∫ (d+ex)3∕2 _____________________

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

Optimal. Leaf size=403 \[ -\frac {\sqrt {2} \sqrt {c} \left (-2 a \left (e \left (d \sqrt {b^2-4 a c}-a e\right )+c d^2\right )+b d \left (d \sqrt {b^2-4 a c}-2 a e\right )+b^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \sqrt {c} \left (-2 a \left (c d^2-e \left (d \sqrt {b^2-4 a c}+a e\right )\right )-b d \left (d \sqrt {b^2-4 a c}+2 a e\right )+b^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {2 \sqrt {d} (b d-2 a e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a^2}-\frac {d \sqrt {d+e x}}{a x}+\frac {\sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a} \]

[Out]

e*arctanh((e*x+d)^(1/2)/d^(1/2))*d^(1/2)/a+2*(-2*a*e+b*d)*arctanh((e*x+d)^(1/2)/d^(1/2))*d^(1/2)/a^2-d*(e*x+d)

^(1/2)/a/x-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)*c^(1/2)*(b^2*

d^2+b*d*(-2*a*e+d*(-4*a*c+b^2)^(1/2))-2*a*(c*d^2+e*(-a*e+d*(-4*a*c+b^2)^(1/2))))/a^2/(-4*a*c+b^2)^(1/2)/(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)*c^(1/2)*(b^2*d^2-b*d*(2*a*e+d*(-4*a*c+b^2)^(1/2))-2*a*(c*d^2-e*(a*e+d*(-4*a*c+b^2)^(1/2))))/a^2/(-4*a

*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

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Rubi [A] time = 3.07, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {897, 1287, 199, 206, 1166, 208} \[ -\frac {\sqrt {2} \sqrt {c} \left (b d \left (d \sqrt {b^2-4 a c}-2 a e\right )-2 a e \left (d \sqrt {b^2-4 a c}-a e\right )-2 a c d^2+b^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \sqrt {c} \left (-b d \left (d \sqrt {b^2-4 a c}+2 a e\right )+2 a e \left (d \sqrt {b^2-4 a c}+a e\right )-2 a c d^2+b^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {2 \sqrt {d} (b d-2 a e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a^2}-\frac {d \sqrt {d+e x}}{a x}+\frac {\sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((d*Sqrt[d + e*x])/(a*x)) + (Sqrt[d]*e*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/a + (2*Sqrt[d]*(b*d - 2*a*e)*ArcTanh[S

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

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

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

a*e*(Sqrt[b^2 - 4*a*c]*d + a*e) - b*d*(Sqrt[b^2 - 4*a*c]*d + 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/S

qrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(a^2*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +

1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In

tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]

)

Rule 206

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

/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 897

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 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]

Rule 1287

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

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Mathematica [A] time = 1.54, size = 393, normalized size = 0.98 \[ \frac {\frac {\sqrt {2} \sqrt {c} \left (2 a \left (e \left (d \sqrt {b^2-4 a c}-a e\right )+c d^2\right )+b d \left (2 a e-d \sqrt {b^2-4 a c}\right )-b^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{\sqrt {b^2-4 a c} \sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}-\frac {\sqrt {2} \sqrt {c} \left (b d \left (d \sqrt {b^2-4 a c}+2 a e\right )-2 a e \left (d \sqrt {b^2-4 a c}+a e\right )+2 a c d^2-b^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+2 \sqrt {d} (b d-2 a e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-\frac {a d \sqrt {d+e x}}{x}+a \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a^2} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-((a*d*Sqrt[d + e*x])/x) + a*Sqrt[d]*e*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + 2*Sqrt[d]*(b*d - 2*a*e)*ArcTanh[Sqrt[

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

[b^2 - 4*a*c]*d - a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sq

rt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*Sqrt[c]*(-(b^2*d^2) + 2*a*c*d^2 - 2*a*e*(

