2\right )}{2 a \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}+\frac {a (B d-A e)+(A c d-a B e) x}{2 a \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a-c x^2\right )}-\frac {\left (\sqrt {c} \left (2 A c d+3 a B e-5 \sqrt {a} A \sqrt {c} e\right )\right ) \operatorname {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 (\sqrt {c} \left (2 A c d+3 a B e+5 \sqrt {a} A \sqrt {c} e\right )\right ) \operatorname {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 (A c d^2-6 a B d e+5 a A e^2\right )}{2 a \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}+\frac {a (B d-A e)+(A c d-a B e) x}{2 a \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a-c x^2\right )}-\frac {\left (2 A c d+3 a B e-5 \sqrt {a} A \sqrt {c} 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} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {\left (2 A c d+3 a B e+5 \sqrt {a} A \sqrt {c} 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} \sqrt [4]{c} \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.61, size = 365, normalized size = 1.20 \[ \frac {-\frac {3 c^{3/4} (A c d-a B e) \left (\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {\sqrt {a} e+\sqrt {c} d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a}}+\frac {c \left (5 a A e^2-6 a B d e+A c d^2\right ) \left (\left (\sqrt {a} e+\sqrt {c} d\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {a} e}\right )+\left (\sqrt {a} e-\sqrt {c} d\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}\right )\right )}{2 \sqrt {a} \sqrt {d+e x} \left (c d^2-a e^2\right )}+\frac {c (-a A e+a B (d-e x)+A c d x)}{\left (c x^2-a\right ) \sqrt {d+e x}}}{2 a c \left (a e^2-c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c*(-(a*A*e) + A*c*d*x + a*B*(d - e*x)))/(Sqrt[d + e*x]*(-a + c*x^2)) - (3*c^(3/4)*(A*c*d - a*B*e)*(-(ArcTanh
[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]/Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ArcTanh[(c^(1/4)*Sqrt[d +
e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]/Sqrt[Sqrt[c]*d + Sqrt[a]*e]))/(2*Sqrt[a]) + (c*(A*c*d^2 - 6*a*B*d*e + 5*a*
A*e^2)*((Sqrt[c]*d + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)] +
(-(Sqrt[c]*d) + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]))/(2*
Sqrt[a]*(c*d^2 - a*e^2)*Sqrt[d + e*x]))/(2*a*c*(-(c*d^2) + a*e^2))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [B]  time = 2.07, size = 1895, normalized size = 6.25 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(6*(a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5)^2*B*a*c*d*abs(c)*e^2 - (a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e
^5)^2*(c^2*d^2*e + 5*a*c*e^3)*A*abs(c) - (sqrt(a*c)*c^4*d^7*e - 15*sqrt(a*c)*a*c^3*d^5*e^3 + 27*sqrt(a*c)*a^2*
c^2*d^3*e^5 - 13*sqrt(a*c)*a^3*c*d*e^7)*A*abs(a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5)*abs(c) - 3*(3*sqrt(a*c)
*a*c^3*d^6*e^2 - 5*sqrt(a*c)*a^2*c^2*d^4*e^4 + sqrt(a*c)*a^3*c*d^2*e^6 + sqrt(a*c)*a^4*e^8)*B*abs(a*c^2*d^4*e
- 2*a^2*c*d^2*e^3 + a^3*e^5)*abs(c) + 2*(a*c^7*d^12*e - 8*a^2*c^6*d^10*e^3 + 22*a^3*c^5*d^8*e^5 - 28*a^4*c^4*d
^6*e^7 + 17*a^5*c^3*d^4*e^9 - 4*a^6*c^2*d^2*e^11)*A*abs(c) + 3*(a^2*c^6*d^11*e^2 - 3*a^3*c^5*d^9*e^4 + 2*a^4*c
^4*d^7*e^6 + 2*a^5*c^3*d^5*e^8 - 3*a^6*c^2*d^3*e^10 + a^7*c*d*e^12)*B*abs(c))*arctan(sqrt(x*e + 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^5*d^8*e - sqrt(a*c)*a*c^5*d^9 + 4*sqrt(a*c)*a^2*c^4*d^7*e^2 - 4*a^3*c^4*d
^6*e^3 - 6*sqrt(a*c)*a^3*c^3*d^5*e^4 + 6*a^4*c^3*d^4*e^5 + 4*sqrt(a*c)*a^4*c^2*d^3*e^6 - 4*a^5*c^2*d^2*e^7 - s
qrt(a*c)*a^5*c*d*e^8 + a^6*c*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)) +
1/4*(6*(a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5)^2*sqrt(a*c)*B*a*d*abs(c)*e^2 - (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)*A*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)*A*abs(a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5)*abs(c) + 3*(3*a^2*c^3*d^6*e^2 - 5*
a^3*c^2*d^4*e^4 + a^4*c*d^2*e^6 + a^5*e^8)*B*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)*A*abs(c) + 3*(sqrt(a*c)*a^2*c^5*d^11*e^2 - 3*sqrt(a
*c)*a^3*c^4*d^9*e^4 + 2*sqrt(a*c)*a^4*c^3*d^7*e^6 + 2*sqrt(a*c)*a^5*c^2*d^5*e^8 - 3*sqrt(a*c)*a^6*c*d^3*e^10 +
sqrt(a*c)*a^7*d*e^12)*B*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4 - sqr
t((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 +
sqrt(a*c)*a^2*c^4*d^8*e - 4*a^3*c^4*d^7*e^2 - 4*sqrt(a*c)*a^3*c^3*d^6*e^3 + 6*a^4*c^3*d^5*e^4 + 6*sqrt(a*c)*a^
4*c^2*d^4*e^5 - 4*a^5*c^2*d^3*e^6 - 4*sqrt(a*c)*a^5*c*d^2*e^7 + a^6*c*d*e^8 + 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*((x*e + d)^2*A*c^2*d^2*e - (x*e + d)*A*c^2
*d^3*e - 6*(x*e + d)^2*B*a*c*d*e^2 + 11*(x*e + d)*B*a*c*d^2*e^2 - 4*B*a*c*d^3*e^2 + 5*(x*e + d)^2*A*a*c*e^3 -
11*(x*e + d)*A*a*c*d*e^3 + 4*A*a*c*d^2*e^3 + (x*e + d)*B*a^2*e^4 + 4*B*a^2*d*e^4 - 4*A*a^2*e^5)/((a*c^2*d^4 -
