\(\int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx\) [525]

Optimal result

Integrand size = 25, antiderivative size = 490 \[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=-\frac {2 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) (d+e x)^{3/2}}{3 c^3 e^3}-\frac {2 (2 c d+b e) (d+e x)^{5/2}}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3}+\frac {\sqrt {2} \left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e-\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{9/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e+\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{9/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

2/3*(c^2*d^2+b^2*e^2+c*e*(-a*e+b*d))*(e*x+d)^(3/2)/c^3/e^3-2/5*(b*e+2*c*d)*(e*x+d)^(5/2)/c^2/e^3+2/7*(e*x+d)^(
7/2)/c/e^3-2*b*(-2*a*c+b^2)*(e*x+d)^(1/2)/c^4+arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(
1/2)))^(1/2))*2^(1/2)*(b^3*c*d-2*a*b*c^2*d-b^4*e+3*a*b^2*c*e-a^2*c^2*e+(5*a^2*b*c^2*e-2*a^2*c^3*d-5*a*b^3*c*e+
4*a*b^2*c^2*d+b^5*e-b^4*c*d)/(-4*a*c+b^2)^(1/2))/c^(9/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)+arctanh(2^(1/2
)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*2^(1/2)*(b^3*c*d-2*a*b*c^2*d-b^4*e+3*a*b^2*c*e
-a^2*c^2*e+(-5*a^2*b*c^2*e+2*a^2*c^3*d+5*a*b^3*c*e-4*a*b^2*c^2*d-b^5*e+b^4*c*d)/(-4*a*c+b^2)^(1/2))/c^(9/2)/(2
*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

Rubi [A] (verified)

Time = 11.99 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {911, 1301, 1180, 214} \[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\frac {\sqrt {2} \left (-\frac {-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d}{\sqrt {b^2-4 a c}}-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{c^{9/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \left (\frac {-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d}{\sqrt {b^2-4 a c}}-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c^{9/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {2 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 (d+e x)^{3/2} \left (c e (b d-a e)+b^2 e^2+c^2 d^2\right )}{3 c^3 e^3}-\frac {2 (d+e x)^{5/2} (b e+2 c d)}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3} \]

[In]

Int[(x^4*Sqrt[d + e*x])/(a + b*x + c*x^2),x]

[Out]

(-2*b*(b^2 - 2*a*c)*Sqrt[d + e*x])/c^4 + (2*(c^2*d^2 + b^2*e^2 + c*e*(b*d - a*e))*(d + e*x)^(3/2))/(3*c^3*e^3)
- (2*(2*c*d + b*e)*(d + e*x)^(5/2))/(5*c^2*e^3) + (2*(d + e*x)^(7/2))/(7*c*e^3) + (Sqrt[2]*(b^3*c*d - 2*a*b*c
^2*d - b^4*e + 3*a*b^2*c*e - a^2*c^2*e - (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*
b*c^2*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/
(c^(9/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*(b^3*c*d - 2*a*b*c^2*d - b^4*e + 3*a*b^2*c*e - a^
2*c^2*e + (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)/Sqrt[b^2 - 4*a*c])*Arc
Tanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(c^(9/2)*Sqrt[2*c*d - (b + Sqrt
[b^2 - 4*a*c])*e])