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

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

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fricas [B] time = 73.17, size = 8653, normalized size = 21.47 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(sqrt(2)*a^2*x*sqrt(-(a^3*b*e^3 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3 + 3*(a*b^3 - 3*a^2*b*c)*d^2*e - 3*(a

^2*b^2 - 2*a^3*c)*d*e^2 + (a^4*b^2 - 4*a^5*c)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^

2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(1

0*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))/(a^4*b^2 - 4*a^5*c))*

log(sqrt(2)*((b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*d^4 - (4*a*b^5 - 21*a^2*b^3*c + 20*a^3*b*c^2)*d^3*e + 3*(2*a^2*

b^4 - 9*a^3*b^2*c + 4*a^4*c^2)*d^2*e^2 - 4*(a^3*b^3 - 4*a^4*b*c)*d*e^3 + (a^4*b^2 - 4*a^5*c)*e^4 + ((a^4*b^4 -

6*a^5*b^2*c + 8*a^6*c^2)*d - (a^5*b^3 - 4*a^6*b*c)*e)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a

^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e

^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))*sqrt(-(a^3*b

*e^3 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3 + 3*(a*b^3 - 3*a^2*b*c)*d^2*e - 3*(a^2*b^2 - 2*a^3*c)*d*e^2 + (a^4*b^

2 - 4*a^5*c)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c +

2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 -

3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))/(a^4*b^2 - 4*a^5*c)) - 4*(4*a^3*b*c*d*e^4 - a^4*c*e^5

+ (b^3*c^2 - 2*a*b*c^3)*d^5 - (b^4*c + a*b^2*c^2 - 3*a^2*c^3)*d^4*e + 2*(2*a*b^3*c - a^2*b*c^2)*d^3*e^2 - 2*(3

*a^2*b^2*c - a^3*c^2)*d^2*e^3)*sqrt(e*x + d)) - sqrt(2)*a^2*x*sqrt(-(a^3*b*e^3 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)

*d^3 + 3*(a*b^3 - 3*a^2*b*c)*d^2*e - 3*(a^2*b^2 - 2*a^3*c)*d*e^2 + (a^4*b^2 - 4*a^5*c)*sqrt(-(6*a^5*b*d*e^5 -

a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 -

10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a

^8*b^2 - 4*a^9*c)))/(a^4*b^2 - 4*a^5*c))*log(-sqrt(2)*((b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*d^4 - (4*a*b^5 - 21*a

^2*b^3*c + 20*a^3*b*c^2)*d^3*e + 3*(2*a^2*b^4 - 9*a^3*b^2*c + 4*a^4*c^2)*d^2*e^2 - 4*(a^3*b^3 - 4*a^4*b*c)*d*e

^3 + (a^4*b^2 - 4*a^5*c)*e^4 + ((a^4*b^4 - 6*a^5*b^2*c + 8*a^6*c^2)*d - (a^5*b^3 - 4*a^6*b*c)*e)*sqrt(-(6*a^5*

b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5

*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d

^2*e^4)/(a^8*b^2 - 4*a^9*c)))*sqrt(-(a^3*b*e^3 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3 + 3*(a*b^3 - 3*a^2*b*c)*d^2

*e - 3*(a^2*b^2 - 2*a^3*c)*d*e^2 + (a^4*b^2 - 4*a^5*c)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a

^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e

^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))/(a^4*b^2 - 4

*a^5*c)) - 4*(4*a^3*b*c*d*e^4 - a^4*c*e^5 + (b^3*c^2 - 2*a*b*c^3)*d^5 - (b^4*c + a*b^2*c^2 - 3*a^2*c^3)*d^4*e

+ 2*(2*a*b^3*c - a^2*b*c^2)*d^3*e^2 - 2*(3*a^2*b^2*c - a^3*c^2)*d^2*e^3)*sqrt(e*x + d)) + sqrt(2)*a^2*x*sqrt(-

