3.20.74 \(\int \frac {\sqrt {d+e x}}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=634 \[ -\frac {\sqrt {c} \left (-8 c^2 d e \left (-3 d \sqrt {b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (-2 b \left (6 d \sqrt {b^2-4 a c}+13 a e\right )+10 a e \sqrt {b^2-4 a c}+23 b^2 d\right )+b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {c} \left (-8 c^2 d e \left (3 d \sqrt {b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (12 b d \sqrt {b^2-4 a c}-10 a e \sqrt {b^2-4 a c}-26 a b e+23 b^2 d\right )+b^2 e^3 \left (b-\sqrt {b^2-4 a c}\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (-c x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-4 b c \left (2 a e^2+3 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+13 b^2 c d e\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

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Rubi [A]  time = 4.97, antiderivative size = 634, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {736, 822, 826, 1166, 208} \begin {gather*} -\frac {\sqrt {d+e x} \left (-c x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-4 b c \left (2 a e^2+3 c d^2\right )-4 a c^2 d e+13 b^2 c d e+b^3 \left (-e^2\right )\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {c} \left (-8 c^2 d e \left (-3 d \sqrt {b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (-2 b \left (6 d \sqrt {b^2-4 a c}+13 a e\right )+10 a e \sqrt {b^2-4 a c}+23 b^2 d\right )+b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {c} \left (-8 c^2 d e \left (3 d \sqrt {b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (12 b d \sqrt {b^2-4 a c}-10 a e \sqrt {b^2-4 a c}-26 a b e+23 b^2 d\right )+b^2 e^3 \left (b-\sqrt {b^2-4 a c}\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b + 2*c*x)*Sqrt[d + e*x])/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (Sqrt[d + e*x]*(13*b^2*c*d*e - 4*a*c^2*d*
e - b^3*e^2 - 4*b*c*(3*c*d^2 + 2*a*e^2) - c*(24*c^2*d^2 + b^2*e^2 - 4*c*e*(6*b*d - 5*a*e))*x))/(4*(b^2 - 4*a*c
)^2*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)) - (Sqrt[c]*(96*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 8*c^
2*d*e*(18*b*d - 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d + 10*a*Sqrt[b^2 - 4*a*c]*e - 2*b*(6*Sqrt[b
^2 - 4*a*c]*d + 13*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(4
*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)) + (Sqrt[c]*(96*c
^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(18*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b
^2*d + 12*b*Sqrt[b^2 - 4*a*c]*d - 26*a*b*e - 10*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x]
)/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c]
)*e]*(c*d^2 - b*d*e + a*e^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 736

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

Rule 822

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

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + 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 {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\int \frac {-6 c d+\frac {b e}{2}-5 c e x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (13 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (3 c d^2+2 a e^2\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{4} \left (-48 c^3 d^3-b^3 e^3-b c e^2 (11 b d-16 a e)+4 c^2 d e (15 b d-13 a e)\right )-\frac {1}{4} c e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (13 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (3 c d^2+2 a e^2\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{4} e \left (-48 c^3 d^3-b^3 e^3-b c e^2 (11 b d-16 a e)+4 c^2 d e (15 b d-13 a e)\right )+\frac {1}{4} c d e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right )-\frac {1}{4} c e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (13 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (3 c d^2+2 a e^2\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {\left (c \left (96 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d+3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+12 b \sqrt {b^2-4 a c} d-26 a b e-10 a \sqrt {b^2-4 a c} e\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )}+\frac {\left (c \left (96 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d-3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+10 a \sqrt {b^2-4 a c} e-2 b \left (6 \sqrt {b^2-4 a c} d+13 a e\right )\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (13 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (3 c d^2+2 a e^2\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {\sqrt {c} \left (96 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d-3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+10 a \sqrt {b^2-4 a c} e-2 b \left (6 \sqrt {b^2-4 a c} d+13 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}+\frac {\sqrt {c} \left (96 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d+3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+12 b \sqrt {b^2-4 a c} d-26 a b e-10 a \sqrt {b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}\\ \end {align*}

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Mathematica [A]  time = 4.02, size = 580, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {c} \left (\frac {\left (8 c^2 d e \left (3 d \sqrt {b^2-4 a c}+13 a e-18 b d\right )+2 c e^2 \left (-2 b \left (6 d \sqrt {b^2-4 a c}+13 a e\right )+10 a e \sqrt {b^2-4 a c}+23 b^2 d\right )+b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{\sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}-\frac {\left (-8 c^2 d e \left (3 d \sqrt {b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (12 b d \sqrt {b^2-4 a c}-10 a e \sqrt {b^2-4 a c}-26 a b e+23 b^2 d\right )+b^2 e^3 \left (b-\sqrt {b^2-4 a c}\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \left (e (a e-b d)+c d^2\right )}-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) (a+x (b+c x))^2}-\frac {\sqrt {d+e x} \left (4 b c \left (2 a e^2+3 c d (d-2 e x)\right )+4 c^2 \left (a e (d+5 e x)+6 c d^2 x\right )+b^3 e^2+b^2 c e (e x-13 d)\right )}{4 \left (b^2-4 a c\right )^2 (a+x (b+c x)) \left (e (b d-a e)-c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((b + 2*c*x)*Sqrt[d + e*x])/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) - (Sqrt[d + e*x]*(b^3*e^2 + b^2*c*e*(-13*
d + e*x) + 4*b*c*(2*a*e^2 + 3*c*d*(d - 2*e*x)) + 4*c^2*(6*c*d^2*x + a*e*(d + 5*e*x))))/(4*(b^2 - 4*a*c)^2*(-(c
*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))) - (Sqrt[c]*(((96*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 + 8*c^2*d
*e*(-18*b*d + 3*Sqrt[b^2 - 4*a*c]*d + 13*a*e) + 2*c*e^2*(23*b^2*d + 10*a*Sqrt[b^2 - 4*a*c]*e - 2*b*(6*Sqrt[b^2
- 4*a*c]*d + 13*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt
[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] - ((96*c^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(18*b*d + 3*
Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d + 12*b*Sqrt[b^2 - 4*a*c]*d - 26*a*b*e - 10*a*Sqrt[b^2 - 4*a*
c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqr
t[b^2 - 4*a*c])*e]))/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*(c*d^2 + e*(-(b*d) + a*e)))

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IntegrateAlgebraic [C]  time = 15.17, size = 1251, normalized size = 1.97 \begin {gather*} -\frac {e \sqrt {d+e x} \left (-24 c^4 d^5+60 b c^3 e d^4+72 c^4 (d+e x) d^4-56 a c^3 e^2 d^3-46 b^2 c^2 e^2 d^3-72 c^4 (d+e x)^2 d^3-144 b c^3 e (d+e x) d^3+84 a b c^2 e^3 d^2+9 b^3 c e^3 d^2+24 c^4 (d+e x)^3 d^2+108 b c^3 e (d+e x)^2 d^2+92 a c^3 e^2 (d+e x) d^2+85 b^2 c^2 e^2 (d+e x) d^2+b^4 e^4 d-32 a^2 c^2 e^4 d-26 a b^2 c e^4 d-24 b c^3 e (d+e x)^3 d-56 a c^3 e^2 (d+e x)^2 d-40 b^2 c^2 e^2 (d+e x)^2 d-92 a b c^2 e^3 (d+e x) d-13 b^3 c e^3 (d+e x) d-a b^3 e^5+16 a^2 b c e^5+20 a c^3 e^2 (d+e x)^3+b^2 c^2 e^2 (d+e x)^3+28 a b c^2 e^3 (d+e x)^2+2 b^3 c e^3 (d+e x)^2+b^4 e^4 (d+e x)+36 a^2 c^2 e^4 (d+e x)+5 a b^2 c e^4 (d+e x)\right )}{4 \left (b^2-4 a c\right )^2 \left (-c d^2+b e d-a e^2\right ) \left (-c d^2+b e d+2 c (d+e x) d-a e^2-c (d+e x)^2-b e (d+e x)\right )^2}+\frac {\left (96 i \sqrt {2} d^3 c^{7/2}+104 i \sqrt {2} a d e^2 c^{5/2}-144 i \sqrt {2} b d^2 e c^{5/2}-24 \sqrt {2} \sqrt {4 a c-b^2} d^2 e c^{5/2}-52 i \sqrt {2} a b e^3 c^{3/2}-20 \sqrt {2} a \sqrt {4 a c-b^2} e^3 c^{3/2}+46 i \sqrt {2} b^2 d e^2 c^{3/2}+24 \sqrt {2} b \sqrt {4 a c-b^2} d e^2 c^{3/2}+i \sqrt {2} b^3 e^3 \sqrt {c}-\sqrt {2} b^2 \sqrt {4 a c-b^2} e^3 \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {4 a c-b^2} e}}\right )}{8 \left (b^2-4 a c\right )^2 \sqrt {4 a c-b^2} \sqrt {-2 c d+b e-i \sqrt {4 a c-b^2} e} \left (-c d^2+b e d-a e^2\right )}+\frac {\left (-96 i \sqrt {2} d^3 c^{7/2}-104 i \sqrt {2} a d e^2 c^{5/2}+144 i \sqrt {2} b d^2 e c^{5/2}-24 \sqrt {2} \sqrt {4 a c-b^2} d^2 e c^{5/2}+52 i \sqrt {2} a b e^3 c^{3/2}-20 \sqrt {2} a \sqrt {4 a c-b^2} e^3 c^{3/2}-46 i \sqrt {2} b^2 d e^2 c^{3/2}+24 \sqrt {2} b \sqrt {4 a c-b^2} d e^2 c^{3/2}-i \sqrt {2} b^3 e^3 \sqrt {c}-\sqrt {2} b^2 \sqrt {4 a c-b^2} e^3 \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {4 a c-b^2} e}}\right )}{8 \left (b^2-4 a c\right )^2 \sqrt {4 a c-b^2} \sqrt {-2 c d+b e+i \sqrt {4 a c-b^2} e} \left (-c d^2+b e d-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(e*Sqrt[d + e*x]*(-24*c^4*d^5 + 60*b*c^3*d^4*e - 46*b^2*c^2*d^3*e^2 - 56*a*c^3*d^3*e^2 + 9*b^3*c*d^2*e^3
