3.11.80 \(\int \frac {x^{15/2}}{(a+b x^2+c x^4)^3} \, dx\) [1080]

Optimal. Leaf size=621 \[ -\frac {3 \left (b^2+12 a c\right ) \sqrt {x}}{16 c \left (b^2-4 a c\right )^2}+\frac {x^{9/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 x^{5/2} \left (8 a b+\left (b^2+12 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {3 \left (b^3-28 a b c+\frac {b^4-30 a b^2 c-24 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 \left (b^3-28 a b c-\frac {b^4-30 a b^2 c-24 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 \left (b^3-28 a b c+\frac {b^4-30 a b^2 c-24 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 \left (b^3-28 a b c-\frac {b^4-30 a b^2 c-24 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}} \]

[Out]

1/4*x^(9/2)*(b*x^2+2*a)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+3/16*x^(5/2)*(8*a*b+(12*a*c+b^2)*x^2)/(-4*a*c+b^2)^2/(c
*x^4+b*x^2+a)-3/64*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*(b^3-28*a*b*c+(-24*a^2*c^2-30
*a*b^2*c+b^4)/(-4*a*c+b^2)^(1/2))*2^(3/4)/c^(5/4)/(-4*a*c+b^2)^2/(-b-(-4*a*c+b^2)^(1/2))^(3/4)-3/64*arctanh(2^
(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*(b^3-28*a*b*c+(-24*a^2*c^2-30*a*b^2*c+b^4)/(-4*a*c+b^2)^(
1/2))*2^(3/4)/c^(5/4)/(-4*a*c+b^2)^2/(-b-(-4*a*c+b^2)^(1/2))^(3/4)-3/64*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4
*a*c+b^2)^(1/2))^(1/4))*(b^3-28*a*b*c+(24*a^2*c^2+30*a*b^2*c-b^4)/(-4*a*c+b^2)^(1/2))*2^(3/4)/c^(5/4)/(-4*a*c+
b^2)^2/(-b+(-4*a*c+b^2)^(1/2))^(3/4)-3/64*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(b^3-
28*a*b*c+(24*a^2*c^2+30*a*b^2*c-b^4)/(-4*a*c+b^2)^(1/2))*2^(3/4)/c^(5/4)/(-4*a*c+b^2)^2/(-b+(-4*a*c+b^2)^(1/2)
)^(3/4)-3/16*(12*a*c+b^2)*x^(1/2)/c/(-4*a*c+b^2)^2

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Rubi [A]
time = 1.30, antiderivative size = 621, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1129, 1379, 1512, 1516, 1436, 218, 214, 211} \begin {gather*} -\frac {3 \left (\frac {-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt {b^2-4 a c}}-28 a b c+b^3\right ) \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 \left (-\frac {-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt {b^2-4 a c}}-28 a b c+b^3\right ) \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 \left (\frac {-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt {b^2-4 a c}}-28 a b c+b^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 \left (-\frac {-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt {b^2-4 a c}}-28 a b c+b^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}+\frac {x^{9/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 x^{5/2} \left (x^2 \left (12 a c+b^2\right )+8 a b\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {3 \sqrt {x} \left (12 a c+b^2\right )}{16 c \left (b^2-4 a c\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(15/2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

(-3*(b^2 + 12*a*c)*Sqrt[x])/(16*c*(b^2 - 4*a*c)^2) + (x^(9/2)*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x
^4)^2) + (3*x^(5/2)*(8*a*b + (b^2 + 12*a*c)*x^2))/(16*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (3*(b^3 - 28*a*b*
c + (b^4 - 30*a*b^2*c - 24*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c
])^(1/4)])/(32*2^(1/4)*c^(5/4)*(b^2 - 4*a*c)^2*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) - (3*(b^3 - 28*a*b*c - (b^4 - 3
0*a*b^2*c - 24*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(
32*2^(1/4)*c^(5/4)*(b^2 - 4*a*c)^2*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) - (3*(b^3 - 28*a*b*c + (b^4 - 30*a*b^2*c -
24*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2^(1/4)*
c^(5/4)*(b^2 - 4*a*c)^2*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) - (3*(b^3 - 28*a*b*c - (b^4 - 30*a*b^2*c - 24*a^2*c^2)
/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2^(1/4)*c^(5/4)*(b^
2 - 4*a*c)^2*(-b + Sqrt[b^2 - 4*a*c])^(3/4))