(a^3*b*e^3 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3 + 3*(a*b^3 - 3*a^2*b*c)*d^2*e - 3*(a^2*b^2 - 2*a^3*c)*d*e^2 - (

a^4*b^2 - 4*a^5*c)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b

^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3

*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))/(a^4*b^2 - 4*a^5*c))*log(sqrt(2)*((b^6 - 6*a*b^4

*c + 8*a^2*b^2*c^2)*d^4 - (4*a*b^5 - 21*a^2*b^3*c + 20*a^3*b*c^2)*d^3*e + 3*(2*a^2*b^4 - 9*a^3*b^2*c + 4*a^4*c

^2)*d^2*e^2 - 4*(a^3*b^3 - 4*a^4*b*c)*d*e^3 + (a^4*b^2 - 4*a^5*c)*e^4 - ((a^4*b^4 - 6*a^5*b^2*c + 8*a^6*c^2)*d

- (a^5*b^3 - 4*a^6*b*c)*e)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5

- 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4

*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))*sqrt(-(a^3*b*e^3 - (b^4 - 4*a*b^2*c + 2

*a^2*c^2)*d^3 + 3*(a*b^3 - 3*a^2*b*c)*d^2*e - 3*(a^2*b^2 - 2*a^3*c)*d*e^2 - (a^4*b^2 - 4*a^5*c)*sqrt(-(6*a^5*b

*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*

a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^

2*e^4)/(a^8*b^2 - 4*a^9*c)))/(a^4*b^2 - 4*a^5*c)) - 4*(4*a^3*b*c*d*e^4 - a^4*c*e^5 + (b^3*c^2 - 2*a*b*c^3)*d^5

- (b^4*c + a*b^2*c^2 - 3*a^2*c^3)*d^4*e + 2*(2*a*b^3*c - a^2*b*c^2)*d^3*e^2 - 2*(3*a^2*b^2*c - a^3*c^2)*d^2*e

^3)*sqrt(e*x + d)) - sqrt(2)*a^2*x*sqrt(-(a^3*b*e^3 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3 + 3*(a*b^3 - 3*a^2*b*c

)*d^2*e - 3*(a^2*b^2 - 2*a^3*c)*d*e^2 - (a^4*b^2 - 4*a^5*c)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c

+ 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*

d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))/(a^4*b^

2 - 4*a^5*c))*log(-sqrt(2)*((b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*d^4 - (4*a*b^5 - 21*a^2*b^3*c + 20*a^3*b*c^2)*d^

3*e + 3*(2*a^2*b^4 - 9*a^3*b^2*c + 4*a^4*c^2)*d^2*e^2 - 4*(a^3*b^3 - 4*a^4*b*c)*d*e^3 + (a^4*b^2 - 4*a^5*c)*e^

4 - ((a^4*b^4 - 6*a^5*b^2*c + 8*a^6*c^2)*d - (a^5*b^3 - 4*a^6*b*c)*e)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 -

4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3

*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)

))*sqrt(-(a^3*b*e^3 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3 + 3*(a*b^3 - 3*a^2*b*c)*d^2*e - 3*(a^2*b^2 - 2*a^3*c)*

d*e^2 - (a^4*b^2 - 4*a^5*c)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5

- 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4

*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))/(a^4*b^2 - 4*a^5*c)) - 4*(4*a^3*b*c*d*e

^4 - a^4*c*e^5 + (b^3*c^2 - 2*a*b*c^3)*d^5 - (b^4*c + a*b^2*c^2 - 3*a^2*c^3)*d^4*e + 2*(2*a*b^3*c - a^2*b*c^2)

*d^3*e^2 - 2*(3*a^2*b^2*c - a^3*c^2)*d^2*e^3)*sqrt(e*x + d)) + (2*b*d - 3*a*e)*sqrt(d)*x*log((e*x - 2*sqrt(e*x