+ 84*a*b*c^2*d^2*e^3 + b^4*d*e^4 - 26*a*b^2*c*d*e^4 - 32*a^2*c^2*d*e^4 - a*b^3*e^5 + 16*a^2*b*c*e^5 + 72*c^4*d
^4*(d + e*x) - 144*b*c^3*d^3*e*(d + e*x) + 85*b^2*c^2*d^2*e^2*(d + e*x) + 92*a*c^3*d^2*e^2*(d + e*x) - 13*b^3*
c*d*e^3*(d + e*x) - 92*a*b*c^2*d*e^3*(d + e*x) + b^4*e^4*(d + e*x) + 5*a*b^2*c*e^4*(d + e*x) + 36*a^2*c^2*e^4*
(d + e*x) - 72*c^4*d^3*(d + e*x)^2 + 108*b*c^3*d^2*e*(d + e*x)^2 - 40*b^2*c^2*d*e^2*(d + e*x)^2 - 56*a*c^3*d*e
^2*(d + e*x)^2 + 2*b^3*c*e^3*(d + e*x)^2 + 28*a*b*c^2*e^3*(d + e*x)^2 + 24*c^4*d^2*(d + e*x)^3 - 24*b*c^3*d*e*
(d + e*x)^3 + b^2*c^2*e^2*(d + e*x)^3 + 20*a*c^3*e^2*(d + e*x)^3))/((b^2 - 4*a*c)^2*(-(c*d^2) + b*d*e - a*e^2)
*(-(c*d^2) + b*d*e - a*e^2 + 2*c*d*(d + e*x) - b*e*(d + e*x) - c*(d + e*x)^2)^2) + (((96*I)*Sqrt[2]*c^(7/2)*d^
3 - (144*I)*Sqrt[2]*b*c^(5/2)*d^2*e - 24*Sqrt[2]*c^(5/2)*Sqrt[-b^2 + 4*a*c]*d^2*e + (46*I)*Sqrt[2]*b^2*c^(3/2)
*d*e^2 + (104*I)*Sqrt[2]*a*c^(5/2)*d*e^2 + 24*Sqrt[2]*b*c^(3/2)*Sqrt[-b^2 + 4*a*c]*d*e^2 + I*Sqrt[2]*b^3*Sqrt[
c]*e^3 - (52*I)*Sqrt[2]*a*b*c^(3/2)*e^3 - Sqrt[2]*b^2*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*e^3 - 20*Sqrt[2]*a*c^(3/2)*Sq
rt[-b^2 + 4*a*c]*e^3)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(8*
(b^2 - 4*a*c)^2*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]*(-(c*d^2) + b*d*e - a*e^2)) + (
((-96*I)*Sqrt[2]*c^(7/2)*d^3 + (144*I)*Sqrt[2]*b*c^(5/2)*d^2*e - 24*Sqrt[2]*c^(5/2)*Sqrt[-b^2 + 4*a*c]*d^2*e -
(46*I)*Sqrt[2]*b^2*c^(3/2)*d*e^2 - (104*I)*Sqrt[2]*a*c^(5/2)*d*e^2 + 24*Sqrt[2]*b*c^(3/2)*Sqrt[-b^2 + 4*a*c]*
d*e^2 - I*Sqrt[2]*b^3*Sqrt[c]*e^3 + (52*I)*Sqrt[2]*a*b*c^(3/2)*e^3 - Sqrt[2]*b^2*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*e^
3 - 20*Sqrt[2]*a*c^(3/2)*Sqrt[-b^2 + 4*a*c]*e^3)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*
Sqrt[-b^2 + 4*a*c]*e]])/(8*(b^2 - 4*a*c)^2*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]*(-(c
*d^2) + b*d*e - a*e^2))

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

[Out]

Timed out

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giac [B]  time = 7.98, size = 5117, normalized size = 8.07

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*((b^4*c*d^2*e - 8*a*b^2*c^2*d^2*e + 16*a^2*c^3*d^2*e - b^5*d*e^2 + 8*a*b^3*c*d*e^2 - 16*a^2*b*c^2*d*e^2 +
a*b^4*e^3 - 8*a^2*b^2*c*e^3 + 16*a^3*c^2*e^3)^2*(24*c^2*d^2*e - 24*b*c*d*e^2 + (b^2 + 20*a*c)*e^3)*sqrt(-4*c^
2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e) + 2*(24*(b^2*c^4 - 4*a*c^5)*sqrt(b^2 - 4*a*c)*d^5*e - 60*(b^3*c^3 - 4*a
*b*c^4)*sqrt(b^2 - 4*a*c)*d^4*e^2 + 2*(23*b^4*c^2 - 64*a*b^2*c^3 - 112*a^2*c^4)*sqrt(b^2 - 4*a*c)*d^3*e^3 - 3*
(3*b^5*c + 16*a*b^3*c^2 - 112*a^2*b*c^3)*sqrt(b^2 - 4*a*c)*d^2*e^4 - (b^6 - 30*a*b^4*c + 72*a^2*b^2*c^2 + 128*
a^3*c^3)*sqrt(b^2 - 4*a*c)*d*e^5 + (a*b^5 - 20*a^2*b^3*c + 64*a^3*b*c^2)*sqrt(b^2 - 4*a*c)*e^6)*sqrt(-4*c^2*d
+ 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(b^4*c*d^2*e - 8*a*b^2*c^2*d^2*e + 16*a^2*c^3*d^2*e - b^5*d*e^2 + 8*a*b^
3*c*d*e^2 - 16*a^2*b*c^2*d*e^2 + a*b^4*e^3 - 8*a^2*b^2*c*e^3 + 16*a^3*c^2*e^3) - (192*(b^6*c^6 - 12*a*b^4*c^7
+ 48*a^2*b^2*c^8 - 64*a^3*c^9)*d^8*e - 768*(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*d^7*e^2 +
4*(299*b^8*c^4 - 3440*a*b^6*c^5 + 12576*a^2*b^4*c^6 - 12032*a^3*b^2*c^7 - 9472*a^4*c^8)*d^6*e^3 - 12*(75*b^9*c
^3 - 752*a*b^7*c^4 + 1824*a^2*b^5*c^5 + 2304*a^3*b^3*c^6 - 9472*a^4*b*c^7)*d^5*e^4 + (323*b^10*c^2 - 1960*a*b^
8*c^3 - 6880*a^2*b^6*c^4 + 64000*a^3*b^4*c^5 - 93440*a^4*b^2*c^6 - 38912*a^5*c^7)*d^4*e^5 - 2*(21*b^11*c + 184
*a*b^9*c^2 - 3616*a^2*b^7*c^3 + 12288*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 38912*a^5*b*c^6)*d^3*e^6 - (b^12 - 150*
a*b^10*c + 948*a^2*b^8*c^2 + 2176*a^3*b^6*c^3 - 24960*a^4*b^4*c^4 + 38400*a^5*b^2*c^5 + 13312*a^6*c^6)*d^2*e^7
+ 2*(a*b^11 - 86*a^2*b^9*c + 832*a^3*b^7*c^2 - 2368*a^4*b^5*c^3 - 256*a^5*b^3*c^4 + 6656*a^6*b*c^5)*d*e^8 - (
a^2*b^10 - 64*a^3*b^8*c + 672*a^4*b^6*c^2 - 2560*a^5*b^4*c^3 + 3328*a^6*b^2*c^4)*e^9)*sqrt(-4*c^2*d + 2*(b*c -
sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^4*c^2*d^3 - 16*a*b^2*c^3*d^3 + 32*a^2*c^
4*d^3 - 3*b^5*c*d^2*e + 24*a*b^3*c^2*d^2*e - 48*a^2*b*c^3*d^2*e + b^6*d*e^2 - 6*a*b^4*c*d*e^2 + 32*a^3*c^3*d*e
^2 - a*b^5*e^3 + 8*a^2*b^3*c*e^3 - 16*a^3*b*c^2*e^3 + sqrt((2*b^4*c^2*d^3 - 16*a*b^2*c^3*d^3 + 32*a^2*c^4*d^3
- 3*b^5*c*d^2*e + 24*a*b^3*c^2*d^2*e - 48*a^2*b*c^3*d^2*e + b^6*d*e^2 - 6*a*b^4*c*d*e^2 + 32*a^3*c^3*d*e^2 - a
*b^5*e^3 + 8*a^2*b^3*c*e^3 - 16*a^3*b*c^2*e^3)^2 - 4*(b^4*c^2*d^4 - 8*a*b^2*c^3*d^4 + 16*a^2*c^4*d^4 - 2*b^5*c
*d^3*e + 16*a*b^3*c^2*d^3*e - 32*a^2*b*c^3*d^3*e + b^6*d^2*e^2 - 6*a*b^4*c*d^2*e^2 + 32*a^3*c^3*d^2*e^2 - 2*a*
b^5*d*e^3 + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3 + a^2*b^4*e^4 - 8*a^3*b^2*c*e^4 + 16*a^4*c^2*e^4)*(b^4*c^2
*d^2 - 8*a*b^2*c^3*d^2 + 16*a^2*c^4*d^2 - b^5*c*d*e + 8*a*b^3*c^2*d*e - 16*a^2*b*c^3*d*e + a*b^4*c*e^2 - 8*a^2
*b^2*c^2*e^2 + 16*a^3*c^3*e^2)))/(b^4*c^2*d^2 - 8*a*b^2*c^3*d^2 + 16*a^2*c^4*d^2 - b^5*c*d*e + 8*a*b^3*c^2*d*e
- 16*a^2*b*c^3*d*e + a*b^4*c*e^2 - 8*a^2*b^2*c^2*e^2 + 16*a^3*c^3*e^2)))/(((b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b
^2*c^5 - 64*a^3*c^6)*sqrt(b^2 - 4*a*c)*d^6 - 3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*sqrt(b
^2 - 4*a*c)*d^5*e + 3*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*sqrt(b^2 - 4*a*c)*
d^4*e^2 - (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*sqrt(b^2 - 4*a*c)*d^3*e^3 + 3*(
a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*sqrt(b^2 - 4*a*c)*d^2*e^4 - 3*(a^2*b^7 -
12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*sqrt(b^2 - 4*a*c)*d*e^5 + (a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c
^2 - 64*a^6*c^3)*sqrt(b^2 - 4*a*c)*e^6)*abs(b^4*c*d^2*e - 8*a*b^2*c^2*d^2*e + 16*a^2*c^3*d^2*e - b^5*d*e^2 + 8
*a*b^3*c*d*e^2 - 16*a^2*b*c^2*d*e^2 + a*b^4*e^3 - 8*a^2*b^2*c*e^3 + 16*a^3*c^2*e^3)*abs(c)) - 1/32*((b^4*c*d^2
*e - 8*a*b^2*c^2*d^2*e + 16*a^2*c^3*d^2*e - b^5*d*e^2 + 8*a*b^3*c*d*e^2 - 16*a^2*b*c^2*d*e^2 + a*b^4*e^3 - 8*a
^2*b^2*c*e^3 + 16*a^3*c^2*e^3)^2*(24*c^2*d^2*e - 24*b*c*d*e^2 + (b^2 + 20*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c + s
qrt(b^2 - 4*a*c)*c)*e) - 2*(24*(b^2*c^4 - 4*a*c^5)*sqrt(b^2 - 4*a*c)*d^5*e - 60*(b^3*c^3 - 4*a*b*c^4)*sqrt(b^2
- 4*a*c)*d^4*e^2 + 2*(23*b^4*c^2 - 64*a*b^2*c^3 - 112*a^2*c^4)*sqrt(b^2 - 4*a*c)*d^3*e^3 - 3*(3*b^5*c + 16*a*
b^3*c^2 - 112*a^2*b*c^3)*sqrt(b^2 - 4*a*c)*d^2*e^4 - (b^6 - 30*a*b^4*c + 72*a^2*b^2*c^2 + 128*a^3*c^3)*sqrt(b^
2 - 4*a*c)*d*e^5 + (a*b^5 - 20*a^2*b^3*c + 64*a^3*b*c^2)*sqrt(b^2 - 4*a*c)*e^6)*sqrt(-4*c^2*d + 2*(b*c + sqrt(
b^2 - 4*a*c)*c)*e)*abs(b^4*c*d^2*e - 8*a*b^2*c^2*d^2*e + 16*a^2*c^3*d^2*e - b^5*d*e^2 + 8*a*b^3*c*d*e^2 - 16*a
^2*b*c^2*d*e^2 + a*b^4*e^3 - 8*a^2*b^2*c*e^3 + 16*a^3*c^2*e^3) - (192*(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8
- 64*a^3*c^9)*d^8*e - 768*(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*d^7*e^2 + 4*(299*b^8*c^4 -
3440*a*b^6*c^5 + 12576*a^2*b^4*c^6 - 12032*a^3*b^2*c^7 - 9472*a^4*c^8)*d^6*e^3 - 12*(75*b^9*c^3 - 752*a*b^7*c
^4 + 1824*a^2*b^5*c^5 + 2304*a^3*b^3*c^6 - 9472*a^4*b*c^7)*d^5*e^4 + (323*b^10*c^2 - 1960*a*b^8*c^3 - 6880*a^2
*b^6*c^4 + 64000*a^3*b^4*c^5 - 93440*a^4*b^2*c^6 - 38912*a^5*c^7)*d^4*e^5 - 2*(21*b^11*c + 184*a*b^9*c^2 - 361
6*a^2*b^7*c^3 + 12288*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 38912*a^5*b*c^6)*d^3*e^6 - (b^12 - 150*a*b^10*c + 948*a
^2*b^8*c^2 + 2176*a^3*b^6*c^3 - 24960*a^4*b^4*c^4 + 38400*a^5*b^2*c^5 + 13312*a^6*c^6)*d^2*e^7 + 2*(a*b^11 - 8
