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

Optimal. Leaf size=417 \[ -\frac {\left (3 A \left (-10 \sqrt {a} c d e+7 a \sqrt {c} e^2+4 c^{3/2} d^2\right )+a B e \left (2 \sqrt {c} d-5 \sqrt {a} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {\left (3 A \left (10 \sqrt {a} c d e+7 a \sqrt {c} e^2+4 c^{3/2} d^2\right )+a B e \left (5 \sqrt {a} e+2 \sqrt {c} d\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{3/4} \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2}}-\frac {\sqrt {d+e x} \left (a e \left (-7 a A e^2+6 a B d e+A c d^2\right )-x \left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (5 a e^2+c d^2\right )\right )\right )}{16 a^2 \left (a-c x^2\right ) \left (c d^2-a e^2\right )^2}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )} \]

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Rubi [A]  time = 0.93, antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {823, 827, 1166, 208} \begin {gather*} -\frac {\left (3 A \left (-10 \sqrt {a} c d e+7 a \sqrt {c} e^2+4 c^{3/2} d^2\right )+a B e \left (2 \sqrt {c} d-5 \sqrt {a} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {\left (3 A \left (10 \sqrt {a} c d e+7 a \sqrt {c} e^2+4 c^{3/2} d^2\right )+a B e \left (5 \sqrt {a} e+2 \sqrt {c} d\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{3/4} \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2}}-\frac {\sqrt {d+e x} \left (a e \left (-7 a A e^2+6 a B d e+A c d^2\right )-x \left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (5 a e^2+c d^2\right )\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} (x (A c d-a B e)+a (B d-A e))}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 1.06, size = 536, normalized size = 1.29 \begin {gather*} \frac {\frac {c^2 \sqrt {d+e x} \left (a^2 e^2 (7 A e-6 B d+5 B e x)+a c d e (B d x-A (d+12 e x))+6 A c^2 d^3 x\right )}{2 \left (a-c x^2\right )}+\frac {c^{7/4} \left (3 A \left (7 a^2 e^4-5 a c d^2 e^2+2 c^2 d^4\right )+a B d e \left (c d^2-13 a e^2\right )\right ) \left (\sqrt {\sqrt {c} d-\sqrt {a} e} \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} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )\right )}{4 \sqrt {a} \sqrt {\sqrt {c} d-\sqrt {a} e} \sqrt {\sqrt {a} e+\sqrt {c} d}}-\frac {c^{5/4} \left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (5 a e^2+c d^2\right )\right ) \left (\sqrt {\sqrt {c} d-\sqrt {a} e} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )-\sqrt {\sqrt {a} e+\sqrt {c} d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )\right )}{4 \sqrt {a}}+\frac {2 a c^2 \sqrt {d+e x} \left (c d^2-a e^2\right ) (-a A e+a B (d-e x)+A c d x)}{\left (a-c x^2\right )^2}}{8 a^2 c^2 \left (c d^2-a e^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a*c^2*(c*d^2 - a*e^2)*Sqrt[d + e*x]*(-(a*A*e) + A*c*d*x + a*B*(d - e*x)))/(a - c*x^2)^2 + (c^2*Sqrt[d + e*
x]*(6*A*c^2*d^3*x + a^2*e^2*(-6*B*d + 7*A*e + 5*B*e*x) + a*c*d*e*(B*d*x - A*(d + 12*e*x))))/(2*(a - c*x^2)) +
(c^(7/4)*(a*B*d*e*(c*d^2 - 13*a*e^2) + 3*A*(2*c^2*d^4 - 5*a*c*d^2*e^2 + 7*a^2*e^4))*(-(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]*ArcTanh[(c^(1
/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]))/(4*Sqrt[a]*Sqrt[Sqrt[c]*d - Sqrt[a]*e]*Sqrt[Sqrt[c]*d + Sqrt
[a]*e]) - (c^(5/4)*(6*A*c*d*(c*d^2 - 2*a*e^2) + a*B*e*(c*d^2 + 5*a*e^2))*(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]*ArcTanh[(c^(1/4)*Sqrt[d + e
*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]))/(4*Sqrt[a]))/(8*a^2*c^2*(c*d^2 - a*e^2)^2)

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IntegrateAlgebraic [A]  time = 2.26, size = 727, normalized size = 1.74 \begin {gather*} \frac {\left (5 a^{3/2} B e^2+30 \sqrt {a} A c d e+21 a A \sqrt {c} e^2+2 a B \sqrt {c} d e+12 A c^{3/2} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{32 a^{5/2} \sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )^2 \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}+\frac {\left (5 a^{3/2} B e^2+30 \sqrt {a} A c d e-21 a A \sqrt {c} e^2-2 a B \sqrt {c} d e-12 A c^{3/2} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{32 a^{5/2} \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )^2 \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}}+\frac {e \sqrt {d+e x} \left (11 a^3 A e^6+9 a^3 B e^5 (d+e x)-19 a^3 B d e^5+4 a^2 A c d^2 e^4-2 a^2 A c d e^4 (d+e x)-7 a^2 A c e^4 (d+e x)^2+18 a^2 B c d^3 e^3-30 a^2 B c d^2 e^3 (d+e x)+21 a^2 B c d e^3 (d+e x)^2-5 a^2 B c e^3 (d+e x)^3-21 a A c^2 d^4 e^2+44 a A c^2 d^3 e^2 (d+e x)-35 a A c^2 d^2 e^2 (d+e x)^2+12 a A c^2 d e^2 (d+e x)^3+a B c^2 d^5 e-3 a B c^2 d^4 e (d+e x)+3 a B c^2 d^3 e (d+e x)^2-a B c^2 d^2 e (d+e x)^3+6 A c^3 d^6-18 A c^3 d^5 (d+e x)+18 A c^3 d^4 (d+e x)^2-6 A c^3 d^3 (d+e x)^3\right )}{16 a^2 \left (a e^2-c d^2\right )^2 \left (a e^2-c d^2+2 c d (d+e x)-c (d+e x)^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Sqrt[d + e*x]*(6*A*c^3*d^6 + a*B*c^2*d^5*e - 21*a*A*c^2*d^4*e^2 + 18*a^2*B*c*d^3*e^3 + 4*a^2*A*c*d^2*e^4 -