Rule 214

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

Rule 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1301

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^2 \left (-\frac {d}{e}+\frac {x^2}{e}\right )^4}{\frac {c d^2-b d e+a e^2}{e^2}-\frac {(2 c d-b e) x^2}{e^2}+\frac {c x^4}{e^2}} \, dx,x,\sqrt {d+e x}\right )}{e} \\ & = \frac {2 \text {Subst}\left (\int \left (-\frac {\left (b^3-2 a b c\right ) e}{c^4}+\frac {\left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) x^2}{c^3 e^2}-\frac {(2 c d+b e) x^4}{c^2 e^2}+\frac {x^6}{c e^2}+\frac {b \left (b^2-2 a c\right ) \left (c d^2-b d e+a e^2\right )-\left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e\right ) x^2}{c^4 e \left (\frac {c d^2-b d e+a e^2}{e^2}-\frac {(2 c d-b e) x^2}{e^2}+\frac {c x^4}{e^2}\right )}\right ) \, dx,x,\sqrt {d+e x}\right )}{e} \\ & = -\frac {2 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) (d+e x)^{3/2}}{3 c^3 e^3}-\frac {2 (2 c d+b e) (d+e x)^{5/2}}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3}+\frac {2 \text {Subst}\left (\int \frac {b \left (b^2-2 a c\right ) \left (c d^2-b d e+a e^2\right )+\left (-b^3 c d+2 a b c^2 d+b^4 e-3 a b^2 c e+a^2 c^2 e\right ) x^2}{\frac {c d^2-b d e+a e^2}{e^2}-\frac {(2 c d-b e) x^2}{e^2}+\frac {c x^4}{e^2}} \, dx,x,\sqrt {d+e x}\right )}{c^4 e^2} \\ & = -\frac {2 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) (d+e x)^{3/2}}{3 c^3 e^3}-\frac {2 (2 c d+b e) (d+e x)^{5/2}}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3}-\frac {\left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e-\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {\sqrt {b^2-4 a c}}{2 e}-\frac {2 c d-b e}{2 e^2}+\frac {c x^2}{e^2}} \, dx,x,\sqrt {d+e x}\right )}{c^4 e^2}-\frac {\left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e+\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b^2-4 a c}}{2 e}-\frac {2 c d-b e}{2 e^2}+\frac {c x^2}{e^2}} \, dx,x,\sqrt {d+e x}\right )}{c^4 e^2} \\ & = -\frac {2 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) (d+e x)^{3/2}}{3 c^3 e^3}-\frac {2 (2 c d+b e) (d+e x)^{5/2}}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3}+\frac {\sqrt {2} \left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e-\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\sqrt {b^2-4 a c}}\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 )}{c^{9/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e+\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\sqrt {b^2-4 a c}}\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 )}{c^{9/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.96 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.28 \[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\frac {2 \sqrt {d+e x} \left (-105 b^3 e^3-7 c^2 e (d+e x) (-2 b d+5 a e+3 b e x)+c^3 \left (8 d^3-4 d^2 e x+3 d e^2 x^2+15 e^3 x^3\right )+35 b c e^2 (6 a e+b (d+e x))\right )}{105 c^4 e^3}+\frac {\left (i b^5 e-b^3 c \left (\sqrt {-b^2+4 a c} d+5 i a e\right )+a b c^2 \left (2 \sqrt {-b^2+4 a c} d+5 i a e\right )+a b^2 c \left (4 i c d-3 \sqrt {-b^2+4 a c} e\right )+b^4 \left (-i c d+\sqrt {-b^2+4 a c} e\right )+a^2 c^2 \left (-2 i c d+\sqrt {-b^2+4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{c^{9/2} \sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\left (-i b^5 e+a b c^2 \left (2 \sqrt {-b^2+4 a c} d-5 i a e\right )+b^3 c \left (-\sqrt {-b^2+4 a c} d+5 i a e\right )+a b^2 c \left (-4 i c d-3 \sqrt {-b^2+4 a c} e\right )+b^4 \left (i c d+\sqrt {-b^2+4 a c} e\right )+a^2 c^2 \left (2 i c d+\sqrt {-b^2+4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{c^{9/2} \sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}} \]

[In]

Integrate[(x^4*Sqrt[d + e*x])/(a + b*x + c*x^2),x]

[Out]