6*a^2*b^9*c + 832*a^3*b^7*c^2 - 2368*a^4*b^5*c^3 - 256*a^5*b^3*c^4 + 6656*a^6*b*c^5)*d*e^8 - (a^2*b^10 - 64*a^
3*b^8*c + 672*a^4*b^6*c^2 - 2560*a^5*b^4*c^3 + 3328*a^6*b^2*c^4)*e^9)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*
c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^4*c^2*d^3 - 16*a*b^2*c^3*d^3 + 32*a^2*c^4*d^3 - 3*b^5*c*
d^2*e + 24*a*b^3*c^2*d^2*e - 48*a^2*b*c^3*d^2*e + b^6*d*e^2 - 6*a*b^4*c*d*e^2 + 32*a^3*c^3*d*e^2 - a*b^5*e^3 +
8*a^2*b^3*c*e^3 - 16*a^3*b*c^2*e^3 - sqrt((2*b^4*c^2*d^3 - 16*a*b^2*c^3*d^3 + 32*a^2*c^4*d^3 - 3*b^5*c*d^2*e
+ 24*a*b^3*c^2*d^2*e - 48*a^2*b*c^3*d^2*e + b^6*d*e^2 - 6*a*b^4*c*d*e^2 + 32*a^3*c^3*d*e^2 - a*b^5*e^3 + 8*a^2
*b^3*c*e^3 - 16*a^3*b*c^2*e^3)^2 - 4*(b^4*c^2*d^4 - 8*a*b^2*c^3*d^4 + 16*a^2*c^4*d^4 - 2*b^5*c*d^3*e + 16*a*b^
3*c^2*d^3*e - 32*a^2*b*c^3*d^3*e + b^6*d^2*e^2 - 6*a*b^4*c*d^2*e^2 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 + 16*a
^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3 + a^2*b^4*e^4 - 8*a^3*b^2*c*e^4 + 16*a^4*c^2*e^4)*(b^4*c^2*d^2 - 8*a*b^2*c
^3*d^2 + 16*a^2*c^4*d^2 - b^5*c*d*e + 8*a*b^3*c^2*d*e - 16*a^2*b*c^3*d*e + a*b^4*c*e^2 - 8*a^2*b^2*c^2*e^2 + 1
6*a^3*c^3*e^2)))/(b^4*c^2*d^2 - 8*a*b^2*c^3*d^2 + 16*a^2*c^4*d^2 - b^5*c*d*e + 8*a*b^3*c^2*d*e - 16*a^2*b*c^3*
d*e + a*b^4*c*e^2 - 8*a^2*b^2*c^2*e^2 + 16*a^3*c^3*e^2)))/(((b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*
c^6)*sqrt(b^2 - 4*a*c)*d^6 - 3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*sqrt(b^2 - 4*a*c)*d^5*
e + 3*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*sqrt(b^2 - 4*a*c)*d^4*e^2 - (b^9 -
6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*sqrt(b^2 - 4*a*c)*d^3*e^3 + 3*(a*b^8 - 11*a^2*b
^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*sqrt(b^2 - 4*a*c)*d^2*e^4 - 3*(a^2*b^7 - 12*a^3*b^5*c + 4
8*a^4*b^3*c^2 - 64*a^5*b*c^3)*sqrt(b^2 - 4*a*c)*d*e^5 + (a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)
*sqrt(b^2 - 4*a*c)*e^6)*abs(b^4*c*d^2*e - 8*a*b^2*c^2*d^2*e + 16*a^2*c^3*d^2*e - b^5*d*e^2 + 8*a*b^3*c*d*e^2 -
16*a^2*b*c^2*d*e^2 + a*b^4*e^3 - 8*a^2*b^2*c*e^3 + 16*a^3*c^2*e^3)*abs(c)) + 1/4*(24*(x*e + d)^(7/2)*c^4*d^2*
e - 72*(x*e + d)^(5/2)*c^4*d^3*e + 72*(x*e + d)^(3/2)*c^4*d^4*e - 24*sqrt(x*e + d)*c^4*d^5*e - 24*(x*e + d)^(7
/2)*b*c^3*d*e^2 + 108*(x*e + d)^(5/2)*b*c^3*d^2*e^2 - 144*(x*e + d)^(3/2)*b*c^3*d^3*e^2 + 60*sqrt(x*e + d)*b*c
^3*d^4*e^2 + (x*e + d)^(7/2)*b^2*c^2*e^3 + 20*(x*e + d)^(7/2)*a*c^3*e^3 - 40*(x*e + d)^(5/2)*b^2*c^2*d*e^3 - 5
6*(x*e + d)^(5/2)*a*c^3*d*e^3 + 85*(x*e + d)^(3/2)*b^2*c^2*d^2*e^3 + 92*(x*e + d)^(3/2)*a*c^3*d^2*e^3 - 46*sqr
t(x*e + d)*b^2*c^2*d^3*e^3 - 56*sqrt(x*e + d)*a*c^3*d^3*e^3 + 2*(x*e + d)^(5/2)*b^3*c*e^4 + 28*(x*e + d)^(5/2)
*a*b*c^2*e^4 - 13*(x*e + d)^(3/2)*b^3*c*d*e^4 - 92*(x*e + d)^(3/2)*a*b*c^2*d*e^4 + 9*sqrt(x*e + d)*b^3*c*d^2*e
^4 + 84*sqrt(x*e + d)*a*b*c^2*d^2*e^4 + (x*e + d)^(3/2)*b^4*e^5 + 5*(x*e + d)^(3/2)*a*b^2*c*e^5 + 36*(x*e + d)
^(3/2)*a^2*c^2*e^5 + sqrt(x*e + d)*b^4*d*e^5 - 26*sqrt(x*e + d)*a*b^2*c*d*e^5 - 32*sqrt(x*e + d)*a^2*c^2*d*e^5
- sqrt(x*e + d)*a*b^3*e^6 + 16*sqrt(x*e + d)*a^2*b*c*e^6)/((b^4*c*d^2 - 8*a*b^2*c^2*d^2 + 16*a^2*c^3*d^2 - b^
5*d*e + 8*a*b^3*c*d*e - 16*a^2*b*c^2*d*e + a*b^4*e^2 - 8*a^2*b^2*c*e^2 + 16*a^3*c^2*e^2)*((x*e + d)^2*c - 2*(x
*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e + a*e^2)^2)

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maple [B]  time = 0.98, size = 3360, normalized size = 5.30 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*e*c^2/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b/c*e-1/2*((-4*a*c+b^2)*e^2)^(1/2)/c)^2*(-4*a*c*e^2+b^
2*e^2)^(1/2)/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*d-3*e^2*c/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b
^2)^2/(e*x+1/2*b/c*e-1/2*((-4*a*c+b^2)*e^2)^(1/2)/c)^2*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-b*e+2*c*d+(-4*a*c*e^2+b^2*
e^2)^(1/2))*(e*x+d)^(3/2)*b+7/4*e^3/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b/c*e+1/2*((-4*a*c+b^2)*e^
2)^(1/2)/c)^2*(e*x+d)^(1/2)*b^2+40*e^3*c^3/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(-4*b*e+8*c*d+4*(-4*a*c*e^2+
b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2
*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a-34*e^3*c^2/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(-4*b*e+8*c*d+4
*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(
1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2-96*e*c^4/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(-
4*b*e+8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x
+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d^2-3*e^2*c/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a
*c-b^2)^2/(e*x+1/2*b/c*e+1/2*((-4*a*c+b^2)*e^2)^(1/2)/c)^2*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-b*e+2*c*d-(-4*a*c*e^2+
b^2*e^2)^(1/2))*(e*x+d)^(3/2)*b+6*e*c^2/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b/c*e+1/2*((-4*a*c+b^2
)*e^2)^(1/2)/c)^2*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*d-40*e^3*c^
3/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(4*b*e-8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-(4*a
*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a+
34*e^3*c^2/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(4*b*e-8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c
*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1
/2)*c)*b^2+96*e*c^4/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(4*b*e-8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/
((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/
2))*c)^(1/2)*c)*d^2-10*e^3*c^2/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b/c*e-1/2*((-4*a*c+b^2)*e^2)^(1
/2)/c)^2/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*a+5/2*e^3*c/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2
)^2/(e*x+1/2*b/c*e-1/2*((-4*a*c+b^2)*e^2)^(1/2)/c)^2/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*b^2
+10*e^3*c^2/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b/c*e+1/2*((-4*a*c+b^2)*e^2)^(1/2)/c)^2/(-b*e+2*c*
d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*a-5/2*e^3*c/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b/c*e+
1/2*((-4*a*c+b^2)*e^2)^(1/2)/c)^2/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*b^2-3*e*c/(-(4*a*c-b^2
)*e^2)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b/c*e-1/2*((-4*a*c+b^2)*e^2)^(1/2)/c)^2*(-4*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)
^(1/2)*d-3*e*c/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b/c*e+1/2*((-4*a*c+b^2)*e^2)^(1/2)/c)^2*(-4*a*c
*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)*d+36*e^2*c^2/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(-4*b*e+8*c*d+4*(-4*a*c*
e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b
*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b*(-4*a*c*e^2+b^2*e^2)^(1/2)-72*e*c^3/(-(4*a*c-b^2)*e^2)^(1/2)/
(4*a*c-b^2)^2/(-4*b*e+8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1
/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d*(-4*a*c*e^2+b^2*e^2)^(1
/2)-96*e^2*c^3/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(4*b*e-8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e
-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c
)^(1/2)*c)*b*d+36*e^2*c^2/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(4*b*e-8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^
(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^
2)^(1/2))*c)^(1/2)*c)*b*(-4*a*c*e^2+b^2*e^2)^(1/2)-72*e*c^3/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(4*b*e-8*c*
d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^
(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d*(-4*a*c*e^2+b^2*e^2)^(1/2)+96*e^2*c^3/(-(4*a*c-b^2)*
e^2)^(1/2)/(4*a*c-b^2)^2/(-4*b*e+8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(
1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b*d-7/4*e^3/(-
(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b/c*e-1/2*((-4*a*c+b^2)*e^2)^(1/2)/c)^2*(e*x+d)^(1/2)*b^2+3/2*e^
2/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b/c*e+1/2*((-4*a*c+b^2)*e^2)^(1/2)/c)^2*(-4*a*c*e^2+b^2*e^2)
^(1/2)*(e*x+d)^(1/2)*b+3/2*e^2/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b/c*e-1/2*((-4*a*c+b^2)*e^2)^(1
/2)/c)^2*(-4*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)*b+7*e^3*c/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b/