2*a^2*c*d^2*e^2 + a^3*e^4)*((x*e + d)^(5/2)*c - 2*(x*e + d)^(3/2)*c*d + sqrt(x*e + d)*c*d^2 - sqrt(x*e + d)*a*
e^2))

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maple [B]  time = 0.10, size = 1392, normalized size = 4.59 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^3/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)*c*(e*x+d)^(3/2)*A-1/2*e/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)*c^2/a*(e*
x+d)^(3/2)*A*d^2+e^2/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)*c*(e*x+d)^(3/2)*B*d+3/2*e^3/(a*e^2-c*d^2)^2/(c*e^2*x^2-
a*e^2)*(e*x+d)^(1/2)*A*c*d+1/2*e/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)/a*(e*x+d)^(1/2)*A*c^2*d^3-1/2*e^4/(a*e^2-c*
d^2)^2/(c*e^2*x^2-a*e^2)*a*(e*x+d)^(1/2)*B-3/2*e^2/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)*(e*x+d)^(1/2)*B*c*d^2-2*e
^3/(a*e^2-c*d^2)^2*c^2/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(
1/2))*c)^(1/2)*c)*A*d+1/2*e/(a*e^2-c*d^2)^2/a*c^3/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x
+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*d^3+3/4*e^4/(a*e^2-c*d^2)^2*a*c/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)
^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B+3/4*e^2/(a*e^2-c*d^2)^2*c^2/(a*c*e
^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B*d^2+5/4*e
^3/(a*e^2-c*d^2)^2*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*
A+1/4*e/(a*e^2-c*d^2)^2/a*c^2/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^
(1/2)*c)*A*d^2-3/2*e^2/(a*e^2-c*d^2)^2*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)
^(1/2))*c)^(1/2)*c)*B*d-2*e^3/(a*e^2-c*d^2)^2*c^2/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x
+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*d+1/2*e/(a*e^2-c*d^2)^2/a*c^3/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)
^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*d^3+3/4*e^4/(a*e^2-c*d^2)^2*a*c/(a
*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B+3/4*
e^2/(a*e^2-c*d^2)^2*c^2/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)
^(1/2))*c)^(1/2)*c)*B*d^2-5/4*e^3/(a*e^2-c*d^2)^2*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c
*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A-1/4*e/(a*e^2-c*d^2)^2/a*c^2/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^
(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*d^2+3/2*e^2/(a*e^2-c*d^2)^2*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arc
tan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B*d-2*e^3/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)*A+2*e^2/(a*e^2-c
*d^2)^2/(e*x+d)^(1/2)*B*d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{{\left (c x^{2} - a\right )}^{2} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 8.12, size = 19787, normalized size = 65.30 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((d + e*x)*(B*a^2*e^4 - A*c^2*d^3*e - 11*A*a*c*d*e^3 + 11*B*a*c*d^2*e^2))/(2*a*(a*e^2 - c*d^2)^2) - (2*(A*e^3
- B*d*e^2))/(a*e^2 - c*d^2) + (c*(d + e*x)^2*(5*A*a*e^3 - 6*B*a*d*e^2 + A*c*d^2*e))/(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((((d + e*x)^(1/2)*(800*A^
2*a^12*c^4*e^20 + 288*B^2*a^13*c^3*e^20 + 128*A^2*a^3*c^13*d^18*e^2 - 1760*A^2*a^4*c^12*d^16*e^4 + 10240*A^2*a
^5*c^11*d^14*e^6 - 30848*A^2*a^6*c^10*d^12*e^8 + 52480*A^2*a^7*c^9*d^10*e^10 - 51008*A^2*a^8*c^8*d^8*e^12 + 25
600*A^2*a^9*c^7*d^6*e^14 - 3200*A^2*a^10*c^6*d^4*e^16 - 2432*A^2*a^11*c^5*d^2*e^18 + 288*B^2*a^5*c^11*d^16*e^4
- 5760*B^2*a^7*c^9*d^12*e^8 + 18432*B^2*a^8*c^8*d^10*e^10 - 25920*B^2*a^9*c^7*d^8*e^12 + 18432*B^2*a^10*c^6*d
^6*e^14 - 5760*B^2*a^11*c^5*d^4*e^16 - 3456*A*B*a^12*c^4*d*e^19 + 384*A*B*a^4*c^12*d^17*e^3 - 3840*A*B*a^5*c^1
1*d^15*e^5 + 11520*A*B*a^6*c^10*d^13*e^7 - 9984*A*B*a^7*c^9*d^11*e^9 - 15360*A*B*a^8*c^8*d^9*e^11 + 43776*A*B*
a^9*c^7*d^7*e^13 - 42240*A*B*a^10*c^6*d^5*e^15 + 19200*A*B*a^11*c^5*d^3*e^17) + (-(4*A^2*a^3*c^5*d^7 + 9*B^2*a
^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^
2*d^3*e^4 - 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 + 25*A^2*a^2*c*e^7*(
a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 + 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) + 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*
A*B*a^4*c^4*d^6*e + 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) + 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d
^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 - 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) - 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(6
4*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e
^8)))^(1/2)*(768*B*a^15*c^3*e^22 - (d + e*x)^(1/2)*(-(4*A^2*a^3*c^5*d^7 + 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2
*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 - 35*A^2*c^3*d^4*e^
3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 + 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e
