3.20.65 \(\int \frac {(d+e x)^{5/2}}{(a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=504 \[ \frac {\left (-2 c^2 d e \left (-d \sqrt {b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (-b \left (d \sqrt {b^2-4 a c}+4 a e\right )+3 a e \sqrt {b^2-4 a c}+b^2 d\right )+b^2 e^3 \left (b-\sqrt {b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-2 c^2 d e \left (d \sqrt {b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (b d \sqrt {b^2-4 a c}-3 a e \sqrt {b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e \sqrt {d+e x} (2 c d-b e)}{c \left (b^2-4 a c\right )} \]

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Rubi [A]  time = 4.62, antiderivative size = 504, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {738, 824, 826, 1166, 208} \begin {gather*} \frac {\left (-2 c^2 d e \left (-d \sqrt {b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (-b \left (d \sqrt {b^2-4 a c}+4 a e\right )+3 a e \sqrt {b^2-4 a c}+b^2 d\right )+b^2 e^3 \left (b-\sqrt {b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-2 c^2 d e \left (d \sqrt {b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (b d \sqrt {b^2-4 a c}-3 a e \sqrt {b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e \sqrt {d+e x} (2 c d-b e)}{c \left (b^2-4 a c\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)/(a + b*x + c*x^2)^2,x]

[Out]

(e*(2*c*d - b*e)*Sqrt[d + e*x])/(c*(b^2 - 4*a*c)) - ((d + e*x)^(3/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 -
4*a*c)*(a + b*x + c*x^2)) + ((8*c^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d - Sqrt[b^2 - 4*a*
c]*d - 8*a*e) + 2*c*e^2*(b^2*d + 3*a*Sqrt[b^2 - 4*a*c]*e - b*(Sqrt[b^2 - 4*a*c]*d + 4*a*e)))*ArcTanh[(Sqrt[2]*
Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c
*d - (b - Sqrt[b^2 - 4*a*c])*e]) - ((8*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d + Sqrt[b^2
- 4*a*c]*d - 8*a*e) + 2*c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 4*a*b*e - 3*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(S
qrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(3/2)*S
qrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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 738

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

Rule 824

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

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1166

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (4 c d^2-5 b d e+6 a e^2\right )-\frac {1}{2} e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=\frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (4 c^2 d^3-a b e^3-c d e (5 b d-8 a e)\right )+\frac {1}{2} e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \operatorname {Subst}\left (\int \frac {\frac {1}{2} e \left (4 c^2 d^3-a b e^3-c d e (5 b d-8 a e)\right )-\frac {1}{2} d e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right )+\frac {1}{2} e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{c \left (b^2-4 a c\right )}\\ &=\frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (8 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+b \sqrt {b^2-4 a c} d-4 a b e-3 a \sqrt {b^2-4 a c} e\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 )}{2 c \left (b^2-4 a c\right )^{3/2}}-\frac {\left (8 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+3 a \sqrt {b^2-4 a c} e-b \left (\sqrt {b^2-4 a c} d+4 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 )}{2 c \left (b^2-4 a c\right )^{3/2}}\\ &=\frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (8 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+3 a \sqrt {b^2-4 a c} e-b \left (\sqrt {b^2-4 a c} d+4 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (8 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+b \sqrt {b^2-4 a c} d-4 a b e-3 a \sqrt {b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

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Mathematica [A]  time = 5.27, size = 549, normalized size = 1.09 \begin {gather*} \frac {\frac {1}{2} \left (e (a e-b d)+c d^2\right ) \left (\frac {\sqrt {2} \left (\frac {\left (2 c^2 d e \left (d \sqrt {b^2-4 a c}+8 a e-6 b d\right )+2 c e^2 \left (-b \left (d \sqrt {b^2-4 a c}+4 a e\right )+3 a e \sqrt {b^2-4 a c}+b^2 d\right )+b^2 e^3 \left (b-\sqrt {b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{\sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}-\frac {\left (-2 c^2 d e \left (d \sqrt {b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (b d \sqrt {b^2-4 a c}-3 a e \sqrt {b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c^{3/2} \sqrt {b^2-4 a c}}-\frac {2 e \sqrt {d+e x} (b e-2 c d)}{c}+4 e (d+e x)^{3/2}\right )+\frac {(d+e x)^{7/2} \left (-2 c (a e+c d x)+b^2 e+b c (e x-d)\right )}{a+x (b+c x)}+e (d+e x)^{5/2} (2 c d-b e)}{\left (b^2-4 a c\right ) \left (e (a e-b d)+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)/(a + b*x + c*x^2)^2,x]