c*e-1/2*((-4*a*c+b^2)*e^2)^(1/2)/c)^2*(e*x+d)^(1/2)*a-7*e^3*c/(-(4*a*c-b^2)*e^2)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*
b/c*e+1/2*((-4*a*c+b^2)*e^2)^(1/2)/c)^2*(e*x+d)^(1/2)*a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 69.78, size = 23750, normalized size = 37.46

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(- (2^(1/2)*((2^(1/2)*((c^2*e^3*(b*e - 2*c*d)*(b^2*e^2 - 12*c^2*d^2 - 16*a*c*e^2 + 12*b*c*d*e))/((4*a*c - b
^2)*(a*e^2 + c*d^2 - b*d*e)) - (2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-(b^17*e^7 + 4718
592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 + b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8*b
*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 -
5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^5*
b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10
*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 - 21*
c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 - 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e
^6 + 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 + 1444
80*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d^4
*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a^4*b
^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 + 3
0965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^2*c^
9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*c^11
*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^3*d^
2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 10321920*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c^4*d
*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b*c^1
0*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2*c^8*d*e^6)/((4*a*c - b^2)
^10*(a*e^2 + c*d^2 - b*d*e)^3))^(1/2))/2)*(-(b^17*e^7 + 4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 + b^2*e^7*(-(
4*a*c - b^2)^15)^(1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*
d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 - 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 -
10160*a^3*b^11*c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7
*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4
*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 - 21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*
c*e^7 - 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e^6 + 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064
320*a^2*b^8*c^7*d^5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 + 144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*
e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^
9*c^5*d^2*e^5 - 18063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 19
08480*a^4*b^7*c^6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 + 30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8
*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*
a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*c^11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11
*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^3*d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c
^3*d*e^6 - 10321920*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c^4*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^
8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b*c^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a
^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2*c^8*d*e^6)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)^3))^(1/2))/16 + (c^3*e
^2*(d + e*x)^(1/2)*(b^6*e^6 + 4608*c^6*d^6 - 800*a^3*c^3*e^6 + 8832*a*c^5*d^4*e^2 + 1472*a^2*b^2*c^2*e^6 + 348
8*a^2*c^4*d^2*e^4 + 15072*b^2*c^4*d^4*e^2 - 7104*b^3*c^3*d^3*e^3 + 1226*b^4*c^2*d^2*e^4 - 34*a*b^4*c*e^6 - 138
24*b*c^5*d^5*e + 22*b^5*c*d*e^5 - 17664*a*b*c^4*d^3*e^3 - 2672*a*b^3*c^2*d*e^5 - 3488*a^2*b*c^3*d*e^5 + 11504*
a*b^2*c^3*d^2*e^4))/(8*(4*a*c - b^2)^4*(a*e^2 + c*d^2 - b*d*e)^2))*(-(b^17*e^7 + 4718592*a^5*c^12*d^7 - 4608*b
^10*c^7*d^7 + b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^
9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 - 5898240*a^4*b^2*c^11*d^7
+ 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*
b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10*d^3*e^4 - 20832*b^12*c^5
*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 - 21*c^2*d^2*e^5*(-(4*a*c - b^
2)^15)^(1/2) - 55*a*b^15*c*e^7 - 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e^6 + 21*b*c*d*e^6*(-(4*a*
c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 + 144480*a^2*b^10*c^5*d^3*e^4 -
136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d^4*e^3 - 2576000*a^3*b^8*c^
6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a^4*b^5*c^8*d^4*e^3 + 12454400
*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 + 30965760*a^5*b^3*c^9*d^4*e
^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^2*c^9*d^3*e^4 + 15912960*a^6*
b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*c^11*d^6*e + 403200*a*b^10*c^
6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^3*d^2*e^5 + 2580480*a^2*b^7*c
^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 10321920*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c^4*d*e^6 + 20643840*a^4*b^3*c
^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b*c^10*d^4*e^3 - 6236160*a^6*b
^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2*c^8*d*e^6)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*
e)^3))^(1/2))/16 - (c^4*e^3*(55296*c^6*d^6 - 35*b^6*e^6 + 8000*a^3*c^3*e^6 + 124416*a*c^5*d^4*e^2 + 12720*a^2*
b^2*c^2*e^6 + 74880*a^2*c^4*d^2*e^4 + 176256*b^2*c^4*d^4*e^2 - 76032*b^3*c^3*d^3*e^3 + 9864*b^4*c^2*d^2*e^4 -
84*a*b^4*c*e^6 - 165888*b*c^5*d^5*e + 504*b^5*c*d*e^5 - 248832*a*b*c^4*d^3*e^3 - 24768*a*b^3*c^2*d*e^5 - 74880
*a^2*b*c^3*d*e^5 + 149184*a*b^2*c^3*d^2*e^4))/(64*(4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)^2))*(-(b^17*e^7 + 47
18592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 + b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8
*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7
- 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^
5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^
10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 - 2
1*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 - 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d
*e^6 + 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 + 14
4480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d
^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a^4
*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 +
30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^2*
c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*c^
11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^3*
d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 10321920*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c^4
*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b*c
^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2*c^8*d*e^6)/(128*(a^3*b^
20*e^6 + 1048576*a^10*c^13*d^6 + 1048576*a^13*c^10*e^6 + b^20*c^3*d^6 - b^23*d^3*e^3 - 40*a*b^18*c^4*d^6 - 40*
a^4*b^18*c*e^6 + 3*a*b^22*d^2*e^4 - 3*a^2*b^21*d*e^5 - 3*b^21*c^2*d^5*e + 3*b^22*c*d^4*e^2 + 720*a^2*b^16*c^5*
d^6 - 7680*a^3*b^14*c^6*d^6 + 53760*a^4*b^12*c^7*d^6 - 258048*a^5*b^10*c^8*d^6 + 860160*a^6*b^8*c^9*d^6 - 1966
080*a^7*b^6*c^10*d^6 + 2949120*a^8*b^4*c^11*d^6 - 2621440*a^9*b^2*c^12*d^6 + 720*a^5*b^16*c^2*e^6 - 7680*a^6*b
^14*c^3*e^6 + 53760*a^7*b^12*c^4*e^6 - 258048*a^8*b^10*c^5*e^6 + 860160*a^9*b^8*c^6*e^6 - 1966080*a^10*b^6*c^7
*e^6 + 2949120*a^11*b^4*c^8*e^6 - 2621440*a^12*b^2*c^9*e^6 + 3145728*a^11*c^12*d^4*e^2 + 3145728*a^12*c^11*d^2
*e^4 + 2040*a^2*b^18*c^3*d^4*e^2 - 480*a^2*b^19*c^2*d^3*e^3 - 20880*a^3*b^16*c^4*d^4*e^2 + 3360*a^3*b^17*c^3*d
^3*e^3 + 2040*a^3*b^18*c^2*d^2*e^4 + 138240*a^4*b^14*c^5*d^4*e^2 - 7680*a^4*b^15*c^4*d^3*e^3 - 20880*a^4*b^16*
c^3*d^2*e^4 - 612864*a^5*b^12*c^6*d^4*e^2 - 64512*a^5*b^13*c^5*d^3*e^3 + 138240*a^5*b^14*c^4*d^2*e^4 + 1806336
*a^6*b^10*c^7*d^4*e^2 + 688128*a^6*b^11*c^6*d^3*e^3 - 612864*a^6*b^12*c^5*d^2*e^4 - 3317760*a^7*b^8*c^8*d^4*e^
2 - 3194880*a^7*b^9*c^7*d^3*e^3 + 1806336*a^7*b^10*c^6*d^2*e^4 + 2949120*a^8*b^6*c^9*d^4*e^2 + 8847360*a^8*b^7
*c^8*d^3*e^3 - 3317760*a^8*b^8*c^7*d^2*e^4 + 983040*a^9*b^4*c^10*d^4*e^2 - 15073280*a^9*b^5*c^9*d^3*e^3 + 2949
120*a^9*b^6*c^8*d^2*e^4 - 4718592*a^10*b^2*c^11*d^4*e^2 + 14680064*a^10*b^3*c^10*d^3*e^3 + 983040*a^10*b^4*c^9
