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

Optimal. Leaf size=372 \[ \frac {\left (a B e \left (2 \sqrt {c} d-3 \sqrt {a} e\right )-A \left (-18 \sqrt {a} c d e+5 a \sqrt {c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{5/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}-\frac {\left (a B e \left (3 \sqrt {a} e+2 \sqrt {c} d\right )-A \left (18 \sqrt {a} c d e+5 a \sqrt {c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{5/4} \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2}}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-5 a A e^2-a B d e+6 A c d^2\right )\right )}{16 a^2 c \left (a-c x^2\right ) \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (a B+A c x)}{4 a c \left (a-c x^2\right )^2} \]

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Rubi [A]  time = 0.76, antiderivative size = 372, 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 = {821, 823, 827, 1166, 208} \begin {gather*} \frac {\left (a B e \left (2 \sqrt {c} d-3 \sqrt {a} e\right )-A \left (-18 \sqrt {a} c d e+5 a \sqrt {c} e^2+12 c^{3/2} d^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^{5/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}-\frac {\left (a B e \left (3 \sqrt {a} e+2 \sqrt {c} d\right )-A \left (18 \sqrt {a} c d e+5 a \sqrt {c} e^2+12 c^{3/2} d^2\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^{5/4} \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2}}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-5 a A e^2-a B d e+6 A c d^2\right )\right )}{16 a^2 c \left (a-c x^2\right ) \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (a B+A c x)}{4 a c \left (a-c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*B + A*c*x)*Sqrt[d + e*x])/(4*a*c*(a - c*x^2)^2) - (Sqrt[d + e*x]*(a*e*(A*c*d - a*B*e) - c*(6*A*c*d^2 - a*B
*d*e - 5*a*A*e^2)*x))/(16*a^2*c*(c*d^2 - a*e^2)*(a - c*x^2)) + ((a*B*e*(2*Sqrt[c]*d - 3*Sqrt[a]*e) - A*(12*c^(
3/2)*d^2 - 18*Sqrt[a]*c*d*e + 5*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^(5/4)*(Sqrt[c]*d - Sqrt[a]*e)^(3/2)) - ((a*B*e*(2*Sqrt[c]*d + 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 +
 18*Sqrt[a]*c*d*e + 5*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^(5/4)*(Sqrt[c]*d + Sqrt[a]*e)^(3/2))

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 821

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

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 {(a B+A c x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} (-6 A c d+a B e)-\frac {5}{2} A c e x}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx}{4 a c}\\ &=\frac {(a B+A c x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c \left (6 A c d^2-a B d e-5 a A e^2\right ) x\right )}{16 a^2 c \left (c d^2-a e^2\right ) \left (a-c x^2\right )}+\frac {\int \frac {\frac {1}{4} c \left (A c d \left (12 c d^2-13 a e^2\right )-a B e \left (2 c d^2-3 a e^2\right )\right )+\frac {1}{4} c^2 e \left (6 A c d^2-a B d e-5 a A e^2\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 )}\\ &=\frac {(a B+A c x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c \left (6 A c d^2-a B d e-5 a A e^2\right ) x\right )}{16 a^2 c \left (c d^2-a e^2\right ) \left (a-c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{4} c^2 d e \left (6 A c d^2-a B d e-5 a A e^2\right )+\frac {1}{4} c e \left (A c d \left (12 c d^2-13 a e^2\right )-a B e \left (2 c d^2-3 a e^2\right )\right )+\frac {1}{4} c^2 e \left (6 A c d^2-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 )}{4 a^2 c^2 \left (c d^2-a e^2\right )}\\ &=\frac {(a B+A c x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c \left (6 A c d^2-a B d e-5 a A e^2\right ) x\right )}{16 a^2 c \left (c d^2-a e^2\right ) \left (a-c x^2\right )}+\frac {\left (a B e \left (2 \sqrt {c} d-3 \sqrt {a} e\right )-A \left (12 c^{3/2} d^2-18 \sqrt {a} c d e+5 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} \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}-\frac {\left (a B e \left (2 \sqrt {c} d+3 \sqrt {a} e\right )-A \left (12 c^{3/2} d^2+18 \sqrt {a} c d e+5 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} \sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}\\ &=\frac {(a B+A c x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c \left (6 A c d^2-a B d e-5 a A e^2\right ) x\right )}{16 a^2 c \left (c d^2-a e^2\right ) \left (a-c x^2\right )}+\frac {\left (a B e \left (2 \sqrt {c} d-3 \sqrt {a} e\right )-A \left (12 c^{3/2} d^2-18 \sqrt {a} c d e+5 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^{5/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}-\frac {\left (a B e \left (2 \sqrt {c} d+3 \sqrt {a} e\right )-A \left (12 c^{3/2} d^2+18 \sqrt {a} c d e+5 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^{5/4} \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 1.04, size = 522, normalized size = 1.40 \begin {gather*} \frac {\frac {c^2 (d+e x)^{3/2} \left (a^2 e^2 (5 A e-2 B d+3 B e x)-a c d e (3 A d+8 A e x+B d x)+6 A c^2 d^3 x\right )}{2 \left (a-c x^2\right )}-\frac {c^{5/4} \left (A \left (5 a^2 e^4-27 a c d^2 e^2+18 c^2 d^4\right )+a B d e \left (7 a e^2-3 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 {c^{3/4} \left (2 A c d \left (3 c d^2-4 a e^2\right )+a B e \left (3 a e^2-c d^2\right )\right ) \left (2 \sqrt {a} \sqrt [4]{c} e \sqrt {d+e x}+\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )-\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \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 (d+e x)^{3/2} \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)*(d + e*x)^(3/2)*(-(a*A*e) + A*c*d*x + a*B*(d - e*x)))/(a - c*x^2)^2 + (c^2*(d + e*x)
^(3/2)*(6*A*c^2*d^3*x - a*c*d*e*(3*A*d + B*d*x + 8*A*e*x) + a^2*e^2*(-2*B*d + 5*A*e + 3*B*e*x)))/(2*(a - c*x^2
)) - (c^(5/4)*(a*B*d*e*(-3*c*d^2 + 7*a*e^2) + A*(18*c^2*d^4 - 27*a*c*d^2*e^2 + 5*a^2*e^4))*(Sqrt[Sqrt[c]*d - S
qrt[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]) + (c^(3/4)*(2*A*c*d*(3*c*d^2 - 4*a*e^2) + a*
B*e*(-(c*d^2) + 3*a*e^2))*(2*Sqrt[a]*c^(1/4)*e*Sqrt[d + e*x] + (Sqrt[c]*d - Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*
Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] - (Sqrt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sq
rt[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 = 2.61, size = 656, normalized size = 1.76 \begin {gather*} \frac {\left (-3 a^{3/2} B e^2+18 \sqrt {a} A c d e+5 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} c \left (\sqrt {a} e+\sqrt {c} d\right ) \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}+\frac {\left (-3 a^{3/2} B e^2+18 \sqrt {a} A c d e-5 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} c \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}}+\frac {e \sqrt {d+e x} \left (3 a^3 B e^5+9 a^2 A c e^4 (d+e x)-8 a^2 A c d e^4-4 a^2 B c d^2 e^3-a^2 B c d e^3 (d+e x)+a^2 B c e^3 (d+e x)^2+14 a A c^2 d^3 e^2-23 a A c^2 d^2 e^2 (d+e x)+14 a A c^2 d e^2 (d+e x)^2-5 a A c^2 e^2 (d+e x)^3+a B c^2 d^4 e-3 a B c^2 d^3 e (d+e x)+3 a B c^2 d^2 e (d+e x)^2-a B c^2 d e (d+e x)^3-6 A c^3 d^5+18 A c^3 d^4 (d+e x)-18 A c^3 d^3 (d+e x)^2+6 A c^3 d^2 (d+e x)^3\right )}{16 a^2 c \left (a e^2-c d^2\right ) \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^5 + a*B*c^2*d^4*e + 14*a*A*c^2*d^3*e^2 - 4*a^2*B*c*d^2*e^3 - 8*a^2*A*c*d*e^4 + 3*
a^3*B*e^5 + 18*A*c^3*d^4*(d + e*x) - 3*a*B*c^2*d^3*e*(d + e*x) - 23*a*A*c^2*d^2*e^2*(d + e*x) - a^2*B*c*d*e^3*
(d + e*x) + 9*a^2*A*c*e^4*(d + e*x) - 18*A*c^3*d^3*(d + e*x)^2 + 3*a*B*c^2*d^2*e*(d + e*x)^2 + 14*a*A*c^2*d*e^
2*(d + e*x)^2 + a^2*B*c*e^3*(d + e*x)^2 + 6*A*c^3*d^2*(d + e*x)^3 - a*B*c^2*d*e*(d + e*x)^3 - 5*a*A*c^2*e^2*(d
 + e*x)^3))/(16*a^2*c*(-(c*d^2) + a*e^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 + 18*Sqrt[a]*A*c*d*e - 3*a^(3/2)*B*e^2 + 5*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)*c*(Sqrt[c]*d + Sqrt[a]*e)*Sqrt[-(Sqrt[
c]*(Sqrt[c]*d + Sqrt[a]*e))]) + ((-12*A*c^(3/2)*d^2 + 2*a*B*Sqrt[c]*d*e + 18*Sqrt[a]*A*c*d*e - 3*a^(3/2)*B*e^2
 - 5*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)*c*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))])