[Out]

(e*(2*c*d - b*e)*(d + e*x)^(5/2) + ((d + e*x)^(7/2)*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/(a + x*(b +
c*x)) + ((c*d^2 + e*(-(b*d) + a*e))*((-2*e*(-2*c*d + b*e)*Sqrt[d + e*x])/c + 4*e*(d + e*x)^(3/2) + (Sqrt[2]*((
(8*c^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 + 2*c^2*d*e*(-6*b*d + Sqrt[b^2 - 4*a*c]*d + 8*a*e) + 2*c*e^2*(b^2
*d + 3*a*Sqrt[b^2 - 4*a*c]*e - b*(Sqrt[b^2 - 4*a*c]*d + 4*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[
2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] - ((8*c^3*d^3 + b^2*(b + Sqrt[b^
2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d - 8*a*e) + 2*c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 4
*a*b*e - 3*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c]
)*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(c^(3/2)*Sqrt[b^2 - 4*a*c])))/2)/((b^2 - 4*a*c)*(c*d^2 + e*(-
(b*d) + a*e)))

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IntegrateAlgebraic [C]  time = 8.13, size = 1055, normalized size = 2.09 \begin {gather*} \frac {e \sqrt {d+e x} \left (-2 c^2 d^3+3 b c e d^2+2 c^2 (d+e x) d^2-b^2 e^2 d-2 a c e^2 d-2 b c e (d+e x) d+a b e^3+b^2 e^2 (d+e x)-2 a c e^2 (d+e x)\right )}{c \left (4 a c-b^2\right ) \left (c d^2-b e d-2 c (d+e x) d+a e^2+c (d+e x)^2+b e (d+e x)\right )}+\frac {\sqrt {2} \left (-3 b e^3+\sqrt {b^2-4 a c} e^3+6 c d e^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}+\frac {\sqrt {2} \left (3 b e^3+\sqrt {b^2-4 a c} e^3-6 c d e^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+\sqrt {b^2-4 a c} e}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {-2 c d+b e+\sqrt {b^2-4 a c} e}}+\frac {\left (8 i \sqrt {2} c^3 d^3-12 i \sqrt {2} b c^2 e d^2-2 \sqrt {2} c^2 \sqrt {4 a c-b^2} e d^2-32 i \sqrt {2} a c^2 e^2 d+14 i \sqrt {2} b^2 c e^2 d+2 \sqrt {2} b c \sqrt {4 a c-b^2} e^2 d-5 i \sqrt {2} b^3 e^3+16 i \sqrt {2} a b c e^3-\sqrt {2} b^2 \sqrt {4 a c-b^2} e^3+2 \sqrt {2} a c \sqrt {4 a c-b^2} e^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {4 a c-b^2} e}}\right )}{2 c^{3/2} \left (b^2-4 a c\right ) \sqrt {4 a c-b^2} \sqrt {-2 c d+b e-i \sqrt {4 a c-b^2} e}}+\frac {\left (-8 i \sqrt {2} c^3 d^3+12 i \sqrt {2} b c^2 e d^2-2 \sqrt {2} c^2 \sqrt {4 a c-b^2} e d^2+32 i \sqrt {2} a c^2 e^2 d-14 i \sqrt {2} b^2 c e^2 d+2 \sqrt {2} b c \sqrt {4 a c-b^2} e^2 d+5 i \sqrt {2} b^3 e^3-16 i \sqrt {2} a b c e^3-\sqrt {2} b^2 \sqrt {4 a c-b^2} e^3+2 \sqrt {2} a c \sqrt {4 a c-b^2} e^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {4 a c-b^2} e}}\right )}{2 c^{3/2} \left (b^2-4 a c\right ) \sqrt {4 a c-b^2} \sqrt {-2 c d+b e+i \sqrt {4 a c-b^2} e}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(5/2)/(a + b*x + c*x^2)^2,x]