*d^2*e^4 - 4718592*a^11*b^2*c^10*d^2*e^4 + 120*a*b^19*c^3*d^5*e + 34*a*b^21*c*d^3*e^3 + 120*a^3*b^19*c*d*e^5 -
3145728*a^10*b*c^12*d^5*e - 3145728*a^12*b*c^10*d*e^5 - 117*a*b^20*c^2*d^4*e^2 - 2160*a^2*b^17*c^4*d^5*e - 11
7*a^2*b^20*c*d^2*e^4 + 23040*a^3*b^15*c^5*d^5*e - 161280*a^4*b^13*c^6*d^5*e - 2160*a^4*b^17*c^2*d*e^5 + 774144
*a^5*b^11*c^7*d^5*e + 23040*a^5*b^15*c^3*d*e^5 - 2580480*a^6*b^9*c^8*d^5*e - 161280*a^6*b^13*c^4*d*e^5 + 58982
40*a^7*b^7*c^9*d^5*e + 774144*a^7*b^11*c^5*d*e^5 - 8847360*a^8*b^5*c^10*d^5*e - 2580480*a^8*b^9*c^6*d*e^5 + 78
64320*a^9*b^3*c^11*d^5*e + 5898240*a^9*b^7*c^7*d*e^5 - 8847360*a^10*b^5*c^8*d*e^5 - 6291456*a^11*b*c^11*d^3*e^
3 + 7864320*a^11*b^3*c^9*d*e^5)))^(1/2) + log(- (2^(1/2)*((2^(1/2)*((c^2*e^3*(b*e - 2*c*d)*(b^2*e^2 - 12*c^2*d
^2 - 16*a*c*e^2 + 12*b*c*d*e))/((4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)) - (2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e -
2*c*d)*(d + e*x)^(1/2)*(-(b^17*e^7 + 4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 - b^2*e^7*(-(4*a*c - b^2)^15)^(
1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2
*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 - 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^
3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 137
62560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^
14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 + 21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 + 25*a*c*e^7
*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e^6 - 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^
5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 + 144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3
*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18
063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*
d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 + 30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896
*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e -
630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*c^11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 215
60*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^3*d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 1032192
0*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c^4*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472
896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b*c^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 +
2580480*a^7*b^2*c^8*d*e^6)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)^3))^(1/2))/2)*(-(b^17*e^7 + 4718592*a^5*
c^12*d^7 - 4608*b^10*c^7*d^7 - b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7
+ 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 - 5898240*
a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*
e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10*d^3*e^4
- 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 + 21*c^2*d^2*
e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 + 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e^6 - 21*
b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 + 144480*a^2*b
^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d^4*e^3 - 2
576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a^4*b^5*c^8*d
^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 + 30965760*
a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^2*c^9*d^3*e^
4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*c^11*d^6*e +
403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^3*d^2*e^5 +
2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 10321920*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c^4*d*e^6 + 2
0643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b*c^10*d^4*e^
3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2*c^8*d*e^6)/((4*a*c - b^2)^10*(a*e
^2 + c*d^2 - b*d*e)^3))^(1/2))/16 + (c^3*e^2*(d + e*x)^(1/2)*(b^6*e^6 + 4608*c^6*d^6 - 800*a^3*c^3*e^6 + 8832*
a*c^5*d^4*e^2 + 1472*a^2*b^2*c^2*e^6 + 3488*a^2*c^4*d^2*e^4 + 15072*b^2*c^4*d^4*e^2 - 7104*b^3*c^3*d^3*e^3 + 1
226*b^4*c^2*d^2*e^4 - 34*a*b^4*c*e^6 - 13824*b*c^5*d^5*e + 22*b^5*c*d*e^5 - 17664*a*b*c^4*d^3*e^3 - 2672*a*b^3
*c^2*d*e^5 - 3488*a^2*b*c^3*d*e^5 + 11504*a*b^2*c^3*d^2*e^4))/(8*(4*a*c - b^2)^4*(a*e^2 + c*d^2 - b*d*e)^2))*(
-(b^17*e^7 + 4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 - b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b^8*c^8*d^
7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^
3*b^4*c^10*d^7 - 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34880*a^4*b^9*c^4
*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 +
12615680*a^7*c^10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 21*b^15*
c^2*d^2*e^5 + 21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 + 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2
) + 21*b^16*c*d*e^6 - 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^2 + 1209600*a^2*b^9*c
^6*d^4*e^3 + 144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 - 215040
0*a^3*b^7*c^7*d^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18063360*a^4*b^4*c^9*d^5*e^
2 - 6451200*a^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*d^2*e^5 + 4128768*a^5*b^2
*c^10*d^5*e^2 + 30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e^5 + 12
328960*a^6*b^2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d*e^6 - 16
515072*a^5*b*c^11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^3*e^4 + 7
980*a*b^13*c^3*d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 10321920*a^3*b^5*c^9*d^6*e + 840
00*a^3*b^10*c^4*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*d*e^6 - 3
4406400*a^6*b*c^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2*c^8*d*e^
6)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)^3))^(1/2))/16 - (c^4*e^3*(55296*c^6*d^6 - 35*b^6*e^6 + 8000*a^3*c
^3*e^6 + 124416*a*c^5*d^4*e^2 + 12720*a^2*b^2*c^2*e^6 + 74880*a^2*c^4*d^2*e^4 + 176256*b^2*c^4*d^4*e^2 - 76032
*b^3*c^3*d^3*e^3 + 9864*b^4*c^2*d^2*e^4 - 84*a*b^4*c*e^6 - 165888*b*c^5*d^5*e + 504*b^5*c*d*e^5 - 248832*a*b*c
^4*d^3*e^3 - 24768*a*b^3*c^2*d*e^5 - 74880*a^2*b*c^3*d*e^5 + 149184*a*b^2*c^3*d^2*e^4))/(64*(4*a*c - b^2)^6*(a
*e^2 + c*d^2 - b*d*e)^2))*(-(b^17*e^7 + 4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 - b^2*e^7*(-(4*a*c - b^2)^15)
^(1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a
^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 - 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*
c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 1
3762560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*
b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 + 21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 + 25*a*c*e
^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e^6 - 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*
d^5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 + 144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a
^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 -
18063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^
6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 + 30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 21288
96*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e
- 630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*c^11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 2
1560*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^3*d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 10321
920*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c^4*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 34
72896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b*c^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5
+ 2580480*a^7*b^2*c^8*d*e^6)/(128*(a^3*b^20*e^6 + 1048576*a^10*c^13*d^6 + 1048576*a^13*c^10*e^6 + b^20*c^3*d^
6 - b^23*d^3*e^3 - 40*a*b^18*c^4*d^6 - 40*a^4*b^18*c*e^6 + 3*a*b^22*d^2*e^4 - 3*a^2*b^21*d*e^5 - 3*b^21*c^2*d^
5*e + 3*b^22*c*d^4*e^2 + 720*a^2*b^16*c^5*d^6 - 7680*a^3*b^14*c^6*d^6 + 53760*a^4*b^12*c^7*d^6 - 258048*a^5*b^
10*c^8*d^6 + 860160*a^6*b^8*c^9*d^6 - 1966080*a^7*b^6*c^10*d^6 + 2949120*a^8*b^4*c^11*d^6 - 2621440*a^9*b^2*c^
12*d^6 + 720*a^5*b^16*c^2*e^6 - 7680*a^6*b^14*c^3*e^6 + 53760*a^7*b^12*c^4*e^6 - 258048*a^8*b^10*c^5*e^6 + 860
160*a^9*b^8*c^6*e^6 - 1966080*a^10*b^6*c^7*e^6 + 2949120*a^11*b^4*c^8*e^6 - 2621440*a^12*b^2*c^9*e^6 + 3145728