Rule 211

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

Rule 214

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

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 1129

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[
k/d, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/d^2) + c*(x^(4*k)/d^4))^p, x], x, (d*x)^(1/k)], x]] /; FreeQ[
{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]

Rule 1379

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-d^(2*n - 1))*(d*
x)^(m - 2*n + 1)*(2*a + b*x^n)*((a + b*x^n + c*x^(2*n))^(p + 1)/(n*(p + 1)*(b^2 - 4*a*c))), x] + Dist[d^(2*n)/
(n*(p + 1)*(b^2 - 4*a*c)), Int[(d*x)^(m - 2*n)*(2*a*(m - 2*n + 1) + b*(m + n*(2*p + 1) + 1)*x^n)*(a + b*x^n +
c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && I
LtQ[p, -1] && GtQ[m, 2*n - 1]

Rule 1436

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] ||  !IGtQ[n/2, 0])

Rule 1512

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :
> Simp[f^(n - 1)*(f*x)^(m - n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)*((b*d - 2*a*e - (b*e - 2*c*d)*x^n)/(n*(p +
1)*(b^2 - 4*a*c))), x] + Dist[f^n/(n*(p + 1)*(b^2 - 4*a*c)), Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^(p + 1)
*Simp[(n - m - 1)*(b*d - 2*a*e) + (2*n*p + 2*n + m + 1)*(b*e - 2*c*d)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, n - 1] && IntegerQ[p]