[Out]

(e*Sqrt[d + e*x]*(-2*c^2*d^3 + 3*b*c*d^2*e - b^2*d*e^2 - 2*a*c*d*e^2 + a*b*e^3 + 2*c^2*d^2*(d + e*x) - 2*b*c*d
*e*(d + e*x) + b^2*e^2*(d + e*x) - 2*a*c*e^2*(d + e*x)))/(c*(-b^2 + 4*a*c)*(c*d^2 - b*d*e + a*e^2 - 2*c*d*(d +
e*x) + b*e*(d + e*x) + c*(d + e*x)^2)) + (Sqrt[2]*(6*c*d*e^2 - 3*b*e^3 + Sqrt[b^2 - 4*a*c]*e^3)*ArcTan[(Sqrt[
2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d +
b*e - Sqrt[b^2 - 4*a*c]*e]) + (Sqrt[2]*(-6*c*d*e^2 + 3*b*e^3 + Sqrt[b^2 - 4*a*c]*e^3)*ArcTan[(Sqrt[2]*Sqrt[c]*
Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e]])/(c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + b*e + Sqrt[
b^2 - 4*a*c]*e]) + (((8*I)*Sqrt[2]*c^3*d^3 - (12*I)*Sqrt[2]*b*c^2*d^2*e - 2*Sqrt[2]*c^2*Sqrt[-b^2 + 4*a*c]*d^2
*e + (14*I)*Sqrt[2]*b^2*c*d*e^2 - (32*I)*Sqrt[2]*a*c^2*d*e^2 + 2*Sqrt[2]*b*c*Sqrt[-b^2 + 4*a*c]*d*e^2 - (5*I)*
Sqrt[2]*b^3*e^3 + (16*I)*Sqrt[2]*a*b*c*e^3 - Sqrt[2]*b^2*Sqrt[-b^2 + 4*a*c]*e^3 + 2*Sqrt[2]*a*c*Sqrt[-b^2 + 4*
a*c]*e^3)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(2*c^(3/2)*(b^2
- 4*a*c)*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]) + (((-8*I)*Sqrt[2]*c^3*d^3 + (12*I)*
Sqrt[2]*b*c^2*d^2*e - 2*Sqrt[2]*c^2*Sqrt[-b^2 + 4*a*c]*d^2*e - (14*I)*Sqrt[2]*b^2*c*d*e^2 + (32*I)*Sqrt[2]*a*c
^2*d*e^2 + 2*Sqrt[2]*b*c*Sqrt[-b^2 + 4*a*c]*d*e^2 + (5*I)*Sqrt[2]*b^3*e^3 - (16*I)*Sqrt[2]*a*b*c*e^3 - Sqrt[2]
*b^2*Sqrt[-b^2 + 4*a*c]*e^3 + 2*Sqrt[2]*a*c*Sqrt[-b^2 + 4*a*c]*e^3)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqr
t[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/(2*c^(3/2)*(b^2 - 4*a*c)*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e + I*S
qrt[-b^2 + 4*a*c]*e])