19*a^3*B*d*e^5 + 11*a^3*A*e^6 - 18*A*c^3*d^5*(d + e*x) - 3*a*B*c^2*d^4*e*(d + e*x) + 44*a*A*c^2*d^3*e^2*(d + e
*x) - 30*a^2*B*c*d^2*e^3*(d + e*x) - 2*a^2*A*c*d*e^4*(d + e*x) + 9*a^3*B*e^5*(d + e*x) + 18*A*c^3*d^4*(d + e*x
)^2 + 3*a*B*c^2*d^3*e*(d + e*x)^2 - 35*a*A*c^2*d^2*e^2*(d + e*x)^2 + 21*a^2*B*c*d*e^3*(d + e*x)^2 - 7*a^2*A*c*
e^4*(d + e*x)^2 - 6*A*c^3*d^3*(d + e*x)^3 - a*B*c^2*d^2*e*(d + e*x)^3 + 12*a*A*c^2*d*e^2*(d + e*x)^3 - 5*a^2*B
*c*e^3*(d + e*x)^3))/(16*a^2*(-(c*d^2) + a*e^2)^2*(-(c*d^2) + a*e^2 + 2*c*d*(d + e*x) - c*(d + e*x)^2)^2) + ((
12*A*c^(3/2)*d^2 + 2*a*B*Sqrt[c]*d*e + 30*Sqrt[a]*A*c*d*e + 5*a^(3/2)*B*e^2 + 21*a*A*Sqrt[c]*e^2)*ArcTan[(Sqrt
[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(32*a^(5/2)*Sqrt[c]*(Sqrt[c]*d + Sqrt[a]
*e)^2*Sqrt[-(Sqrt[c]*(Sqrt[c]*d + Sqrt[a]*e))]) + ((-12*A*c^(3/2)*d^2 - 2*a*B*Sqrt[c]*d*e + 30*Sqrt[a]*A*c*d*e
 + 5*a^(3/2)*B*e^2 - 21*a*A*Sqrt[c]*e^2)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d -
Sqrt[a]*e)])/(32*a^(5/2)*Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e)^2*Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:inde
x.cc index_m operator + Error: Bad Argument Value

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maple [B]  time = 0.71, size = 1778, normalized size = 4.26

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15/16*e*c/a^2/(-a*e^2-c*d^2+2*(a*c*e^2)^(1/2)*d)/((-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+15/16*e*c/a^2/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2)*d)/((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+9/32*e/c*(a*c*e^2)^(1/2)/a^2/(e*x+(a*c*e^2)^(1
/2)/c)^2/(a*e^2+c*d^2-2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)*A-11/32*e/c*(a*c*e^2)^(1/2)/a^2/(e*x+(a*c*e^2)^(1/2)/
c)^2/(c*d-(a*c*e^2)^(1/2))*(e*x+d)^(1/2)*A-21/32*e^3*c/(a*c*e^2)^(1/2)/a/(-a*e^2-c*d^2+2*(a*c*e^2)^(1/2)*d)/((
-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-9/32*e/c*(a*c*e^2)^(
1/2)/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)*A+11/32*e/c*(a*c*e^2)^(1/2)
/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2/(c*d+(a*c*e^2)^(1/2))*(e*x+d)^(1/2)*A+21/32*e^3*c/(a*c*e^2)^(1/2)/a/(a*e^2+c*d^
2+2*(a*c*e^2)^(1/2)*d)/((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+1/16*e^2*c/(a*c*e^2)^(1/2)/a/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2)*d)/((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+3/8*e*c^2/(a*c*e^2)^(1/2)/a^2/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2)
*d)/((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^2-3/8*e*c^2/(
a*c*e^2)^(1/2)/a^2/(-a*e^2-c*d^2+2*(a*c*e^2)^(1/2)*d)/((-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^2-1/16*e^2*c/(a*c*e^2)^(1/2)/a/(-a*e^2-c*d^2+2*(a*c*e^2)^(1/2)*d)/((-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-1/16/c*(a*c*e^2)^(1/2)
/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)*B*d+1/16/c*(a*c*e^2)^(1/2)/a^2/
(e*x-(a*c*e^2)^(1/2)/c)^2/(c*d+(a*c*e^2)^(1/2))*(e*x+d)^(1/2)*B*d+1/16/c*(a*c*e^2)^(1/2)/a^2/(e*x+(a*c*e^2)^(1
/2)/c)^2/(a*e^2+c*d^2-2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)*B*d-1/16/c*(a*c*e^2)^(1/2)/a^2/(e*x+(a*c*e^2)^(1/2)/c
)^2/(c*d-(a*c*e^2)^(1/2))*(e*x+d)^(1/2)*B*d+3/16*e/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2/(c*d+(a*c*e^2)^(1/2))*(e*x+d)
^(1/2)*A*d-3/16*e/a^2/(e*x+(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2-2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)*A*d+3/16*e/a^2
/(e*x+(a*c*e^2)^(1/2)/c)^2/(c*d-(a*c*e^2)^(1/2))*(e*x+d)^(1/2)*A*d-3/16*e/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2/(a*e^2
+c*d^2+2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)*A*d+7/32*e^2/c/a/(e*x+(a*c*e^2)^(1/2)/c)^2/(c*d-(a*c*e^2)^(1/2))*(e*
x+d)^(1/2)*B-5/32*e^2/c/a/(e*x+(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2-2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)*B+7/32*e^2
/c/a/(e*x-(a*c*e^2)^(1/2)/c)^2/(c*d+(a*c*e^2)^(1/2))*(e*x+d)^(1/2)*B-5/32*e^2/c/a/(e*x-(a*c*e^2)^(1/2)/c)^2/(a
*e^2+c*d^2+2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)*B+5/32*e^2/a/(-a*e^2-c*d^2+2*(a*c*e^2)^(1/2)*d)/((-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+5/32*e^2/a/(a*e^2+c*d^2+2*(a*c*e^2