*a^11*c^12*d^4*e^2 + 3145728*a^12*c^11*d^2*e^4 + 2040*a^2*b^18*c^3*d^4*e^2 - 480*a^2*b^19*c^2*d^3*e^3 - 20880*
a^3*b^16*c^4*d^4*e^2 + 3360*a^3*b^17*c^3*d^3*e^3 + 2040*a^3*b^18*c^2*d^2*e^4 + 138240*a^4*b^14*c^5*d^4*e^2 - 7
680*a^4*b^15*c^4*d^3*e^3 - 20880*a^4*b^16*c^3*d^2*e^4 - 612864*a^5*b^12*c^6*d^4*e^2 - 64512*a^5*b^13*c^5*d^3*e
^3 + 138240*a^5*b^14*c^4*d^2*e^4 + 1806336*a^6*b^10*c^7*d^4*e^2 + 688128*a^6*b^11*c^6*d^3*e^3 - 612864*a^6*b^1
2*c^5*d^2*e^4 - 3317760*a^7*b^8*c^8*d^4*e^2 - 3194880*a^7*b^9*c^7*d^3*e^3 + 1806336*a^7*b^10*c^6*d^2*e^4 + 294
9120*a^8*b^6*c^9*d^4*e^2 + 8847360*a^8*b^7*c^8*d^3*e^3 - 3317760*a^8*b^8*c^7*d^2*e^4 + 983040*a^9*b^4*c^10*d^4
*e^2 - 15073280*a^9*b^5*c^9*d^3*e^3 + 2949120*a^9*b^6*c^8*d^2*e^4 - 4718592*a^10*b^2*c^11*d^4*e^2 + 14680064*a
^10*b^3*c^10*d^3*e^3 + 983040*a^10*b^4*c^9*d^2*e^4 - 4718592*a^11*b^2*c^10*d^2*e^4 + 120*a*b^19*c^3*d^5*e + 34
*a*b^21*c*d^3*e^3 + 120*a^3*b^19*c*d*e^5 - 3145728*a^10*b*c^12*d^5*e - 3145728*a^12*b*c^10*d*e^5 - 117*a*b^20*
c^2*d^4*e^2 - 2160*a^2*b^17*c^4*d^5*e - 117*a^2*b^20*c*d^2*e^4 + 23040*a^3*b^15*c^5*d^5*e - 161280*a^4*b^13*c^
6*d^5*e - 2160*a^4*b^17*c^2*d*e^5 + 774144*a^5*b^11*c^7*d^5*e + 23040*a^5*b^15*c^3*d*e^5 - 2580480*a^6*b^9*c^8
*d^5*e - 161280*a^6*b^13*c^4*d*e^5 + 5898240*a^7*b^7*c^9*d^5*e + 774144*a^7*b^11*c^5*d*e^5 - 8847360*a^8*b^5*c
^10*d^5*e - 2580480*a^8*b^9*c^6*d*e^5 + 7864320*a^9*b^3*c^11*d^5*e + 5898240*a^9*b^7*c^7*d*e^5 - 8847360*a^10*
b^5*c^8*d*e^5 - 6291456*a^11*b*c^11*d^3*e^3 + 7864320*a^11*b^3*c^9*d*e^5)))^(1/2) - log(- (((c^2*e^3*(b*e - 2*
c*d)*(b^2*e^2 - 12*c^2*d^2 - 16*a*c*e^2 + 12*b*c*d*e))/((4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)) + 8*c^2*e^2*(4*
a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-((b^17*e^7)/128 + 36864*a^5*c^12*d^7 - 36*b^10*c^7*d^7 - (b^2*e^7*(
-(4*a*c - b^2)^15)^(1/2))/128 + 720*a*b^8*c^8*d^7 - 13440*a^8*b*c^8*e^7 + 26880*a^8*c^9*d*e^6 + 126*b^11*c^6*d
^6*e - 5760*a^2*b^6*c^9*d^7 + 23040*a^3*b^4*c^10*d^7 - 46080*a^4*b^2*c^11*d^7 + (285*a^2*b^13*c^2*e^7)/32 - (6
35*a^3*b^11*c^3*e^7)/8 + (545*a^4*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*e^7 - 5320*a^6*b^5*c^6*e^7 + 14560*a^7*b^3*
c^7*e^7 + 107520*a^6*c^11*d^5*e^2 + 98560*a^7*c^10*d^3*e^4 - (651*b^12*c^5*d^5*e^2)/4 + (735*b^13*c^4*d^4*e^3)
/8 - (1225*b^14*c^3*d^3*e^4)/64 - (21*b^15*c^2*d^2*e^5)/128 + (21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 -
(55*a*b^15*c*e^7)/128 + (25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + (21*b^16*c*d*e^6)/128 - (21*b*c*d*e^6*(-
(4*a*c - b^2)^15)^(1/2))/128 - 23940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2*b^9*c^6*d^4*e^3 + (4515*a^2*b^10*c^5*d^3*e
^4)/4 - (8505*a^2*b^11*c^4*d^2*e^5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 - 16800*a^3*b^7*c^7*d^4*e^3 - 20125*a^3*b^8*
c^6*d^3*e^4 + (13335*a^3*b^9*c^5*d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^5*e^2 - 50400*a^4*b^5*c^8*d^4*e^3 + 97300*a
^4*b^6*c^7*d^3*e^4 - 14910*a^4*b^7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10*d^5*e^2 + 241920*a^5*b^3*c^9*d^4*e^3 - 193
200*a^5*b^4*c^8*d^3*e^4 - 16632*a^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2*c^9*d^3*e^4 + 124320*a^6*b^3*c^8*d^2*e^5 -
2520*a*b^9*c^7*d^6*e - (315*a*b^14*c^2*d*e^6)/64 - 129024*a^5*b*c^11*d^6*e + 3150*a*b^10*c^6*d^5*e^2 - 1575*a
*b^11*c^5*d^4*e^3 + (2695*a*b^12*c^4*d^3*e^4)/16 + (1995*a*b^13*c^3*d^2*e^5)/32 + 20160*a^2*b^7*c^8*d^6*e + (1
05*a^2*b^12*c^3*d*e^6)/16 - 80640*a^3*b^5*c^9*d^6*e + (2625*a^3*b^10*c^4*d*e^6)/4 + 161280*a^4*b^3*c^10*d^6*e
- 6615*a^4*b^8*c^5*d*e^6 + 27132*a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c^10*d^4*e^3 - 48720*a^6*b^4*c^7*d*e^6 - 147
840*a^7*b*c^9*d^2*e^5 + 20160*a^7*b^2*c^8*d*e^6)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)^3))^(1/2))*(-((b^17
*e^7)/128 + 36864*a^5*c^12*d^7 - 36*b^10*c^7*d^7 - (b^2*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + 720*a*b^8*c^8*d^7
- 13440*a^8*b*c^8*e^7 + 26880*a^8*c^9*d*e^6 + 126*b^11*c^6*d^6*e - 5760*a^2*b^6*c^9*d^7 + 23040*a^3*b^4*c^10*
d^7 - 46080*a^4*b^2*c^11*d^7 + (285*a^2*b^13*c^2*e^7)/32 - (635*a^3*b^11*c^3*e^7)/8 + (545*a^4*b^9*c^4*e^7)/2
+ 342*a^5*b^7*c^5*e^7 - 5320*a^6*b^5*c^6*e^7 + 14560*a^7*b^3*c^7*e^7 + 107520*a^6*c^11*d^5*e^2 + 98560*a^7*c^1
0*d^3*e^4 - (651*b^12*c^5*d^5*e^2)/4 + (735*b^13*c^4*d^4*e^3)/8 - (1225*b^14*c^3*d^3*e^4)/64 - (21*b^15*c^2*d^
2*e^5)/128 + (21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (55*a*b^15*c*e^7)/128 + (25*a*c*e^7*(-(4*a*c - b
^2)^15)^(1/2))/128 + (21*b^16*c*d*e^6)/128 - (21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2))/128 - 23940*a^2*b^8*c^7*
d^5*e^2 + 9450*a^2*b^9*c^6*d^4*e^3 + (4515*a^2*b^10*c^5*d^3*e^4)/4 - (8505*a^2*b^11*c^4*d^2*e^5)/8 + 87360*a^3
*b^6*c^8*d^5*e^2 - 16800*a^3*b^7*c^7*d^4*e^3 - 20125*a^3*b^8*c^6*d^3*e^4 + (13335*a^3*b^9*c^5*d^2*e^5)/2 - 141
120*a^4*b^4*c^9*d^5*e^2 - 50400*a^4*b^5*c^8*d^4*e^3 + 97300*a^4*b^6*c^7*d^3*e^4 - 14910*a^4*b^7*c^6*d^2*e^5 +
32256*a^5*b^2*c^10*d^5*e^2 + 241920*a^5*b^3*c^9*d^4*e^3 - 193200*a^5*b^4*c^8*d^3*e^4 - 16632*a^5*b^5*c^7*d^2*e
^5 + 96320*a^6*b^2*c^9*d^3*e^4 + 124320*a^6*b^3*c^8*d^2*e^5 - 2520*a*b^9*c^7*d^6*e - (315*a*b^14*c^2*d*e^6)/64
- 129024*a^5*b*c^11*d^6*e + 3150*a*b^10*c^6*d^5*e^2 - 1575*a*b^11*c^5*d^4*e^3 + (2695*a*b^12*c^4*d^3*e^4)/16
+ (1995*a*b^13*c^3*d^2*e^5)/32 + 20160*a^2*b^7*c^8*d^6*e + (105*a^2*b^12*c^3*d*e^6)/16 - 80640*a^3*b^5*c^9*d^6
*e + (2625*a^3*b^10*c^4*d*e^6)/4 + 161280*a^4*b^3*c^10*d^6*e - 6615*a^4*b^8*c^5*d*e^6 + 27132*a^5*b^6*c^6*d*e^
6 - 268800*a^6*b*c^10*d^4*e^3 - 48720*a^6*b^4*c^7*d*e^6 - 147840*a^7*b*c^9*d^2*e^5 + 20160*a^7*b^2*c^8*d*e^6)/
((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)^3))^(1/2) - (c^3*e^2*(d + e*x)^(1/2)*(b^6*e^6 + 4608*c^6*d^6 - 800*a
^3*c^3*e^6 + 8832*a*c^5*d^4*e^2 + 1472*a^2*b^2*c^2*e^6 + 3488*a^2*c^4*d^2*e^4 + 15072*b^2*c^4*d^4*e^2 - 7104*b
^3*c^3*d^3*e^3 + 1226*b^4*c^2*d^2*e^4 - 34*a*b^4*c*e^6 - 13824*b*c^5*d^5*e + 22*b^5*c*d*e^5 - 17664*a*b*c^4*d^
3*e^3 - 2672*a*b^3*c^2*d*e^5 - 3488*a^2*b*c^3*d*e^5 + 11504*a*b^2*c^3*d^2*e^4))/(8*(4*a*c - b^2)^4*(a*e^2 + c*
d^2 - b*d*e)^2))*(-((b^17*e^7)/128 + 36864*a^5*c^12*d^7 - 36*b^10*c^7*d^7 - (b^2*e^7*(-(4*a*c - b^2)^15)^(1/2)
)/128 + 720*a*b^8*c^8*d^7 - 13440*a^8*b*c^8*e^7 + 26880*a^8*c^9*d*e^6 + 126*b^11*c^6*d^6*e - 5760*a^2*b^6*c^9*
d^7 + 23040*a^3*b^4*c^10*d^7 - 46080*a^4*b^2*c^11*d^7 + (285*a^2*b^13*c^2*e^7)/32 - (635*a^3*b^11*c^3*e^7)/8 +
(545*a^4*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*e^7 - 5320*a^6*b^5*c^6*e^7 + 14560*a^7*b^3*c^7*e^7 + 107520*a^6*c^1
1*d^5*e^2 + 98560*a^7*c^10*d^3*e^4 - (651*b^12*c^5*d^5*e^2)/4 + (735*b^13*c^4*d^4*e^3)/8 - (1225*b^14*c^3*d^3*
e^4)/64 - (21*b^15*c^2*d^2*e^5)/128 + (21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (55*a*b^15*c*e^7)/128 +
(25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + (21*b^16*c*d*e^6)/128 - (21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2))
/128 - 23940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2*b^9*c^6*d^4*e^3 + (4515*a^2*b^10*c^5*d^3*e^4)/4 - (8505*a^2*b^11*c
^4*d^2*e^5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 - 16800*a^3*b^7*c^7*d^4*e^3 - 20125*a^3*b^8*c^6*d^3*e^4 + (13335*a^3
*b^9*c^5*d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^5*e^2 - 50400*a^4*b^5*c^8*d^4*e^3 + 97300*a^4*b^6*c^7*d^3*e^4 - 149
10*a^4*b^7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10*d^5*e^2 + 241920*a^5*b^3*c^9*d^4*e^3 - 193200*a^5*b^4*c^8*d^3*e^4
- 16632*a^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2*c^9*d^3*e^4 + 124320*a^6*b^3*c^8*d^2*e^5 - 2520*a*b^9*c^7*d^6*e -
(315*a*b^14*c^2*d*e^6)/64 - 129024*a^5*b*c^11*d^6*e + 3150*a*b^10*c^6*d^5*e^2 - 1575*a*b^11*c^5*d^4*e^3 + (269
5*a*b^12*c^4*d^3*e^4)/16 + (1995*a*b^13*c^3*d^2*e^5)/32 + 20160*a^2*b^7*c^8*d^6*e + (105*a^2*b^12*c^3*d*e^6)/1
6 - 80640*a^3*b^5*c^9*d^6*e + (2625*a^3*b^10*c^4*d*e^6)/4 + 161280*a^4*b^3*c^10*d^6*e - 6615*a^4*b^8*c^5*d*e^6
+ 27132*a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c^10*d^4*e^3 - 48720*a^6*b^4*c^7*d*e^6 - 147840*a^7*b*c^9*d^2*e^5 +
20160*a^7*b^2*c^8*d*e^6)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)^3))^(1/2) - (c^4*e^3*(55296*c^6*d^6 - 35*b^
6*e^6 + 8000*a^3*c^3*e^6 + 124416*a*c^5*d^4*e^2 + 12720*a^2*b^2*c^2*e^6 + 74880*a^2*c^4*d^2*e^4 + 176256*b^2*c
^4*d^4*e^2 - 76032*b^3*c^3*d^3*e^3 + 9864*b^4*c^2*d^2*e^4 - 84*a*b^4*c*e^6 - 165888*b*c^5*d^5*e + 504*b^5*c*d*