^7 + 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) + 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e + 45*B^2*a*
c^2*d^4*e^3*(a^9*c)^(1/2) + 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*
e^5 - 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) - 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10
+ 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(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*a^14*c^4*d*e^21 + 256*A*a^5*c^13*d^19*e^3 - 5376
*A*a^6*c^12*d^17*e^5 + 33792*A*a^7*c^11*d^15*e^7 - 107520*A*a^8*c^10*d^13*e^9 + 204288*A*a^9*c^9*d^11*e^11 - 2
47296*A*a^10*c^8*d^9*e^13 + 193536*A*a^11*c^7*d^7*e^15 - 95232*A*a^12*c^6*d^5*e^17 + 26880*A*a^13*c^5*d^3*e^19
+ 2304*B*a^6*c^12*d^18*e^4 - 17664*B*a^7*c^11*d^16*e^6 + 58368*B*a^8*c^10*d^14*e^8 - 107520*B*a^9*c^9*d^12*e^
10 + 118272*B*a^10*c^8*d^10*e^12 - 75264*B*a^11*c^7*d^8*e^14 + 21504*B*a^12*c^6*d^6*e^16 + 3072*B*a^13*c^5*d^4
*e^18 - 3840*B*a^14*c^4*d^2*e^20))*(-(4*A^2*a^3*c^5*d^7 + 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2
+ 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 - 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2)
+ 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 + 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 + 30*A*B*c^3*
d^5*e^2*(a^9*c)^(1/2) + 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e + 45*B^2*a*c^2*d^4*e^3*(a^9
*c)^(1/2) + 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 - 138*A*B*a^
2*c*d*e^6*(a^9*c)^(1/2) - 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8
*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2)*1i + ((d + e*x)^(1/2)*(800*A^2*a^
12*c^4*e^20 + 288*B^2*a^13*c^3*e^20 + 128*A^2*a^3*c^13*d^18*e^2 - 1760*A^2*a^4*c^12*d^16*e^4 + 10240*A^2*a^5*c
^11*d^14*e^6 - 30848*A^2*a^6*c^10*d^12*e^8 + 52480*A^2*a^7*c^9*d^10*e^10 - 51008*A^2*a^8*c^8*d^8*e^12 + 25600*
A^2*a^9*c^7*d^6*e^14 - 3200*A^2*a^10*c^6*d^4*e^16 - 2432*A^2*a^11*c^5*d^2*e^18 + 288*B^2*a^5*c^11*d^16*e^4 - 5
760*B^2*a^7*c^9*d^12*e^8 + 18432*B^2*a^8*c^8*d^10*e^10 - 25920*B^2*a^9*c^7*d^8*e^12 + 18432*B^2*a^10*c^6*d^6*e
^14 - 5760*B^2*a^11*c^5*d^4*e^16 - 3456*A*B*a^12*c^4*d*e^19 + 384*A*B*a^4*c^12*d^17*e^3 - 3840*A*B*a^5*c^11*d^
15*e^5 + 11520*A*B*a^6*c^10*d^13*e^7 - 9984*A*B*a^7*c^9*d^11*e^9 - 15360*A*B*a^8*c^8*d^9*e^11 + 43776*A*B*a^9*
c^7*d^7*e^13 - 42240*A*B*a^10*c^6*d^5*e^15 + 19200*A*B*a^11*c^5*d^3*e^17) - (-(4*A^2*a^3*c^5*d^7 + 9*B^2*a^3*e
^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^
3*e^4 - 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 + 25*A^2*a^2*c*e^7*(a^9*
c)^(1/2) - 30*A*B*a^7*c*e^7 + 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) + 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*
a^4*c^4*d^6*e + 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) + 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e
^3 - 240*A*B*a^6*c^2*d^2*e^5 - 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) - 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a
^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8))
)^(1/2)*((d + e*x)^(1/2)*(-(4*A^2*a^3*c^5*d^7 + 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*
a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 - 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a
^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 + 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 + 30*A*B*c^3*d^5*e^2*(a
^9*c)^(1/2) + 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e + 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2)
+ 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 - 138*A*B*a^2*c*d*e^6*
(a^9*c)^(1/2) - 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*
a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(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^1
0 - 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) + 768*B*a^15*c^3*e^22 - 3328*A*a^14*c^4*d*e^21 + 256*A*a^5*c^13*d^19*e^3 - 5376*A*a
^6*c^12*d^17*e^5 + 33792*A*a^7*c^11*d^15*e^7 - 107520*A*a^8*c^10*d^13*e^9 + 204288*A*a^9*c^9*d^11*e^11 - 24729
6*A*a^10*c^8*d^9*e^13 + 193536*A*a^11*c^7*d^7*e^15 - 95232*A*a^12*c^6*d^5*e^17 + 26880*A*a^13*c^5*d^3*e^19 + 2
304*B*a^6*c^12*d^18*e^4 - 17664*B*a^7*c^11*d^16*e^6 + 58368*B*a^8*c^10*d^14*e^8 - 107520*B*a^9*c^9*d^12*e^10 +
118272*B*a^10*c^8*d^10*e^12 - 75264*B*a^11*c^7*d^8*e^14 + 21504*B*a^12*c^6*d^6*e^16 + 3072*B*a^13*c^5*d^4*e^1
8 - 3840*B*a^14*c^4*d^2*e^20))*(-(4*A^2*a^3*c^5*d^7 + 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 7
0*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 - 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45
*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 + 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 + 30*A*B*c^3*d^5*
e^2*(a^9*c)^(1/2) + 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e + 45*B^2*a*c^2*d^4*e^3*(a^9*c)^
(1/2) + 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 - 138*A*B*a^2*c*