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fricas [B]  time = 60.10, size = 8803, normalized size = 23.66

result too large to display

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]

1/64*((a^4*c^2*d^2 - a^5*c*e^2 + (a^2*c^4*d^2 - a^3*c^3*e^2)*x^4 - 2*(a^3*c^3*d^2 - a^4*c^2*e^2)*x^2)*sqrt((14
4*A^2*c^4*d^7 - 48*A*B*a*c^3*d^6*e + 160*A*B*a^2*c^2*d^4*e^3 - 150*A*B*a^3*c*d^2*e^5 + 30*A*B*a^4*e^7 + 4*(B^2
*a^2*c^2 - 105*A^2*a*c^3)*d^5*e^2 - 5*(3*B^2*a^3*c - 77*A^2*a^2*c^2)*d^3*e^4 + 15*(B^2*a^4 - 7*A^2*a^3*c)*d*e^
6 + (a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6)*sqrt((900*A^2*B^2*c^4*d^6*e^8 - 60*(5*
A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(115*A*
B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*e^12 - 76
*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^1
1*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^2*e
^10 + a^11*c^5*e^12)))/(a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6))*log((4320*A^3*B*c^
5*d^7*e^4 - 432*(5*A^2*B^2*a*c^4 + 7*A^4*c^5)*d^6*e^5 + 72*(5*A*B^3*a^2*c^3 - 147*A^3*B*a*c^4)*d^5*e^6 - 4*(5*
B^4*a^3*c^2 - 1674*A^2*B^2*a^2*c^3 - 1971*A^4*a*c^4)*d^4*e^7 - 2*(647*A*B^3*a^3*c^2 - 2727*A^3*B*a^2*c^3)*d^3*
e^8 + 3*(27*B^4*a^4*c - 1672*A^2*B^2*a^3*c^2 - 1875*A^4*a^2*c^3)*d^2*e^9 + 2*(567*A*B^3*a^4*c + 625*A^3*B*a^3*
c^2)*d*e^10 - (81*B^4*a^5 - 625*A^4*a^3*c^2)*e^11)*sqrt(e*x + d) + (180*A^2*B*a^3*c^5*d^6*e^5 - 6*(10*A*B^2*a^
4*c^4 + 21*A^3*a^3*c^5)*d^5*e^6 + (5*B^3*a^5*c^3 - 447*A^2*B*a^4*c^4)*d^4*e^7 + 6*(37*A*B^2*a^5*c^3 + 53*A^3*a
^4*c^4)*d^3*e^8 - 24*(B^3*a^6*c^2 - 9*A^2*B*a^5*c^3)*d^2*e^9 - 2*(93*A*B^2*a^6*c^2 + 100*A^3*a^5*c^3)*d*e^10 +
 3*(9*B^3*a^7*c + 25*A^2*B*a^6*c^2)*e^11 - (12*A*a^5*c^9*d^10 - 2*B*a^6*c^8*d^9*e - 55*A*a^6*c^8*d^8*e^2 + 10*
B*a^7*c^7*d^7*e^3 + 98*A*a^7*c^7*d^6*e^4 - 18*B*a^8*c^6*d^5*e^5 - 84*A*a^8*c^6*d^4*e^6 + 14*B*a^9*c^5*d^3*e^7
+ 34*A*a^9*c^5*d^2*e^8 - 4*B*a^10*c^4*d*e^9 - 5*A*a^10*c^4*e^10)*sqrt((900*A^2*B^2*c^4*d^6*e^8 - 60*(5*A*B^3*a
*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(115*A*B^3*a^2
*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*e^12 - 76*(9*A*B
^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11*d^12
- 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^2*e^10 + a
^11*c^5*e^12)))*sqrt((144*A^2*c^4*d^7 - 48*A*B*a*c^3*d^6*e + 160*A*B*a^2*c^2*d^4*e^3 - 150*A*B*a^3*c*d^2*e^5 +
 30*A*B*a^4*e^7 + 4*(B^2*a^2*c^2 - 105*A^2*a*c^3)*d^5*e^2 - 5*(3*B^2*a^3*c - 77*A^2*a^2*c^2)*d^3*e^4 + 15*(B^2
*a^4 - 7*A^2*a^3*c)*d*e^6 + (a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6)*sqrt((900*A^2*
B^2*c^4*d^6*e^8 - 60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c
^4)*d^4*e^10 + 8*(115*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*
A^4*a*c^3)*d^2*e^12 - 76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4
*a^2*c^2)*e^14)/(a^5*c^11*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^
4*e^8 - 6*a^10*c^6*d^2*e^10 + a^11*c^5*e^12)))/(a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*
e^6))) - (a^4*c^2*d^2 - a^5*c*e^2 + (a^2*c^4*d^2 - a^3*c^3*e^2)*x^4 - 2*(a^3*c^3*d^2 - a^4*c^2*e^2)*x^2)*sqrt(
(144*A^2*c^4*d^7 - 48*A*B*a*c^3*d^6*e + 160*A*B*a^2*c^2*d^4*e^3 - 150*A*B*a^3*c*d^2*e^5 + 30*A*B*a^4*e^7 + 4*(
B^2*a^2*c^2 - 105*A^2*a*c^3)*d^5*e^2 - 5*(3*B^2*a^3*c - 77*A^2*a^2*c^2)*d^3*e^4 + 15*(B^2*a^4 - 7*A^2*a^3*c)*d
*e^6 + (a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6)*sqrt((900*A^2*B^2*c^4*d^6*e^8 - 60*
(5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(115
*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*e^12 -
 76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*
c^11*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^
2*e^10 + a^11*c^5*e^12)))/(a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6))*log((4320*A^3*B
*c^5*d^7*e^4 - 432*(5*A^2*B^2*a*c^4 + 7*A^4*c^5)*d^6*e^5 + 72*(5*A*B^3*a^2*c^3 - 147*A^3*B*a*c^4)*d^5*e^6 - 4*