(2*Sqrt[d + e*x]*(-105*b^3*e^3 - 7*c^2*e*(d + e*x)*(-2*b*d + 5*a*e + 3*b*e*x) + c^3*(8*d^3 - 4*d^2*e*x + 3*d*e
^2*x^2 + 15*e^3*x^3) + 35*b*c*e^2*(6*a*e + b*(d + e*x))))/(105*c^4*e^3) + ((I*b^5*e - b^3*c*(Sqrt[-b^2 + 4*a*c
]*d + (5*I)*a*e) + a*b*c^2*(2*Sqrt[-b^2 + 4*a*c]*d + (5*I)*a*e) + a*b^2*c*((4*I)*c*d - 3*Sqrt[-b^2 + 4*a*c]*e)
+ b^4*((-I)*c*d + Sqrt[-b^2 + 4*a*c]*e) + a^2*c^2*((-2*I)*c*d + Sqrt[-b^2 + 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c
]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(c^(9/2)*Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (
b - I*Sqrt[-b^2 + 4*a*c])*e]) + (((-I)*b^5*e + a*b*c^2*(2*Sqrt[-b^2 + 4*a*c]*d - (5*I)*a*e) + b^3*c*(-(Sqrt[-b
^2 + 4*a*c]*d) + (5*I)*a*e) + a*b^2*c*((-4*I)*c*d - 3*Sqrt[-b^2 + 4*a*c]*e) + b^4*(I*c*d + Sqrt[-b^2 + 4*a*c]*
e) + a^2*c^2*((2*I)*c*d + Sqrt[-b^2 + 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*
Sqrt[-b^2 + 4*a*c]*e]])/(c^(9/2)*Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e])

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.21

method	result	size
pseudoelliptic	\(\frac {-\sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, e^{3} \left (\left (\left (a^{2} c^{2}-3 a \,b^{2} c +b^{4}\right ) e +2 a b \,c^{2} d -b^{3} c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-5 e \left (b \left (a^{2} c^{2}-a \,b^{2} c +\frac {1}{5} b^{4}\right ) e -\frac {2 d \left (a^{2} c^{2}-2 a \,b^{2} c +\frac {1}{2} b^{4}\right ) c}{5}\right )\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (\left (\left (\left (a^{2} c^{2}-3 a \,b^{2} c +b^{4}\right ) e +2 a b \,c^{2} d -b^{3} c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+5 e \left (b \left (a^{2} c^{2}-a \,b^{2} c +\frac {1}{5} b^{4}\right ) e -\frac {2 d \left (a^{2} c^{2}-2 a \,b^{2} c +\frac {1}{2} b^{4}\right ) c}{5}\right )\right ) \sqrt {2}\, e^{3} \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+4 \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (\frac {c^{3} x^{3}}{14}-\frac {\left (\frac {3 b x}{5}+a \right ) x \,c^{2}}{6}+b \left (\frac {b x}{6}+a \right ) c -\frac {b^{3}}{2}\right ) e^{3}-\frac {d \left (-\frac {3 c^{2} x^{2}}{35}+\left (\frac {b x}{5}+a \right ) c -b^{2}\right ) c \,e^{2}}{6}+\frac {d^{2} \left (-\frac {2 c x}{7}+b \right ) c^{2} e}{15}+\frac {4 c^{3} d^{3}}{105}\right ) \sqrt {e x +d}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}{\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, e^{3} c^{4}}\)	\(594\)
derivativedivides	\(\frac {\frac {2 \left (\frac {\left (e x +d \right )^{\frac {7}{2}} c^{3}}{7}-\frac {b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {b \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a b c \,e^{3} \sqrt {e x +d}-b^{3} e^{3} \sqrt {e x +d}\right )}{c^{4}}-\frac {8 e^{3} \left (-\frac {\left (5 a^{2} b \,c^{2} e^{2}-2 a^{2} c^{3} d e -5 a \,b^{3} e^{2} c +4 a \,b^{2} c^{2} d e +e^{2} b^{5}-b^{4} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-5 a^{2} b \,c^{2} e^{2}+2 a^{2} c^{3} d e +5 a \,b^{3} e^{2} c -4 a \,b^{2} c^{2} d e -e^{2} b^{5}+b^{4} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c^{3}}}{e^{3}}\)	\(708\)
default	\(\frac {\frac {2 \left (\frac {\left (e x +d \right )^{\frac {7}{2}} c^{3}}{7}-\frac {b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {b \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a b c \,e^{3} \sqrt {e x +d}-b^{3} e^{3} \sqrt {e x +d}\right )}{c^{4}}-\frac {8 e^{3} \left (-\frac {\left (5 a^{2} b \,c^{2} e^{2}-2 a^{2} c^{3} d e -5 a \,b^{3} e^{2} c +4 a \,b^{2} c^{2} d e +e^{2} b^{5}-b^{4} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-5 a^{2} b \,c^{2} e^{2}+2 a^{2} c^{3} d e +5 a \,b^{3} e^{2} c -4 a \,b^{2} c^{2} d e -e^{2} b^{5}+b^{4} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c^{3}}}{e^{3}}\)	\(708\)
risch	\(\frac {2 \left (15 c^{3} e^{3} x^{3}-21 c^{2} e^{3} b \,x^{2}+3 c^{3} d \,e^{2} x^{2}-35 a \,c^{2} e^{3} x +35 b^{2} c \,e^{3} x -7 b \,c^{2} d \,e^{2} x -4 c^{3} d^{2} e x +210 c \,e^{3} b a -35 a \,c^{2} d \,e^{2}-105 b^{3} e^{3}+35 b^{2} c d \,e^{2}+14 b \,c^{2} d^{2} e +8 c^{3} d^{3}\right ) \sqrt {e x +d}}{105 e^{3} c^{4}}+\frac {-\frac {\left (-5 a^{2} b \,c^{2} e^{2}+2 a^{2} c^{3} d e +5 a \,b^{3} e^{2} c -4 a \,b^{2} c^{2} d e -e^{2} b^{5}+b^{4} c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e -3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (5 a^{2} b \,c^{2} e^{2}-2 a^{2} c^{3} d e -5 a \,b^{3} e^{2} c +4 a \,b^{2} c^{2} d e +e^{2} b^{5}-b^{4} c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e -3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c^{3}}\)	\(708\)