)^(1/2)*d)/((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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 9.23, size = 19125, normalized size = 45.86

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- atan(((((86016*A*a^9*c^3*e^11 - 53248*B*a^9*c^3*d*e^10 + 24576*A*a^5*c^7*d^8*e^3 - 110592*A*a^6*c^6*d^6*e^5
+ 233472*A*a^7*c^5*d^4*e^7 - 233472*A*a^8*c^4*d^2*e^9 + 4096*B*a^6*c^6*d^7*e^4 - 61440*B*a^7*c^5*d^5*e^6 + 110
592*B*a^8*c^4*d^3*e^8)/(4096*(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)*((144*A^2*a^5*c^7*d^9 - 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701
*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*
c^3*d^3*e^6 - 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 - 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2)
 + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 + 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*
e + 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c
^3*d^2*e^7 + 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) - 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) + 666*A*B*a^2*c*d*
e^8*(a^15*c^3)^(1/2) - 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c
^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 40
96*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)))*((144*A^2*a^5*c^7*d^9 - 25*B^2*a^3*e^9*
(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c
^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 - 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^
10*c^2*e^9 - 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 + 210*A*B*c
^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e + 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*
d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 + 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) - 154*B^2*
a^2*c*d^2*e^7*(a^15*c^3)^(1/2) + 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) - 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2)
)/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^
14*c^4*d^2*e^8)))^(1/2) + ((d + e*x)^(1/2)*(441*A^2*a^4*c^3*e^10 + 25*B^2*a^5*c^2*e^10 + 144*A^2*c^7*d^8*e^2 +
 1089*A^2*a^2*c^5*d^4*e^6 - 990*A^2*a^3*c^4*d^2*e^8 + 4*B^2*a^2*c^5*d^6*e^4 - 31*B^2*a^3*c^4*d^4*e^6 + 74*B^2*
a^4*c^3*d^2*e^8 - 612*A^2*a*c^6*d^6*e^4 + 48*A*B*a*c^6*d^7*e^3 - 456*A*B*a^4*c^3*d*e^9 - 288*A*B*a^2*c^5*d^5*e
^5 + 552*A*B*a^3*c^4*d^3*e^7))/(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)))*((144*A^2*a^5*c^7*d^9 - 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*
d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 -
 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 - 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^
9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 + 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e + 486*A^2*a
*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 +
35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) - 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) + 666*A*B*a^2*c*d*e^8*(a^15*c^3
)^(1/2) - 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 +
10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8)))^(1/2)*1i - (((86016*A*a^9*c^3*e^11 - 53248*B
*a^9*c^3*d*e^10 + 24576*A*a^5*c^7*d^8*e^3 - 110592*A*a^6*c^6*d^6*e^5 + 233472*A*a^7*c^5*d^4*e^7 - 233472*A*a^8
*c^4*d^2*e^9 + 4096*B*a^6*c^6*d^7*e^4 - 61440*B*a^7*c^5*d^5*e^6 + 110592*B*a^8*c^4*d^3*e^8)/(4096*(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)*((144*A^2*a^5*c^7*d
^9 - 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d
^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 - 441*A^2*a^2*c*e^9*(a^15*c^3
)^(1/2) - 210*A*B*a^10*c^2*e^9 - 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c
^2*d*e^8 + 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e + 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2
) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 + 35*B^2*a*c^2*d^4*e^5*(a^15*c
^3)^(1/2) - 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) + 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) - 588*A*B*a*c^2*d^3*
e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^1
3*c^5*d^4*e^6 + 5*a^14*c^4*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)))*((144*A^2*a^5*c^7*d^9 - 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^
2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*
B^2*a^9*c^3*d^3*e^6 - 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 - 189*A^2*c^3*d^4*e^5*(a^15*c^