(5*B^4*a^3*c^2 - 1674*A^2*B^2*a^2*c^3 - 1971*A^4*a*c^4)*d^4*e^7 - 2*(647*A*B^3*a^3*c^2 - 2727*A^3*B*a^2*c^3)*d
^3*e^8 + 3*(27*B^4*a^4*c - 1672*A^2*B^2*a^3*c^2 - 1875*A^4*a^2*c^3)*d^2*e^9 + 2*(567*A*B^3*a^4*c + 625*A^3*B*a
^3*c^2)*d*e^10 - (81*B^4*a^5 - 625*A^4*a^3*c^2)*e^11)*sqrt(e*x + d) - (180*A^2*B*a^3*c^5*d^6*e^5 - 6*(10*A*B^2
*a^4*c^4 + 21*A^3*a^3*c^5)*d^5*e^6 + (5*B^3*a^5*c^3 - 447*A^2*B*a^4*c^4)*d^4*e^7 + 6*(37*A*B^2*a^5*c^3 + 53*A^
3*a^4*c^4)*d^3*e^8 - 24*(B^3*a^6*c^2 - 9*A^2*B*a^5*c^3)*d^2*e^9 - 2*(93*A*B^2*a^6*c^2 + 100*A^3*a^5*c^3)*d*e^1
0 + 3*(9*B^3*a^7*c + 25*A^2*B*a^6*c^2)*e^11 - (12*A*a^5*c^9*d^10 - 2*B*a^6*c^8*d^9*e - 55*A*a^6*c^8*d^8*e^2 +
10*B*a^7*c^7*d^7*e^3 + 98*A*a^7*c^7*d^6*e^4 - 18*B*a^8*c^6*d^5*e^5 - 84*A*a^8*c^6*d^4*e^6 + 14*B*a^9*c^5*d^3*e
^7 + 34*A*a^9*c^5*d^2*e^8 - 4*B*a^10*c^4*d*e^9 - 5*A*a^10*c^4*e^10)*sqrt((900*A^2*B^2*c^4*d^6*e^8 - 60*(5*A*B^
3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(115*A*B^3*
a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*e^12 - 76*(9*
A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11*d^
12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^2*e^10
+ a^11*c^5*e^12)))*sqrt((144*A^2*c^4*d^7 - 48*A*B*a*c^3*d^6*e + 160*A*B*a^2*c^2*d^4*e^3 - 150*A*B*a^3*c*d^2*e^
5 + 30*A*B*a^4*e^7 + 4*(B^2*a^2*c^2 - 105*A^2*a*c^3)*d^5*e^2 - 5*(3*B^2*a^3*c - 77*A^2*a^2*c^2)*d^3*e^4 + 15*(
B^2*a^4 - 7*A^2*a^3*c)*d*e^6 + (a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6)*sqrt((900*A
^2*B^2*c^4*d^6*e^8 - 60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^
4*c^4)*d^4*e^10 + 8*(115*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 1
75*A^4*a*c^3)*d^2*e^12 - 76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*
A^4*a^2*c^2)*e^14)/(a^5*c^11*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7
*d^4*e^8 - 6*a^10*c^6*d^2*e^10 + a^11*c^5*e^12)))/(a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c
^2*e^6))) + (a^4*c^2*d^2 - a^5*c*e^2 + (a^2*c^4*d^2 - a^3*c^3*e^2)*x^4 - 2*(a^3*c^3*d^2 - a^4*c^2*e^2)*x^2)*sq
rt((144*A^2*c^4*d^7 - 48*A*B*a*c^3*d^6*e + 160*A*B*a^2*c^2*d^4*e^3 - 150*A*B*a^3*c*d^2*e^5 + 30*A*B*a^4*e^7 +
4*(B^2*a^2*c^2 - 105*A^2*a*c^3)*d^5*e^2 - 5*(3*B^2*a^3*c - 77*A^2*a^2*c^2)*d^3*e^4 + 15*(B^2*a^4 - 7*A^2*a^3*c
)*d*e^6 - (a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6)*sqrt((900*A^2*B^2*c^4*d^6*e^8 -
60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(
115*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*e^1
2 - 76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a
^5*c^11*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*c^6
*d^2*e^10 + a^11*c^5*e^12)))/(a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6))*log((4320*A^
3*B*c^5*d^7*e^4 - 432*(5*A^2*B^2*a*c^4 + 7*A^4*c^5)*d^6*e^5 + 72*(5*A*B^3*a^2*c^3 - 147*A^3*B*a*c^4)*d^5*e^6 -
 4*(5*B^4*a^3*c^2 - 1674*A^2*B^2*a^2*c^3 - 1971*A^4*a*c^4)*d^4*e^7 - 2*(647*A*B^3*a^3*c^2 - 2727*A^3*B*a^2*c^3
)*d^3*e^8 + 3*(27*B^4*a^4*c - 1672*A^2*B^2*a^3*c^2 - 1875*A^4*a^2*c^3)*d^2*e^9 + 2*(567*A*B^3*a^4*c + 625*A^3*
B*a^3*c^2)*d*e^10 - (81*B^4*a^5 - 625*A^4*a^3*c^2)*e^11)*sqrt(e*x + d) + (180*A^2*B*a^3*c^5*d^6*e^5 - 6*(10*A*
B^2*a^4*c^4 + 21*A^3*a^3*c^5)*d^5*e^6 + (5*B^3*a^5*c^3 - 447*A^2*B*a^4*c^4)*d^4*e^7 + 6*(37*A*B^2*a^5*c^3 + 53
*A^3*a^4*c^4)*d^3*e^8 - 24*(B^3*a^6*c^2 - 9*A^2*B*a^5*c^3)*d^2*e^9 - 2*(93*A*B^2*a^6*c^2 + 100*A^3*a^5*c^3)*d*
e^10 + 3*(9*B^3*a^7*c + 25*A^2*B*a^6*c^2)*e^11 + (12*A*a^5*c^9*d^10 - 2*B*a^6*c^8*d^9*e - 55*A*a^6*c^8*d^8*e^2
 + 10*B*a^7*c^7*d^7*e^3 + 98*A*a^7*c^7*d^6*e^4 - 18*B*a^8*c^6*d^5*e^5 - 84*A*a^8*c^6*d^4*e^6 + 14*B*a^9*c^5*d^
3*e^7 + 34*A*a^9*c^5*d^2*e^8 - 4*B*a^10*c^4*d*e^9 - 5*A*a^10*c^4*e^10)*sqrt((900*A^2*B^2*c^4*d^6*e^8 - 60*(5*A
*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(115*A*B
^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*e^12 - 76*
(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11
*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^2*e^