e^5 - 248832*a*b*c^4*d^3*e^3 - 24768*a*b^3*c^2*d*e^5 - 74880*a^2*b*c^3*d*e^5 + 149184*a*b^2*c^3*d^2*e^4))/(64*
(4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)^2))*(-((b^17*e^7)/128 + 36864*a^5*c^12*d^7 - 36*b^10*c^7*d^7 - (b^2*e^
7*(-(4*a*c - b^2)^15)^(1/2))/128 + 720*a*b^8*c^8*d^7 - 13440*a^8*b*c^8*e^7 + 26880*a^8*c^9*d*e^6 + 126*b^11*c^
6*d^6*e - 5760*a^2*b^6*c^9*d^7 + 23040*a^3*b^4*c^10*d^7 - 46080*a^4*b^2*c^11*d^7 + (285*a^2*b^13*c^2*e^7)/32 -
(635*a^3*b^11*c^3*e^7)/8 + (545*a^4*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*e^7 - 5320*a^6*b^5*c^6*e^7 + 14560*a^7*b
^3*c^7*e^7 + 107520*a^6*c^11*d^5*e^2 + 98560*a^7*c^10*d^3*e^4 - (651*b^12*c^5*d^5*e^2)/4 + (735*b^13*c^4*d^4*e
^3)/8 - (1225*b^14*c^3*d^3*e^4)/64 - (21*b^15*c^2*d^2*e^5)/128 + (21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2))/12
8 - (55*a*b^15*c*e^7)/128 + (25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + (21*b^16*c*d*e^6)/128 - (21*b*c*d*e^6
*(-(4*a*c - b^2)^15)^(1/2))/128 - 23940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2*b^9*c^6*d^4*e^3 + (4515*a^2*b^10*c^5*d^
3*e^4)/4 - (8505*a^2*b^11*c^4*d^2*e^5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 - 16800*a^3*b^7*c^7*d^4*e^3 - 20125*a^3*b
^8*c^6*d^3*e^4 + (13335*a^3*b^9*c^5*d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^5*e^2 - 50400*a^4*b^5*c^8*d^4*e^3 + 9730
0*a^4*b^6*c^7*d^3*e^4 - 14910*a^4*b^7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10*d^5*e^2 + 241920*a^5*b^3*c^9*d^4*e^3 -
193200*a^5*b^4*c^8*d^3*e^4 - 16632*a^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2*c^9*d^3*e^4 + 124320*a^6*b^3*c^8*d^2*e^
5 - 2520*a*b^9*c^7*d^6*e - (315*a*b^14*c^2*d*e^6)/64 - 129024*a^5*b*c^11*d^6*e + 3150*a*b^10*c^6*d^5*e^2 - 157
5*a*b^11*c^5*d^4*e^3 + (2695*a*b^12*c^4*d^3*e^4)/16 + (1995*a*b^13*c^3*d^2*e^5)/32 + 20160*a^2*b^7*c^8*d^6*e +
(105*a^2*b^12*c^3*d*e^6)/16 - 80640*a^3*b^5*c^9*d^6*e + (2625*a^3*b^10*c^4*d*e^6)/4 + 161280*a^4*b^3*c^10*d^6
*e - 6615*a^4*b^8*c^5*d*e^6 + 27132*a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c^10*d^4*e^3 - 48720*a^6*b^4*c^7*d*e^6 -
147840*a^7*b*c^9*d^2*e^5 + 20160*a^7*b^2*c^8*d*e^6)/(a^3*b^20*e^6 + 1048576*a^10*c^13*d^6 + 1048576*a^13*c^10*
e^6 + b^20*c^3*d^6 - b^23*d^3*e^3 - 40*a*b^18*c^4*d^6 - 40*a^4*b^18*c*e^6 + 3*a*b^22*d^2*e^4 - 3*a^2*b^21*d*e^
5 - 3*b^21*c^2*d^5*e + 3*b^22*c*d^4*e^2 + 720*a^2*b^16*c^5*d^6 - 7680*a^3*b^14*c^6*d^6 + 53760*a^4*b^12*c^7*d^
6 - 258048*a^5*b^10*c^8*d^6 + 860160*a^6*b^8*c^9*d^6 - 1966080*a^7*b^6*c^10*d^6 + 2949120*a^8*b^4*c^11*d^6 - 2
621440*a^9*b^2*c^12*d^6 + 720*a^5*b^16*c^2*e^6 - 7680*a^6*b^14*c^3*e^6 + 53760*a^7*b^12*c^4*e^6 - 258048*a^8*b
^10*c^5*e^6 + 860160*a^9*b^8*c^6*e^6 - 1966080*a^10*b^6*c^7*e^6 + 2949120*a^11*b^4*c^8*e^6 - 2621440*a^12*b^2*
c^9*e^6 + 3145728*a^11*c^12*d^4*e^2 + 3145728*a^12*c^11*d^2*e^4 + 2040*a^2*b^18*c^3*d^4*e^2 - 480*a^2*b^19*c^2
*d^3*e^3 - 20880*a^3*b^16*c^4*d^4*e^2 + 3360*a^3*b^17*c^3*d^3*e^3 + 2040*a^3*b^18*c^2*d^2*e^4 + 138240*a^4*b^1
4*c^5*d^4*e^2 - 7680*a^4*b^15*c^4*d^3*e^3 - 20880*a^4*b^16*c^3*d^2*e^4 - 612864*a^5*b^12*c^6*d^4*e^2 - 64512*a
^5*b^13*c^5*d^3*e^3 + 138240*a^5*b^14*c^4*d^2*e^4 + 1806336*a^6*b^10*c^7*d^4*e^2 + 688128*a^6*b^11*c^6*d^3*e^3
- 612864*a^6*b^12*c^5*d^2*e^4 - 3317760*a^7*b^8*c^8*d^4*e^2 - 3194880*a^7*b^9*c^7*d^3*e^3 + 1806336*a^7*b^10*
c^6*d^2*e^4 + 2949120*a^8*b^6*c^9*d^4*e^2 + 8847360*a^8*b^7*c^8*d^3*e^3 - 3317760*a^8*b^8*c^7*d^2*e^4 + 983040
*a^9*b^4*c^10*d^4*e^2 - 15073280*a^9*b^5*c^9*d^3*e^3 + 2949120*a^9*b^6*c^8*d^2*e^4 - 4718592*a^10*b^2*c^11*d^4
*e^2 + 14680064*a^10*b^3*c^10*d^3*e^3 + 983040*a^10*b^4*c^9*d^2*e^4 - 4718592*a^11*b^2*c^10*d^2*e^4 + 120*a*b^
19*c^3*d^5*e + 34*a*b^21*c*d^3*e^3 + 120*a^3*b^19*c*d*e^5 - 3145728*a^10*b*c^12*d^5*e - 3145728*a^12*b*c^10*d*
e^5 - 117*a*b^20*c^2*d^4*e^2 - 2160*a^2*b^17*c^4*d^5*e - 117*a^2*b^20*c*d^2*e^4 + 23040*a^3*b^15*c^5*d^5*e - 1
61280*a^4*b^13*c^6*d^5*e - 2160*a^4*b^17*c^2*d*e^5 + 774144*a^5*b^11*c^7*d^5*e + 23040*a^5*b^15*c^3*d*e^5 - 25
80480*a^6*b^9*c^8*d^5*e - 161280*a^6*b^13*c^4*d*e^5 + 5898240*a^7*b^7*c^9*d^5*e + 774144*a^7*b^11*c^5*d*e^5 -
8847360*a^8*b^5*c^10*d^5*e - 2580480*a^8*b^9*c^6*d*e^5 + 7864320*a^9*b^3*c^11*d^5*e + 5898240*a^9*b^7*c^7*d*e^
5 - 8847360*a^10*b^5*c^8*d*e^5 - 6291456*a^11*b*c^11*d^3*e^3 + 7864320*a^11*b^3*c^9*d*e^5))^(1/2) - log(- (((c
^2*e^3*(b*e - 2*c*d)*(b^2*e^2 - 12*c^2*d^2 - 16*a*c*e^2 + 12*b*c*d*e))/((4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e))
+ 8*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-((b^17*e^7)/128 + 36864*a^5*c^12*d^7 - 36*b^10*c^7*
d^7 + (b^2*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + 720*a*b^8*c^8*d^7 - 13440*a^8*b*c^8*e^7 + 26880*a^8*c^9*d*e^6
+ 126*b^11*c^6*d^6*e - 5760*a^2*b^6*c^9*d^7 + 23040*a^3*b^4*c^10*d^7 - 46080*a^4*b^2*c^11*d^7 + (285*a^2*b^13*
c^2*e^7)/32 - (635*a^3*b^11*c^3*e^7)/8 + (545*a^4*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*e^7 - 5320*a^6*b^5*c^6*e^7
+ 14560*a^7*b^3*c^7*e^7 + 107520*a^6*c^11*d^5*e^2 + 98560*a^7*c^10*d^3*e^4 - (651*b^12*c^5*d^5*e^2)/4 + (735*b
^13*c^4*d^4*e^3)/8 - (1225*b^14*c^3*d^3*e^4)/64 - (21*b^15*c^2*d^2*e^5)/128 - (21*c^2*d^2*e^5*(-(4*a*c - b^2)^
15)^(1/2))/128 - (55*a*b^15*c*e^7)/128 - (25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + (21*b^16*c*d*e^6)/128 +
(21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2))/128 - 23940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2*b^9*c^6*d^4*e^3 + (4515*a^
2*b^10*c^5*d^3*e^4)/4 - (8505*a^2*b^11*c^4*d^2*e^5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 - 16800*a^3*b^7*c^7*d^4*e^3
- 20125*a^3*b^8*c^6*d^3*e^4 + (13335*a^3*b^9*c^5*d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^5*e^2 - 50400*a^4*b^5*c^8*d
^4*e^3 + 97300*a^4*b^6*c^7*d^3*e^4 - 14910*a^4*b^7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10*d^5*e^2 + 241920*a^5*b^3*c
^9*d^4*e^3 - 193200*a^5*b^4*c^8*d^3*e^4 - 16632*a^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2*c^9*d^3*e^4 + 124320*a^6*b
^3*c^8*d^2*e^5 - 2520*a*b^9*c^7*d^6*e - (315*a*b^14*c^2*d*e^6)/64 - 129024*a^5*b*c^11*d^6*e + 3150*a*b^10*c^6*
d^5*e^2 - 1575*a*b^11*c^5*d^4*e^3 + (2695*a*b^12*c^4*d^3*e^4)/16 + (1995*a*b^13*c^3*d^2*e^5)/32 + 20160*a^2*b^
7*c^8*d^6*e + (105*a^2*b^12*c^3*d*e^6)/16 - 80640*a^3*b^5*c^9*d^6*e + (2625*a^3*b^10*c^4*d*e^6)/4 + 161280*a^4
*b^3*c^10*d^6*e - 6615*a^4*b^8*c^5*d*e^6 + 27132*a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c^10*d^4*e^3 - 48720*a^6*b^4
*c^7*d*e^6 - 147840*a^7*b*c^9*d^2*e^5 + 20160*a^7*b^2*c^8*d*e^6)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)^3))
^(1/2))*(-((b^17*e^7)/128 + 36864*a^5*c^12*d^7 - 36*b^10*c^7*d^7 + (b^2*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + 7
20*a*b^8*c^8*d^7 - 13440*a^8*b*c^8*e^7 + 26880*a^8*c^9*d*e^6 + 126*b^11*c^6*d^6*e - 5760*a^2*b^6*c^9*d^7 + 230
40*a^3*b^4*c^10*d^7 - 46080*a^4*b^2*c^11*d^7 + (285*a^2*b^13*c^2*e^7)/32 - (635*a^3*b^11*c^3*e^7)/8 + (545*a^4
*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*e^7 - 5320*a^6*b^5*c^6*e^7 + 14560*a^7*b^3*c^7*e^7 + 107520*a^6*c^11*d^5*e^2
+ 98560*a^7*c^10*d^3*e^4 - (651*b^12*c^5*d^5*e^2)/4 + (735*b^13*c^4*d^4*e^3)/8 - (1225*b^14*c^3*d^3*e^4)/64 -
(21*b^15*c^2*d^2*e^5)/128 - (21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (55*a*b^15*c*e^7)/128 - (25*a*c*
e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + (21*b^16*c*d*e^6)/128 + (21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2))/128 - 23
940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2*b^9*c^6*d^4*e^3 + (4515*a^2*b^10*c^5*d^3*e^4)/4 - (8505*a^2*b^11*c^4*d^2*e^
5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 - 16800*a^3*b^7*c^7*d^4*e^3 - 20125*a^3*b^8*c^6*d^3*e^4 + (13335*a^3*b^9*c^5*
d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^5*e^2 - 50400*a^4*b^5*c^8*d^4*e^3 + 97300*a^4*b^6*c^7*d^3*e^4 - 14910*a^4*b^
7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10*d^5*e^2 + 241920*a^5*b^3*c^9*d^4*e^3 - 193200*a^5*b^4*c^8*d^3*e^4 - 16632*a
^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2*c^9*d^3*e^4 + 124320*a^6*b^3*c^8*d^2*e^5 - 2520*a*b^9*c^7*d^6*e - (315*a*b^
14*c^2*d*e^6)/64 - 129024*a^5*b*c^11*d^6*e + 3150*a*b^10*c^6*d^5*e^2 - 1575*a*b^11*c^5*d^4*e^3 + (2695*a*b^12*
c^4*d^3*e^4)/16 + (1995*a*b^13*c^3*d^2*e^5)/32 + 20160*a^2*b^7*c^8*d^6*e + (105*a^2*b^12*c^3*d*e^6)/16 - 80640
*a^3*b^5*c^9*d^6*e + (2625*a^3*b^10*c^4*d*e^6)/4 + 161280*a^4*b^3*c^10*d^6*e - 6615*a^4*b^8*c^5*d*e^6 + 27132*
