∫ 1______ x3∕2(a+bx+cx2)3 dx

3.20.100 \(\int \frac {1}{x^{3/2} (a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=458 \[ -\frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{4 a^3 \sqrt {x} \left (b^2-4 a c\right )^2}+\frac {36 a^2 c^2+b c x \left (5 b^2-32 a c\right )-35 a b^2 c+5 b^4}{4 a^2 \sqrt {x} \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \sqrt {c} \left (\sqrt {b^2-4 a c} \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )+124 a^2 b c^2-47 a b^3 c+5 b^5\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {c} \left (-60 a^2 c^2 \sqrt {b^2-4 a c}+124 a^2 b c^2-47 a b^3 c+37 a b^2 c \sqrt {b^2-4 a c}-5 b^4 \sqrt {b^2-4 a c}+5 b^5\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {-2 a c+b^2+b c x}{2 a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

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Rubi [A] time = 9.72, antiderivative size = 458, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {740, 822, 828, 826, 1166, 205} \begin {gather*} \frac {36 a^2 c^2+b c x \left (5 b^2-32 a c\right )-35 a b^2 c+5 b^4}{4 a^2 \sqrt {x} \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \sqrt {c} \left (\sqrt {b^2-4 a c} \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )+124 a^2 b c^2-47 a b^3 c+5 b^5\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {c} \left (-60 a^2 c^2 \sqrt {b^2-4 a c}+124 a^2 b c^2-5 b^4 \sqrt {b^2-4 a c}-47 a b^3 c+37 a b^2 c \sqrt {b^2-4 a c}+5 b^5\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{4 a^3 \sqrt {x} \left (b^2-4 a c\right )^2}+\frac {-2 a c+b^2+b c x}{2 a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x^(3/2)*(a + b*x + c*x^2)^3),x]

[Out]

(-3*(5*b^2 - 12*a*c)*(b^2 - 5*a*c))/(4*a^3*(b^2 - 4*a*c)^2*Sqrt[x]) + (b^2 - 2*a*c + b*c*x)/(2*a*(b^2 - 4*a*c)

*Sqrt[x]*(a + b*x + c*x^2)^2) + (5*b^4 - 35*a*b^2*c + 36*a^2*c^2 + b*c*(5*b^2 - 32*a*c)*x)/(4*a^2*(b^2 - 4*a*c

)^2*Sqrt[x]*(a + b*x + c*x^2)) - (3*Sqrt[c]*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 + Sqrt[b^2 - 4*a*c]*(5*b^4 - 3

7*a*b^2*c + 60*a^2*c^2))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*a^3*(b^2 -

4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[c]*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 - 5*b^4*Sqrt[b^2 -

4*a*c] + 37*a*b^2*c*Sqrt[b^2 - 4*a*c] - 60*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b

+ Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

Rule 205

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

&& PosQ[a/b]

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(

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

+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m

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

, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp

[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x

)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*

c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2

*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -

b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,

c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||

IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,

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

; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((

e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d

+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,

d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

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 {1}{x^{3/2} \left (a+b x+c x^2\right )^3} \, dx &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (-5 b^2+18 a c\right )-\frac {7 b c x}{2}}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )^2}+\frac {5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt {x} \left (a+b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )+\frac {3}{4} b c \left (5 b^2-32 a c\right ) x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx}{2 a^2 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt {x}}+\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )^2}+\frac {5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt {x} \left (a+b x+c x^2\right )}+\frac {\int \frac {-\frac {3}{4} b \left (5 b^4-42 a b^2 c+92 a^2 c^2\right )-\frac {3}{4} c \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right ) x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx}{2 a^3 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt {x}}+\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )^2}+\frac {5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt {x} \left (a+b x+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{4} b \left (5 b^4-42 a b^2 c+92 a^2 c^2\right )-\frac {3}{4} c \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right ) x^2}{a+b x^2+c x^4} \, dx,x,\sqrt {x}\right )}{a^3 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt {x}}+\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )^2}+\frac {5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt {x} \left (a+b x+c x^2\right )}+\frac {\left (3 c \left (5 b^5-47 a b^3 c+124 a^2 b c^2-\sqrt {b^2-4 a c} \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,\sqrt {x}\right )}{8 a^3 \left (b^2-4 a c\right )^{5/2}}-\frac {\left (3 c \left (5 b^5-47 a b^3 c+124 a^2 b c^2+\sqrt {b^2-4 a c} \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,\sqrt {x}\right )}{8 a^3 \left (b^2-4 a c\right )^{5/2}}\\ &=-\frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt {x}}+\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )^2}+\frac {5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt {x} \left (a+b x+c x^2\right )}-\frac {3 \sqrt {c} \left (5 b^5-47 a b^3 c+124 a^2 b c^2+\sqrt {b^2-4 a c} \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {c} \left (5 b^5-47 a b^3 c+124 a^2 b c^2-\sqrt {b^2-4 a c} \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A] time = 1.21, size = 440, normalized size = 0.96 \begin {gather*} \frac {\frac {-36 a^2 c^2+35 a b^2 c+32 a b c^2 x-5 b^4-5 b^3 c x}{2 a \sqrt {x} \left (4 a c-b^2\right ) (a+x (b+c x))}-\frac {\frac {3 \sqrt {c} \left (\frac {b \left (124 a^2 c^2-47 a b^2 c+5 b^4\right )}{\sqrt {b^2-4 a c}}+\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {c} \left (60 a^2 c^2 \sqrt {b^2-4 a c}-124 a^2 b c^2+47 a b^3 c-37 a b^2 c \sqrt {b^2-4 a c}+5 b^4 \sqrt {b^2-4 a c}-5 b^5\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{\sqrt {x}}}{2 a^2 \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{\sqrt {x} (a+x (b+c x))^2}}{2 a \left (b^2-4 a c\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(x^(3/2)*(a + b*x + c*x^2)^3),x]