________________________________________________________________________________________

fricas [B]  time = 0.82, size = 5256, normalized size = 10.43

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

1/2*(sqrt(1/2)*(a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c - 4*a*b*c^2)*x)*sqrt((32*c^5*d^5 - 80*b
*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 2
4*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c
^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 +
(b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a
*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(sqrt(1/2)*(10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e^4 - 15*(
b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^5 + 3*(b^6*c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^6 +
(b^7 - 17*a*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^7 - (8*(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3
*c^8)*d^2 - 8*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*d*e - (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2
*b^4*c^5 - 640*a^3*b^2*c^6 + 768*a^4*c^7)*e^2)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3
)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*
a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5*(b^3*c^2
- 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 +
(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 +
6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c
^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)) + 2*(80*c^5*d^6*e^
3 - 240*b*c^4*d^5*e^4 + (199*b^2*c^3 + 404*a*c^4)*d^4*e^5 + 2*(b^3*c^2 - 404*a*b*c^3)*d^3*e^6 - 6*(6*b^4*c - 4
7*a*b^2*c^2 - 108*a^2*c^3)*d^2*e^7 - (5*b^5 - 122*a*b^3*c + 648*a^2*b*c^2)*d*e^8 + (5*a*b^4 - 81*a^2*b^2*c + 3
24*a^3*c^2)*e^9)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c - 4*a*b*c^
2)*x)*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3
- 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 + (b^6*c^3 - 12*a*b^4*c^
4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10
*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 -
64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(-sqrt(1/2)*(10*(b^4*c^3 - 8*a*b^2*c^
4 + 16*a^2*c^5)*d^3*e^4 - 15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^5 + 3*(b^6*c - 2*a*b^4*c^2 - 32*a^2*
b^2*c^3 + 96*a^3*c^4)*d*e^6 + (b^7 - 17*a*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^7 - (8*(b^6*c^5 - 12*a*b^4
*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*d^2 - 8*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*d*e - (b^
8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b^2*c^6 + 768*a^4*c^7)*e^2)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d
^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)
/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2*c^3 +
12*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*
a*b^3*c + 60*a^2*b*c^2)*e^5 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50
*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^
2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 6
4*a^3*c^6)) + 2*(80*c^5*d^6*e^3 - 240*b*c^4*d^5*e^4 + (199*b^2*c^3 + 404*a*c^4)*d^4*e^5 + 2*(b^3*c^2 - 404*a*b
*c^3)*d^3*e^6 - 6*(6*b^4*c - 47*a*b^2*c^2 - 108*a^2*c^3)*d^2*e^7 - (5*b^5 - 122*a*b^3*c + 648*a^2*b*c^2)*d*e^8
+ (5*a*b^4 - 81*a^2*b^2*c + 324*a^3*c^2)*e^9)*sqrt(e*x + d)) + sqrt(1/2)*(a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*
a*c^3)*x^2 + (b^3*c - 4*a*b*c^2)*x)*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5*
(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2
)*e^5 - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b
^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12
*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(sqrt(1
/2)*(10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e^4 - 15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^5 + 3*(
b^6*c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^6 + (b^7 - 17*a*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3)
*e^7 + (8*(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*d^2 - 8*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*
c^6 - 64*a^3*b*c^7)*d*e - (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b^2*c^6 + 768*a^4*c^7)*e^2)*sqrt
((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 1
8*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt((32*c^5*d^5 - 80*b
*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 2
4*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c
^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 +
(b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a
*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)) + 2*(80*c^5*d^6*e^3 - 240*b*c^4*d^5*e^4 + (199*b^2*c^3 + 404*a*c^4)*d
^4*e^5 + 2*(b^3*c^2 - 404*a*b*c^3)*d^3*e^6 - 6*(6*b^4*c - 47*a*b^2*c^2 - 108*a^2*c^3)*d^2*e^7 - (5*b^5 - 122*a
*b^3*c + 648*a^2*b*c^2)*d*e^8 + (5*a*b^4 - 81*a^2*b^2*c + 324*a^3*c^2)*e^9)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2*
c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c - 4*a*b*c^2)*x)*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2
*c^3 + 12*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^
5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e
^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81
*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*
c^5 - 64*a^3*c^6))*log(-sqrt(1/2)*(10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e^4 - 15*(b^5*c^2 - 8*a*b^3*c^3
+ 16*a^2*b*c^4)*d^2*e^5 + 3*(b^6*c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^6 + (b^7 - 17*a*b^5*c + 8
8*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^7 + (8*(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*d^2 - 8*(b^7*c^
4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*d*e - (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b^
2*c^6 + 768*a^4*c^7)*e^2)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c
- 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*
c^9)))*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^
3 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 - (b^6*c^3 - 12*a*b^4*c
^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 1
0*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 -
64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)) + 2*(80*c^5*d^6*e^3 - 240*b*c^4*d^5*e^4
+ (199*b^2*c^3 + 404*a*c^4)*d^4*e^5 + 2*(b^3*c^2 - 404*a*b*c^3)*d^3*e^6 - 6*(6*b^4*c - 47*a*b^2*c^2 - 108*a^2
*c^3)*d^2*e^7 - (5*b^5 - 122*a*b^3*c + 648*a^2*b*c^2)*d*e^8 + (5*a*b^4 - 81*a^2*b^2*c + 324*a^3*c^2)*e^9)*sqrt
(e*x + d)) - 2*(b*c*d^2 - 4*a*c*d*e + a*b*e^2 + (2*c^2*d^2 - 2*b*c*d*e + (b^2 - 2*a*c)*e^2)*x)*sqrt(e*x + d))/
(a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c - 4*a*b*c^2)*x)