+ d)*sqrt(d) + 2*d)/x) + 2*sqrt(e*x + d)*a*d)/(a^2*x), -1/2*(sqrt(2)*a^2*x*sqrt(-(a^3*b*e^3 - (b^4 - 4*a*b^2*

c + 2*a^2*c^2)*d^3 + 3*(a*b^3 - 3*a^2*b*c)*d^2*e - 3*(a^2*b^2 - 2*a^3*c)*d*e^2 + (a^4*b^2 - 4*a^5*c)*sqrt(-(6*

a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e -

3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*

c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))/(a^4*b^2 - 4*a^5*c))*log(sqrt(2)*((b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*d^4 - (4

*a*b^5 - 21*a^2*b^3*c + 20*a^3*b*c^2)*d^3*e + 3*(2*a^2*b^4 - 9*a^3*b^2*c + 4*a^4*c^2)*d^2*e^2 - 4*(a^3*b^3 - 4

*a^4*b*c)*d*e^3 + (a^4*b^2 - 4*a^5*c)*e^4 + ((a^4*b^4 - 6*a^5*b^2*c + 8*a^6*c^2)*d - (a^5*b^3 - 4*a^6*b*c)*e)*

sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)

*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2

- 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))*sqrt(-(a^3*b*e^3 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3 + 3*(a*b^3 - 3

*a^2*b*c)*d^2*e - 3*(a^2*b^2 - 2*a^3*c)*d*e^2 + (a^4*b^2 - 4*a^5*c)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*

a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a

^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))

/(a^4*b^2 - 4*a^5*c)) - 4*(4*a^3*b*c*d*e^4 - a^4*c*e^5 + (b^3*c^2 - 2*a*b*c^3)*d^5 - (b^4*c + a*b^2*c^2 - 3*a^

2*c^3)*d^4*e + 2*(2*a*b^3*c - a^2*b*c^2)*d^3*e^2 - 2*(3*a^2*b^2*c - a^3*c^2)*d^2*e^3)*sqrt(e*x + d)) - sqrt(2)

*a^2*x*sqrt(-(a^3*b*e^3 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3 + 3*(a*b^3 - 3*a^2*b*c)*d^2*e - 3*(a^2*b^2 - 2*a^3

*c)*d*e^2 + (a^4*b^2 - 4*a^5*c)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*

b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11

*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))/(a^4*b^2 - 4*a^5*c))*log(-sqrt(2)*(

(b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*d^4 - (4*a*b^5 - 21*a^2*b^3*c + 20*a^3*b*c^2)*d^3*e + 3*(2*a^2*b^4 - 9*a^3*b

^2*c + 4*a^4*c^2)*d^2*e^2 - 4*(a^3*b^3 - 4*a^4*b*c)*d*e^3 + (a^4*b^2 - 4*a^5*c)*e^4 + ((a^4*b^4 - 6*a^5*b^2*c

+ 8*a^6*c^2)*d - (a^5*b^3 - 4*a^6*b*c)*e)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d

^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^

3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))*sqrt(-(a^3*b*e^3 - (b^4 -

4*a*b^2*c + 2*a^2*c^2)*d^3 + 3*(a*b^3 - 3*a^2*b*c)*d^2*e - 3*(a^2*b^2 - 2*a^3*c)*d*e^2 + (a^4*b^2 - 4*a^5*c)*

sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)

*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2

- 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))/(a^4*b^2 - 4*a^5*c)) - 4*(4*a^3*b*c*d*e^4 - a^4*c*e^5 + (b^3*c^2 -

2*a*b*c^3)*d^5 - (b^4*c + a*b^2*c^2 - 3*a^2*c^3)*d^4*e + 2*(2*a*b^3*c - a^2*b*c^2)*d^3*e^2 - 2*(3*a^2*b^2*c -

a^3*c^2)*d^2*e^3)*sqrt(e*x + d)) + sqrt(2)*a^2*x*sqrt(-(a^3*b*e^3 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3 + 3*(a*b