[Out]

((b^2 - 2*a*c + b*c*x)/(Sqrt[x]*(a + x*(b + c*x))^2) + (-5*b^4 + 35*a*b^2*c - 36*a^2*c^2 - 5*b^3*c*x + 32*a*b*

c^2*x)/(2*a*(-b^2 + 4*a*c)*Sqrt[x]*(a + x*(b + c*x))) - ((3*(5*b^2 - 12*a*c)*(b^2 - 5*a*c))/Sqrt[x] + (3*Sqrt[

c]*((5*b^2 - 12*a*c)*(b^2 - 5*a*c) + (b*(5*b^4 - 47*a*b^2*c + 124*a^2*c^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]

*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[c]*(-5*b^5 + 4

7*a*b^3*c - 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a

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

^2 - 4*a*c]]))/(2*a^2*(b^2 - 4*a*c)))/(2*a*(b^2 - 4*a*c))

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IntegrateAlgebraic [A] time = 8.23, size = 558, normalized size = 1.22 \begin {gather*} -\frac {3 \left (60 a^2 c^{5/2} \sqrt {b^2-4 a c}+124 a^2 b c^{5/2}-47 a b^3 c^{3/2}-37 a b^2 c^{3/2} \sqrt {b^2-4 a c}+5 b^4 \sqrt {c} \sqrt {b^2-4 a c}+5 b^5 \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \left (60 a^2 c^{5/2} \sqrt {b^2-4 a c}-124 a^2 b c^{5/2}+47 a b^3 c^{3/2}-37 a b^2 c^{3/2} \sqrt {b^2-4 a c}+5 b^4 \sqrt {c} \sqrt {b^2-4 a c}-5 b^5 \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {-128 a^4 c^2+64 a^3 b^2 c-364 a^3 b c^2 x-324 a^3 c^3 x^2-8 a^2 b^4+194 a^2 b^3 c x-25 a^2 b^2 c^2 x^2-392 a^2 b c^3 x^3-180 a^2 c^4 x^4-25 a b^5 x+91 a b^4 c x^2+227 a b^3 c^2 x^3+111 a b^2 c^3 x^4-15 b^6 x^2-30 b^5 c x^3-15 b^4 c^2 x^4}{4 a^3 \sqrt {x} \left (4 a c-b^2\right )^2 \left (a+b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

IntegrateAlgebraic[1/(x^(3/2)*(a + b*x + c*x^2)^3),x]

[Out]

(-8*a^2*b^4 + 64*a^3*b^2*c - 128*a^4*c^2 - 25*a*b^5*x + 194*a^2*b^3*c*x - 364*a^3*b*c^2*x - 15*b^6*x^2 + 91*a*

b^4*c*x^2 - 25*a^2*b^2*c^2*x^2 - 324*a^3*c^3*x^2 - 30*b^5*c*x^3 + 227*a*b^3*c^2*x^3 - 392*a^2*b*c^3*x^3 - 15*b

^4*c^2*x^4 + 111*a*b^2*c^3*x^4 - 180*a^2*c^4*x^4)/(4*a^3*(-b^2 + 4*a*c)^2*Sqrt[x]*(a + b*x + c*x^2)^2) - (3*(5

*b^5*Sqrt[c] - 47*a*b^3*c^(3/2) + 124*a^2*b*c^(5/2) + 5*b^4*Sqrt[c]*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c^(3/2)*Sqrt[

b^2 - 4*a*c] + 60*a^2*c^(5/2)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]]

)/(4*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*(-5*b^5*Sqrt[c] + 47*a*b^3*c^(3/2) - 12

4*a^2*b*c^(5/2) + 5*b^4*Sqrt[c]*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c^(3/2)*Sqrt[b^2 - 4*a*c] + 60*a^2*c^(5/2)*Sqrt[b

^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)

*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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fricas [B] time = 2.12, size = 4933, normalized size = 10.77

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

3)*x^4 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^3 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^2 + (a^5*b^4 -

8*a^6*b^2*c + 16*a^7*c^2)*x)*sqrt(-(25*b^11 - 495*a*b^9*c + 3894*a^2*b^7*c^2 - 15015*a^3*b^5*c^3 + 27720*a^4*b

^3*c^4 - 18480*a^5*b*c^5 + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 -

1024*a^12*c^5)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4

- 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*

a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*

c^4 - 1024*a^12*c^5))*log(27/2*sqrt(1/2)*(125*b^17 - 3775*a*b^15*c + 49360*a^2*b^13*c^2 - 362733*a^3*b^11*c^3

+ 1623534*a^4*b^9*c^4 - 4463140*a^5*b^7*c^5 + 7146736*a^6*b^5*c^6 - 5684672*a^7*b^3*c^7 + 1324800*a^8*b*c^8 -