d*e^6*(a^9*c)^(1/2) - 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2
- 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2)*1i)/(((d + e*x)^(1/2)*(800*A^2*a^12*c
^4*e^20 + 288*B^2*a^13*c^3*e^20 + 128*A^2*a^3*c^13*d^18*e^2 - 1760*A^2*a^4*c^12*d^16*e^4 + 10240*A^2*a^5*c^11*
d^14*e^6 - 30848*A^2*a^6*c^10*d^12*e^8 + 52480*A^2*a^7*c^9*d^10*e^10 - 51008*A^2*a^8*c^8*d^8*e^12 + 25600*A^2*
a^9*c^7*d^6*e^14 - 3200*A^2*a^10*c^6*d^4*e^16 - 2432*A^2*a^11*c^5*d^2*e^18 + 288*B^2*a^5*c^11*d^16*e^4 - 5760*
B^2*a^7*c^9*d^12*e^8 + 18432*B^2*a^8*c^8*d^10*e^10 - 25920*B^2*a^9*c^7*d^8*e^12 + 18432*B^2*a^10*c^6*d^6*e^14
- 5760*B^2*a^11*c^5*d^4*e^16 - 3456*A*B*a^12*c^4*d*e^19 + 384*A*B*a^4*c^12*d^17*e^3 - 3840*A*B*a^5*c^11*d^15*e
^5 + 11520*A*B*a^6*c^10*d^13*e^7 - 9984*A*B*a^7*c^9*d^11*e^9 - 15360*A*B*a^8*c^8*d^9*e^11 + 43776*A*B*a^9*c^7*
d^7*e^13 - 42240*A*B*a^10*c^6*d^5*e^15 + 19200*A*B*a^11*c^5*d^3*e^17) + (-(4*A^2*a^3*c^5*d^7 + 9*B^2*a^3*e^7*(
a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^
4 - 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 + 25*A^2*a^2*c*e^7*(a^9*c)^(
1/2) - 30*A*B*a^7*c*e^7 + 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) + 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*
c^4*d^6*e + 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) + 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 -
240*A*B*a^6*c^2*d^2*e^5 - 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) - 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*
c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1
/2)*(768*B*a^15*c^3*e^22 - (d + e*x)^(1/2)*(-(4*A^2*a^3*c^5*d^7 + 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4
*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 - 35*A^2*c^3*d^4*e^3*(a^9*c
)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 + 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 + 30*
A*B*c^3*d^5*e^2*(a^9*c)^(1/2) + 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e + 45*B^2*a*c^2*d^4*
e^3*(a^9*c)^(1/2) + 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 - 13
8*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) - 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7
*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2
048*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*a^14*c^4*d*e^21 + 256*A*a^5*c^13*d^19*e^3 - 5376*A*a^6*c
^12*d^17*e^5 + 33792*A*a^7*c^11*d^15*e^7 - 107520*A*a^8*c^10*d^13*e^9 + 204288*A*a^9*c^9*d^11*e^11 - 247296*A*
a^10*c^8*d^9*e^13 + 193536*A*a^11*c^7*d^7*e^15 - 95232*A*a^12*c^6*d^5*e^17 + 26880*A*a^13*c^5*d^3*e^19 + 2304*
B*a^6*c^12*d^18*e^4 - 17664*B*a^7*c^11*d^16*e^6 + 58368*B*a^8*c^10*d^14*e^8 - 107520*B*a^9*c^9*d^12*e^10 + 118
272*B*a^10*c^8*d^10*e^12 - 75264*B*a^11*c^7*d^8*e^14 + 21504*B*a^12*c^6*d^6*e^16 + 3072*B*a^13*c^5*d^4*e^18 -
3840*B*a^14*c^4*d^2*e^20))*(-(4*A^2*a^3*c^5*d^7 + 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^
2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 - 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2
*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 + 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 + 30*A*B*c^3*d^5*e^2*
(a^9*c)^(1/2) + 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e + 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2
) + 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 - 138*A*B*a^2*c*d*e^
6*(a^9*c)^(1/2) - 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 1
0*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2) - ((d + e*x)^(1/2)*(800*A^2*a^12*c^4*e^20
+ 288*B^2*a^13*c^3*e^20 + 128*A^2*a^3*c^13*d^18*e^2 - 1760*A^2*a^4*c^12*d^16*e^4 + 10240*A^2*a^5*c^11*d^14*e^
6 - 30848*A^2*a^6*c^10*d^12*e^8 + 52480*A^2*a^7*c^9*d^10*e^10 - 51008*A^2*a^8*c^8*d^8*e^12 + 25600*A^2*a^9*c^7
*d^6*e^14 - 3200*A^2*a^10*c^6*d^4*e^16 - 2432*A^2*a^11*c^5*d^2*e^18 + 288*B^2*a^5*c^11*d^16*e^4 - 5760*B^2*a^7
*c^9*d^12*e^8 + 18432*B^2*a^8*c^8*d^10*e^10 - 25920*B^2*a^9*c^7*d^8*e^12 + 18432*B^2*a^10*c^6*d^6*e^14 - 5760*
B^2*a^11*c^5*d^4*e^16 - 3456*A*B*a^12*c^4*d*e^19 + 384*A*B*a^4*c^12*d^17*e^3 - 3840*A*B*a^5*c^11*d^15*e^5 + 11
520*A*B*a^6*c^10*d^13*e^7 - 9984*A*B*a^7*c^9*d^11*e^9 - 15360*A*B*a^8*c^8*d^9*e^11 + 43776*A*B*a^9*c^7*d^7*e^1
3 - 42240*A*B*a^10*c^6*d^5*e^15 + 19200*A*B*a^11*c^5*d^3*e^17) - (-(4*A^2*a^3*c^5*d^7 + 9*B^2*a^3*e^7*(a^9*c)^
(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 - 35*
A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 + 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) -
30*A*B*a^7*c*e^7 + 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) + 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6
*e + 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) + 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*
B*a^6*c^2*d^2*e^5 - 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) - 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10
- a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2)*((d