10 + a^11*c^5*e^12)))*sqrt((144*A^2*c^4*d^7 - 48*A*B*a*c^3*d^6*e + 160*A*B*a^2*c^2*d^4*e^3 - 150*A*B*a^3*c*d^2
*e^5 + 30*A*B*a^4*e^7 + 4*(B^2*a^2*c^2 - 105*A^2*a*c^3)*d^5*e^2 - 5*(3*B^2*a^3*c - 77*A^2*a^2*c^2)*d^3*e^4 + 1
5*(B^2*a^4 - 7*A^2*a^3*c)*d*e^6 - (a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6)*sqrt((90
0*A^2*B^2*c^4*d^6*e^8 - 60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441
*A^4*c^4)*d^4*e^10 + 8*(115*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2
+ 175*A^4*a*c^3)*d^2*e^12 - 76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 6
25*A^4*a^2*c^2)*e^14)/(a^5*c^11*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*
c^7*d^4*e^8 - 6*a^10*c^6*d^2*e^10 + a^11*c^5*e^12)))/(a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^
8*c^2*e^6))) - (a^4*c^2*d^2 - a^5*c*e^2 + (a^2*c^4*d^2 - a^3*c^3*e^2)*x^4 - 2*(a^3*c^3*d^2 - a^4*c^2*e^2)*x^2)
*sqrt((144*A^2*c^4*d^7 - 48*A*B*a*c^3*d^6*e + 160*A*B*a^2*c^2*d^4*e^3 - 150*A*B*a^3*c*d^2*e^5 + 30*A*B*a^4*e^7
 + 4*(B^2*a^2*c^2 - 105*A^2*a*c^3)*d^5*e^2 - 5*(3*B^2*a^3*c - 77*A^2*a^2*c^2)*d^3*e^4 + 15*(B^2*a^4 - 7*A^2*a^
3*c)*d*e^6 - (a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6)*sqrt((900*A^2*B^2*c^4*d^6*e^8
 - 60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 +
8*(115*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*
e^12 - 76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)
/(a^5*c^11*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*
c^6*d^2*e^10 + a^11*c^5*e^12)))/(a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6))*log((4320
*A^3*B*c^5*d^7*e^4 - 432*(5*A^2*B^2*a*c^4 + 7*A^4*c^5)*d^6*e^5 + 72*(5*A*B^3*a^2*c^3 - 147*A^3*B*a*c^4)*d^5*e^
6 - 4*(5*B^4*a^3*c^2 - 1674*A^2*B^2*a^2*c^3 - 1971*A^4*a*c^4)*d^4*e^7 - 2*(647*A*B^3*a^3*c^2 - 2727*A^3*B*a^2*
c^3)*d^3*e^8 + 3*(27*B^4*a^4*c - 1672*A^2*B^2*a^3*c^2 - 1875*A^4*a^2*c^3)*d^2*e^9 + 2*(567*A*B^3*a^4*c + 625*A
^3*B*a^3*c^2)*d*e^10 - (81*B^4*a^5 - 625*A^4*a^3*c^2)*e^11)*sqrt(e*x + d) - (180*A^2*B*a^3*c^5*d^6*e^5 - 6*(10
*A*B^2*a^4*c^4 + 21*A^3*a^3*c^5)*d^5*e^6 + (5*B^3*a^5*c^3 - 447*A^2*B*a^4*c^4)*d^4*e^7 + 6*(37*A*B^2*a^5*c^3 +
 53*A^3*a^4*c^4)*d^3*e^8 - 24*(B^3*a^6*c^2 - 9*A^2*B*a^5*c^3)*d^2*e^9 - 2*(93*A*B^2*a^6*c^2 + 100*A^3*a^5*c^3)
*d*e^10 + 3*(9*B^3*a^7*c + 25*A^2*B*a^6*c^2)*e^11 + (12*A*a^5*c^9*d^10 - 2*B*a^6*c^8*d^9*e - 55*A*a^6*c^8*d^8*
e^2 + 10*B*a^7*c^7*d^7*e^3 + 98*A*a^7*c^7*d^6*e^4 - 18*B*a^8*c^6*d^5*e^5 - 84*A*a^8*c^6*d^4*e^6 + 14*B*a^9*c^5
*d^3*e^7 + 34*A*a^9*c^5*d^2*e^8 - 4*B*a^10*c^4*d*e^9 - 5*A*a^10*c^4*e^10)*sqrt((900*A^2*B^2*c^4*d^6*e^8 - 60*(
5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(115*
A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*e^12 -
76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c
^11*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^2
*e^10 + a^11*c^5*e^12)))*sqrt((144*A^2*c^4*d^7 - 48*A*B*a*c^3*d^6*e + 160*A*B*a^2*c^2*d^4*e^3 - 150*A*B*a^3*c*
d^2*e^5 + 30*A*B*a^4*e^7 + 4*(B^2*a^2*c^2 - 105*A^2*a*c^3)*d^5*e^2 - 5*(3*B^2*a^3*c - 77*A^2*a^2*c^2)*d^3*e^4
+ 15*(B^2*a^4 - 7*A^2*a^3*c)*d*e^6 - (a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6)*sqrt(
(900*A^2*B^2*c^4*d^6*e^8 - 60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 +
441*A^4*c^4)*d^4*e^10 + 8*(115*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c
^2 + 175*A^4*a*c^3)*d^2*e^12 - 76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c
+ 625*A^4*a^2*c^2)*e^14)/(a^5*c^11*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a
^9*c^7*d^4*e^8 - 6*a^10*c^6*d^2*e^10 + a^11*c^5*e^12)))/(a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 -
 a^8*c^2*e^6))) + 4*(4*B*a^2*c*d^2 - A*a^2*c*d*e - 3*B*a^3*e^2 - (6*A*c^3*d^2 - B*a*c^2*d*e - 5*A*a*c^2*e^2)*x
^3 + (A*a*c^2*d*e - B*a^2*c*e^2)*x^2 + (10*A*a*c^2*d^2 - B*a^2*c*d*e - 9*A*a^2*c*e^2)*x)*sqrt(e*x + d))/(a^4*c
^2*d^2 - a^5*c*e^2 + (a^2*c^4*d^2 - a^3*c^3*e^2)*x^4 - 2*(a^3*c^3*d^2 - a^4*c^2*e^2)*x^2)