3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 + 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*
c^6*d^8*e + 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A
*B*a^9*c^3*d^2*e^7 + 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) - 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) + 666*A*B*
a^2*c*d*e^8*(a^15*c^3)^(1/2) - 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 -
5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8)))^(1/2) - ((d + e*x)^(1/2
)*(441*A^2*a^4*c^3*e^10 + 25*B^2*a^5*c^2*e^10 + 144*A^2*c^7*d^8*e^2 + 1089*A^2*a^2*c^5*d^4*e^6 - 990*A^2*a^3*c
^4*d^2*e^8 + 4*B^2*a^2*c^5*d^6*e^4 - 31*B^2*a^3*c^4*d^4*e^6 + 74*B^2*a^4*c^3*d^2*e^8 - 612*A^2*a*c^6*d^6*e^4 +
 48*A*B*a*c^6*d^7*e^3 - 456*A*B*a^4*c^3*d*e^9 - 288*A*B*a^2*c^5*d^5*e^5 + 552*A*B*a^3*c^4*d^3*e^7))/(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)))*((144*A^2*a^5*c^7*d^9 - 25*B^2*a
^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^
2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 - 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210
*A*B*a^10*c^2*e^9 - 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 + 21
0*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e + 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a
^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 + 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) - 1
54*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) + 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) - 588*A*B*a*c^2*d^3*e^6*(a^15*c^3
)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6
 + 5*a^14*c^4*d^2*e^8)))^(1/2)*1i)/((((86016*A*a^9*c^3*e^11 - 53248*B*a^9*c^3*d*e^10 + 24576*A*a^5*c^7*d^8*e^3
 - 110592*A*a^6*c^6*d^6*e^5 + 233472*A*a^7*c^5*d^4*e^7 - 233472*A*a^8*c^4*d^2*e^9 + 4096*B*a^6*c^6*d^7*e^4 - 6
1440*B*a^7*c^5*d^5*e^6 + 110592*B*a^8*c^4*d^3*e^8)/(4096*(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)*((144*A^2*a^5*c^7*d^9 - 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 75
6*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a
^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 - 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 - 189*A^2*
c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 + 210*A*B*c^3*d^5*e^4*(a^15*c^3)
^(1/2) + 48*A*B*a^6*c^6*d^8*e + 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8
*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 + 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) - 154*B^2*a^2*c*d^2*e^7*(a^15*c
^3)^(1/2) + 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) - 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^1
0 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*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)))*((144*A^2*a
^5*c^7*d^9 - 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a
^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 - 441*A^2*a^2*c*e^9*(
a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 - 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^
2*a^10*c^2*d*e^8 + 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e + 486*A^2*a*c^2*d^2*e^7*(a^15*c
^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 + 35*B^2*a*c^2*d^4*e^5
*(a^15*c^3)^(1/2) - 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) + 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) - 588*A*B*a*
c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4
- 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8)))^(1/2) + ((d + e*x)^(1/2)*(441*A^2*a^4*c^3*e^10 + 25*B^2*a^5*c^2*
e^10 + 144*A^2*c^7*d^8*e^2 + 1089*A^2*a^2*c^5*d^4*e^6 - 990*A^2*a^3*c^4*d^2*e^8 + 4*B^2*a^2*c^5*d^6*e^4 - 31*B
^2*a^3*c^4*d^4*e^6 + 74*B^2*a^4*c^3*d^2*e^8 - 612*A^2*a*c^6*d^6*e^4 + 48*A*B*a*c^6*d^7*e^3 - 456*A*B*a^4*c^3*d
*e^9 - 288*A*B*a^2*c^5*d^5*e^5 + 552*A*B*a^3*c^4*d^3*e^7))/(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)))*((144*A^2*a^5*c^7*d^9 - 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6
*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^
4 + 70*B^2*a^9*c^3*d^3*e^6 - 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 - 189*A^2*c^3*d^4*e^5*(
a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 + 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A
*B*a^6*c^6*d^8*e + 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5
- 420*A*B*a^9*c^3*d^2*e^7 + 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) - 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) + 6
66*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) - 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*