[In]

int(x^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/(-4*e^2*(a*c-1/4*b^2))^(1/2)/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*(-2^(1/2)*((b*e-2*c*d+(-4*e^
2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*e^3*(((a^2*c^2-3*a*b^2*c+b^4)*e+2*a*b*c^2*d-b^3*c*d)*(-4*e^2*(a*c-1/4*b^2))^(
1/2)-5*e*(b*(a^2*c^2-a*b^2*c+1/5*b^4)*e-2/5*d*(a^2*c^2-2*a*b^2*c+1/2*b^4)*c))*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/
((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+((((a^2*c^2-3*a*b^2*c+b^4)*e+2*a*b*c^2*d-b^3*c*d)*(-4*e^2
*(a*c-1/4*b^2))^(1/2)+5*e*(b*(a^2*c^2-a*b^2*c+1/5*b^4)*e-2/5*d*(a^2*c^2-2*a*b^2*c+1/2*b^4)*c))*2^(1/2)*e^3*arc
tan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+4*(-4*e^2*(a*c-1/4*b^2))^(1/2)
*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*((1/14*c^3*x^3-1/6*(3/5*b*x+a)*x*c^2+b*(1/6*b*x+a)*c-1/2*b
^3)*e^3-1/6*d*(-3/35*c^2*x^2+(1/5*b*x+a)*c-b^2)*c*e^2+1/15*d^2*(-2/7*c*x+b)*c^2*e+4/105*c^3*d^3)*(e*x+d)^(1/2)
)*((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))/((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)/e^3
/c^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5507 vs. \(2 (436) = 872\).

Time = 0.99 (sec) , antiderivative size = 5507, normalized size of antiderivative = 11.24 \[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Too large to display} \]

[In]

integrate(x^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Timed out} \]

[In]

integrate(x**4*(e*x+d)**(1/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\int { \frac {\sqrt {e x + d} x^{4}}{c x^{2} + b x + a} \,d x } \]

[In]

integrate(x^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x + d)*x^4/(c*x^2 + b*x + a), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1200 vs. \(2 (436) = 872\).