(5*a^7*b^16 - 152*a^8*b^14*c + 2006*a^9*b^12*c^2 - 14960*a^10*b^10*c^3 + 68640*a^11*b^8*c^4 - 197120*a^12*b^6*

c^5 + 342528*a^13*b^4*c^6 - 323584*a^14*b^2*c^7 + 122880*a^15*c^8)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2

*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*

b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))*sqrt(-(25*b^11 - 495*a*b^9*

c + 3894*a^2*b^7*c^2 - 15015*a^3*b^5*c^3 + 27720*a^4*b^3*c^4 - 18480*a^5*b*c^5 + (a^7*b^10 - 20*a^8*b^8*c + 16

0*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*

a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^

15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c

+ 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)) - 27*(4125*b^10*c^4 - 77825*a*b^8*

c^5 + 571030*a^2*b^6*c^6 - 1957349*a^3*b^4*c^7 + 2835000*a^4*b^2*c^8 - 810000*a^5*c^9)*sqrt(x)) - 3*sqrt(1/2)*

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

- 6*a^4*b^4*c + 32*a^6*c^3)*x^3 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a

^7*c^2)*x)*sqrt(-(25*b^11 - 495*a*b^9*c + 3894*a^2*b^7*c^2 - 15015*a^3*b^5*c^3 + 27720*a^4*b^3*c^4 - 18480*a^5

*b*c^5 + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sq

rt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c

^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 102

4*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^

5))*log(-27/2*sqrt(1/2)*(125*b^17 - 3775*a*b^15*c + 49360*a^2*b^13*c^2 - 362733*a^3*b^11*c^3 + 1623534*a^4*b^9

*c^4 - 4463140*a^5*b^7*c^5 + 7146736*a^6*b^5*c^6 - 5684672*a^7*b^3*c^7 + 1324800*a^8*b*c^8 - (5*a^7*b^16 - 152

*a^8*b^14*c + 2006*a^9*b^12*c^2 - 14960*a^10*b^10*c^3 + 68640*a^11*b^8*c^4 - 197120*a^12*b^6*c^5 + 342528*a^13

*b^4*c^6 - 323584*a^14*b^2*c^7 + 122880*a^15*c^8)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310

*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*

b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))*sqrt(-(25*b^11 - 495*a*b^9*c + 3894*a^2*b^7*

c^2 - 15015*a^3*b^5*c^3 + 27720*a^4*b^3*c^4 - 18480*a^5*b*c^5 + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 6

40*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351

310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^

16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^

2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)) - 27*(4125*b^10*c^4 - 77825*a*b^8*c^5 + 571030*a^2*

b^6*c^6 - 1957349*a^3*b^4*c^7 + 2835000*a^4*b^2*c^8 - 810000*a^5*c^9)*sqrt(x)) + 3*sqrt(1/2)*((a^3*b^4*c^2 - 8

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

2*a^6*c^3)*x^3 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x)*sqrt(-

(25*b^11 - 495*a*b^9*c + 3894*a^2*b^7*c^2 - 15015*a^3*b^5*c^3 + 27720*a^4*b^3*c^4 - 18480*a^5*b*c^5 - (a^7*b^1

0 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((625*b^12 - 12

250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^

6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^

7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5))*log(27/2*sqrt

(1/2)*(125*b^17 - 3775*a*b^15*c + 49360*a^2*b^13*c^2 - 362733*a^3*b^11*c^3 + 1623534*a^4*b^9*c^4 - 4463140*a^5

*b^7*c^5 + 7146736*a^6*b^5*c^6 - 5684672*a^7*b^3*c^7 + 1324800*a^8*b*c^8 + (5*a^7*b^16 - 152*a^8*b^14*c + 2006

*a^9*b^12*c^2 - 14960*a^10*b^10*c^3 + 68640*a^11*b^8*c^4 - 197120*a^12*b^6*c^5 + 342528*a^13*b^4*c^6 - 323584*

a^14*b^2*c^7 + 122880*a^15*c^8)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591

886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17

*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))*sqrt(-(25*b^11 - 495*a*b^9*c + 3894*a^2*b^7*c^2 - 15015*a^3*b^

5*c^3 + 27720*a^4*b^3*c^4 - 18480*a^5*b*c^5 - (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 +

1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 +

591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a

^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c

^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)) - 27*(4125*b^10*c^4 - 77825*a*b^8*c^5 + 571030*a^2*b^6*c^6 - 1957349*

a^3*b^4*c^7 + 2835000*a^4*b^2*c^8 - 810000*a^5*c^9)*sqrt(x)) - 3*sqrt(1/2)*((a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*

a^5*c^4)*x^5 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^4 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^3 + 2

*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x)*sqrt(-(25*b^11 - 495*a*b

^9*c + 3894*a^2*b^7*c^2 - 15015*a^3*b^5*c^3 + 27720*a^4*b^3*c^4 - 18480*a^5*b*c^5 - (a^7*b^10 - 20*a^8*b^8*c +

160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((625*b^12 - 12250*a*b^10*c + 947

25*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20

*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^

8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5))*log(-27/2*sqrt(1/2)*(125*b^17 -