Rule 1516

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Simp[e*f^(n - 1)*(f*x)^(m - n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + n*(2*p + 1) + 1))), x] - Dist[f^n
/(c*(m + n*(2*p + 1) + 1)), Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m - n + 1) + (b*e*(m + n*p +
1) - c*d*(m + n*(2*p + 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2
- 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {x^{15/2}}{\left (a+b x^2+c x^4\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {x^{16}}{\left (a+b x^4+c x^8\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{9/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\text {Subst}\left (\int \frac {x^8 \left (18 a-3 b x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx,x,\sqrt {x}\right )}{4 \left (b^2-4 a c\right )}\\ &=\frac {x^{9/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 x^{5/2} \left (8 a b+\left (b^2+12 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\text {Subst}\left (\int \frac {x^4 \left (-120 a b-3 \left (b^2+12 a c\right ) x^4\right )}{a+b x^4+c x^8} \, dx,x,\sqrt {x}\right )}{16 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (b^2+12 a c\right ) \sqrt {x}}{16 c \left (b^2-4 a c\right )^2}+\frac {x^{9/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 x^{5/2} \left (8 a b+\left (b^2+12 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\text {Subst}\left (\int \frac {-3 a \left (b^2+12 a c\right )-3 b \left (b^2-28 a c\right ) x^4}{a+b x^4+c x^8} \, dx,x,\sqrt {x}\right )}{16 c \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (b^2+12 a c\right ) \sqrt {x}}{16 c \left (b^2-4 a c\right )^2}+\frac {x^{9/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 x^{5/2} \left (8 a b+\left (b^2+12 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (3 \left (b^3-28 a b c-\frac {b^4-30 a b^2 c-24 a^2 c^2}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^4} \, dx,x,\sqrt {x}\right )}{32 c \left (b^2-4 a c\right )^2}+\frac {\left (3 \left (b^3-28 a b c+\frac {b^4-30 a b^2 c-24 a^2 c^2}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^4} \, dx,x,\sqrt {x}\right )}{32 c \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (b^2+12 a c\right ) \sqrt {x}}{16 c \left (b^2-4 a c\right )^2}+\frac {x^{9/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 x^{5/2} \left (8 a b+\left (b^2+12 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (3 \left (b^3-28 a b c-\frac {b^4-30 a b^2 c-24 a^2 c^2}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b+\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {x}\right )}{32 c \left (b^2-4 a c\right )^2 \sqrt {-b+\sqrt {b^2-4 a c}}}-\frac {\left (3 \left (b^3-28 a b c-\frac {b^4-30 a b^2 c-24 a^2 c^2}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b+\sqrt {b^2-4 a c}}+\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {x}\right )}{32 c \left (b^2-4 a c\right )^2 \sqrt {-b+\sqrt {b^2-4 a c}}}-\frac {\left (3 \left (b^3-28 a b c+\frac {b^4-30 a b^2 c-24 a^2 c^2}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {x}\right )}{32 c \left (b^2-4 a c\right )^2 \sqrt {-b-\sqrt {b^2-4 a c}}}-\frac {\left (3 \left (b^3-28 a b c+\frac {b^4-30 a b^2 c-24 a^2 c^2}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}+\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {x}\right )}{32 c \left (b^2-4 a c\right )^2 \sqrt {-b-\sqrt {b^2-4 a c}}}\\ &=-\frac {3 \left (b^2+12 a c\right ) \sqrt {x}}{16 c \left (b^2-4 a c\right )^2}+\frac {x^{9/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 x^{5/2} \left (8 a b+\left (b^2+12 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {3 \left (b^3-28 a b c+\frac {b^4-30 a b^2 c-24 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 \left (b^3-28 a b c-\frac {b^4-30 a b^2 c-24 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 \left (b^3-28 a b c+\frac {b^4-30 a b^2 c-24 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 \left (b^3-28 a b c-\frac {b^4-30 a b^2 c-24 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.84, size = 506, normalized size = 0.81 \begin {gather*} \frac {-\frac {4 c^2 \sqrt {x} \left (36 a^3 c+b^3 x^4 \left (3 b-c x^2\right )+a b x^2 \left (6 b^2+7 b c x^2+28 c^2 x^4\right )+a^2 \left (3 b^2+48 b c x^2+68 c^2 x^4\right )\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^2}+32 c \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {\log \left (\sqrt {x}-\text {$\#$1}\right )}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]+\frac {8 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {3 b^4 \log \left (\sqrt {x}-\text {$\#$1}\right )-22 a b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right )+28 a^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+3 b^3 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4+6 a b c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{a \left (b^2-4 a c\right )}-\frac {3 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {8 b^6 \log \left (\sqrt {x}-\text {$\#$1}\right )-80 a b^4 c \log \left (\sqrt {x}-\text {$\#$1}\right )+223 a^2 b^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )-140 a^3 c^3 \log \left (\sqrt {x}-\text {$\#$1}\right )+8 b^5 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-17 a b^3 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-36 a^2 b c^3 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{a \left (b^2-4 a c\right )^2}}{64 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(15/2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

((-4*c^2*Sqrt[x]*(36*a^3*c + b^3*x^4*(3*b - c*x^2) + a*b*x^2*(6*b^2 + 7*b*c*x^2 + 28*c^2*x^4) + a^2*(3*b^2 + 4
8*b*c*x^2 + 68*c^2*x^4)))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2) + 32*c*RootSum[a + b*#1^4 + c*#1^8 & , Log[S
qrt[x] - #1]/(b*#1^3 + 2*c*#1^7) & ] + (8*RootSum[a + b*#1^4 + c*#1^8 & , (3*b^4*Log[Sqrt[x] - #1] - 22*a*b^2*
c*Log[Sqrt[x] - #1] + 28*a^2*c^2*Log[Sqrt[x] - #1] + 3*b^3*c*Log[Sqrt[x] - #1]*#1^4 + 6*a*b*c^2*Log[Sqrt[x] -
#1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(a*(b^2 - 4*a*c)) - (3*RootSum[a + b*#1^4 + c*#1^8 & , (8*b^6*Log[Sqrt[x] -
#1] - 80*a*b^4*c*Log[Sqrt[x] - #1] + 223*a^2*b^2*c^2*Log[Sqrt[x] - #1] - 140*a^3*c^3*Log[Sqrt[x] - #1] + 8*b^
5*c*Log[Sqrt[x] - #1]*#1^4 - 17*a*b^3*c^2*Log[Sqrt[x] - #1]*#1^4 - 36*a^2*b*c^3*Log[Sqrt[x] - #1]*#1^4)/(b*#1^
3 + 2*c*#1^7) & ])/(a*(b^2 - 4*a*c)^2))/(64*c^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.10, size = 275, normalized size = 0.44