e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8)))^(1/2) - (864*A^3
*c^6*d^7*e^3 - 125*B^3*a^5*c*e^10 + 7398*A^3*a^2*c^4*d^3*e^7 + 4*B^3*a^3*c^3*d^4*e^6 - 5*B^3*a^4*c^2*d^2*e^8 +
 2205*A^2*B*a^4*c^2*e^10 - 4104*A^3*a*c^5*d^5*e^5 - 5292*A^3*a^3*c^3*d*e^9 + 72*A*B^2*a^2*c^4*d^5*e^5 - 174*A*
B^2*a^3*c^3*d^3*e^7 - 1548*A^2*B*a^2*c^4*d^4*e^6 + 1053*A^2*B*a^3*c^3*d^2*e^8 - 780*A*B^2*a^4*c^2*d*e^9 + 432*
A^2*B*a*c^5*d^6*e^4)/(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))
 + (((86016*A*a^9*c^3*e^11 - 53248*B*a^9*c^3*d*e^10 + 24576*A*a^5*c^7*d^8*e^3 - 110592*A*a^6*c^6*d^6*e^5 + 233
472*A*a^7*c^5*d^4*e^7 - 233472*A*a^8*c^4*d^2*e^9 + 4096*B*a^6*c^6*d^7*e^4 - 61440*B*a^7*c^5*d^5*e^6 + 110592*B
*a^8*c^4*d^3*e^8)/(4096*(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)*((144*A^2*a^5*c^7*d^9 - 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*
a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d
^3*e^6 - 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 - 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 94
5*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 + 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e + 4
86*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^
2*e^7 + 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) - 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) + 666*A*B*a^2*c*d*e^8*(
a^15*c^3)^(1/2) - 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^
8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*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)))*((144*A^2*a^5*c^7*d^9 - 25*B^2*a^3*e^9*(a^15
*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^
7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 - 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^
2*e^9 - 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 + 210*A*B*c^3*d^
5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e + 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e
^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 + 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) - 154*B^2*a^2*c
*d^2*e^7*(a^15*c^3)^(1/2) + 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) - 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(40
96*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^
4*d^2*e^8)))^(1/2) - ((d + e*x)^(1/2)*(441*A^2*a^4*c^3*e^10 + 25*B^2*a^5*c^2*e^10 + 144*A^2*c^7*d^8*e^2 + 1089
*A^2*a^2*c^5*d^4*e^6 - 990*A^2*a^3*c^4*d^2*e^8 + 4*B^2*a^2*c^5*d^6*e^4 - 31*B^2*a^3*c^4*d^4*e^6 + 74*B^2*a^4*c
^3*d^2*e^8 - 612*A^2*a*c^6*d^6*e^4 + 48*A*B*a*c^6*d^7*e^3 - 456*A*B*a^4*c^3*d*e^9 - 288*A*B*a^2*c^5*d^5*e^5 +
552*A*B*a^3*c^4*d^3*e^7))/(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
)))*((144*A^2*a^5*c^7*d^9 - 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e
^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 - 441*
A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 - 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3
*d*e^8 + 105*B^2*a^10*c^2*d*e^8 + 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e + 486*A^2*a*c^2*
d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 + 35*B^
2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) - 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) + 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/
2) - 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^
12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8)))^(1/2)))*((144*A^2*a^5*c^7*d^9 - 25*B^2*a^3*e^9*(a
^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5
*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 - 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10
*c^2*e^9 - 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 + 210*A*B*c^3
*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e + 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^
6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 + 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) - 154*B^2*a^
2*c*d^2*e^7*(a^15*c^3)^(1/2) + 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) - 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/
(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14
*c^4*d^2*e^8)))^(1/2)*2i - atan(((((86016*A*a^9*c^3*e^11 - 53248*B*a^9*c^3*d*e^10 + 24576*A*a^5*c^7*d^8*e^3 -
110592*A*a^6*c^6*d^6*e^5 + 233472*A*a^7*c^5*d^4*e^7 - 233472*A*a^8*c^4*d^2*e^9 + 4096*B*a^6*c^6*d^7*e^4 - 6144
0*B*a^7*c^5*d^5*e^6 + 110592*B*a^8*c^4*d^3*e^8)/(4096*(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)*((144*A^2*a^5*c^7*d^9 + 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A