+ e*x)^(1/2)*(-(4*A^2*a^3*c^5*d^7 + 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3
*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 - 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6
+ 105*A^2*a^6*c^2*d*e^6 + 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 + 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2)
+ 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e + 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) + 90*B^2*a^
2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 - 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/
2) - 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6
*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 - 20480
*a^7*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 - 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 - 516096*
a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 - 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 - 20480*a^1
5*c^5*d^3*e^20) + 768*B*a^15*c^3*e^22 - 3328*A*a^14*c^4*d*e^21 + 256*A*a^5*c^13*d^19*e^3 - 5376*A*a^6*c^12*d^1
7*e^5 + 33792*A*a^7*c^11*d^15*e^7 - 107520*A*a^8*c^10*d^13*e^9 + 204288*A*a^9*c^9*d^11*e^11 - 247296*A*a^10*c^
8*d^9*e^13 + 193536*A*a^11*c^7*d^7*e^15 - 95232*A*a^12*c^6*d^5*e^17 + 26880*A*a^13*c^5*d^3*e^19 + 2304*B*a^6*c
^12*d^18*e^4 - 17664*B*a^7*c^11*d^16*e^6 + 58368*B*a^8*c^10*d^14*e^8 - 107520*B*a^9*c^9*d^12*e^10 + 118272*B*a
^10*c^8*d^10*e^12 - 75264*B*a^11*c^7*d^8*e^14 + 21504*B*a^12*c^6*d^6*e^16 + 3072*B*a^13*c^5*d^4*e^18 - 3840*B*
a^14*c^4*d^2*e^20))*(-(4*A^2*a^3*c^5*d^7 + 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c
^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 - 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*
d*e^6 + 105*A^2*a^6*c^2*d*e^6 + 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 + 30*A*B*c^3*d^5*e^2*(a^9*c)
^(1/2) + 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e + 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) + 90*
B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 - 138*A*B*a^2*c*d*e^6*(a^9*
c)^(1/2) - 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c
^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2) + 1000*A^3*a^10*c^4*e^19 - 32*A^3*a^2*c^12*d^16*
e^3 + 232*A^3*a^3*c^11*d^14*e^5 + 280*A^3*a^4*c^10*d^12*e^7 - 4760*A^3*a^5*c^9*d^10*e^9 + 13720*A^3*a^6*c^8*d^
8*e^11 - 19208*A^3*a^7*c^7*d^6*e^13 + 14728*A^3*a^8*c^6*d^4*e^15 - 5960*A^3*a^9*c^5*d^2*e^17 + 432*B^3*a^5*c^9
*d^13*e^6 - 2592*B^3*a^6*c^8*d^11*e^8 + 6480*B^3*a^7*c^7*d^9*e^10 - 8640*B^3*a^8*c^6*d^7*e^12 + 6480*B^3*a^9*c
^5*d^5*e^14 - 2592*B^3*a^10*c^4*d^3*e^16 - 360*A*B^2*a^11*c^3*e^19 + 432*B^3*a^11*c^3*d*e^18 + 504*A*B^2*a^4*c
^10*d^14*e^5 - 3384*A*B^2*a^5*c^9*d^12*e^7 + 9720*A*B^2*a^6*c^8*d^10*e^9 - 15480*A*B^2*a^7*c^7*d^8*e^11 + 1476
0*A*B^2*a^8*c^6*d^6*e^13 - 8424*A*B^2*a^9*c^5*d^4*e^15 + 2664*A*B^2*a^10*c^4*d^2*e^17 + 96*A^2*B*a^3*c^11*d^15
*e^4 - 2256*A^2*B*a^4*c^10*d^13*e^6 + 11520*A^2*B*a^5*c^9*d^11*e^8 - 27120*A^2*B*a^6*c^8*d^9*e^10 + 35040*A^2*
B*a^7*c^7*d^7*e^12 - 25776*A^2*B*a^8*c^6*d^5*e^14 + 10176*A^2*B*a^9*c^5*d^3*e^16 - 1680*A^2*B*a^10*c^4*d*e^18)
)*(-(4*A^2*a^3*c^5*d^7 + 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2
*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 - 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^
6*c^2*d*e^6 + 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 + 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) + 154*A^2*a
*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e + 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) + 90*B^2*a^2*c*d^2*e^5*
(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 - 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) - 180*A*B
*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^
9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2)*2i - atan((((d + e*x)^(1/2)*(800*A^2*a^12*c^4*e^20 + 288*B^2*a^13*
c^3*e^20 + 128*A^2*a^3*c^13*d^18*e^2 - 1760*A^2*a^4*c^12*d^16*e^4 + 10240*A^2*a^5*c^11*d^14*e^6 - 30848*A^2*a^
6*c^10*d^12*e^8 + 52480*A^2*a^7*c^9*d^10*e^10 - 51008*A^2*a^8*c^8*d^8*e^12 + 25600*A^2*a^9*c^7*d^6*e^14 - 3200
*A^2*a^10*c^6*d^4*e^16 - 2432*A^2*a^11*c^5*d^2*e^18 + 288*B^2*a^5*c^11*d^16*e^4 - 5760*B^2*a^7*c^9*d^12*e^8 +
18432*B^2*a^8*c^8*d^10*e^10 - 25920*B^2*a^9*c^7*d^8*e^12 + 18432*B^2*a^10*c^6*d^6*e^14 - 5760*B^2*a^11*c^5*d^4
*e^16 - 3456*A*B*a^12*c^4*d*e^19 + 384*A*B*a^4*c^12*d^17*e^3 - 3840*A*B*a^5*c^11*d^15*e^5 + 11520*A*B*a^6*c^10
*d^13*e^7 - 9984*A*B*a^7*c^9*d^11*e^9 - 15360*A*B*a^8*c^8*d^9*e^11 + 43776*A*B*a^9*c^7*d^7*e^13 - 42240*A*B*a^
10*c^6*d^5*e^15 + 19200*A*B*a^11*c^5*d^3*e^17) + (-(4*A^2*a^3*c^5*d^7 - 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a
^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 + 35*A^2*c^3*d^4*e^3*
(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 - 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7