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giac [B]  time = 2.68, size = 1416, normalized size = 2.81

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

-(2*(x*e + d)^(3/2)*c^2*d^2*e - 2*sqrt(x*e + d)*c^2*d^3*e - 2*(x*e + d)^(3/2)*b*c*d*e^2 + 3*sqrt(x*e + d)*b*c*
d^2*e^2 + (x*e + d)^(3/2)*b^2*e^3 - 2*(x*e + d)^(3/2)*a*c*e^3 - sqrt(x*e + d)*b^2*d*e^3 - 2*sqrt(x*e + d)*a*c*
d*e^3 + sqrt(x*e + d)*a*b*e^4)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e + a*e^2)*(b^2
*c - 4*a*c^2)) - (16*(b^2*c^6 - 4*a*c^7)*d^4*e - 32*(b^3*c^5 - 4*a*b*c^6)*d^3*e^2 + 16*(b^4*c^4 - 2*a*b^2*c^5
- 8*a^2*c^6)*d^2*e^3 - (2*c^2*d^2*e - 2*b*c*d*e^2 - (b^2 - 6*a*c)*e^3)*(b^2*c*e - 4*a*c^2*e)^2 - 32*(a*b^3*c^4
- 4*a^2*b*c^5)*d*e^4 - 2*(2*sqrt(b^2 - 4*a*c)*c^4*d^3*e - 3*sqrt(b^2 - 4*a*c)*b*c^3*d^2*e^2 - sqrt(b^2 - 4*a*
c)*a*b*c^2*e^4 + (b^2*c^2 + 2*a*c^3)*sqrt(b^2 - 4*a*c)*d*e^3)*abs(b^2*c*e - 4*a*c^2*e) - (b^6*c^2 - 12*a*b^4*c
^3 + 32*a^2*b^2*c^4)*e^5)*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^2*c^2*d - 8*a*c^3*d - b^3*c*e + 4*a*b*c^
2*e + sqrt((2*b^2*c^2*d - 8*a*c^3*d - b^3*c*e + 4*a*b*c^2*e)^2 - 4*(b^2*c^2*d^2 - 4*a*c^3*d^2 - b^3*c*d*e + 4*
a*b*c^2*d*e + a*b^2*c*e^2 - 4*a^2*c^2*e^2)*(b^2*c^2 - 4*a*c^3)))/(b^2*c^2 - 4*a*c^3)))/(sqrt(-4*c^2*d + 2*(b*c
- sqrt(b^2 - 4*a*c)*c)*e)*(2*(b^2*c^2 - 4*a*c^3)*sqrt(b^2 - 4*a*c)*d - (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3 + (b
^3*c - 4*a*b*c^2)*sqrt(b^2 - 4*a*c))*e)*abs(b^2*c*e - 4*a*c^2*e)*abs(c)) + (16*(b^2*c^6 - 4*a*c^7)*d^4*e - 32*
(b^3*c^5 - 4*a*b*c^6)*d^3*e^2 + 16*(b^4*c^4 - 2*a*b^2*c^5 - 8*a^2*c^6)*d^2*e^3 - (2*c^2*d^2*e - 2*b*c*d*e^2 -
(b^2 - 6*a*c)*e^3)*(b^2*c*e - 4*a*c^2*e)^2 - 32*(a*b^3*c^4 - 4*a^2*b*c^5)*d*e^4 + 2*(2*sqrt(b^2 - 4*a*c)*c^4*d
^3*e - 3*sqrt(b^2 - 4*a*c)*b*c^3*d^2*e^2 - sqrt(b^2 - 4*a*c)*a*b*c^2*e^4 + (b^2*c^2 + 2*a*c^3)*sqrt(b^2 - 4*a*
c)*d*e^3)*abs(b^2*c*e - 4*a*c^2*e) - (b^6*c^2 - 12*a*b^4*c^3 + 32*a^2*b^2*c^4)*e^5)*arctan(2*sqrt(1/2)*sqrt(x*
e + d)/sqrt(-(2*b^2*c^2*d - 8*a*c^3*d - b^3*c*e + 4*a*b*c^2*e - sqrt((2*b^2*c^2*d - 8*a*c^3*d - b^3*c*e + 4*a*
b*c^2*e)^2 - 4*(b^2*c^2*d^2 - 4*a*c^3*d^2 - b^3*c*d*e + 4*a*b*c^2*d*e + a*b^2*c*e^2 - 4*a^2*c^2*e^2)*(b^2*c^2
- 4*a*c^3)))/(b^2*c^2 - 4*a*c^3)))/(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(2*(b^2*c^2 - 4*a*c^3)*sq
rt(b^2 - 4*a*c)*d + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3 - (b^3*c - 4*a*b*c^2)*sqrt(b^2 - 4*a*c))*e)*abs(b^2*c*e
- 4*a*c^2*e)*abs(c))