^3 - 3*a^2*b*c)*d^2*e - 3*(a^2*b^2 - 2*a^3*c)*d*e^2 - (a^4*b^2 - 4*a^5*c)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^

6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c

+ 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^

9*c)))/(a^4*b^2 - 4*a^5*c))*log(sqrt(2)*((b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*d^4 - (4*a*b^5 - 21*a^2*b^3*c + 20*

a^3*b*c^2)*d^3*e + 3*(2*a^2*b^4 - 9*a^3*b^2*c + 4*a^4*c^2)*d^2*e^2 - 4*(a^3*b^3 - 4*a^4*b*c)*d*e^3 + (a^4*b^2

- 4*a^5*c)*e^4 - ((a^4*b^4 - 6*a^5*b^2*c + 8*a^6*c^2)*d - (a^5*b^3 - 4*a^6*b*c)*e)*sqrt(-(6*a^5*b*d*e^5 - a^6*

e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*

a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b

^2 - 4*a^9*c)))*sqrt(-(a^3*b*e^3 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3 + 3*(a*b^3 - 3*a^2*b*c)*d^2*e - 3*(a^2*b^

2 - 2*a^3*c)*d*e^2 - (a^4*b^2 - 4*a^5*c)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^

6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3

*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))/(a^4*b^2 - 4*a^5*c)) - 4*(

4*a^3*b*c*d*e^4 - a^4*c*e^5 + (b^3*c^2 - 2*a*b*c^3)*d^5 - (b^4*c + a*b^2*c^2 - 3*a^2*c^3)*d^4*e + 2*(2*a*b^3*c

- a^2*b*c^2)*d^3*e^2 - 2*(3*a^2*b^2*c - a^3*c^2)*d^2*e^3)*sqrt(e*x + d)) - sqrt(2)*a^2*x*sqrt(-(a^3*b*e^3 - (

b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3 + 3*(a*b^3 - 3*a^2*b*c)*d^2*e - 3*(a^2*b^2 - 2*a^3*c)*d*e^2 - (a^4*b^2 - 4*a^

5*c)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b

*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^

4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))/(a^4*b^2 - 4*a^5*c))*log(-sqrt(2)*((b^6 - 6*a*b^4*c + 8*a^2*b^

2*c^2)*d^4 - (4*a*b^5 - 21*a^2*b^3*c + 20*a^3*b*c^2)*d^3*e + 3*(2*a^2*b^4 - 9*a^3*b^2*c + 4*a^4*c^2)*d^2*e^2 -

4*(a^3*b^3 - 4*a^4*b*c)*d*e^3 + (a^4*b^2 - 4*a^5*c)*e^4 - ((a^4*b^4 - 6*a^5*b^2*c + 8*a^6*c^2)*d - (a^5*b^3 -

4*a^6*b*c)*e)*sqrt(-(6*a^5*b*d*e^5 - a^6*e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c

+ 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3

- 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b^2 - 4*a^9*c)))*sqrt(-(a^3*b*e^3 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3

+ 3*(a*b^3 - 3*a^2*b*c)*d^2*e - 3*(a^2*b^2 - 2*a^3*c)*d*e^2 - (a^4*b^2 - 4*a^5*c)*sqrt(-(6*a^5*b*d*e^5 - a^6*

e^6 - (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*d^6 + 6*(a*b^5 - 3*a^2*b^3*c + 2*a^3*b*c^2)*d^5*e - 3*(5*a^2*b^4 - 10*

a^3*b^2*c + 3*a^4*c^2)*d^4*e^2 + 2*(10*a^3*b^3 - 11*a^4*b*c)*d^3*e^3 - 3*(5*a^4*b^2 - 2*a^5*c)*d^2*e^4)/(a^8*b

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

*b^2*c^2 - 3*a^2*c^3)*d^4*e + 2*(2*a*b^3*c - a^2*b*c^2)*d^3*e^2 - 2*(3*a^2*b^2*c - a^3*c^2)*d^2*e^3)*sqrt(e*x