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giac [B]  time = 1.25, size = 1637, normalized size = 4.40

result too large to display

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]

1/32*((a^2*c^2*d^2*e - a^3*c*e^3)^2*B*a*d*abs(c)*e^2 - (a^2*c^2*d^2*e - a^3*c*e^3)^2*(6*c*d^2*e - 5*a*e^3)*A*a
bs(c) - 2*(3*sqrt(a*c)*a*c^3*d^5*e - 7*sqrt(a*c)*a^2*c^2*d^3*e^3 + 4*sqrt(a*c)*a^3*c*d*e^5)*A*abs(a^2*c^2*d^2*
e - a^3*c*e^3)*abs(c) + (sqrt(a*c)*a^2*c^2*d^4*e^2 - 4*sqrt(a*c)*a^3*c*d^2*e^4 + 3*sqrt(a*c)*a^4*e^6)*B*abs(a^
2*c^2*d^2*e - a^3*c*e^3)*abs(c) + (12*a^3*c^6*d^8*e - 37*a^4*c^5*d^6*e^3 + 38*a^5*c^4*d^4*e^5 - 13*a^6*c^3*d^2
*e^7)*A*abs(c) - (2*a^4*c^5*d^7*e^2 - 7*a^5*c^4*d^5*e^4 + 8*a^6*c^3*d^3*e^6 - 3*a^7*c^2*d*e^8)*B*abs(c))*arcta
n(sqrt(x*e + d)/sqrt(-(a^2*c^3*d^3 - a^3*c^2*d*e^2 + sqrt((a^2*c^3*d^3 - a^3*c^2*d*e^2)^2 - (a^2*c^3*d^4 - 2*a
^3*c^2*d^2*e^2 + a^4*c*e^4)*(a^2*c^3*d^2 - a^3*c^2*e^2)))/(a^2*c^3*d^2 - a^3*c^2*e^2)))/((a^4*c^4*d^4*e - sqrt
(a*c)*a^3*c^4*d^5 + 2*sqrt(a*c)*a^4*c^3*d^3*e^2 - 2*a^5*c^3*d^2*e^3 - sqrt(a*c)*a^5*c^2*d*e^4 + a^6*c^2*e^5)*s
qrt(-c^2*d - sqrt(a*c)*c*e)*abs(a^2*c^2*d^2*e - a^3*c*e^3)) + 1/32*((a^2*c^2*d^2*e - a^3*c*e^3)^2*sqrt(a*c)*B*
a*d*abs(c)*e^2 - (a^2*c^2*d^2*e - a^3*c*e^3)^2*(6*sqrt(a*c)*c*d^2*e - 5*sqrt(a*c)*a*e^3)*A*abs(c) + 2*(3*a^2*c
^4*d^5*e - 7*a^3*c^3*d^3*e^3 + 4*a^4*c^2*d*e^5)*A*abs(a^2*c^2*d^2*e - a^3*c*e^3)*abs(c) - (a^3*c^3*d^4*e^2 - 4
*a^4*c^2*d^2*e^4 + 3*a^5*c*e^6)*B*abs(a^2*c^2*d^2*e - a^3*c*e^3)*abs(c) + (12*sqrt(a*c)*a^3*c^6*d^8*e - 37*sqr
t(a*c)*a^4*c^5*d^6*e^3 + 38*sqrt(a*c)*a^5*c^4*d^4*e^5 - 13*sqrt(a*c)*a^6*c^3*d^2*e^7)*A*abs(c) - (2*sqrt(a*c)*
a^4*c^5*d^7*e^2 - 7*sqrt(a*c)*a^5*c^4*d^5*e^4 + 8*sqrt(a*c)*a^6*c^3*d^3*e^6 - 3*sqrt(a*c)*a^7*c^2*d*e^8)*B*abs
(c))*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^3*d^3 - a^3*c^2*d*e^2 - sqrt((a^2*c^3*d^3 - a^3*c^2*d*e^2)^2 - (a^2*c^3
*d^4 - 2*a^3*c^2*d^2*e^2 + a^4*c*e^4)*(a^2*c^3*d^2 - a^3*c^2*e^2)))/(a^2*c^3*d^2 - a^3*c^2*e^2)))/((a^4*c^5*d^
5 + sqrt(a*c)*a^4*c^4*d^4*e - 2*a^5*c^4*d^3*e^2 - 2*sqrt(a*c)*a^5*c^3*d^2*e^3 + a^6*c^3*d*e^4 + sqrt(a*c)*a^6*
c^2*e^5)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(a^2*c^2*d^2*e - a^3*c*e^3)) - 1/16*(6*(x*e + d)^(7/2)*A*c^3*d^2*e -
18*(x*e + d)^(5/2)*A*c^3*d^3*e + 18*(x*e + d)^(3/2)*A*c^3*d^4*e - 6*sqrt(x*e + d)*A*c^3*d^5*e - (x*e + d)^(7/2
)*B*a*c^2*d*e^2 + 3*(x*e + d)^(5/2)*B*a*c^2*d^2*e^2 - 3*(x*e + d)^(3/2)*B*a*c^2*d^3*e^2 + sqrt(x*e + d)*B*a*c^
2*d^4*e^2 - 5*(x*e + d)^(7/2)*A*a*c^2*e^3 + 14*(x*e + d)^(5/2)*A*a*c^2*d*e^3 - 23*(x*e + d)^(3/2)*A*a*c^2*d^2*
e^3 + 14*sqrt(x*e + d)*A*a*c^2*d^3*e^3 + (x*e + d)^(5/2)*B*a^2*c*e^4 - (x*e + d)^(3/2)*B*a^2*c*d*e^4 - 4*sqrt(
x*e + d)*B*a^2*c*d^2*e^4 + 9*(x*e + d)^(3/2)*A*a^2*c*e^5 - 8*sqrt(x*e + d)*A*a^2*c*d*e^5 + 3*sqrt(x*e + d)*B*a
^3*e^6)/((a^2*c^2*d^2 - a^3*c*e^2)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 - a*e^2)^2)