^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*
c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 + 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 + 189*A^2*c^3
*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 - 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1
/2) + 48*A*B*a^6*c^6*d^8*e - 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^
4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 - 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) + 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)
^(1/2) - 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) + 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 -
 a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*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)))*((144*A^2*a^5*
c^7*d^9 + 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*
c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 + 441*A^2*a^2*c*e^9*(a^1
5*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 + 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a
^10*c^2*d*e^8 - 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e - 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)
^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 - 35*B^2*a*c^2*d^4*e^5*(a
^15*c^3)^(1/2) + 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) - 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) + 588*A*B*a*c^2
*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 1
0*a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8)))^(1/2) + ((d + e*x)^(1/2)*(441*A^2*a^4*c^3*e^10 + 25*B^2*a^5*c^2*e^1
0 + 144*A^2*c^7*d^8*e^2 + 1089*A^2*a^2*c^5*d^4*e^6 - 990*A^2*a^3*c^4*d^2*e^8 + 4*B^2*a^2*c^5*d^6*e^4 - 31*B^2*
a^3*c^4*d^4*e^6 + 74*B^2*a^4*c^3*d^2*e^8 - 612*A^2*a*c^6*d^6*e^4 + 48*A*B*a*c^6*d^7*e^3 - 456*A*B*a^4*c^3*d*e^
9 - 288*A*B*a^2*c^5*d^5*e^5 + 552*A*B*a^3*c^4*d^3*e^7))/(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)))*((144*A^2*a^5*c^7*d^9 + 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^
7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 +
 70*B^2*a^9*c^3*d^3*e^6 + 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 + 189*A^2*c^3*d^4*e^5*(a^1
5*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 - 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*
a^6*c^6*d^8*e - 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 4
20*A*B*a^9*c^3*d^2*e^7 - 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) + 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) - 666*
A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) + 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^1
0 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8)))^(1/2)*1i - (((86016
*A*a^9*c^3*e^11 - 53248*B*a^9*c^3*d*e^10 + 24576*A*a^5*c^7*d^8*e^3 - 110592*A*a^6*c^6*d^6*e^5 + 233472*A*a^7*c
^5*d^4*e^7 - 233472*A*a^8*c^4*d^2*e^9 + 4096*B*a^6*c^6*d^7*e^4 - 61440*B*a^7*c^5*d^5*e^6 + 110592*B*a^8*c^4*d^
3*e^8)/(4096*(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)*((144*A^2*a^5*c^7*d^9 + 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5
*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 + 44
1*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 + 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c
^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 - 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e - 486*A^2*a*c^
2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 - 35*
B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) + 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) - 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(
1/2) + 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*
a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*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)))*((144*A^2*a^5*c^7*d^9 + 25*B^2*a^3*e^9*(a^15*c^3)^(1/2)
 - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*
B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 + 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 + 189
*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 - 210*A*B*c^3*d^5*e^4*(a^15
*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e - 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*
B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 - 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) + 154*B^2*a^2*c*d^2*e^7*(a
^15*c^3)^(1/2) - 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) + 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^