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maple [B]  time = 0.14, size = 2557, normalized size = 5.07

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x)

[Out]

-4*e/(4*a*c-b^2)*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^3-4*e/(4*a*c-b^2)*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^3+4*e^4/(4*a*c-b^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)*
a*b-1/2*e^4/(4*a*c-b^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)*arct
an((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^3+4*e^4/(4*a*c-b^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)*a*b-8*e^3/(4*a*c-b^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)*a
*d+6*e^2/(4*a*c-b^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*d^2-8*e^3/(4*a*c-b^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)*a*d+6*e^2/(4*a*c-b^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*d^2-e^3/(4*a*c-b^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)*ar
ctan((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-1/2*e^4/(4*a*c-b^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^3-e^3/(4*a*c-b^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)*b^2*d+e^4/(c*e^2*x^2+b*e^2*x+a*e^2)/c/(4*a*c-b^2)*(e*x+d)^(1/2)*a*b-2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)/(
4*a*c-b^2)*(e*x+d)^(3/2)*b*d+2*e/(c*e^2*x^2+b*e^2*x+a*e^2)*c/(4*a*c-b^2)*(e*x+d)^(3/2)*d^2-2*e^3/(c*e^2*x^2+b*
e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(1/2)*a*d+3*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(1/2)*b*d^2-2*e
/(c*e^2*x^2+b*e^2*x+a*e^2)*c/(4*a*c-b^2)*(e*x+d)^(1/2)*d^3+e^3/(c*e^2*x^2+b*e^2*x+a*e^2)/c/(4*a*c-b^2)*(e*x+d)
^(3/2)*b^2-3*e^3/(4*a*c-b^2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(
1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a+3*e^3/(4*a*c-b^2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*
e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a-2*e^3/(c
*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(3/2)*a-1/2*e^3/(4*a*c-b^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^2-e^2/(4*a
*c-b^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-e/(4*a*c-b^2)*c*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*ar
ctanh((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/(4*a*c-b^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)*d^2-e^3/(c*e^2*x^2+b*e^2*x+a*e^2)/c/(4*a*c-b^2)*(e*x+d)^(1/2)*b^2*d+1/2*e^3/(4*a*c-b^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^2+e^2/(4*a*c-b^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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(5/2)/(c*x^2 + b*x + a)^2, x)

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mupad [B]  time = 4.68, size = 21160, normalized size = 41.98

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(5/2)/(a + b*x + c*x^2)^2,x)