method	result	size
derivativedivides	\(\frac {-\frac {3 a^{2} \left (12 a c +b^{2}\right ) \sqrt {x}}{16 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 a b \left (8 a c +b^{2}\right ) x^{\frac {5}{2}}}{8 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (68 a^{2} c^{2}+7 a \,b^{2} c +3 b^{4}\right ) x^{\frac {9}{2}}}{16 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {b \left (28 a c -b^{2}\right ) x^{\frac {13}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (b \left (-28 a c +b^{2}\right ) \textit {\_R}^{4}+12 a^{2} c +a \,b^{2}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}\right )}{64 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\)	\(275\)
default	\(\frac {-\frac {3 a^{2} \left (12 a c +b^{2}\right ) \sqrt {x}}{16 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 a b \left (8 a c +b^{2}\right ) x^{\frac {5}{2}}}{8 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (68 a^{2} c^{2}+7 a \,b^{2} c +3 b^{4}\right ) x^{\frac {9}{2}}}{16 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {b \left (28 a c -b^{2}\right ) x^{\frac {13}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (b \left (-28 a c +b^{2}\right ) \textit {\_R}^{4}+12 a^{2} c +a \,b^{2}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}\right )}{64 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\)	\(275\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(15/2)/(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

2*(-3/32*a^2*(12*a*c+b^2)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)-3/16*a/c*b*(8*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^
4)*x^(5/2)-1/32*(68*a^2*c^2+7*a*b^2*c+3*b^4)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(9/2)-1/32*b*(28*a*c-b^2)/(16*a^2*
c^2-8*a*b^2*c+b^4)*x^(13/2))/(c*x^4+b*x^2+a)^2+3/64/c/(16*a^2*c^2-8*a*b^2*c+b^4)*sum((b*(-28*a*c+b^2)*_R^4+12*
a^2*c+a*b^2)/(2*_R^7*c+_R^3*b)*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+_Z^4*b+a))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(15/2)/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/16*(3*(b^2*c + 12*a*c^2)*x^(17/2) + (7*b^3 + 44*a*b*c)*x^(13/2) + 24*a^2*b*x^(5/2) + (35*a*b^2 + 4*a^2*c)*x^
(9/2))/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^
3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2) - inte
grate(3/32*((b^2 + 12*a*c)*x^(7/2) + 40*a*b*x^(3/2))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2
+ 16*a^2*c^3)*x^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x^2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 18827 vs. \(2 (521) = 1042\).
time = 115.55, size = 18827, normalized size = 30.32 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(15/2)/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")

[Out]