3775*a*b^15*c + 49360*a^2*b^13*c^2 - 362733*a^3*b^11*c^3 + 1623534*a^4*b^9*c^4 - 4463140*a^5*b^7*c^5 + 714673

6*a^6*b^5*c^6 - 5684672*a^7*b^3*c^7 + 1324800*a^8*b*c^8 + (5*a^7*b^16 - 152*a^8*b^14*c + 2006*a^9*b^12*c^2 - 1

4960*a^10*b^10*c^3 + 68640*a^11*b^8*c^4 - 197120*a^12*b^6*c^5 + 342528*a^13*b^4*c^6 - 323584*a^14*b^2*c^7 + 12

2880*a^15*c^8)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 -

312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a

^18*b^2*c^4 - 1024*a^19*c^5)))*sqrt(-(25*b^11 - 495*a*b^9*c + 3894*a^2*b^7*c^2 - 15015*a^3*b^5*c^3 + 27720*a^4

*b^3*c^4 - 18480*a^5*b*c^5 - (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4

- 1024*a^12*c^5)*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^

4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 128

0*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^

2*c^4 - 1024*a^12*c^5)) - 27*(4125*b^10*c^4 - 77825*a*b^8*c^5 + 571030*a^2*b^6*c^6 - 1957349*a^3*b^4*c^7 + 283

5000*a^4*b^2*c^8 - 810000*a^5*c^9)*sqrt(x)) + 2*(8*a^2*b^4 - 64*a^3*b^2*c + 128*a^4*c^2 + 3*(5*b^4*c^2 - 37*a*

b^2*c^3 + 60*a^2*c^4)*x^4 + (30*b^5*c - 227*a*b^3*c^2 + 392*a^2*b*c^3)*x^3 + (15*b^6 - 91*a*b^4*c + 25*a^2*b^2

*c^2 + 324*a^3*c^3)*x^2 + (25*a*b^5 - 194*a^2*b^3*c + 364*a^3*b*c^2)*x)*sqrt(x))/((a^3*b^4*c^2 - 8*a^4*b^2*c^3

+ 16*a^5*c^4)*x^5 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^4 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x

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

________________________________________________________________________________________

giac [B] time = 1.17, size = 5281, normalized size = 11.53

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/32*(10*a^6*b^14*c^2 - 254*a^7*b^12*c^3 + 2712*a^8*b^10*c^4 - 15552*a^9*b^8*c^5 + 50432*a^10*b^6*c^6 - 87552

*a^11*b^4*c^7 + 63488*a^12*b^2*c^8 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^14 + 12

7*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^12*c + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c

+ sqrt(b^2 - 4*a*c)*c)*a^6*b^13*c - 1356*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^10*c

^2 - 214*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^11*c^2 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*

sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^12*c^2 + 7776*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*

a^9*b^8*c^3 + 1856*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^9*c^3 + 107*sqrt(2)*sqrt(b^

2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^10*c^3 - 25216*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2

- 4*a*c)*c)*a^10*b^6*c^4 - 8128*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^7*c^4 - 928*sq

rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^8*c^4 + 43776*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c

+ sqrt(b^2 - 4*a*c)*c)*a^11*b^4*c^5 + 17920*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^10*b^

5*c^5 + 4064*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^6*c^5 - 31744*sqrt(2)*sqrt(b^2 -

4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b^2*c^6 - 15872*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*

a*c)*c)*a^11*b^3*c^6 - 8960*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^10*b^4*c^6 + 7936*sqrt

(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^11*b^2*c^7 - 10*(b^2 - 4*a*c)*a^6*b^12*c^2 + 214*(b^2

- 4*a*c)*a^7*b^10*c^3 - 1856*(b^2 - 4*a*c)*a^8*b^8*c^4 + 8128*(b^2 - 4*a*c)*a^9*b^6*c^5 - 17920*(b^2 - 4*a*c)*

a^10*b^4*c^6 + 15872*(b^2 - 4*a*c)*a^11*b^2*c^7 + (10*b^6*c^2 - 114*a*b^4*c^3 + 416*a^2*b^2*c^4 - 480*a^3*c^5

- 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6 + 57*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq

rt(b^2 - 4*a*c)*c)*a*b^4*c + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c - 208*sqrt(2)*

sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - 74*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b

^2 - 4*a*c)*c)*a*b^3*c^2 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 240*sqrt(2)*s

qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^3 + 120*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -

4*a*c)*c)*a^2*b*c^3 + 37*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 - 60*sqrt(2)*sqr

t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^4 - 10*(b^2 - 4*a*c)*b^4*c^2 + 74*(b^2 - 4*a*c)*a*b^2*c^3

- 120*(b^2 - 4*a*c)*a^2*c^4)*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)^2 + 2*(5*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*

c)*c)*a^3*b^11 - 102*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^9*c - 10*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*

c)*c)*a^3*b^10*c - 10*a^3*b^11*c + 836*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^7*c^2 + 164*sqrt(2)*sqrt(

b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^8*c^2 + 5*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^9*c^2 + 204*a^4*b^9*c

^2 - 3440*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^5*c^3 - 1016*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a

^5*b^6*c^3 - 82*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^7*c^3 - 1672*a^5*b^7*c^3 + 7104*sqrt(2)*sqrt(b*c