+ d)) + 2*(2*b*d - 3*a*e)*sqrt(-d)*x*arctan(sqrt(e*x + d)*sqrt(-d)/d) + 2*sqrt(e*x + d)*a*d)/(a^2*x)]

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giac [A] time = 0.55, size = 425, normalized size = 1.05 \[ -\frac {\sqrt {x e + d} d}{a x} - \frac {{\left (2 \, b d^{2} - 3 \, a d e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{a^{2} \sqrt {-d}} - \frac {\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} b\right )} d - {\left (a b + \sqrt {b^{2} - 4 \, a c} a\right )} e\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, a^{2} c d - a^{2} b e + \sqrt {-4 \, {\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )} a^{2} c + {\left (2 \, a^{2} c d - a^{2} b e\right )}^{2}}}{a^{2} c}}}\right )}{2 \, \sqrt {b^{2} - 4 \, a c} a^{2} {\left | c \right |}} + \frac {\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} b\right )} d - {\left (a b - \sqrt {b^{2} - 4 \, a c} a\right )} e\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, a^{2} c d - a^{2} b e - \sqrt {-4 \, {\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )} a^{2} c + {\left (2 \, a^{2} c d - a^{2} b e\right )}^{2}}}{a^{2} c}}}\right )}{2 \, \sqrt {b^{2} - 4 \, a c} a^{2} {\left | c \right |}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(x*e + d)*d/(a*x) - (2*b*d^2 - 3*a*d*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(a^2*sqrt(-d)) - 1/2*sqrt(-4*c^2*d

+ 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*b)*d - (a*b + sqrt(b^2 - 4*a*c)*a)*e)*ar

ctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*a^2*c*d - a^2*b*e + sqrt(-4*(a^2*c*d^2 - a^2*b*d*e + a^3*e^2)*a^2*c +

(2*a^2*c*d - a^2*b*e)^2))/(a^2*c)))/(sqrt(b^2 - 4*a*c)*a^2*abs(c)) + 1/2*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4

*a*c)*c)*e)*((b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*b)*d - (a*b - sqrt(b^2 - 4*a*c)*a)*e)*arctan(2*sqrt(1/2)*sqrt(x*

e + d)/sqrt(-(2*a^2*c*d - a^2*b*e - sqrt(-4*(a^2*c*d^2 - a^2*b*d*e + a^3*e^2)*a^2*c + (2*a^2*c*d - a^2*b*e)^2)

)/(a^2*c)))/(sqrt(b^2 - 4*a*c)*a^2*abs(c))

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maple [B] time = 0.05, size = 1215, normalized size = 3.01 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d*(e*x+d)^(1/2)/a/x-3*e*arctanh((e*x+d)^(1/2)/d^(1/2))*d^(1/2)/a+2*d^(3/2)/a^2*arctanh((e*x+d)^(1/2)/d^(1/2))

*b-2*e^3*c/(-(4*a*c-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)+2*e^2/a*c/(-(4*a*c-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+2*e/a*c^2/(-(4*a*c-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-e/a^2*c/(-(4*a*c-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*d^2+2*e/a*c*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arc

tanh((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-1/a^2*c*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^2-2*e^3*c/(-(4*a*c-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)+2*e^2/a*c/(-(4*a*c-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+2*e/a*c^2/(-(4*a*c-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-e/a^2*c/(-(4*a*c-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*d^2-2*e/a*c*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+1/a^2*c*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^2

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B] time = 7.36, size = 29890, normalized size = 74.17 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d^(1/2)*atan(((d^(1/2)*((8*(d + e*x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4

*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11 +

8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*

b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 - 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 60*

a^4*b*c^4*d^3*e^13))/a^4 - (d^(1/2)*((8*(56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*

a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3

*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^

3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 +

6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*

e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4 + (d^(1/2)*((8*(d + e*x)^(1/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^