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maple [B]  time = 0.10, size = 1733, normalized size = 4.66

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]

13/32*e^3/a/(a*e^2-c*d^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*c*d-3/8*e/a^2/(a*e^2-c*d^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*c^2*d^3+1/16*e^2/a/(a*e^2-c*d^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*c*d^2+1/16*e^2/a/(a*e^2-c*d^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*c
*d^2+13/32*e^3/a/(a*e^2-c*d^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*c*d-3/8*e/a^2/(a*e^2-c*d^2)/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan
h((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*c^2*d^3+9/8*e/(c*e^2*x^2-a*e^2)^2/a^2/(a*e^2-c*d^2)*(e*x+
d)^(3/2)*A*c^2*d^4+1/16*e^4/(c*e^2*x^2-a*e^2)^2/(a*e^2-c*d^2)*(e*x+d)^(5/2)*B+3/16*e^4/(c*e^2*x^2-a*e^2)^2/c*(
e*x+d)^(1/2)*B-1/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(1/2)*B*d^2-3/32*e^4/(a*e^2-c*d^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-3/32*e^4/(a*e^2-c*d^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+5/3
2*e^3/a/(a*e^2-c*d^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-1/16*e^4/(c*e^2*x^2-a*e^2)^2/(a*e^2-c*d^2)*(e*x+d)^(3/2)*B*d-5/32*e^3/a/(a*e^2-c*d^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-1/2*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(
1/2)*A*d+7/8*e^3/(c*e^2*x^2-a*e^2)^2/a/(a*e^2-c*d^2)*(e*x+d)^(5/2)*A*c*d-3/16*e/a^2/(a*e^2-c*d^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*c*d^2+3/8*e/(c*e^2*x^2-a*e^2)^2
*c^2/a^2/(a*e^2-c*d^2)*(e*x+d)^(7/2)*A*d^2-9/8*e/(c*e^2*x^2-a*e^2)^2/a^2/(a*e^2-c*d^2)*(e*x+d)^(5/2)*A*c^2*d^3
+3/16*e/a^2/(a*e^2-c*d^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*c*d^2-3/16*e^2/(c*e^2*x^2-a*e^2)^2/a/(a*e^2-c*d^2)*(e*x+d)^(3/2)*B*c*d^3+3/16*e^2/(c*e^2*x^2-a*e^2)^2/
a/(a*e^2-c*d^2)*(e*x+d)^(5/2)*B*c*d^2-23/16*e^3/(c*e^2*x^2-a*e^2)^2/a/(a*e^2-c*d^2)*(e*x+d)^(3/2)*A*c*d^2-1/16
*e^2/(c*e^2*x^2-a*e^2)^2*c/a/(a*e^2-c*d^2)*(e*x+d)^(7/2)*B*d+9/16*e^5/(c*e^2*x^2-a*e^2)^2/(a*e^2-c*d^2)*(e*x+d
)^(3/2)*A-5/16*e^3/(c*e^2*x^2-a*e^2)^2*c/a/(a*e^2-c*d^2)*(e*x+d)^(7/2)*A+3/8*e/(c*e^2*x^2-a*e^2)^2/a^2*c*(e*x+
d)^(1/2)*A*d^3-1/32*e^2/a/(a*e^2-c*d^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+1/32*e^2/a/(a*e^2-c*d^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

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

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mupad [B]  time = 7.83, size = 13200, normalized size = 35.48