-1/64*(12*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^8 + a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + 2*(b^5*c^2 - 8*
a*b^3*c^3 + 16*a^2*b*c^4)*x^6 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^4 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b
*c^3)*x^2)*sqrt(sqrt(1/2)*sqrt(-(b^13 - 117*a*b^11*c + 5109*a^2*b^9*c^2 - 97968*a^3*b^7*c^3 + 670176*a^4*b^5*c
^4 + 2895360*a^5*b^3*c^5 - 449280*a^6*b*c^6 + (b^20*c^5 - 40*a*b^18*c^6 + 720*a^2*b^16*c^7 - 7680*a^3*b^14*c^8
+ 53760*a^4*b^12*c^9 - 258048*a^5*b^10*c^10 + 860160*a^6*b^8*c^11 - 1966080*a^7*b^6*c^12 + 2949120*a^8*b^4*c^
13 - 2621440*a^9*b^2*c^14 + 1048576*a^10*c^15)*sqrt((b^16 - 214*a*b^14*c + 19467*a^2*b^12*c^2 - 967222*a^3*b^1
0*c^3 + 27769345*a^4*b^8*c^4 - 438445008*a^5*b^6*c^5 + 2996795232*a^6*b^4*c^6 - 141647616*a^7*b^2*c^7 + 167961
6*a^8*c^8)/(b^30*c^10 - 60*a*b^28*c^11 + 1680*a^2*b^26*c^12 - 29120*a^3*b^24*c^13 + 349440*a^4*b^22*c^14 - 307
5072*a^5*b^20*c^15 + 20500480*a^6*b^18*c^16 - 105431040*a^7*b^16*c^17 + 421724160*a^8*b^14*c^18 - 1312030720*a
^9*b^12*c^19 + 3148873728*a^10*b^10*c^20 - 5725224960*a^11*b^8*c^21 + 7633633280*a^12*b^6*c^22 - 7046430720*a^
13*b^4*c^23 + 4026531840*a^14*b^2*c^24 - 1073741824*a^15*c^25)))/(b^20*c^5 - 40*a*b^18*c^6 + 720*a^2*b^16*c^7
- 7680*a^3*b^14*c^8 + 53760*a^4*b^12*c^9 - 258048*a^5*b^10*c^10 + 860160*a^6*b^8*c^11 - 1966080*a^7*b^6*c^12 +
2949120*a^8*b^4*c^13 - 2621440*a^9*b^2*c^14 + 1048576*a^10*c^15)))*arctan(1/4*(2*sqrt(1/2)*(39*a*b^45 - 14092
*a^2*b^43*c + 2342105*a^3*b^41*c^2 - 236978592*a^4*b^39*c^3 + 16281606995*a^5*b^37*c^4 - 802176194594*a^6*b^35
*c^5 + 29187154410144*a^7*b^33*c^6 - 796345615508224*a^8*b^31*c^7 + 16398991931243520*a^9*b^29*c^8 - 255116875
223562240*a^10*b^27*c^9 + 2991680147781025792*a^11*b^25*c^10 - 26351652213068988416*a^12*b^23*c^11 + 173550335
869159342080*a^13*b^21*c^12 - 848488257540540071936*a^14*b^19*c^13 + 3040685322832896327680*a^15*b^17*c^14 - 7
810647267905472823296*a^16*b^15*c^15 + 13826407321898900783104*a^17*b^13*c^16 - 15695440087743077548032*a^18*b
^11*c^17 + 9831375748320473382912*a^19*b^9*c^18 - 2132270903021870776320*a^20*b^7*c^19 - 302763289379535323136
*a^21*b^5*c^20 + 17492739194823376896*a^22*b^3*c^21 - 219122084616339456*a^23*b*c^22 - (39*a*b^52*c^5 - 11089*
a^2*b^50*c^6 + 1437201*a^3*b^48*c^7 - 112481582*a^4*b^46*c^8 + 5937253574*a^5*b^44*c^9 - 223744431280*a^6*b^42
*c^10 + 6224393237408*a^7*b^40*c^11 - 130617052733440*a^8*b^38*c^12 + 2099519233610240*a^9*b^36*c^13 - 2615144