+ sqrt(b^2 - 4*a*c)*c)*a^7*b^3*c^4 + 2816*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^4 + 508*sqrt(2)*s

qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c^4 + 6880*a^6*b^5*c^4 - 5888*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a

^8*b*c^5 - 2944*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^5 - 1408*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c

)*c)*a^6*b^3*c^5 - 14208*a^7*b^3*c^5 + 1472*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b*c^6 + 11776*a^8*b*c^

6 + 10*(b^2 - 4*a*c)*a^3*b^9*c - 164*(b^2 - 4*a*c)*a^4*b^7*c^2 + 1016*(b^2 - 4*a*c)*a^5*b^5*c^3 - 2816*(b^2 -

4*a*c)*a^6*b^3*c^4 + 2944*(b^2 - 4*a*c)*a^7*b*c^5)*abs(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2))*arctan(2*sqrt(1/2)

*sqrt(x)/sqrt((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2 + sqrt((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)^2 - 4*(a^4*b

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

c^3)))/((a^7*b^10 - 20*a^8*b^8*c - 2*a^7*b^9*c + 160*a^9*b^6*c^2 + 32*a^8*b^7*c^2 + a^7*b^8*c^2 - 640*a^10*b^4

*c^3 - 192*a^9*b^5*c^3 - 16*a^8*b^6*c^3 + 1280*a^11*b^2*c^4 + 512*a^10*b^3*c^4 + 96*a^9*b^4*c^4 - 1024*a^12*c^

5 - 512*a^11*b*c^5 - 256*a^10*b^2*c^5 + 256*a^11*c^6)*abs(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*abs(c)) + 3/32*(

10*a^6*b^14*c^2 - 254*a^7*b^12*c^3 + 2712*a^8*b^10*c^4 - 15552*a^9*b^8*c^5 + 50432*a^10*b^6*c^6 - 87552*a^11*b

^4*c^7 + 63488*a^12*b^2*c^8 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^14 + 127*sqrt(

2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^12*c + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt

(b^2 - 4*a*c)*c)*a^6*b^13*c - 1356*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^10*c^2 - 21

4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^11*c^2 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*

c - sqrt(b^2 - 4*a*c)*c)*a^6*b^12*c^2 + 7776*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b^8

*c^3 + 1856*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^9*c^3 + 107*sqrt(2)*sqrt(b^2 - 4*a

*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^10*c^3 - 25216*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c

)*c)*a^10*b^6*c^4 - 8128*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b^7*c^4 - 928*sqrt(2)*s

qrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^8*c^4 + 43776*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt

(b^2 - 4*a*c)*c)*a^11*b^4*c^5 + 17920*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^10*b^5*c^5 +

4064*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b^6*c^5 - 31744*sqrt(2)*sqrt(b^2 - 4*a*c)*

sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^12*b^2*c^6 - 15872*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)

*a^11*b^3*c^6 - 8960*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^10*b^4*c^6 + 7936*sqrt(2)*sqr

t(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^11*b^2*c^7 - 10*(b^2 - 4*a*c)*a^6*b^12*c^2 + 214*(b^2 - 4*a*c

)*a^7*b^10*c^3 - 1856*(b^2 - 4*a*c)*a^8*b^8*c^4 + 8128*(b^2 - 4*a*c)*a^9*b^6*c^5 - 17920*(b^2 - 4*a*c)*a^10*b^

4*c^6 + 15872*(b^2 - 4*a*c)*a^11*b^2*c^7 + (10*b^6*c^2 - 114*a*b^4*c^3 + 416*a^2*b^2*c^4 - 480*a^3*c^5 - 5*sqr

t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^6 + 57*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2

- 4*a*c)*c)*a*b^4*c + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^5*c - 208*sqrt(2)*sqrt(b^

2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - 74*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*

a*c)*c)*a*b^3*c^2 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 240*sqrt(2)*sqrt(b^2

- 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*c^3 + 120*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)

*c)*a^2*b*c^3 + 37*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 - 60*sqrt(2)*sqrt(b^2 -

4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*c^4 - 10*(b^2 - 4*a*c)*b^4*c^2 + 74*(b^2 - 4*a*c)*a*b^2*c^3 - 120*

(b^2 - 4*a*c)*a^2*c^4)*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)^2 - 2*(5*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a

^3*b^11 - 102*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^9*c - 10*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a

^3*b^10*c + 10*a^3*b^11*c + 836*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^7*c^2 + 164*sqrt(2)*sqrt(b*c - s

qrt(b^2 - 4*a*c)*c)*a^4*b^8*c^2 + 5*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^9*c^2 - 204*a^4*b^9*c^2 - 34

40*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^5*c^3 - 1016*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^6*

c^3 - 82*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^7*c^3 + 1672*a^5*b^7*c^3 + 7104*sqrt(2)*sqrt(b*c - sqrt

(b^2 - 4*a*c)*c)*a^7*b^3*c^4 + 2816*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^4 + 508*sqrt(2)*sqrt(b*c

- sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c^4 - 6880*a^6*b^5*c^4 - 5888*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b*c^

5 - 2944*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^5 - 1408*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^

6*b^3*c^5 + 14208*a^7*b^3*c^5 + 1472*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b*c^6 - 11776*a^8*b*c^6 - 10*