[Out]

atan(((((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 512*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*
a^3*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576
*a^2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^
5 + 768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) - (2*(
d + e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c
^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e
^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e
^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d
*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^
2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 4
80*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^
4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*
a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^
3 - 256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*b^6*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a
^2*b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^
5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e +
1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^
2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e
^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2
+ 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*
b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e
- 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a
^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8))
)^(1/2) - (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3*c^3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3
+ 74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 90*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16
*a*b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3*e^5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d
^2*e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c
- b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b
^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8
*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*
c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2
*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*
e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*
b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b
^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*1i
- (((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 512*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*a^3
*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576*a^
2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^5 +
768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) + (2*(d +
e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*
d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5
+ 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3
- 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^
4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c
^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*
a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d
*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2
*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^3 -
256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*b^6*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a^2*
b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(
-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 153
6*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 +
50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*
(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2
400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8
*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 4
80*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*
c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(
1/2) + (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3*c^3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3 +
74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 90*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16*a*
b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3*e^5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d^2*
e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c -
b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*
c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^
3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c -
b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^
5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4
+ 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6
*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12
*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*1i)/(
(((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 512*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*a^3*b^
3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576*a^2*b
^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^5 + 76
8*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) - (2*(d + e*
x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5
+ 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1
504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5
*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 +
5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*
d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b
^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^
4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^
8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^3 - 25
6*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*b^6*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a^2*b^2
*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4
*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a
^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50
*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(
4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400
*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^
2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*
a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9
+ b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2
) - (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3*c^3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3 + 74*
a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 90*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16*a*b^4
*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3*e^5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d^2*e^6
))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2
)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7
*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d
^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^
2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c
^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 +
5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^
3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^
3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2) - (2*(5*
a^2*b^4*e^11 + 216*a^4*c^2*e^11 + 5*b^6*d^2*e^9 + 32*c^6*d^8*e^3 - 66*a^3*b^2*c*e^11 + 232*a*c^5*d^6*e^5 - 128
*b*c^5*d^7*e^4 + 16*b^5*c*d^3*e^8 + 584*a^2*c^4*d^4*e^7 + 600*a^3*c^3*d^2*e^9 + 166*b^2*c^4*d^6*e^5 - 50*b^3*c
^3*d^5*e^6 - 41*b^4*c^2*d^4*e^7 - 10*a*b^5*d*e^10 + 426*a^2*b^2*c^2*d^2*e^9 - 696*a*b*c^4*d^5*e^6 - 108*a*b^4*
c*d^2*e^9 + 158*a^2*b^3*c*d*e^10 - 600*a^3*b*c^2*d*e^10 + 578*a*b^2*c^3*d^4*e^7 + 4*a*b^3*c^2*d^3*e^8 - 1168*a
^2*b*c^3*d^3*e^8))/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) + (((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e
^6 - 512*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*a^3*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*
b^6*c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576*a^2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5
- 96*a*b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^5 + 768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^
5*d*e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) + (2*(d + e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3
*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6
*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c
^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2
) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2
) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 +
960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e
^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a
^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*
b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^3 - 256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2
- 8*b^6*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a^2*b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8
*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6
*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5
+ 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3
- 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e
^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*
c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480
*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*
d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^
2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2) + (2*(d + e*x)^(1/2)*(b^6*e^8 - 72
*a^3*c^3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3 + 74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6
+ 90*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16*a*b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4
*d^3*e^5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d^2*e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2
))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 384
0*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^
3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d
^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c
*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^
2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4
*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11
520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5
- 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11
*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7
*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a
^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*
e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4
*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*
d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b
^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^
4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*
a^5*b^2*c^8)))^(1/2)*2i - (((d + e*x)^(1/2)*(b^2*d*e^3 + 2*c^2*d^3*e - a*b*e^4 + 2*a*c*d*e^3 - 3*b*c*d^2*e^2))
/(c*(4*a*c - b^2)) - (e*(d + e*x)^(3/2)*(b^2*e^2 + 2*c^2*d^2 - 2*a*c*e^2 - 2*b*c*d*e))/(c*(4*a*c - b^2)))/((b*
e - 2*c*d)*(d + e*x) + c*(d + e*x)^2 + a*e^2 + c*d^2 - b*d*e) + atan(((((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 -
512*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*a^3*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*
c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576*a^2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96
*a*b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^5 + 768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*