0607784960*a^10*b^34*c^14 + 254528845453582336*a^11*b^32*c^15 - 1944123359024644096*a^12*b^30*c^16 + 116394868
70997172224*a^13*b^28*c^17 - 54083721102022934528*a^14*b^26*c^18 + 189937300181956427776*a^15*b^24*c^19 - 4704
43868125682204672*a^16*b^22*c^20 + 634824840388053827584*a^17*b^20*c^21 + 533578961559771676672*a^18*b^18*c^22
- 5318575782259016597504*a^19*b^16*c^23 + 14710673859192740118528*a^20*b^14*c^24 - 24240052148772213358592*a^
21*b^12*c^25 + 25158757634824950775808*a^22*b^10*c^26 - 15081855906511551725568*a^23*b^8*c^27 + 36582540717545
44644096*a^24*b^6*c^28 + 386591606652134227968*a^25*b^4*c^29 - 40205445868698992640*a^26*b^2*c^30 + 6925339958
24480256*a^27*c^31)*sqrt((b^16 - 214*a*b^14*c + 19467*a^2*b^12*c^2 - 967222*a^3*b^10*c^3 + 27769345*a^4*b^8*c^
4 - 438445008*a^5*b^6*c^5 + 2996795232*a^6*b^4*c^6 - 141647616*a^7*b^2*c^7 + 1679616*a^8*c^8)/(b^30*c^10 - 60*
a*b^28*c^11 + 1680*a^2*b^26*c^12 - 29120*a^3*b^24*c^13 + 349440*a^4*b^22*c^14 - 3075072*a^5*b^20*c^15 + 205004
80*a^6*b^18*c^16 - 105431040*a^7*b^16*c^17 + 421724160*a^8*b^14*c^18 - 1312030720*a^9*b^12*c^19 + 3148873728*a
^10*b^10*c^20 - 5725224960*a^11*b^8*c^21 + 7633633280*a^12*b^6*c^22 - 7046430720*a^13*b^4*c^23 + 4026531840*a^
14*b^2*c^24 - 1073741824*a^15*c^25)))*sqrt(x)*sqrt(-(b^13 - 117*a*b^11*c + 5109*a^2*b^9*c^2 - 97968*a^3*b^7*c^
3 + 670176*a^4*b^5*c^4 + 2895360*a^5*b^3*c^5 - 449280*a^6*b*c^6 + (b^20*c^5 - 40*a*b^18*c^6 + 720*a^2*b^16*c^7
- 7680*a^3*b^14*c^8 + 53760*a^4*b^12*c^9 - 258048*a^5*b^10*c^10 + 860160*a^6*b^8*c^11 - 1966080*a^7*b^6*c^12
+ 2949120*a^8*b^4*c^13 - 2621440*a^9*b^2*c^14 + 1048576*a^10*c^15)*sqrt((b^16 - 214*a*b^14*c + 19467*a^2*b^12*
c^2 - 967222*a^3*b^10*c^3 + 27769345*a^4*b^8*c^4 - 438445008*a^5*b^6*c^5 + 2996795232*a^6*b^4*c^6 - 141647616*
a^7*b^2*c^7 + 1679616*a^8*c^8)/(b^30*c^10 - 60*a*b^28*c^11 + 1680*a^2*b^26*c^12 - 29120*a^3*b^24*c^13 + 349440
*a^4*b^22*c^14 - 3075072*a^5*b^20*c^15 + 20500480*a^6*b^18*c^16 - 105431040*a^7*b^16*c^17 + 421724160*a^8*b^14
*c^18 - 1312030720*a^9*b^12*c^19 + 3148873728*a^10*b^10*c^20 - 5725224960*a^11*b^8*c^21 + 7633633280*a^12*b^6*
c^22 - 7046430720*a^13*b^4*c^23 + 4026531840*a^14*b^2*c^24 - 1073741824*a^15*c^25)))/(b^20*c^5 - 40*a*b^18*c^6
+ 720*a^2*b^16*c^7 - 7680*a^3*b^14*c^8 + 53760*a^4*b^12*c^9 - 258048*a^5*b^10*c^10 + 860160*a^6*b^8*c^11 - 19
66080*a^7*b^6*c^12 + 2949120*a^8*b^4*c^13 - 2621440*a^9*b^2*c^14 + 1048576*a^10*c^15)) + (b^33 - 225*a*b^31*c
+ 22235*a^2*b^29*c^2 - 1265758*a^3*b^27*c^3 + 45810016*a^4*b^25*c^4 - 1102080960*a^5*b^23*c^5 + 18046071552*a^
6*b^21*c^6 - 204854161920*a^7*b^19*c^7 + 163725...