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((((12288*B*a^8*c^2*e^8 - 32768*A*a^7*c^3*d*e^7 - 24576*A*a^5*c^5*d^5*e^3 + 57344*A*a^6*c^4*d^3*e^5 + 409
6*B*a^6*c^4*d^4*e^4 - 16384*B*a^7*c^3*d^2*e^6)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2)) - ((d + e*x)^(
1/2)*((144*A^2*a^5*c^7*d^7 - 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^
4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2)
- 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) - 30*A*B*c^2*d^3*e^4*(a^15*c^
5)^(1/2) - 48*A*B*a^6*c^6*d^6*e + 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c
^4*d^2*e^5 + 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^
12*c^6*d^2*e^4)))^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4))/(64*(a^6*e^4 + a^4
*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*A^2*a^5*c^7*d^7 - 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2
 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 + 21*A^2*c^2*
d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) - 30
*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e + 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^
5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 + 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 -
3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2)*(25*A^2*a^3*c^2*e^8 + 144*A^2*c^5*d^6*e^2
+ 9*B^2*a^4*c*e^8 + 109*A^2*a^2*c^3*d^2*e^6 + 4*B^2*a^2*c^3*d^4*e^4 - 11*B^2*a^3*c^2*d^2*e^6 - 276*A^2*a*c^4*d
^4*e^4 - 48*A*B*a*c^4*d^5*e^3 - 68*A*B*a^3*c^2*d*e^7 + 112*A*B*a^2*c^3*d^3*e^5))/(64*(a^6*e^4 + a^4*c^2*d^4 -
2*a^5*c*d^2*e^2)))*((144*A^2*a^5*c^7*d^7 - 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*
a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 + 21*A^2*c^2*d^2*e^5*(a^
15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) - 30*A*B*c^2*d^
3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e + 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 -
 150*A*B*a^8*c^4*d^2*e^5 + 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*
d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2)*1i - (((12288*B*a^8*c^2*e^8 - 32768*A*a^7*c^3*d*e^7 - 24576*A*a^5*c^5*d^
5*e^3 + 57344*A*a^6*c^4*d^3*e^5 + 4096*B*a^6*c^4*d^4*e^4 - 16384*B*a^7*c^3*d^2*e^6)/(4096*(a^8*e^4 + a^6*c^2*d
^4 - 2*a^7*c*d^2*e^2)) + ((d + e*x)^(1/2)*((144*A^2*a^5*c^7*d^7 - 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6
*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 +
 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^5
)^(1/2) - 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e + 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 16
0*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 + 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^1
3*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192
*a^6*c^5*d^3*e^4))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*A^2*a^5*c^7*d^7 - 9*B^2*a^2*e^7*(a^15
*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e
^4 + 30*A*B*a^9*c^3*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 -
 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) - 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e + 5*B^2*a*c*d^2*
e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 + 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/
(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(25*A
^2*a^3*c^2*e^8 + 144*A^2*c^5*d^6*e^2 + 9*B^2*a^4*c*e^8 + 109*A^2*a^2*c^3*d^2*e^6 + 4*B^2*a^2*c^3*d^4*e^4 - 11*
B^2*a^3*c^2*d^2*e^6 - 276*A^2*a*c^4*d^4*e^4 - 48*A*B*a*c^4*d^5*e^3 - 68*A*B*a^3*c^2*d*e^7 + 112*A*B*a^2*c^3*d^
3*e^5))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*A^2*a^5*c^7*d^7 - 9*B^2*a^2*e^7*(a^15*c^5)^(1/2)
 - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B
*a^9*c^3*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 - 25*A^2*a*c
*e^7*(a^15*c^5)^(1/2) - 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e + 5*B^2*a*c*d^2*e^5*(a^15*c
^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 + 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10
*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2)*1i)/((((12288*B*a^8*c^2*e^8 - 32768
*A*a^7*c^3*d*e^7 - 24576*A*a^5*c^5*d^5*e^3 + 57344*A*a^6*c^4*d^3*e^5 + 4096*B*a^6*c^4*d^4*e^4 - 16384*B*a^7*c^
3*d^2*e^6)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2)) - ((d + e*x)^(1/2)*((144*A^2*a^5*c^7*d^7 - 9*B^2*a
^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a
^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9
*c^3*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) - 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e + 5*
B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 + 38*A*B*a*c*d*e^6*(a^15*
c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2)*(4096*a^7*c^
4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*
A^2*a^5*c^7*d^7 - 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a
^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a
^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) - 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) -
48*A*B*a^6*c^6*d^6*e + 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5
+ 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*
e^4)))^(1/2) + ((d + e*x)^(1/2)*(25*A^2*a^3*c^2*e^8 + 144*A^2*c^5*d^6*e^2 + 9*B^2*a^4*c*e^8 + 109*A^2*a^2*c^3*
d^2*e^6 + 4*B^2*a^2*c^3*d^4*e^4 - 11*B^2*a^3*c^2*d^2*e^6 - 276*A^2*a*c^4*d^4*e^4 - 48*A*B*a*c^4*d^5*e^3 - 68*A
*B*a^3*c^2*d*e^7 + 112*A*B*a^2*c^3*d^3*e^5))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*A^2*a^5*c^7
*d^7 - 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*
e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^
6 + 15*B^2*a^9*c^3*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) - 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*
c^6*d^6*e + 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 + 38*A*B*a*
c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2
) + (((12288*B*a^8*c^2*e^8 - 32768*A*a^7*c^3*d*e^7 - 24576*A*a^5*c^5*d^5*e^3 + 57344*A*a^6*c^4*d^3*e^5 + 4096*
B*a^6*c^4*d^4*e^4 - 16384*B*a^7*c^3*d^2*e^6)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2)) + ((d + e*x)^(1/
2)*((144*A^2*a^5*c^7*d^7 - 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4
+ 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) -
105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) - 30*A*B*c^2*d^3*e^4*(a^15*c^5)
^(1/2) - 48*A*B*a^6*c^6*d^6*e + 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4
*d^2*e^5 + 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12
*c^6*d^2*e^4)))^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4))/(64*(a^6*e^4 + a^4*c
^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*A^2*a^5*c^7*d^7 - 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 +
 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 + 21*A^2*c^2*d^
2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) - 30*A
*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e + 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*
d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 + 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*
a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(25*A^2*a^3*c^2*e^8 + 144*A^2*c^5*d^6*e^2 +
9*B^2*a^4*c*e^8 + 109*A^2*a^2*c^3*d^2*e^6 + 4*B^2*a^2*c^3*d^4*e^4 - 11*B^2*a^3*c^2*d^2*e^6 - 276*A^2*a*c^4*d^4
*e^4 - 48*A*B*a*c^4*d^5*e^3 - 68*A*B*a^3*c^2*d*e^7 + 112*A*B*a^2*c^3*d^3*e^5))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*
a^5*c*d^2*e^2)))*((144*A^2*a^5*c^7*d^7 - 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^
7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 + 21*A^2*c^2*d^2*e^5*(a^15
*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) - 30*A*B*c^2*d^3*
e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e + 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 1
50*A*B*a^8*c^4*d^2*e^5 + 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^
4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2) + (864*A^3*c^4*d^6*e^3 + 45*A*B^2*a^4*e^9 - 125*A^3*a^3*c*e^9 + 9*B^3*a^4*
d*e^8 + 1170*A^3*a^2*c^2*d^2*e^7 - 1944*A^3*a*c^3*d^4*e^5 - 4*B^3*a^3*c*d^3*e^6 + 72*A*B^2*a^2*c^2*d^4*e^5 + 9
72*A^2*B*a^2*c^2*d^3*e^6 - 465*A^2*B*a^3*c*d*e^8 - 162*A*B^2*a^3*c*d^2*e^7 - 432*A^2*B*a*c^3*d^5*e^4)/(2048*(a
^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2))))*((144*A^2*a^5*c^7*d^7 - 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^
6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7
+ 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^
5)^(1/2) - 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e + 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 1
60*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 + 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^
13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2)*2i + atan(((((12288*B*a^8*c^2*e^8 - 32768*A*a^7*
c^3*d*e^7 - 24576*A*a^5*c^5*d^5*e^3 + 57344*A*a^6*c^4*d^3*e^5 + 4096*B*a^6*c^4*d^4*e^4 - 16384*B*a^7*c^3*d^2*e