6*b^3*c^2*e^13 - 40*a^5*c^6*d^5*e^8 + 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 5

6*a^3*b^4*c^4*d^5*e^8 + 36*a^3*b^5*c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^

3*c^4*d^4*e^9 - 179*a^4*b^4*c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^

3*d^2*e^11 - 224*a^5*b*c^5*d^4*e^9 + 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/

a^4 + (d^(1/2)*(3*a*e - 2*b*d)*((8*(80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4

*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^10 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8

+ 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b*c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11))/a^4 - (4*d^(1/2)*(3*a*e - 2*b*d)*(d

+ e*x)^(1/2)*(64*a^9*c^4*e^10 + 4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^10 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3

*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^8 - 112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^6))/(

2*a^2))*(3*a*e - 2*b*d))/(2*a^2))*(3*a*e - 2*b*d))/(2*a^2))*(3*a*e - 2*b*d)*1i)/(2*a^2) + (d^(1/2)*((8*(d + e*

x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14

+ 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2

*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8

- 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/a^4 + (d^(1/2)

*((8*(56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^

3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4

*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^1

3 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d

^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4 -

(d^(1/2)*((8*(d + e*x)^(1/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2*e^13 - 40*a^5*c^6*d^5*e^8

+ 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4*c^4*d^5*e^8 + 36*a^3*b^5*

c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4*e^9 - 179*a^4*b^4*c^3*d^3

*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*e^11 - 224*a^5*b*c^5*d^4*e^9

+ 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/a^4 - (d^(1/2)*(3*a*e - 2*b*d)*((8*

(80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^

10 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8 + 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b

*c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11))/a^4 + (4*d^(1/2)*(3*a*e - 2*b*d)*(d + e*x)^(1/2)*(64*a^9*c^4*e^10 + 4*

a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^10 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^8 -

112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^6))/(2*a^2))*(3*a*e - 2*b*d))/(2*a^2))*(

3*a*e - 2*b*d))/(2*a^2))*(3*a*e - 2*b*d)*1i)/(2*a^2))/((16*(6*a*c^7*d^9*e^9 + 6*a^5*c^3*d*e^17 - 4*b*c^7*d^10*

e^8 + 6*a^2*c^6*d^7*e^11 + 6*a^4*c^4*d^3*e^15 + 8*b^2*c^6*d^9*e^9 - 4*b^3*c^5*d^8*e^10 + 4*a^2*b^2*c^4*d^5*e^1

3 - 11*a^2*b^3*c^3*d^4*e^14 + 22*a^3*b^2*c^3*d^3*e^15 - 16*a*b*c^6*d^8*e^10 + 8*a*b^2*c^5*d^7*e^11 + 2*a*b^4*c

^3*d^5*e^13 - 3*a^2*b*c^5*d^6*e^12 - 10*a^3*b*c^4*d^4*e^14 - 19*a^4*b*c^3*d^2*e^16))/a^4 - (d^(1/2)*((8*(d + e

*x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14

+ 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^

2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8

- 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/a^4 - (d^(1/2

)*((8*(56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c

^3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^

4*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^

13 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*

d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4

+ (d^(1/2)*((8*(d + e*x)^(1/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2*e^13 - 40*a^5*c^6*d^5*e^

8 + 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4*c^4*d^5*e^8 + 36*a^3*b^5

*c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4*e^9 - 179*a^4*b^4*c^3*d^

3*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*e^11 - 224*a^5*b*c^5*d^4*e^9

+ 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/a^4 + (d^(1/2)*(3*a*e - 2*b*d)*((8

*(80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e

^10 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8 + 2*a^6*b^4*c^2*d*e^11 - 112*a^7*

b*c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11))/a^4 - (4*d^(1/2)*(3*a*e - 2*b*d)*(d + e*x)^(1/2)*(64*a^9*c^4*e^10 + 4

*a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^10 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^8

- 112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^6))/(2*a^2))*(3*a*e - 2*b*d))/(2*a^2))*