(b^2 - 4*a*c)*a^3*b^9*c + 164*(b^2 - 4*a*c)*a^4*b^7*c^2 - 1016*(b^2 - 4*a*c)*a^5*b^5*c^3 + 2816*(b^2 - 4*a*c)*

a^6*b^3*c^4 - 2944*(b^2 - 4*a*c)*a^7*b*c^5)*abs(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2))*arctan(2*sqrt(1/2)*sqrt(x

)/sqrt((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2 - sqrt((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)^2 - 4*(a^4*b^4 - 8*

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

((a^7*b^10 - 20*a^8*b^8*c - 2*a^7*b^9*c + 160*a^9*b^6*c^2 + 32*a^8*b^7*c^2 + a^7*b^8*c^2 - 640*a^10*b^4*c^3 -

192*a^9*b^5*c^3 - 16*a^8*b^6*c^3 + 1280*a^11*b^2*c^4 + 512*a^10*b^3*c^4 + 96*a^9*b^4*c^4 - 1024*a^12*c^5 - 512

*a^11*b*c^5 - 256*a^10*b^2*c^5 + 256*a^11*c^6)*abs(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*abs(c)) - 1/4*(7*b^4*c^

2*x^(7/2) - 47*a*b^2*c^3*x^(7/2) + 52*a^2*c^4*x^(7/2) + 14*b^5*c*x^(5/2) - 99*a*b^3*c^2*x^(5/2) + 136*a^2*b*c^

3*x^(5/2) + 7*b^6*x^(3/2) - 43*a*b^4*c*x^(3/2) + 25*a^2*b^2*c^2*x^(3/2) + 68*a^3*c^3*x^(3/2) + 9*a*b^5*sqrt(x)

- 66*a^2*b^3*c*sqrt(x) + 108*a^3*b*c^2*sqrt(x))/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*(c*x^2 + b*x + a)^2) -

2/(a^3*sqrt(x))

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maple [B] time = 0.19, size = 1571, normalized size = 3.43

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

43/4/a^2/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*b^4*c+33/2/a/(c*x^2+b*x+a)^2*b^3/(16*a^2*c^2-8*a*b

^2*c+b^4)*x^(1/2)*c-7/2/a^3/(c*x^2+b*x+a)^2*c*b^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)+47/4/a^2/(c*x^2+b*x+a)^2*

c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)*b^2-34/a/(c*x^2+b*x+a)^2*c^3*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)+99/4/

a^2/(c*x^2+b*x+a)^2*c^2*b^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)-2/a^3/x^(1/2)-45/2/a/(16*a^2*c^2-8*a*b^2*c+b^4)

*c^3*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))+45/2/

a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(

1/2))*c)^(1/2)*c*x^(1/2))-25/4/a/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*b^2*c^2-7/4/a^3/(c*x^2+b*x

+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)*b^4-17/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*c^3+15/

8/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2

)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*b^5-141/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-4*a*c+b^2)^(1/2)

*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*b^3-14

1/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1

/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*b^3+15/8/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(-4*a*c+b^2)^(1/2)*2

^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*b^5+93/2/a/

(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+

(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*b+93/2/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-

b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*b-27/(c*x^2+b*x+a)

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

a/(c*x^2+b*x+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)-9/4/a^2/(c*x^2+b*x+a)^2*b^5/(16*a^2*c^2-8*a*b^2*c+b^4

)*x^(1/2)-15/8/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-

4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*b^4+15/8/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2

))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*b^4-111/8/a^2/(16*a^2*c^2-8*a*b^2*c+b

^4)*c^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))

*b^2+111/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*

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

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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B] time = 6.48, size = 12164, normalized size = 26.56

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (2/a + (x^2*(15*b^6 + 324*a^3*c^3 + 25*a^2*b^2*c^2 - 91*a*b^4*c))/(4*a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (

b*x*(25*b^4 + 364*a^2*c^2 - 194*a*b^2*c))/(4*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (b*x^3*(30*b^4*c + 392*a^2*

c^3 - 227*a*b^2*c^2))/(4*a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*c*x^4*(5*b^4*c + 60*a^2*c^3 - 37*a*b^2*c^2))

/(4*a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^(5/2)*(2*a*c + b^2) + a^2*x^(1/2) + c^2*x^(9/2) + 2*a*b*x^(3/2) +

2*b*c*x^(7/2)) - atan((((-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10 + 17794*a^2*b^

17*c^2 - 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19905600*a^6*b^9*c^6 - 43904256*a

^7*b^7*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 995*a*b^19*

c - 694*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 10485

76*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*

c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*(x^(1/

2)*(-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10 + 17794*a^2*b^17*c^2 - 188095*a^3*b

^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19905600*a^6*b^9*c^6 - 43904256*a^7*b^7*c^7 + 62684160

*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 995*a*b^19*c - 694*a^2*b^2*c^2*(

-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8

*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8

*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*(34359738368*a^26*b*c^13 -

8192*a^15*b^23*c^2 + 360448*a^16*b^21*c^3 - 7208960*a^17*b^19*c^4 + 86507520*a^18*b^17*c^5 - 692060160*a^19*b^

15*c^6 + 3875536896*a^20*b^13*c^7 - 15502147584*a^21*b^11*c^8 + 44291850240*a^22*b^9*c^9 - 88583700480*a^23*b^