^6)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2)) - ((d + e*x)^(1/2)*((144*A^2*a^5*c^7*d^7 + 9*B^2*a^2*e^7*
(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*
d^3*e^4 + 30*A*B*a^9*c^3*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*
e^6 + 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) + 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e - 5*B^2*a*c
*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 - 38*A*B*a*c*d*e^6*(a^15*c^5)^(1
/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2)*(4096*a^7*c^4*d*e^6
 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*A^2*a^5
*c^7*d^7 + 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*
d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*
d*e^6 + 15*B^2*a^9*c^3*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) + 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*
a^6*c^6*d^6*e - 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 - 38*A*
B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^
(1/2) + ((d + e*x)^(1/2)*(25*A^2*a^3*c^2*e^8 + 144*A^2*c^5*d^6*e^2 + 9*B^2*a^4*c*e^8 + 109*A^2*a^2*c^3*d^2*e^6
 + 4*B^2*a^2*c^3*d^4*e^4 - 11*B^2*a^3*c^2*d^2*e^6 - 276*A^2*a*c^4*d^4*e^4 - 48*A*B*a*c^4*d^5*e^3 - 68*A*B*a^3*
c^2*d*e^7 + 112*A*B*a^2*c^3*d^3*e^5))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*A^2*a^5*c^7*d^7 +
9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 1
5*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*
B^2*a^9*c^3*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) + 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6
*e - 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 - 38*A*B*a*c*d*e^6
*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2)*1i -
(((12288*B*a^8*c^2*e^8 - 32768*A*a^7*c^3*d*e^7 - 24576*A*a^5*c^5*d^5*e^3 + 57344*A*a^6*c^4*d^3*e^5 + 4096*B*a^
6*c^4*d^4*e^4 - 16384*B*a^7*c^3*d^2*e^6)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2)) + ((d + e*x)^(1/2)*(
(144*A^2*a^5*c^7*d^7 + 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*
B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*
A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) + 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/
2) - 48*A*B*a^6*c^6*d^6*e - 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2
*e^5 - 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6
*d^2*e^4)))^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4))/(64*(a^6*e^4 + a^4*c^2*d
^4 - 2*a^5*c*d^2*e^2)))*((144*A^2*a^5*c^7*d^7 + 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385
*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 - 21*A^2*c^2*d^2*e^
5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) + 30*A*B*c
^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e - 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*
e^3 - 150*A*B*a^8*c^4*d^2*e^5 - 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11
*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(25*A^2*a^3*c^2*e^8 + 144*A^2*c^5*d^6*e^2 + 9*B^
2*a^4*c*e^8 + 109*A^2*a^2*c^3*d^2*e^6 + 4*B^2*a^2*c^3*d^4*e^4 - 11*B^2*a^3*c^2*d^2*e^6 - 276*A^2*a*c^4*d^4*e^4
 - 48*A*B*a*c^4*d^5*e^3 - 68*A*B*a^3*c^2*d*e^7 + 112*A*B*a^2*c^3*d^3*e^5))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*
c*d^2*e^2)))*((144*A^2*a^5*c^7*d^7 + 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^
5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^5
)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) + 30*A*B*c^2*d^3*e^4*
(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e - 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A
*B*a^8*c^4*d^2*e^5 - 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^
2 + 3*a^12*c^6*d^2*e^4)))^(1/2)*1i)/((((12288*B*a^8*c^2*e^8 - 32768*A*a^7*c^3*d*e^7 - 24576*A*a^5*c^5*d^5*e^3
+ 57344*A*a^6*c^4*d^3*e^5 + 4096*B*a^6*c^4*d^4*e^4 - 16384*B*a^7*c^3*d^2*e^6)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2
*a^7*c*d^2*e^2)) - ((d + e*x)^(1/2)*((144*A^2*a^5*c^7*d^7 + 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d
^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 - 21*A^
2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^5)^(1/2
) + 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e - 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*
a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 - 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*
e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c
^5*d^3*e^4))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*A^2*a^5*c^7*d^7 + 9*B^2*a^2*e^7*(a^15*c^5)^
(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 3
0*A*B*a^9*c^3*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 + 25*A^
2*a*c*e^7*(a^15*c^5)^(1/2) + 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e - 5*B^2*a*c*d^2*e^5*(a
^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 - 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*
(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2)*(25*A^2*a^3
*c^2*e^8 + 144*A^2*c^5*d^6*e^2 + 9*B^2*a^4*c*e^8 + 109*A^2*a^2*c^3*d^2*e^6 + 4*B^2*a^2*c^3*d^4*e^4 - 11*B^2*a^
3*c^2*d^2*e^6 - 276*A^2*a*c^4*d^4*e^4 - 48*A*B*a*c^4*d^5*e^3 - 68*A*B*a^3*c^2*d*e^7 + 112*A*B*a^2*c^3*d^3*e^5)
)/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*A^2*a^5*c^7*d^7 + 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420
*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c
^3*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 + 25*A^2*a*c*e^7*(
a^15*c^5)^(1/2) + 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e - 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1
/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 - 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d
^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2) + (((12288*B*a^8*c^2*e^8 - 32768*A*a^7*c^
3*d*e^7 - 24576*A*a^5*c^5*d^5*e^3 + 57344*A*a^6*c^4*d^3*e^5 + 4096*B*a^6*c^4*d^4*e^4 - 16384*B*a^7*c^3*d^2*e^6
)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2)) + ((d + e*x)^(1/2)*((144*A^2*a^5*c^7*d^7 + 9*B^2*a^2*e^7*(a
^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^
3*e^4 + 30*A*B*a^9*c^3*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^
6 + 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) + 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e - 5*B^2*a*c*d
^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 - 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2
))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2)*(4096*a^7*c^4*d*e^6 +
 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*A^2*a^5*c
^7*d^7 + 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^
5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*
e^6 + 15*B^2*a^9*c^3*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) + 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^
6*c^6*d^6*e - 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 - 38*A*B*
a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1
/2) - ((d + e*x)^(1/2)*(25*A^2*a^3*c^2*e^8 + 144*A^2*c^5*d^6*e^2 + 9*B^2*a^4*c*e^8 + 109*A^2*a^2*c^3*d^2*e^6 +
 4*B^2*a^2*c^3*d^4*e^4 - 11*B^2*a^3*c^2*d^2*e^6 - 276*A^2*a*c^4*d^4*e^4 - 48*A*B*a*c^4*d^5*e^3 - 68*A*B*a^3*c^
2*d*e^7 + 112*A*B*a^2*c^3*d^3*e^5))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*A^2*a^5*c^7*d^7 + 9*
B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*
B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^
2*a^9*c^3*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) + 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e
 - 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 - 38*A*B*a*c*d*e^6*(
a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2) + (864*
A^3*c^4*d^6*e^3 + 45*A*B^2*a^4*e^9 - 125*A^3*a^3*c*e^9 + 9*B^3*a^4*d*e^8 + 1170*A^3*a^2*c^2*d^2*e^7 - 1944*A^3
*a*c^3*d^4*e^5 - 4*B^3*a^3*c*d^3*e^6 + 72*A*B^2*a^2*c^2*d^4*e^5 + 972*A^2*B*a^2*c^2*d^3*e^6 - 465*A^2*B*a^3*c*
d*e^8 - 162*A*B^2*a^3*c*d^2*e^7 - 432*A^2*B*a*c^3*d^5*e^4)/(2048*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2))))*
((144*A^2*a^5*c^7*d^7 + 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4
*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 105
*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^5)^(1/2) + 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1
/2) - 48*A*B*a^6*c^6*d^6*e - 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^
2*e^5 - 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^
6*d^2*e^4)))^(1/2)*2i - (((d + e*x)^(3/2)*(B*a^2*d*e^4 - 9*A*a^2*e^5 - 18*A*c^2*d^4*e + 23*A*a*c*d^2*e^3 + 3*B
*a*c*d^3*e^2))/(16*a^2*(a*e^2 - c*d^2)) - ((d + e*x)^(1/2)*(3*B*a^2*e^4 + 6*A*c^2*d^3*e - 8*A*a*c*d*e^3 - B*a*
c*d^2*e^2))/(16*a^2*c) - ((d + e*x)^(5/2)*(B*a^2*e^4 - 18*A*c^2*d^3*e + 14*A*a*c*d*e^3 + 3*B*a*c*d^2*e^2))/(16
*a^2*(a*e^2 - c*d^2)) + (c*(d + e*x)^(7/2)*(5*A*a*e^3 + B*a*d*e^2 - 6*A*c*d^2*e))/(16*a^2*(a*e^2 - c*d^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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