a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c^10*d^4*e^3 - 48720*a^6*b^4*c^7*d*e^6 - 147840*a^7*b*c^9*d^2*e^5 + 20160*a^7
*b^2*c^8*d*e^6)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)^3))^(1/2) - (c^3*e^2*(d + e*x)^(1/2)*(b^6*e^6 + 4608
*c^6*d^6 - 800*a^3*c^3*e^6 + 8832*a*c^5*d^4*e^2 + 1472*a^2*b^2*c^2*e^6 + 3488*a^2*c^4*d^2*e^4 + 15072*b^2*c^4*
d^4*e^2 - 7104*b^3*c^3*d^3*e^3 + 1226*b^4*c^2*d^2*e^4 - 34*a*b^4*c*e^6 - 13824*b*c^5*d^5*e + 22*b^5*c*d*e^5 -
17664*a*b*c^4*d^3*e^3 - 2672*a*b^3*c^2*d*e^5 - 3488*a^2*b*c^3*d*e^5 + 11504*a*b^2*c^3*d^2*e^4))/(8*(4*a*c - b^
2)^4*(a*e^2 + c*d^2 - b*d*e)^2))*(-((b^17*e^7)/128 + 36864*a^5*c^12*d^7 - 36*b^10*c^7*d^7 + (b^2*e^7*(-(4*a*c
- b^2)^15)^(1/2))/128 + 720*a*b^8*c^8*d^7 - 13440*a^8*b*c^8*e^7 + 26880*a^8*c^9*d*e^6 + 126*b^11*c^6*d^6*e - 5
760*a^2*b^6*c^9*d^7 + 23040*a^3*b^4*c^10*d^7 - 46080*a^4*b^2*c^11*d^7 + (285*a^2*b^13*c^2*e^7)/32 - (635*a^3*b
^11*c^3*e^7)/8 + (545*a^4*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*e^7 - 5320*a^6*b^5*c^6*e^7 + 14560*a^7*b^3*c^7*e^7
+ 107520*a^6*c^11*d^5*e^2 + 98560*a^7*c^10*d^3*e^4 - (651*b^12*c^5*d^5*e^2)/4 + (735*b^13*c^4*d^4*e^3)/8 - (12
25*b^14*c^3*d^3*e^4)/64 - (21*b^15*c^2*d^2*e^5)/128 - (21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (55*a*b
^15*c*e^7)/128 - (25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + (21*b^16*c*d*e^6)/128 + (21*b*c*d*e^6*(-(4*a*c -
b^2)^15)^(1/2))/128 - 23940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2*b^9*c^6*d^4*e^3 + (4515*a^2*b^10*c^5*d^3*e^4)/4 -
(8505*a^2*b^11*c^4*d^2*e^5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 - 16800*a^3*b^7*c^7*d^4*e^3 - 20125*a^3*b^8*c^6*d^3*
e^4 + (13335*a^3*b^9*c^5*d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^5*e^2 - 50400*a^4*b^5*c^8*d^4*e^3 + 97300*a^4*b^6*c
^7*d^3*e^4 - 14910*a^4*b^7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10*d^5*e^2 + 241920*a^5*b^3*c^9*d^4*e^3 - 193200*a^5*
b^4*c^8*d^3*e^4 - 16632*a^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2*c^9*d^3*e^4 + 124320*a^6*b^3*c^8*d^2*e^5 - 2520*a*
b^9*c^7*d^6*e - (315*a*b^14*c^2*d*e^6)/64 - 129024*a^5*b*c^11*d^6*e + 3150*a*b^10*c^6*d^5*e^2 - 1575*a*b^11*c^
5*d^4*e^3 + (2695*a*b^12*c^4*d^3*e^4)/16 + (1995*a*b^13*c^3*d^2*e^5)/32 + 20160*a^2*b^7*c^8*d^6*e + (105*a^2*b
^12*c^3*d*e^6)/16 - 80640*a^3*b^5*c^9*d^6*e + (2625*a^3*b^10*c^4*d*e^6)/4 + 161280*a^4*b^3*c^10*d^6*e - 6615*a
^4*b^8*c^5*d*e^6 + 27132*a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c^10*d^4*e^3 - 48720*a^6*b^4*c^7*d*e^6 - 147840*a^7*
b*c^9*d^2*e^5 + 20160*a^7*b^2*c^8*d*e^6)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)^3))^(1/2) - (c^4*e^3*(55296
*c^6*d^6 - 35*b^6*e^6 + 8000*a^3*c^3*e^6 + 124416*a*c^5*d^4*e^2 + 12720*a^2*b^2*c^2*e^6 + 74880*a^2*c^4*d^2*e^
4 + 176256*b^2*c^4*d^4*e^2 - 76032*b^3*c^3*d^3*e^3 + 9864*b^4*c^2*d^2*e^4 - 84*a*b^4*c*e^6 - 165888*b*c^5*d^5*
e + 504*b^5*c*d*e^5 - 248832*a*b*c^4*d^3*e^3 - 24768*a*b^3*c^2*d*e^5 - 74880*a^2*b*c^3*d*e^5 + 149184*a*b^2*c^
3*d^2*e^4))/(64*(4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)^2))*(-((b^17*e^7)/128 + 36864*a^5*c^12*d^7 - 36*b^10*c
^7*d^7 + (b^2*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + 720*a*b^8*c^8*d^7 - 13440*a^8*b*c^8*e^7 + 26880*a^8*c^9*d*e
^6 + 126*b^11*c^6*d^6*e - 5760*a^2*b^6*c^9*d^7 + 23040*a^3*b^4*c^10*d^7 - 46080*a^4*b^2*c^11*d^7 + (285*a^2*b^
13*c^2*e^7)/32 - (635*a^3*b^11*c^3*e^7)/8 + (545*a^4*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*e^7 - 5320*a^6*b^5*c^6*e
^7 + 14560*a^7*b^3*c^7*e^7 + 107520*a^6*c^11*d^5*e^2 + 98560*a^7*c^10*d^3*e^4 - (651*b^12*c^5*d^5*e^2)/4 + (73
5*b^13*c^4*d^4*e^3)/8 - (1225*b^14*c^3*d^3*e^4)/64 - (21*b^15*c^2*d^2*e^5)/128 - (21*c^2*d^2*e^5*(-(4*a*c - b^
2)^15)^(1/2))/128 - (55*a*b^15*c*e^7)/128 - (25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + (21*b^16*c*d*e^6)/128
+ (21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2))/128 - 23940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2*b^9*c^6*d^4*e^3 + (4515
*a^2*b^10*c^5*d^3*e^4)/4 - (8505*a^2*b^11*c^4*d^2*e^5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 - 16800*a^3*b^7*c^7*d^4*e
^3 - 20125*a^3*b^8*c^6*d^3*e^4 + (13335*a^3*b^9*c^5*d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^5*e^2 - 50400*a^4*b^5*c^
8*d^4*e^3 + 97300*a^4*b^6*c^7*d^3*e^4 - 14910*a^4*b^7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10*d^5*e^2 + 241920*a^5*b^
3*c^9*d^4*e^3 - 193200*a^5*b^4*c^8*d^3*e^4 - 16632*a^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2*c^9*d^3*e^4 + 124320*a^
6*b^3*c^8*d^2*e^5 - 2520*a*b^9*c^7*d^6*e - (315*a*b^14*c^2*d*e^6)/64 - 129024*a^5*b*c^11*d^6*e + 3150*a*b^10*c
^6*d^5*e^2 - 1575*a*b^11*c^5*d^4*e^3 + (2695*a*b^12*c^4*d^3*e^4)/16 + (1995*a*b^13*c^3*d^2*e^5)/32 + 20160*a^2
*b^7*c^8*d^6*e + (105*a^2*b^12*c^3*d*e^6)/16 - 80640*a^3*b^5*c^9*d^6*e + (2625*a^3*b^10*c^4*d*e^6)/4 + 161280*
a^4*b^3*c^10*d^6*e - 6615*a^4*b^8*c^5*d*e^6 + 27132*a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c^10*d^4*e^3 - 48720*a^6*
b^4*c^7*d*e^6 - 147840*a^7*b*c^9*d^2*e^5 + 20160*a^7*b^2*c^8*d*e^6)/(a^3*b^20*e^6 + 1048576*a^10*c^13*d^6 + 10
48576*a^13*c^10*e^6 + b^20*c^3*d^6 - b^23*d^3*e^3 - 40*a*b^18*c^4*d^6 - 40*a^4*b^18*c*e^6 + 3*a*b^22*d^2*e^4 -
3*a^2*b^21*d*e^5 - 3*b^21*c^2*d^5*e + 3*b^22*c*d^4*e^2 + 720*a^2*b^16*c^5*d^6 - 7680*a^3*b^14*c^6*d^6 + 53760
*a^4*b^12*c^7*d^6 - 258048*a^5*b^10*c^8*d^6 + 860160*a^6*b^8*c^9*d^6 - 1966080*a^7*b^6*c^10*d^6 + 2949120*a^8*
b^4*c^11*d^6 - 2621440*a^9*b^2*c^12*d^6 + 720*a^5*b^16*c^2*e^6 - 7680*a^6*b^14*c^3*e^6 + 53760*a^7*b^12*c^4*e^
6 - 258048*a^8*b^10*c^5*e^6 + 860160*a^9*b^8*c^6*e^6 - 1966080*a^10*b^6*c^7*e^6 + 2949120*a^11*b^4*c^8*e^6 - 2
621440*a^12*b^2*c^9*e^6 + 3145728*a^11*c^12*d^4*e^2 + 3145728*a^12*c^11*d^2*e^4 + 2040*a^2*b^18*c^3*d^4*e^2 -
480*a^2*b^19*c^2*d^3*e^3 - 20880*a^3*b^16*c^4*d^4*e^2 + 3360*a^3*b^17*c^3*d^3*e^3 + 2040*a^3*b^18*c^2*d^2*e^4
+ 138240*a^4*b^14*c^5*d^4*e^2 - 7680*a^4*b^15*c^4*d^3*e^3 - 20880*a^4*b^16*c^3*d^2*e^4 - 612864*a^5*b^12*c^6*d
^4*e^2 - 64512*a^5*b^13*c^5*d^3*e^3 + 138240*a^5*b^14*c^4*d^2*e^4 + 1806336*a^6*b^10*c^7*d^4*e^2 + 688128*a^6*
b^11*c^6*d^3*e^3 - 612864*a^6*b^12*c^5*d^2*e^4 - 3317760*a^7*b^8*c^8*d^4*e^2 - 3194880*a^7*b^9*c^7*d^3*e^3 + 1
806336*a^7*b^10*c^6*d^2*e^4 + 2949120*a^8*b^6*c^9*d^4*e^2 + 8847360*a^8*b^7*c^8*d^3*e^3 - 3317760*a^8*b^8*c^7*
d^2*e^4 + 983040*a^9*b^4*c^10*d^4*e^2 - 15073280*a^9*b^5*c^9*d^3*e^3 + 2949120*a^9*b^6*c^8*d^2*e^4 - 4718592*a
^10*b^2*c^11*d^4*e^2 + 14680064*a^10*b^3*c^10*d^3*e^3 + 983040*a^10*b^4*c^9*d^2*e^4 - 4718592*a^11*b^2*c^10*d^
2*e^4 + 120*a*b^19*c^3*d^5*e + 34*a*b^21*c*d^3*e^3 + 120*a^3*b^19*c*d*e^5 - 3145728*a^10*b*c^12*d^5*e - 314572
8*a^12*b*c^10*d*e^5 - 117*a*b^20*c^2*d^4*e^2 - 2160*a^2*b^17*c^4*d^5*e - 117*a^2*b^20*c*d^2*e^4 + 23040*a^3*b^
15*c^5*d^5*e - 161280*a^4*b^13*c^6*d^5*e - 2160*a^4*b^17*c^2*d*e^5 + 774144*a^5*b^11*c^7*d^5*e + 23040*a^5*b^1
5*c^3*d*e^5 - 2580480*a^6*b^9*c^8*d^5*e - 161280*a^6*b^13*c^4*d*e^5 + 5898240*a^7*b^7*c^9*d^5*e + 774144*a^7*b
^11*c^5*d*e^5 - 8847360*a^8*b^5*c^10*d^5*e - 2580480*a^8*b^9*c^6*d*e^5 + 7864320*a^9*b^3*c^11*d^5*e + 5898240*
a^9*b^7*c^7*d*e^5 - 8847360*a^10*b^5*c^8*d*e^5 - 6291456*a^11*b*c^11*d^3*e^3 + 7864320*a^11*b^3*c^9*d*e^5))^(1
/2) + (((d + e*x)^(3/2)*(b^4*e^5 + 72*c^4*d^4*e + 36*a^2*c^2*e^5 + 92*a*c^3*d^2*e^3 - 144*b*c^3*d^3*e^2 + 85*b
^2*c^2*d^2*e^3 + 5*a*b^2*c*e^5 - 13*b^3*c*d*e^4 - 92*a*b*c^2*d*e^4))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)*(a*e^2
+ c*d^2 - b*d*e)) - ((d + e*x)^(1/2)*(b^3*e^4 + 24*c^3*d^3*e - 36*b*c^2*d^2*e^2 - 16*a*b*c*e^4 + 32*a*c^2*d*e^
3 + 10*b^2*c*d*e^3))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + ((b*e - 2*c*d)*(d + e*x)^(5/2)*(14*a*c^2*e^3 + b^2*c
*e^3 + 18*c^3*d^2*e - 18*b*c^2*d*e^2))/(2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)*(a*e^2 + c*d^2 - b*d*e)) + (c*e*(d +
e*x)^(7/2)*(24*c^3*d^2 + 20*a*c^2*e^2 + b^2*c*e^2 - 24*b*c^2*d*e))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)*(a*e^2 +
c*d^2 - b*d*e)))/(c^2*(d + e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 2*a*b*e^3 + 4*a*c*d*e^2 - 6*b*c*d^2*e
) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 + 2*a*c*e^2 - 6*b*c*d*e) + a^2*e^4 + c^
2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 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((e*x+d)**(1/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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