(3*a*e - 2*b*d))/(2*a^2))*(3*a*e - 2*b*d))/(2*a^2) + (d^(1/2)*((8*(d + e*x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*

d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5

*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^

13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 - 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*

d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/a^4 + (d^(1/2)*((8*(56*a^4*c^6*d^6*e^9 - 44*a^5*c^5

*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11

- 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237

*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6

*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a

^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4 - (d^(1/2)*((8*(d + e*x)^(1/2)*(16*a^7

*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2*e^13 - 40*a^5*c^6*d^5*e^8 + 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c

^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4*c^4*d^5*e^8 + 36*a^3*b^5*c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10

+ 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4*e^9 - 179*a^4*b^4*c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234

*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*e^11 - 224*a^5*b*c^5*d^4*e^9 + 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c

^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/a^4 - (d^(1/2)*(3*a*e - 2*b*d)*((8*(80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e

^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^10 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b

^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8 + 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b*c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11

))/a^4 + (4*d^(1/2)*(3*a*e - 2*b*d)*(d + e*x)^(1/2)*(64*a^9*c^4*e^10 + 4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^1

0 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^8 - 112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*

d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^6))/(2*a^2))*(3*a*e - 2*b*d))/(2*a^2))*(3*a*e - 2*b*d))/(2*a^2))*(3*a*e - 2*b

*d))/(2*a^2)))*(3*a*e - 2*b*d)*1i)/a^2 - atan(((((((8*(80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*

d^4*e^8 - 6*a^5*b^4*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^10 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 3

2*a^6*b*c^5*d^4*e^8 + 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b*c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11))/a^4 - (8*(d + e*

x)^(1/2)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^

3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d

^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c

- b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)

^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*(64*a^9*c^4*e^10 + 4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*

c^3*e^10 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^8 - 112*a^8*b*c^4*d*e^9 - 8*a^6*b

^5*c^2*d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3

)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4

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

3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e

^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(d + e*x)^(1

/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2*e^13 - 40*a^5*c^6*d^5*e^8 + 184*a^6*c^5*d^3*e^10 +

8*a^2*b^6*c^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4*c^4*d^5*e^8 + 36*a^3*b^5*c^3*d^4*e^9 + 28*a^3*b^6*c

^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4*e^9 - 179*a^4*b^4*c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2

*e^11 + 234*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*e^11 - 224*a^5*b*c^5*d^4*e^9 + 16*a^5*b^4*c^2*d*e^12 -

348*a^6*b*c^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a

*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^

3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*

a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^

3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(

56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*

e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11

+ 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 11

1*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^1

0 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4)*((b^6*

d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^

2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*

d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2)

+ 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(

a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) - (8*(d + e*x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c

^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^

2*b^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c

^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 - 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3

*b*c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^

2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a

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

^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c

*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*1i - (((((8*(

80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^1

0 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8 + 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b*

c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11))/a^4 + (8*(d + e*x)^(1/2)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*

e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*

b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d

^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2

*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1

/2)*(64*a^9*c^4*e^10 + 4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^10 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 -

56*a^7*b^2*c^4*d^2*e^8 - 112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^4)*((b^6*d^3 -

a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*

d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-

(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*

a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^

4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) - (8*(d + e*x)^(1/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*

c^2*e^13 - 40*a^5*c^6*d^5*e^8 + 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*

b^4*c^4*d^5*e^8 + 36*a^3*b^5*c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*

d^4*e^9 - 179*a^4*b^4*c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*

e^11 - 224*a^5*b*c^5*d^4*e^9 + 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/a^4)*(

(b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) +

3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a

*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^

(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))

/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*

d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^

9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 -

161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*

c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*

a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c -

b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8

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

b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2

*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(d + e

*x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14

+ 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^

2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8

- 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/a^4)*((b^6*d^

3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*

b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^

3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) +

21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^

4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*1i)/((((((8*(80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*