- 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) - 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e - 45*B^2*a*c^
2*d^4*e^3*(a^9*c)^(1/2) - 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^
5 + 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) + 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 +
5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2)*(768*B*a^15*c^3*e^2
2 - (d + e*x)^(1/2)*(-(4*A^2*a^3*c^5*d^7 - 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c
^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 + 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*
d*e^6 + 105*A^2*a^6*c^2*d*e^6 - 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 - 30*A*B*c^3*d^5*e^2*(a^9*c)
^(1/2) - 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e - 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) - 90*
B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 + 138*A*B*a^2*c*d*e^6*(a^9*
c)^(1/2) + 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c
^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(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 - 5
16096*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 - 204
80*a^15*c^5*d^3*e^20) - 3328*A*a^14*c^4*d*e^21 + 256*A*a^5*c^13*d^19*e^3 - 5376*A*a^6*c^12*d^17*e^5 + 33792*A*
a^7*c^11*d^15*e^7 - 107520*A*a^8*c^10*d^13*e^9 + 204288*A*a^9*c^9*d^11*e^11 - 247296*A*a^10*c^8*d^9*e^13 + 193
536*A*a^11*c^7*d^7*e^15 - 95232*A*a^12*c^6*d^5*e^17 + 26880*A*a^13*c^5*d^3*e^19 + 2304*B*a^6*c^12*d^18*e^4 - 1
7664*B*a^7*c^11*d^16*e^6 + 58368*B*a^8*c^10*d^14*e^8 - 107520*B*a^9*c^9*d^12*e^10 + 118272*B*a^10*c^8*d^10*e^1
2 - 75264*B*a^11*c^7*d^8*e^14 + 21504*B*a^12*c^6*d^6*e^16 + 3072*B*a^13*c^5*d^4*e^18 - 3840*B*a^14*c^4*d^2*e^2
0))*(-(4*A^2*a^3*c^5*d^7 - 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B
^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 + 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*
a^6*c^2*d*e^6 - 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 - 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) - 154*A^2
*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e - 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) - 90*B^2*a^2*c*d^2*e^
5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 + 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) + 180*A
*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*
a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2)*1i + ((d + e*x)^(1/2)*(800*A^2*a^12*c^4*e^20 + 288*B^2*a^13*c^3*
e^20 + 128*A^2*a^3*c^13*d^18*e^2 - 1760*A^2*a^4*c^12*d^16*e^4 + 10240*A^2*a^5*c^11*d^14*e^6 - 30848*A^2*a^6*c^
10*d^12*e^8 + 52480*A^2*a^7*c^9*d^10*e^10 - 51008*A^2*a^8*c^8*d^8*e^12 + 25600*A^2*a^9*c^7*d^6*e^14 - 3200*A^2
*a^10*c^6*d^4*e^16 - 2432*A^2*a^11*c^5*d^2*e^18 + 288*B^2*a^5*c^11*d^16*e^4 - 5760*B^2*a^7*c^9*d^12*e^8 + 1843
2*B^2*a^8*c^8*d^10*e^10 - 25920*B^2*a^9*c^7*d^8*e^12 + 18432*B^2*a^10*c^6*d^6*e^14 - 5760*B^2*a^11*c^5*d^4*e^1
6 - 3456*A*B*a^12*c^4*d*e^19 + 384*A*B*a^4*c^12*d^17*e^3 - 3840*A*B*a^5*c^11*d^15*e^5 + 11520*A*B*a^6*c^10*d^1
3*e^7 - 9984*A*B*a^7*c^9*d^11*e^9 - 15360*A*B*a^8*c^8*d^9*e^11 + 43776*A*B*a^9*c^7*d^7*e^13 - 42240*A*B*a^10*c
^6*d^5*e^15 + 19200*A*B*a^11*c^5*d^3*e^17) - (-(4*A^2*a^3*c^5*d^7 - 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c
^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 + 35*A^2*c^3*d^4*e^3*(a^9
*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 - 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 - 3
0*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) - 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e - 45*B^2*a*c^2*d^
4*e^3*(a^9*c)^(1/2) - 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 +
138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) + 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a
^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2)*((d + e*x)^(1/2)*(-(4*A
^2*a^3*c^5*d^7 - 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3
*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 + 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*
e^6 - 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 - 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) - 154*A^2*a*c^2*d^2
*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e - 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) - 90*B^2*a^2*c*d^2*e^5*(a^9*c)^
(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 + 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) + 180*A*B*a*c^2*d
^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^
4*e^6 - 5*a^10*c^2*d^2*e^8)))^(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) + 76
8*B*a^15*c^3*e^22 - 3328*A*a^14*c^4*d*e^21 + 256*A*a^5*c^13*d^19*e^3 - 5376*A*a^6*c^12*d^17*e^5 + 33792*A*a^7*
c^11*d^15*e^7 - 107520*A*a^8*c^10*d^13*e^9 + 204288*A*a^9*c^9*d^11*e^11 - 247296*A*a^10*c^8*d^9*e^13 + 193536*