Time = 0.38 (sec) , antiderivative size = 1200, normalized size of antiderivative = 2.45 \[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Too large to display} \]

[In]

integrate(x^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

-1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d - (b^6 - 7*a*b^4
*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e)*c^2*e^2 - 2*((b^3*c^3 - 2*a*b*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^4*c^2 - 2*a*
b^2*c^3)*sqrt(b^2 - 4*a*c)*d*e + (a*b^3*c^2 - 2*a^2*b*c^3)*sqrt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqr
t(b^2 - 4*a*c)*c)*e)*abs(c)*abs(e) + (2*(b^4*c^4 - 4*a*b^2*c^5 + 2*a^2*c^6)*d^2*e - (3*b^5*c^3 - 14*a*b^3*c^4
+ 12*a^2*b*c^5)*d*e^2 + (b^6*c^2 - 5*a*b^4*c^3 + 5*a^2*b^2*c^4)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c
)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*c^8*d*e^24 - b*c^7*e^25 + sqrt(-4*(c^8*d^2*e^24 - b*c^7*d*e
^25 + a*c^7*e^26)*c^8*e^24 + (2*c^8*d*e^24 - b*c^7*e^25)^2))/(c^8*e^24)))/((sqrt(b^2 - 4*a*c)*c^7*d^2 - sqrt(b
^2 - 4*a*c)*b*c^6*d*e + sqrt(b^2 - 4*a*c)*a*c^6*e^2)*c^2*abs(e)) + 1/4*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*
a*c)*c)*e)*((b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d - (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e)*c^2*e^2
+ 2*((b^3*c^3 - 2*a*b*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^4*c^2 - 2*a*b^2*c^3)*sqrt(b^2 - 4*a*c)*d*e + (a*b^3*c^2
- 2*a^2*b*c^3)*sqrt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(c)*abs(e) + (2*(b^4
*c^4 - 4*a*b^2*c^5 + 2*a^2*c^6)*d^2*e - (3*b^5*c^3 - 14*a*b^3*c^4 + 12*a^2*b*c^5)*d*e^2 + (b^6*c^2 - 5*a*b^4*c
^3 + 5*a^2*b^2*c^4)*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sq
rt(-(2*c^8*d*e^24 - b*c^7*e^25 - sqrt(-4*(c^8*d^2*e^24 - b*c^7*d*e^25 + a*c^7*e^26)*c^8*e^24 + (2*c^8*d*e^24 -
b*c^7*e^25)^2))/(c^8*e^24)))/((sqrt(b^2 - 4*a*c)*c^7*d^2 - sqrt(b^2 - 4*a*c)*b*c^6*d*e + sqrt(b^2 - 4*a*c)*a*
c^6*e^2)*c^2*abs(e)) + 2/105*(15*(e*x + d)^(7/2)*c^6*e^18 - 42*(e*x + d)^(5/2)*c^6*d*e^18 + 35*(e*x + d)^(3/2)
*c^6*d^2*e^18 - 21*(e*x + d)^(5/2)*b*c^5*e^19 + 35*(e*x + d)^(3/2)*b*c^5*d*e^19 + 35*(e*x + d)^(3/2)*b^2*c^4*e
^20 - 35*(e*x + d)^(3/2)*a*c^5*e^20 - 105*sqrt(e*x + d)*b^3*c^3*e^21 + 210*sqrt(e*x + d)*a*b*c^4*e^21)/(c^7*e^
21)

Mupad [B] (verification not implemented)

Time = 14.52 (sec) , antiderivative size = 13879, normalized size of antiderivative = 28.32 \[ \int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Too large to display} \]

[In]

int((x^4*(d + e*x)^(1/2))/(a + b*x + c*x^2),x)

[Out]