7*c^10 + 118111600640*a^24*b^5*c^11 - 94489280512*a^25*b^3*c^12) - 74088185856*a^23*b*c^13 + 15360*a^12*b^23*c

^2 - 681984*a^13*b^21*c^3 + 13774848*a^14*b^19*c^4 - 167067648*a^15*b^17*c^5 + 1351876608*a^16*b^15*c^6 - 7662

993408*a^17*b^13*c^7 + 31048335360*a^18*b^11*c^8 - 89917489152*a^19*b^9*c^9 + 182401892352*a^20*b^7*c^10 - 246

826401792*a^21*b^5*c^11 + 200521285632*a^22*b^3*c^12) + x^(1/2)*(33973862400*a^20*c^14 - 28800*a^9*b^22*c^3 +

1232640*a^10*b^20*c^4 - 23879808*a^11*b^18*c^5 + 275975424*a^12*b^16*c^6 - 2109763584*a^13*b^14*c^7 + 11171856

384*a^14*b^12*c^8 - 41653370880*a^15*b^10*c^9 + 108726976512*a^16*b^8*c^10 - 192980975616*a^17*b^6*c^11 + 2184

14186496*a^18*b^4*c^12 - 137631891456*a^19*b^2*c^13))*(-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923

520*a^10*b*c^10 + 17794*a^2*b^17*c^2 - 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 199

05600*a^6*b^9*c^6 - 43904256*a^7*b^7*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a*c

- b^2)^15)^(1/2) - 995*a*b^19*c - 694*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2)^15)^

(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^1

1*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 262144

0*a^16*b^2*c^9)))^(1/2)*1i + ((-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10 + 17794*

a^2*b^17*c^2 - 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19905600*a^6*b^9*c^6 - 4390

4256*a^7*b^7*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 995*a

*b^19*c - 694*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 +

1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12

*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*

(x^(1/2)*(-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10 + 17794*a^2*b^17*c^2 - 188095

*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19905600*a^6*b^9*c^6 - 43904256*a^7*b^7*c^7 + 62

684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 995*a*b^19*c - 694*a^2*b^2

*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 -

40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^

13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*(34359738368*a^26*b*c

^13 - 8192*a^15*b^23*c^2 + 360448*a^16*b^21*c^3 - 7208960*a^17*b^19*c^4 + 86507520*a^18*b^17*c^5 - 692060160*a

^19*b^15*c^6 + 3875536896*a^20*b^13*c^7 - 15502147584*a^21*b^11*c^8 + 44291850240*a^22*b^9*c^9 - 88583700480*a

^23*b^7*c^10 + 118111600640*a^24*b^5*c^11 - 94489280512*a^25*b^3*c^12) + 74088185856*a^23*b*c^13 - 15360*a^12*

b^23*c^2 + 681984*a^13*b^21*c^3 - 13774848*a^14*b^19*c^4 + 167067648*a^15*b^17*c^5 - 1351876608*a^16*b^15*c^6

+ 7662993408*a^17*b^13*c^7 - 31048335360*a^18*b^11*c^8 + 89917489152*a^19*b^9*c^9 - 182401892352*a^20*b^7*c^10

+ 246826401792*a^21*b^5*c^11 - 200521285632*a^22*b^3*c^12) + x^(1/2)*(33973862400*a^20*c^14 - 28800*a^9*b^22*

c^3 + 1232640*a^10*b^20*c^4 - 23879808*a^11*b^18*c^5 + 275975424*a^12*b^16*c^6 - 2109763584*a^13*b^14*c^7 + 11

171856384*a^14*b^12*c^8 - 41653370880*a^15*b^10*c^9 + 108726976512*a^16*b^8*c^10 - 192980975616*a^17*b^6*c^11

+ 218414186496*a^18*b^4*c^12 - 137631891456*a^19*b^2*c^13))*(-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) +

18923520*a^10*b*c^10 + 17794*a^2*b^17*c^2 - 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5

+ 19905600*a^6*b^9*c^6 - 43904256*a^7*b^7*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(

4*a*c - b^2)^15)^(1/2) - 995*a*b^19*c - 694*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2

)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 537

60*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 -

2621440*a^16*b^2*c^9)))^(1/2)*1i)/(((-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10 +

17794*a^2*b^17*c^2 - 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19905600*a^6*b^9*c^6

- 43904256*a^7*b^7*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) -

995*a*b^19*c - 694*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*

b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 25804

8*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^

(1/2)*(x^(1/2)*(-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10 + 17794*a^2*b^17*c^2 -

188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19905600*a^6*b^9*c^6 - 43904256*a^7*b^7*c^

7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 995*a*b^19*c - 694*a

^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c

^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860

160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*(34359738368*a^

26*b*c^13 - 8192*a^15*b^23*c^2 + 360448*a^16*b^21*c^3 - 7208960*a^17*b^19*c^4 + 86507520*a^18*b^17*c^5 - 69206

0160*a^19*b^15*c^6 + 3875536896*a^20*b^13*c^7 - 15502147584*a^21*b^11*c^8 + 44291850240*a^22*b^9*c^9 - 8858370

0480*a^23*b^7*c^10 + 118111600640*a^24*b^5*c^11 - 94489280512*a^25*b^3*c^12) + 74088185856*a^23*b*c^13 - 15360