A*a^11*c^7*d^7*e^15 - 95232*A*a^12*c^6*d^5*e^17 + 26880*A*a^13*c^5*d^3*e^19 + 2304*B*a^6*c^12*d^18*e^4 - 17664
*B*a^7*c^11*d^16*e^6 + 58368*B*a^8*c^10*d^14*e^8 - 107520*B*a^9*c^9*d^12*e^10 + 118272*B*a^10*c^8*d^10*e^12 -
75264*B*a^11*c^7*d^8*e^14 + 21504*B*a^12*c^6*d^6*e^16 + 3072*B*a^13*c^5*d^4*e^18 - 3840*B*a^14*c^4*d^2*e^20))*
(-(4*A^2*a^3*c^5*d^7 - 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a
^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 + 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*
c^2*d*e^6 - 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 - 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) - 154*A^2*a*c
^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e - 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) - 90*B^2*a^2*c*d^2*e^5*(a
^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 + 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) + 180*A*B*a
*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*
c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2)*1i)/(((d + e*x)^(1/2)*(800*A^2*a^12*c^4*e^20 + 288*B^2*a^13*c^3*e^20
+ 128*A^2*a^3*c^13*d^18*e^2 - 1760*A^2*a^4*c^12*d^16*e^4 + 10240*A^2*a^5*c^11*d^14*e^6 - 30848*A^2*a^6*c^10*d
^12*e^8 + 52480*A^2*a^7*c^9*d^10*e^10 - 51008*A^2*a^8*c^8*d^8*e^12 + 25600*A^2*a^9*c^7*d^6*e^14 - 3200*A^2*a^1
0*c^6*d^4*e^16 - 2432*A^2*a^11*c^5*d^2*e^18 + 288*B^2*a^5*c^11*d^16*e^4 - 5760*B^2*a^7*c^9*d^12*e^8 + 18432*B^
2*a^8*c^8*d^10*e^10 - 25920*B^2*a^9*c^7*d^8*e^12 + 18432*B^2*a^10*c^6*d^6*e^14 - 5760*B^2*a^11*c^5*d^4*e^16 -
3456*A*B*a^12*c^4*d*e^19 + 384*A*B*a^4*c^12*d^17*e^3 - 3840*A*B*a^5*c^11*d^15*e^5 + 11520*A*B*a^6*c^10*d^13*e^
7 - 9984*A*B*a^7*c^9*d^11*e^9 - 15360*A*B*a^8*c^8*d^9*e^11 + 43776*A*B*a^9*c^7*d^7*e^13 - 42240*A*B*a^10*c^6*d
^5*e^15 + 19200*A*B*a^11*c^5*d^3*e^17) + (-(4*A^2*a^3*c^5*d^7 - 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d
^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 + 35*A^2*c^3*d^4*e^3*(a^9*c)^
(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 - 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 - 30*A*
B*c^3*d^5*e^2*(a^9*c)^(1/2) - 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e - 45*B^2*a*c^2*d^4*e^
3*(a^9*c)^(1/2) - 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 + 138*
A*B*a^2*c*d*e^6*(a^9*c)^(1/2) + 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c
^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2)*(768*B*a^15*c^3*e^22 - (d +
e*x)^(1/2)*(-(4*A^2*a^3*c^5*d^7 - 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e
^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 + 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 +
105*A^2*a^6*c^2*d*e^6 - 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 - 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) -
154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e - 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) - 90*B^2*a^2*
c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 + 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2)
+ 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e
^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(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*a^14*c^4*d*e^21 + 256*A*a^5*c^13*d^19*e^3 - 5376*A*a^6*c^12*d^17*e^5 + 33792*A*a^7*c^11
*d^15*e^7 - 107520*A*a^8*c^10*d^13*e^9 + 204288*A*a^9*c^9*d^11*e^11 - 247296*A*a^10*c^8*d^9*e^13 + 193536*A*a^
11*c^7*d^7*e^15 - 95232*A*a^12*c^6*d^5*e^17 + 26880*A*a^13*c^5*d^3*e^19 + 2304*B*a^6*c^12*d^18*e^4 - 17664*B*a
^7*c^11*d^16*e^6 + 58368*B*a^8*c^10*d^14*e^8 - 107520*B*a^9*c^9*d^12*e^10 + 118272*B*a^10*c^8*d^10*e^12 - 7526
4*B*a^11*c^7*d^8*e^14 + 21504*B*a^12*c^6*d^6*e^16 + 3072*B*a^13*c^5*d^4*e^18 - 3840*B*a^14*c^4*d^2*e^20))*(-(4
*A^2*a^3*c^5*d^7 - 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c
^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 + 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*
d*e^6 - 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 - 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) - 154*A^2*a*c^2*d
^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e - 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) - 90*B^2*a^2*c*d^2*e^5*(a^9*c
)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 + 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) + 180*A*B*a*c^2
*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*
d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2) - ((d + e*x)^(1/2)*(800*A^2*a^12*c^4*e^20 + 288*B^2*a^13*c^3*e^20 + 128*
A^2*a^3*c^13*d^18*e^2 - 1760*A^2*a^4*c^12*d^16*e^4 + 10240*A^2*a^5*c^11*d^14*e^6 - 30848*A^2*a^6*c^10*d^12*e^8
+ 52480*A^2*a^7*c^9*d^10*e^10 - 51008*A^2*a^8*c^8*d^8*e^12 + 25600*A^2*a^9*c^7*d^6*e^14 - 3200*A^2*a^10*c^6*d
^4*e^16 - 2432*A^2*a^11*c^5*d^2*e^18 + 288*B^2*a^5*c^11*d^16*e^4 - 5760*B^2*a^7*c^9*d^12*e^8 + 18432*B^2*a^8*c