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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(x**(15/2)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(15/2)/(c*x^4+b*x^2+a)^3,x, algorithm="giac")

[Out]

integrate(x^(15/2)/(c*x^4 + b*x^2 + a)^3, x)

________________________________________________________________________________________

Mupad [B]
time = 9.35, size = 2500, normalized size = 4.03 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(15/2)/(a + b*x^2 + c*x^4)^3,x)

[Out]

atan(((((3*(3159*a^3*b^14 - 20155392*a^10*c^7 - 367497*a^4*b^12*c + 15900219*a^5*b^10*c^2 - 299549340*a^6*b^8*
c^3 + 1945179360*a^7*b^6*c^4 + 2840323968*a^8*b^4*c^5 + 164042496*a^9*b^2*c^6))/(65536*(b^18*c - 262144*a^9*c^
10 - 36*a*b^16*c^2 + 576*a^2*b^14*c^3 - 5376*a^3*b^12*c^4 + 32256*a^4*b^10*c^5 - 129024*a^5*b^8*c^6 + 344064*a
^6*b^6*c^7 - 589824*a^7*b^4*c^8 + 589824*a^8*b^2*c^9)) + ((3*(-(81*(b^33 + b^8*(-(4*a*c - b^2)^25)^(1/2) - 471
104225280*a^16*b*c^16 + 10509*a^2*b^29*c^2 - 394248*a^3*b^27*c^3 + 9219696*a^4*b^25*c^4 - 140233728*a^5*b^23*c
^5 + 1424368896*a^6*b^21*c^6 - 9732052992*a^7*b^19*c^7 + 43376799744*a^8*b^17*c^8 - 108493078528*a^9*b^15*c^9
+ 13151174656*a^10*b^13*c^10 + 986354024448*a^11*b^11*c^11 - 3840358219776*a^12*b^9*c^12 + 7562531438592*a^13*
b^7*c^13 - 8212262682624*a^14*b^5*c^14 + 4213765570560*a^15*b^3*c^15 + 1296*a^4*c^4*(-(4*a*c - b^2)^25)^(1/2)
- 157*a*b^31*c + 4009*a^2*b^4*c^2*(-(4*a*c - b^2)^25)^(1/2) - 54648*a^3*b^2*c^3*(-(4*a*c - b^2)^25)^(1/2) - 10
7*a*b^6*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(1099511627776*a^20*c^25 + b^40*c^5 - 80*a*b^38*c^6 + 3040*a^2
*b^36*c^7 - 72960*a^3*b^34*c^8 + 1240320*a^4*b^32*c^9 - 15876096*a^5*b^30*c^10 + 158760960*a^6*b^28*c^11 - 127
0087680*a^7*b^26*c^12 + 8255569920*a^8*b^24*c^13 - 44029706240*a^9*b^22*c^14 + 193730707456*a^10*b^20*c^15 - 7
04475299840*a^11*b^18*c^16 + 2113425899520*a^12*b^16*c^17 - 5202279137280*a^13*b^14*c^18 + 10404558274560*a^14
*b^12*c^19 - 16647293239296*a^15*b^10*c^20 + 20809116549120*a^16*b^8*c^21 - 19585050869760*a^17*b^6*c^22 + 130
56700579840*a^18*b^4*c^23 - 5497558138880*a^19*b^2*c^24)))^(1/4)*(703687441776640*a^13*b*c^15 + 671088640*a^3*
b^21*c^5 - 26843545600*a^4*b^19*c^6 + 483183820800*a^5*b^17*c^7 - 5153960755200*a^6*b^15*c^8 + 36077725286400*
a^7*b^13*c^9 - 173173081374720*a^8*b^11*c^10 + 577243604582400*a^9*b^9*c^11 - 1319413953331200*a^10*b^7*c^12 +
1979120929996800*a^11*b^5*c^13 - 1759218604441600*a^12*b^3*c^14))/(65536*(b^18*c - 262144*a^9*c^10 - 36*a*b^1
6*c^2 + 576*a^2*b^14*c^3 - 5376*a^3*b^12*c^4 + 32256*a^4*b^10*c^5 - 129024*a^5*b^8*c^6 + 344064*a^6*b^6*c^7 -
589824*a^7*b^4*c^8 + 589824*a^8*b^2*c^9)) - (9*x^(1/2)*(16777216*a^3*b^25*c^4 - 31243722414882816*a^15*b*c^16
+ 23890755584*a^4*b^23*c^5 - 1000190509056*a^5*b^21*c^6 + 18747532247040*a^6*b^19*c^7 - 209186382151680*a^7*b^