8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8))
)^(1/2) - ((d + e*x)^(1/2)*(441*A^2*a^4*c^3*e^10 + 25*B^2*a^5*c^2*e^10 + 144*A^2*c^7*d^8*e^2 + 1089*A^2*a^2*c^
5*d^4*e^6 - 990*A^2*a^3*c^4*d^2*e^8 + 4*B^2*a^2*c^5*d^6*e^4 - 31*B^2*a^3*c^4*d^4*e^6 + 74*B^2*a^4*c^3*d^2*e^8
- 612*A^2*a*c^6*d^6*e^4 + 48*A*B*a*c^6*d^7*e^3 - 456*A*B*a^4*c^3*d*e^9 - 288*A*B*a^2*c^5*d^5*e^5 + 552*A*B*a^3
*c^4*d^3*e^7))/(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)))*((144*A
^2*a^5*c^7*d^9 + 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A
^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 + 441*A^2*a^2*c*e
^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 + 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 10
5*B^2*a^10*c^2*d*e^8 - 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e - 486*A^2*a*c^2*d^2*e^7*(a^
15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 - 35*B^2*a*c^2*d^4
*e^5*(a^15*c^3)^(1/2) + 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) - 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) + 588*A*
B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*
e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8)))^(1/2)*1i)/((((86016*A*a^9*c^3*e^11 - 53248*B*a^9*c^3*d*e^10
+ 24576*A*a^5*c^7*d^8*e^3 - 110592*A*a^6*c^6*d^6*e^5 + 233472*A*a^7*c^5*d^4*e^7 - 233472*A*a^8*c^4*d^2*e^9 + 4
096*B*a^6*c^6*d^7*e^4 - 61440*B*a^7*c^5*d^5*e^6 + 110592*B*a^8*c^4*d^3*e^8)/(4096*(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)*((144*A^2*a^5*c^7*d^9 + 25*B^2*a^3*
e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a
^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 + 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*
B*a^10*c^2*e^9 + 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 - 210*A
*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e - 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*
c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 - 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) + 154*
B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) - 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) + 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(
1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 +
5*a^14*c^4*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)))*((144*A^2*a^5*c^7*d^9 + 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7
*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*
e^6 + 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 + 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A
^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 - 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e - 486*
A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e
^7 - 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) + 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) - 666*A*B*a^2*c*d*e^8*(a^1
5*c^3)^(1/2) + 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e
^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8)))^(1/2) + ((d + e*x)^(1/2)*(441*A^2*a^4*c
^3*e^10 + 25*B^2*a^5*c^2*e^10 + 144*A^2*c^7*d^8*e^2 + 1089*A^2*a^2*c^5*d^4*e^6 - 990*A^2*a^3*c^4*d^2*e^8 + 4*B
^2*a^2*c^5*d^6*e^4 - 31*B^2*a^3*c^4*d^4*e^6 + 74*B^2*a^4*c^3*d^2*e^8 - 612*A^2*a*c^6*d^6*e^4 + 48*A*B*a*c^6*d^
7*e^3 - 456*A*B*a^4*c^3*d*e^9 - 288*A*B*a^2*c^5*d^5*e^5 + 552*A*B*a^3*c^4*d^3*e^7))/(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)))*((144*A^2*a^5*c^7*d^9 + 25*B^2*a^3*e^9*(a^15*c^3
)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^
2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 + 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^
9 + 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 - 210*A*B*c^3*d^5*e^
4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e - 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 +
 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 - 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) + 154*B^2*a^2*c*d^2
*e^7*(a^15*c^3)^(1/2) - 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) + 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(
a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^
2*e^8)))^(1/2) - (864*A^3*c^6*d^7*e^3 - 125*B^3*a^5*c*e^10 + 7398*A^3*a^2*c^4*d^3*e^7 + 4*B^3*a^3*c^3*d^4*e^6
- 5*B^3*a^4*c^2*d^2*e^8 + 2205*A^2*B*a^4*c^2*e^10 - 4104*A^3*a*c^5*d^5*e^5 - 5292*A^3*a^3*c^3*d*e^9 + 72*A*B^2
*a^2*c^4*d^5*e^5 - 174*A*B^2*a^3*c^3*d^3*e^7 - 1548*A^2*B*a^2*c^4*d^4*e^6 + 1053*A^2*B*a^3*c^3*d^2*e^8 - 780*A