(d + e*x)^(3/2)*((4*d^2)/(c*e^3) - (2*(a*e^5 + c*d^2*e^3 - b*d*e^4))/(3*c^2*e^6) + (((8*d)/(c*e^3) + (2*(b*e^4
- 2*c*d*e^3))/(c^2*e^6))*(b*e^4 - 2*c*d*e^3))/(3*c*e^3)) - atan(((((8*(a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*
c^5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^
2 - 8*a^2*b^2*c^7*d*e^3))/c^7 - (8*(d + e*x)^(1/2)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) -
b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129
*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d
*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*
a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c -
b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*
(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2 - 4*a*b*c^10*e^3 + 8*a*c^11*d*e^2))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*
a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 1
38*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36
*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*
c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^
2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 -
8*a*b^2*c^10)))^(1/2) - (8*(d + e*x)^(1/2)*(b^10*e^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4
+ 25*a^4*b^2*c^4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*
d^2*e^2 - 16*a^3*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*
c^3*d*e^3 + 60*a^3*b^3*c^4*d*e^3))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d -
52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^
4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c -
b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d
*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1
/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*1i - (((8*(
a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*c^5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*
a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^2 - 8*a^2*b^2*c^7*d*e^3))/c^7 + (8*(d + e*x)^(1/2)*(-(b^11*e + 8*a^5*c^6
*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a
^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*
a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b
^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^
3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c
^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2 - 4*a*b*c^10*e^3 + 8*a*c^11*d*e^2))/c^7)
*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 6
6*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2
) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c -
b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*
c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^
3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2) + (8*(d + e*x)^(1/2)*(b^10*e^4 - 2*a^5*c^5*e^4 + 3
5*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4 + 25*a^4*b^2*c^4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c
*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*d^2*e^2 - 16*a^3*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e
^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*c^3*d*e^3 + 60*a^3*b^3*c^4*d*e^3))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*
(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*
e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d
- 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-
(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) +
15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c
^9 - 8*a*b^2*c^10)))^(1/2)*1i)/((((8*(a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*c^5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^
5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^2 - 8*a^2*b^2*c^7*d*e^3))/c^7 - (8
*(d + e*x)^(1/2)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a
^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*
c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^
6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/
2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*
(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2 -
4*a*b*c^10*e^3 + 8*a*c^11*d*e^2))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 5
2*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4
*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c -
b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*
(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/
2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2) - (8*(d + e*
x)^(1/2)*(b^10*e^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4 + 25*a^4*b^2*c^4*e^4 + 2*a^4*c^6*
d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*d^2*e^2 - 16*a^3*b^2*c^5*d^2*e^2 +
18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*c^3*d*e^3 + 60*a^3*b^3*c^4*d*e^3))
/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*
d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)
^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*
a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2
*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c -
b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2) - (16*(a^5*b^4*e^5 + a^7*c^2*e^5 - 3*a^6*b^2*
c*e^5 - a^4*b^5*d*e^4 + a^6*c^3*d^2*e^3 - a^4*b^3*c^2*d^3*e^2 - 5*a^5*b^2*c^2*d^2*e^3 + 2*a^5*b^3*c*d*e^4 + a^
6*b*c^2*d*e^4 + 2*a^4*b^4*c*d^2*e^3 + 2*a^5*b*c^3*d^3*e^2))/c^7 + (((8*(a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*
c^5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^
2 - 8*a^2*b^2*c^7*d*e^3))/c^7 + (8*(d + e*x)^(1/2)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) -
b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129
*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d
*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*
a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c -
b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*
(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2 - 4*a*b*c^10*e^3 + 8*a*c^11*d*e^2))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*
a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 1
38*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36
*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*
c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^
2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 -
8*a*b^2*c^10)))^(1/2) + (8*(d + e*x)^(1/2)*(b^10*e^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4
+ 25*a^4*b^2*c^4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*
d^2*e^2 - 16*a^3*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*
c^3*d*e^3 + 60*a^3*b^3*c^4*d*e^3))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d -
52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^
4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c -
b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d
*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1
/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)))*(-(b^11*e
+ 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*
c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b
^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(
1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4
*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/
(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*2i - atan(((((8*(a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*c^5*d
*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^2 - 8