e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) - (2*(d + e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8
*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e
^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e
^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) +
27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) +
960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960
*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 -
3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b
^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*
c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^3 - 256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*
b^6*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a^2*b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b
^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5
+ 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1
504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5
*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 -
5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*
d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b
^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^
4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^
8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2) - (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3
*c^3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3 + 74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 9
0*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16*a*b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3
*e^5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d^2*e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*(
(32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^
5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^
5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e
^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e
^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 -
8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3
*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*
a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 12
80*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*1i - (((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 51
2*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*a^3*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*c^4
*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576*a^2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*
b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^5 + 768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*e^5
)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) + (2*(d + e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^
5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4
- 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5
- 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*
a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960
*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*
b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 38
40*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*
c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7
- 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^3 - 256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*b^6
*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a^2*b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*
c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 +
3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504
*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^
2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*
b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3
*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*
c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 +
11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c
^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2) + (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3*c^
3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3 + 74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 90*b
^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16*a*b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3*e^
5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d^2*e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32
*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b
*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c
^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*
(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*
(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 896
0*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^
2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4
*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*
a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*1i)/((((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 512*a
^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*a^3*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*c^4*d^
3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576*a^2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*b^4
*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^5 + 768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*e^5)/(
b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) - (2*(d + e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 -
b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 8
0*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7
680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b
^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^
2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5
*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*
a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5
*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 -
6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^3 - 256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*b^6*c^
4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a^2*b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2
))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 384
0*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^
3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d
^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c
*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^
2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4
*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11
520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5
- 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2) - (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3*c^3*e
^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3 + 74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 90*b^2*
c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16*a*b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3*e^5 -
20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d^2*e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^
6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^
5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*
e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*(-(
4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*(-(
4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a
^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 -
240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*
c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3
*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2) - (2*(5*a^2*b^4*e^11 + 216*a^4*c^2*e^11 + 5*b^6*d^2*e^
9 + 32*c^6*d^8*e^3 - 66*a^3*b^2*c*e^11 + 232*a*c^5*d^6*e^5 - 128*b*c^5*d^7*e^4 + 16*b^5*c*d^3*e^8 + 584*a^2*c^
4*d^4*e^7 + 600*a^3*c^3*d^2*e^9 + 166*b^2*c^4*d^6*e^5 - 50*b^3*c^3*d^5*e^6 - 41*b^4*c^2*d^4*e^7 - 10*a*b^5*d*e
^10 + 426*a^2*b^2*c^2*d^2*e^9 - 696*a*b*c^4*d^5*e^6 - 108*a*b^4*c*d^2*e^9 + 158*a^2*b^3*c*d*e^10 - 600*a^3*b*c
^2*d*e^10 + 578*a*b^2*c^3*d^4*e^7 + 4*a*b^3*c^2*d^3*e^8 - 1168*a^2*b*c^3*d^3*e^8))/(b^6*c - 64*a^3*c^4 - 12*a*
b^4*c^2 + 48*a^2*b^2*c^3) + (((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 512*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*
a^2*b^5*c^3*e^6 - 192*a^3*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2
*b^2*c^6*d^3*e^3 - 576*a^2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4
+ 96*a^2*b^4*c^4*d*e^5 + 768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 +
48*a^2*b^2*c^3) + (2*(d + e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^
9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d
^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3
*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)
^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4
*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 51
20*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*
d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3
- 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3
*e^3 - 48*a*b^5*c^4*e^3 - 256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*b^6*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a
*b^4*c^5*d*e^2 - 384*a^2*b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d
^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4
- 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5
- 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27
*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 96
0*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a
*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3
840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2
*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^
7 - 6144*a^5*b^2*c^8)))^(1/2) + (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3*c^3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*
e^4 - 96*b*c^5*d^5*e^3 + 74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 90*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 1
0*b^4*c^2*d^2*e^6 - 16*a*b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3*e^5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d
*e^7 + 140*a*b^2*c^3*d^2*e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*
e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*
c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^
4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e
^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*
c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d
^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^
3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4
)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a
^5*b^2*c^8)))^(1/2)))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*
a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^
7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^
2*d^2*e^3 + 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b
^10*c*d*e^4 - 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560
*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^
4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3
*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4
+ 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*2i

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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