^8*d^10*e^10 - 25920*B^2*a^9*c^7*d^8*e^12 + 18432*B^2*a^10*c^6*d^6*e^14 - 5760*B^2*a^11*c^5*d^4*e^16 - 3456*A*
B*a^12*c^4*d*e^19 + 384*A*B*a^4*c^12*d^17*e^3 - 3840*A*B*a^5*c^11*d^15*e^5 + 11520*A*B*a^6*c^10*d^13*e^7 - 998
4*A*B*a^7*c^9*d^11*e^9 - 15360*A*B*a^8*c^8*d^9*e^11 + 43776*A*B*a^9*c^7*d^7*e^13 - 42240*A*B*a^10*c^6*d^5*e^15
+ 19200*A*B*a^11*c^5*d^3*e^17) - (-(4*A^2*a^3*c^5*d^7 - 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2
+ 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c^2*d^3*e^4 + 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) +
45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 - 25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 - 30*A*B*c^3*d
^5*e^2*(a^9*c)^(1/2) - 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e - 45*B^2*a*c^2*d^4*e^3*(a^9*
c)^(1/2) - 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 + 138*A*B*a^2
*c*d*e^6*(a^9*c)^(1/2) + 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*
e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*e^8)))^(1/2)*((d + e*x)^(1/2)*(-(4*A^2*a^3*c^5*
d^7 - 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 +
90*B^2*a^6*c^2*d^3*e^4 + 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 - 25*A^
2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 - 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) - 154*A^2*a*c^2*d^2*e^5*(a^9*c
)^(1/2) + 12*A*B*a^4*c^4*d^6*e - 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) - 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*
A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 + 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) + 180*A*B*a*c^2*d^3*e^4*(a^9
*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a
^10*c^2*d^2*e^8)))^(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^1
2*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) + 768*B*a^15*c^
3*e^22 - 3328*A*a^14*c^4*d*e^21 + 256*A*a^5*c^13*d^19*e^3 - 5376*A*a^6*c^12*d^17*e^5 + 33792*A*a^7*c^11*d^15*e
^7 - 107520*A*a^8*c^10*d^13*e^9 + 204288*A*a^9*c^9*d^11*e^11 - 247296*A*a^10*c^8*d^9*e^13 + 193536*A*a^11*c^7*
d^7*e^15 - 95232*A*a^12*c^6*d^5*e^17 + 26880*A*a^13*c^5*d^3*e^19 + 2304*B*a^6*c^12*d^18*e^4 - 17664*B*a^7*c^11
*d^16*e^6 + 58368*B*a^8*c^10*d^14*e^8 - 107520*B*a^9*c^9*d^12*e^10 + 118272*B*a^10*c^8*d^10*e^12 - 75264*B*a^1
1*c^7*d^8*e^14 + 21504*B*a^12*c^6*d^6*e^16 + 3072*B*a^13*c^5*d^4*e^18 - 3840*B*a^14*c^4*d^2*e^20))*(-(4*A^2*a^
3*c^5*d^7 - 9*B^2*a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*
e^2 + 90*B^2*a^6*c^2*d^3*e^4 + 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 -
25*A^2*a^2*c*e^7*(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 - 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) - 154*A^2*a*c^2*d^2*e^5*
(a^9*c)^(1/2) + 12*A*B*a^4*c^4*d^6*e - 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) - 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2)
- 30*A*B*a^5*c^3*d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 + 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) + 180*A*B*a*c^2*d^3*e^
4*(a^9*c)^(1/2))/(64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6
- 5*a^10*c^2*d^2*e^8)))^(1/2) + 1000*A^3*a^10*c^4*e^19 - 32*A^3*a^2*c^12*d^16*e^3 + 232*A^3*a^3*c^11*d^14*e^5
+ 280*A^3*a^4*c^10*d^12*e^7 - 4760*A^3*a^5*c^9*d^10*e^9 + 13720*A^3*a^6*c^8*d^8*e^11 - 19208*A^3*a^7*c^7*d^6*
e^13 + 14728*A^3*a^8*c^6*d^4*e^15 - 5960*A^3*a^9*c^5*d^2*e^17 + 432*B^3*a^5*c^9*d^13*e^6 - 2592*B^3*a^6*c^8*d^
11*e^8 + 6480*B^3*a^7*c^7*d^9*e^10 - 8640*B^3*a^8*c^6*d^7*e^12 + 6480*B^3*a^9*c^5*d^5*e^14 - 2592*B^3*a^10*c^4
*d^3*e^16 - 360*A*B^2*a^11*c^3*e^19 + 432*B^3*a^11*c^3*d*e^18 + 504*A*B^2*a^4*c^10*d^14*e^5 - 3384*A*B^2*a^5*c
^9*d^12*e^7 + 9720*A*B^2*a^6*c^8*d^10*e^9 - 15480*A*B^2*a^7*c^7*d^8*e^11 + 14760*A*B^2*a^8*c^6*d^6*e^13 - 8424
*A*B^2*a^9*c^5*d^4*e^15 + 2664*A*B^2*a^10*c^4*d^2*e^17 + 96*A^2*B*a^3*c^11*d^15*e^4 - 2256*A^2*B*a^4*c^10*d^13
*e^6 + 11520*A^2*B*a^5*c^9*d^11*e^8 - 27120*A^2*B*a^6*c^8*d^9*e^10 + 35040*A^2*B*a^7*c^7*d^7*e^12 - 25776*A^2*
B*a^8*c^6*d^5*e^14 + 10176*A^2*B*a^9*c^5*d^3*e^16 - 1680*A^2*B*a^10*c^4*d*e^18))*(-(4*A^2*a^3*c^5*d^7 - 9*B^2*
a^3*e^7*(a^9*c)^(1/2) - 35*A^2*a^4*c^4*d^5*e^2 + 70*A^2*a^5*c^3*d^3*e^4 + 9*B^2*a^5*c^3*d^5*e^2 + 90*B^2*a^6*c
^2*d^3*e^4 + 35*A^2*c^3*d^4*e^3*(a^9*c)^(1/2) + 45*B^2*a^7*c*d*e^6 + 105*A^2*a^6*c^2*d*e^6 - 25*A^2*a^2*c*e^7*
(a^9*c)^(1/2) - 30*A*B*a^7*c*e^7 - 30*A*B*c^3*d^5*e^2*(a^9*c)^(1/2) - 154*A^2*a*c^2*d^2*e^5*(a^9*c)^(1/2) + 12
*A*B*a^4*c^4*d^6*e - 45*B^2*a*c^2*d^4*e^3*(a^9*c)^(1/2) - 90*B^2*a^2*c*d^2*e^5*(a^9*c)^(1/2) - 30*A*B*a^5*c^3*
d^4*e^3 - 240*A*B*a^6*c^2*d^2*e^5 + 138*A*B*a^2*c*d*e^6*(a^9*c)^(1/2) + 180*A*B*a*c^2*d^3*e^4*(a^9*c)^(1/2))/(
64*(a^11*c*e^10 - a^6*c^6*d^10 + 5*a^7*c^5*d^8*e^2 - 10*a^8*c^4*d^6*e^4 + 10*a^9*c^3*d^4*e^6 - 5*a^10*c^2*d^2*
e^8)))^(1/2)*2i

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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