*a^2*b^2*c^7*d*e^3))/c^7 - (8*(d + e*x)^(1/2)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c
*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b
^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*
a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*
c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^
3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*(b^3*c
^9*e^3 - 2*b^2*c^10*d*e^2 - 4*a*b*c^10*e^3 + 8*a*c^11*d*e^2))/c^7)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^
6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*
b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*
c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2
)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c
^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2
*c^10)))^(1/2) - (8*(d + e*x)^(1/2)*(b^10*e^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4 + 25*a
^4*b^2*c^4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*d^2*e^2
- 16*a^3*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*c^3*d*e
^3 + 60*a^3*b^3*c^4*d*e^3))/c^7)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b
^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^
4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)
^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*
c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10
*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*1i - (((8*(a*b^5*c^
5*e^4 + 8*a^3*b*c^7*e^4 - b^6*c^5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^
7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^2 - 8*a^2*b^2*c^7*d*e^3))/c^7 + (8*(d + e*x)^(1/2)*((b^8*e*(-(4*a*c - b^2)^3)^(1
/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^
2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2
*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*
(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2)
+ 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4
*c^9 - 8*a*b^2*c^10)))^(1/2)*(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2 - 4*a*b*c^10*e^3 + 8*a*c^11*d*e^2))/c^7)*((b^8*e*
(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*
c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b
^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(
1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4
*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/
(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2) + (8*(d + e*x)^(1/2)*(b^10*e^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*
c^2*e^4 - 50*a^3*b^4*c^3*e^4 + 25*a^4*b^2*c^4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b
^9*c*d*e^3 + 20*a^2*b^4*c^4*d^2*e^2 - 16*a^3*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b
^6*c^3*d^2*e^2 - 54*a^2*b^5*c^3*d*e^3 + 60*a^3*b^3*c^4*d*e^3))/c^7)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c
^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3
*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b
*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^
2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*
c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^
2*c^10)))^(1/2)*1i)/((((8*(a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*c^5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e
^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^2 - 8*a^2*b^2*c^7*d*e^3))/c^7 - (8*(d + e*x)^
(1/2)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d
+ 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^
(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a
*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*
b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b
^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2 - 4*a*b*c^10*e
^3 + 8*a*c^11*d*e^2))/c^7)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3
*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*
e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2)
- 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^
2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b
^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2) - (8*(d + e*x)^(1/2)*(b^1
0*e^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4 + 25*a^4*b^2*c^4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8
*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*d^2*e^2 - 16*a^3*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2
*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*c^3*d*e^3 + 60*a^3*b^3*c^4*d*e^3))/c^7)*((b^8*e
*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2
*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*
b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^
(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(
4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))
/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2) - (16*(a^5*b^4*e^5 + a^7*c^2*e^5 - 3*a^6*b^2*c*e^5 - a^4*b^
5*d*e^4 + a^6*c^3*d^2*e^3 - a^4*b^3*c^2*d^3*e^2 - 5*a^5*b^2*c^2*d^2*e^3 + 2*a^5*b^3*c*d*e^4 + a^6*b*c^2*d*e^4
+ 2*a^4*b^4*c*d^2*e^3 + 2*a^5*b*c^3*d^3*e^2))/c^7 + (((8*(a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*c^5*d*e^3 - 6*
a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^2 - 8*a^2*b^2*
c^7*d*e^3))/c^7 + (8*(d + e*x)^(1/2)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a
^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e
+ a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2
)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(
4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2)
- 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*(b^3*c^9*e^3 -
2*b^2*c^10*d*e^2 - 4*a*b*c^10*e^3 + 8*a*c^11*d*e^2))/c^7)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^1
1*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e
- 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b
^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2
) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4
*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^
(1/2) + (8*(d + e*x)^(1/2)*(b^10*e^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4 + 25*a^4*b^2*c^
4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*d^2*e^2 - 16*a^3
*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*c^3*d*e^3 + 60*a
^3*b^3*c^4*d*e^3))/c^7)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d
- 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(
-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) -
7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^
3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*
c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)))*((b^8*e*(-(4*a*c - b^2)^3)
^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7
*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*
c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2
*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/
2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 +
b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*2i - ((8*d)/(5*c*e^3) + (2*(b*e^4 - 2*c*d*e^3))/(5*c^2*e^6))*(d + e*x)^(5/2) -
(d + e*x)^(1/2)*((8*d^3)/(c*e^3) - (((8*d)/(c*e^3) + (2*(b*e^4 - 2*c*d*e^3))/(c^2*e^6))*(a*e^5 + c*d^2*e^3 -
b*d*e^4))/(c*e^3) + ((b*e^4 - 2*c*d*e^3)*((12*d^2)/(c*e^3) - (2*(a*e^5 + c*d^2*e^3 - b*d*e^4))/(c^2*e^6) + (((
8*d)/(c*e^3) + (2*(b*e^4 - 2*c*d*e^3))/(c^2*e^6))*(b*e^4 - 2*c*d*e^3))/(c*e^3)))/(c*e^3)) + (2*(d + e*x)^(7/2)
)/(7*c*e^3)