*B^2*a^4*c^2*d*e^9 + 432*A^2*B*a*c^5*d^6*e^4)/(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)) + (((86016*A*a^9*c^3*e^11 - 53248*B*a^9*c^3*d*e^10 + 24576*A*a^5*c^7*d^8*e^3 - 11059
2*A*a^6*c^6*d^6*e^5 + 233472*A*a^7*c^5*d^4*e^7 - 233472*A*a^8*c^4*d^2*e^9 + 4096*B*a^6*c^6*d^7*e^4 - 61440*B*a
^7*c^5*d^5*e^6 + 110592*B*a^8*c^4*d^3*e^8)/(4096*(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)*((144*A^2*a^5*c^7*d^9 + 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^
6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d
^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 + 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 + 189*A^2*c^3*d^4*
e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 - 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) +
 48*A*B*a^6*c^6*d^8*e - 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4
*e^5 - 420*A*B*a^9*c^3*d^2*e^7 - 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) + 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2
) - 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) + 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15
*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*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)))*((144*A^2*a^5*c^7*d
^9 + 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d
^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 + 441*A^2*a^2*c*e^9*(a^15*c^3
)^(1/2) - 210*A*B*a^10*c^2*e^9 + 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c
^2*d*e^8 - 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e - 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2
) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 - 35*B^2*a*c^2*d^4*e^5*(a^15*c
^3)^(1/2) + 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) - 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) + 588*A*B*a*c^2*d^3*
e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^1
3*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8)))^(1/2) - ((d + e*x)^(1/2)*(441*A^2*a^4*c^3*e^10 + 25*B^2*a^5*c^2*e^10 + 1
44*A^2*c^7*d^8*e^2 + 1089*A^2*a^2*c^5*d^4*e^6 - 990*A^2*a^3*c^4*d^2*e^8 + 4*B^2*a^2*c^5*d^6*e^4 - 31*B^2*a^3*c
^4*d^4*e^6 + 74*B^2*a^4*c^3*d^2*e^8 - 612*A^2*a*c^6*d^6*e^4 + 48*A*B*a*c^6*d^7*e^3 - 456*A*B*a^4*c^3*d*e^9 - 2
88*A*B*a^2*c^5*d^5*e^5 + 552*A*B*a^3*c^4*d^3*e^7))/(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)))*((144*A^2*a^5*c^7*d^9 + 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2
 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B
^2*a^9*c^3*d^3*e^6 + 441*A^2*a^2*c*e^9*(a^15*c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 + 189*A^2*c^3*d^4*e^5*(a^15*c^3
)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 - 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c
^6*d^8*e - 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*
B*a^9*c^3*d^2*e^7 - 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) + 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) - 666*A*B*a
^2*c*d*e^8*(a^15*c^3)^(1/2) + 588*A*B*a*c^2*d^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5
*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8)))^(1/2)))*((144*A^2*a^5*c^
7*d^9 + 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2*a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^
4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 + 441*A^2*a^2*c*e^9*(a^15*
c^3)^(1/2) - 210*A*B*a^10*c^2*e^9 + 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945*A^2*a^9*c^3*d*e^8 + 105*B^2*a^1
0*c^2*d*e^8 - 210*A*B*c^3*d^5*e^4*(a^15*c^3)^(1/2) + 48*A*B*a^6*c^6*d^8*e - 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(
1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e^7 - 35*B^2*a*c^2*d^4*e^5*(a^1
5*c^3)^(1/2) + 154*B^2*a^2*c*d^2*e^7*(a^15*c^3)^(1/2) - 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) + 588*A*B*a*c^2*d
^3*e^6*(a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*
a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8)))^(1/2)*2i - (((d + e*x)^(3/2)*(18*A*c^3*d^5*e - 9*B*a^3*e^6 - 44*A*a*c
^2*d^3*e^3 + 3*B*a*c^2*d^4*e^2 + 30*B*a^2*c*d^2*e^4 + 2*A*a^2*c*d*e^5))/(16*a^2*(a*e^2 - c*d^2)^2) + ((d + e*x
)^(1/2)*(19*B*a^2*d*e^4 - 11*A*a^2*e^5 + 6*A*c^2*d^4*e - 15*A*a*c*d^2*e^3 + B*a*c*d^3*e^2))/(16*a^2*(a*e^2 - c
*d^2)) - (c*(d + e*x)^(5/2)*(21*B*a^2*d*e^4 - 7*A*a^2*e^5 + 18*A*c^2*d^4*e - 35*A*a*c*d^2*e^3 + 3*B*a*c*d^3*e^
2))/(16*a^2*(a*e^2 - c*d^2)^2) + (c*(d + e*x)^(7/2)*(5*B*a^2*e^4 + 6*A*c^2*d^3*e - 12*A*a*c*d*e^3 + B*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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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