17*c^8 + 1544951275978752*a^8*b^15*c^9 - 7925554690916352*a^9*b^13*c^10 + 28783015391920128*a^10*b^11*c^11 - 7
3870688712130560*a^11*b^9*c^12 + 130973825100677120*a^12*b^7*c^13 - 152242778028376064*a^13*b^5*c^14 + 1038642
66406232064*a^14*b^3*c^15))/(4194304*(b^24*c + 16777216*a^12*c^13 - 48*a*b^22*c^2 + 1056*a^2*b^20*c^3 - 14080*
a^3*b^18*c^4 + 126720*a^4*b^16*c^5 - 811008*a^5*b^14*c^6 + 3784704*a^6*b^12*c^7 - 12976128*a^7*b^10*c^8 + 3244
0320*a^8*b^8*c^9 - 57671680*a^9*b^6*c^10 + 69206016*a^10*b^4*c^11 - 50331648*a^11*b^2*c^12)))*(-(81*(b^33 + b^
8*(-(4*a*c - b^2)^25)^(1/2) - 471104225280*a^16*b*c^16 + 10509*a^2*b^29*c^2 - 394248*a^3*b^27*c^3 + 9219696*a^
4*b^25*c^4 - 140233728*a^5*b^23*c^5 + 1424368896*a^6*b^21*c^6 - 9732052992*a^7*b^19*c^7 + 43376799744*a^8*b^17
*c^8 - 108493078528*a^9*b^15*c^9 + 13151174656*a^10*b^13*c^10 + 986354024448*a^11*b^11*c^11 - 3840358219776*a^
12*b^9*c^12 + 7562531438592*a^13*b^7*c^13 - 8212262682624*a^14*b^5*c^14 + 4213765570560*a^15*b^3*c^15 + 1296*a
^4*c^4*(-(4*a*c - b^2)^25)^(1/2) - 157*a*b^31*c + 4009*a^2*b^4*c^2*(-(4*a*c - b^2)^25)^(1/2) - 54648*a^3*b^2*c
^3*(-(4*a*c - b^2)^25)^(1/2) - 107*a*b^6*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(1099511627776*a^20*c^25 + b^
40*c^5 - 80*a*b^38*c^6 + 3040*a^2*b^36*c^7 - 72960*a^3*b^34*c^8 + 1240320*a^4*b^32*c^9 - 15876096*a^5*b^30*c^1
0 + 158760960*a^6*b^28*c^11 - 1270087680*a^7*b^26*c^12 + 8255569920*a^8*b^24*c^13 - 44029706240*a^9*b^22*c^14
+ 193730707456*a^10*b^20*c^15 - 704475299840*a^11*b^18*c^16 + 2113425899520*a^12*b^16*c^17 - 5202279137280*a^1
3*b^14*c^18 + 10404558274560*a^14*b^12*c^19 - 16647293239296*a^15*b^10*c^20 + 20809116549120*a^16*b^8*c^21 - 1
9585050869760*a^17*b^6*c^22 + 13056700579840*a^18*b^4*c^23 - 5497558138880*a^19*b^2*c^24)))^(3/4))*(-(81*(b^33
+ b^8*(-(4*a*c - b^2)^25)^(1/2) - 471104225280*a^16*b*c^16 + 10509*a^2*b^29*c^2 - 394248*a^3*b^27*c^3 + 92196
96*a^4*b^25*c^4 - 140233728*a^5*b^23*c^5 + 1424368896*a^6*b^21*c^6 - 9732052992*a^7*b^19*c^7 + 43376799744*a^8
*b^17*c^8 - 108493078528*a^9*b^15*c^9 + 13151174656*a^10*b^13*c^10 + 986354024448*a^11*b^11*c^11 - 38403582197
76*a^12*b^9*c^12 + 7562531438592*a^13*b^7*c^13 - 8212262682624*a^14*b^5*c^14 + 4213765570560*a^15*b^3*c^15 + 1
296*a^4*c^4*(-(4*a*c - b^2)^25)^(1/2) - 157*a*b^31*c + 4009*a^2*b^4*c^2*(-(4*a*c - b^2)^25)^(1/2) - 54648*a^3*
b^2*c^3*(-(4*a*c - b^2)^25)^(1/2) - 107*a*b^6*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(1099511627776*a^20*c^25
+ b^40*c^5 - 80*a*b^38*c^6 + 3040*a^2*b^36*c^7 - 72960*a^3*b^34*c^8 + 1240320*a^4*b^32*c^9 - 15876096*a^5*b^3
0*c^10 + 158760960*a^6*b^28*c^11 - 1270087680*a^7*b^26*c^12 + 8255569920*a^8*b^24*c^13 - 44029706240*a^9*b^22*
c^14 + 193730707456*a^10*b^20*c^15 - 7044752998...

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