*a^12*b^23*c^2 + 681984*a^13*b^21*c^3 - 13774848*a^14*b^19*c^4 + 167067648*a^15*b^17*c^5 - 1351876608*a^16*b^1

5*c^6 + 7662993408*a^17*b^13*c^7 - 31048335360*a^18*b^11*c^8 + 89917489152*a^19*b^9*c^9 - 182401892352*a^20*b^

7*c^10 + 246826401792*a^21*b^5*c^11 - 200521285632*a^22*b^3*c^12) + x^(1/2)*(33973862400*a^20*c^14 - 28800*a^9

*b^22*c^3 + 1232640*a^10*b^20*c^4 - 23879808*a^11*b^18*c^5 + 275975424*a^12*b^16*c^6 - 2109763584*a^13*b^14*c^

7 + 11171856384*a^14*b^12*c^8 - 41653370880*a^15*b^10*c^9 + 108726976512*a^16*b^8*c^10 - 192980975616*a^17*b^6

*c^11 + 218414186496*a^18*b^4*c^12 - 137631891456*a^19*b^2*c^13))*(-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(

1/2) + 18923520*a^10*b*c^10 + 17794*a^2*b^17*c^2 - 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^

11*c^5 + 19905600*a^6*b^9*c^6 - 43904256*a^7*b^7*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c

^3*(-(4*a*c - b^2)^15)^(1/2) - 995*a*b^19*c - 694*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c

- b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3

+ 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*

c^8 - 2621440*a^16*b^2*c^9)))^(1/2) - ((-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10

+ 17794*a^2*b^17*c^2 - 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19905600*a^6*b^9*c

^6 - 43904256*a^7*b^7*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2

) - 995*a*b^19*c - 694*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a

^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 25

8048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)

))^(1/2)*(x^(1/2)*(-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10 + 17794*a^2*b^17*c^2

- 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19905600*a^6*b^9*c^6 - 43904256*a^7*b^7

*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 995*a*b^19*c - 69

4*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^1

7*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 +

860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*(34359738368

*a^26*b*c^13 - 8192*a^15*b^23*c^2 + 360448*a^16*b^21*c^3 - 7208960*a^17*b^19*c^4 + 86507520*a^18*b^17*c^5 - 69

2060160*a^19*b^15*c^6 + 3875536896*a^20*b^13*c^7 - 15502147584*a^21*b^11*c^8 + 44291850240*a^22*b^9*c^9 - 8858

3700480*a^23*b^7*c^10 + 118111600640*a^24*b^5*c^11 - 94489280512*a^25*b^3*c^12) - 74088185856*a^23*b*c^13 + 15

360*a^12*b^23*c^2 - 681984*a^13*b^21*c^3 + 13774848*a^14*b^19*c^4 - 167067648*a^15*b^17*c^5 + 1351876608*a^16*

b^15*c^6 - 7662993408*a^17*b^13*c^7 + 31048335360*a^18*b^11*c^8 - 89917489152*a^19*b^9*c^9 + 182401892352*a^20

*b^7*c^10 - 246826401792*a^21*b^5*c^11 + 200521285632*a^22*b^3*c^12) + x^(1/2)*(33973862400*a^20*c^14 - 28800*

a^9*b^22*c^3 + 1232640*a^10*b^20*c^4 - 23879808*a^11*b^18*c^5 + 275975424*a^12*b^16*c^6 - 2109763584*a^13*b^14

*c^7 + 11171856384*a^14*b^12*c^8 - 41653370880*a^15*b^10*c^9 + 108726976512*a^16*b^8*c^10 - 192980975616*a^17*

b^6*c^11 + 218414186496*a^18*b^4*c^12 - 137631891456*a^19*b^2*c^13))*(-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15

)^(1/2) + 18923520*a^10*b*c^10 + 17794*a^2*b^17*c^2 - 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5

*b^11*c^5 + 19905600*a^6*b^9*c^6 - 43904256*a^7*b^7*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^

3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 995*a*b^19*c - 694*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*

a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*

c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b

^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2) + 47775744000*a^17*c^14 + 712800*a^9*b^16*c^6 - 23142240*a^10*b^14*c^7

+ 328157568*a^11*b^12*c^8 - 2652784128*a^12*b^10*c^9 + 13361338368*a^13*b^8*c^10 - 42897973248*a^14*b^6*c^11 +

85645099008*a^15*b^4*c^12 - 97090928640*a^16*b^2*c^13))*(-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18

923520*a^10*b*c^10 + 17794*a^2*b^17*c^2 - 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 +

19905600*a^6*b^9*c^6 - 43904256*a^7*b^7*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a

*c - b^2)^15)^(1/2) - 995*a*b^19*c - 694*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2)^1

5)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*

a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 262

1440*a^16*b^2*c^9)))^(1/2)*2i - atan((((-(9*(25*b^21 + 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10

+ 17794*a^2*b^17*c^2 - 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19905600*a^6*b^9*c

^6 - 43904256*a^7*b^7*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 - 225*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2

) - 995*a*b^19*c + 694*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 245*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a

^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 25

8048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)

))^(1/2)*(x^(1/2)*(-(9*(25*b^21 + 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10 + 17794*a^2*b^17*c^2

- 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19905600*a^6*b^9*c^6 - 43904256*a^7*b^7

*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 - 225*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 995*a*b^19*c + 69

4*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 245*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^1