∫ x15∕2 ____________________

3.9.47 $\int \frac {x^{15/2}}{(a+b x^2+c x^4)^3} \, dx$

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

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Rubi [A] time = 1.77, 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 = {1115, 1365, 1498, 1502, 1422, 212, 208, 205} \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 ) \tan ^{-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 ) \tan ^{-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 {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 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 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 212

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] &&

!GtQ[a/b, 0]

Rule 1115

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 1365

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] && ILt

Q[p, -1] && GtQ[m, 2*n - 1]

Rule 1422

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 1498

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 1502

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 \operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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] time = 0.44, size = 254, normalized size = 0.41 \begin {gather*} \frac {3 c \left (a+b x^2+c x^4\right )^2 \text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {-28 \text {$\#$1}^4 a b c \log \left (\sqrt {x}-\text {$\#$1}\right )+\text {$\#$1}^4 b^3 \log \left (\sqrt {x}-\text {$\#$1}\right )+12 a^2 c \log \left (\sqrt {x}-\text {$\#$1}\right )+a b^2 \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^7 c+\text {$\#$1}^3 b}\&\right ]+16 \sqrt {x} \left (b^2-4 a c\right ) \left (-2 a^2 c+a b \left (b-3 c x^2\right )+b^3 x^2\right )+4 \sqrt {x} \left (-68 a^2 c^2+21 a b^2 c-28 a b c^2 x^2-4 b^4+b^3 c x^2\right ) \left (a+b x^2+c x^4\right )}{64 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*Sqrt[x]*(-4*b^4 + 21*a*b^2*c - 68*a^2*c^2 + b^3*c*x^2 - 28*a*b*c^2*x^2)*(a + b*x^2 + c*x^4) + 16*(b^2 - 4*a

*c)*Sqrt[x]*(-2*a^2*c + b^3*x^2 + a*b*(b - 3*c*x^2)) + 3*c*(a + b*x^2 + c*x^4)^2*RootSum[a + b*#1^4 + c*#1^8 &

, (a*b^2*Log[Sqrt[x] - #1] + 12*a^2*c*Log[Sqrt[x] - #1] + b^3*Log[Sqrt[x] - #1]*#1^4 - 28*a*b*c*Log[Sqrt[x] -

#1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(64*c^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2)

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IntegrateAlgebraic [C] time = 1.19, size = 527, normalized size = 0.85 \begin {gather*} -\frac {\text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {6 \text {$\#$1}^4 a b c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+3 \text {$\#$1}^4 b^3 c \log \left (\sqrt {x}-\text {$\#$1}\right )+28 a^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )-22 a b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right )+3 b^4 \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^7 c+\text {$\#$1}^3 b}\&\right ]}{8 a c^3 \left (4 a c-b^2\right )}-\frac {3 \text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {-36 \text {$\#$1}^4 a^2 b c^3 \log \left (\sqrt {x}-\text {$\#$1}\right )-17 \text {$\#$1}^4 a b^3 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+8 \text {$\#$1}^4 b^5 c \log \left (\sqrt {x}-\text {$\#$1}\right )-140 a^3 c^3 \log \left (\sqrt {x}-\text {$\#$1}\right )+223 a^2 b^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )-80 a b^4 c \log \left (\sqrt {x}-\text {$\#$1}\right )+8 b^6 \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^7 c+\text {$\#$1}^3 b}\&\right ]}{64 a c^3 \left (4 a c-b^2\right )^2}+\frac {\text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {\log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^7 c+\text {$\#$1}^3 b}\&\right ]}{2 c^2}-\frac {\sqrt {x} \left (36 a^3 c+3 a^2 b^2+48 a^2 b c x^2+68 a^2 c^2 x^4+6 a b^3 x^2+7 a b^2 c x^4+28 a b c^2 x^6+3 b^4 x^4-b^3 c x^6\right )}{16 c \left (4 a c-b^2\right )^2 \left (a+b x^2+c x^4\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/16*(Sqrt[x]*(3*a^2*b^2 + 36*a^3*c + 6*a*b^3*x^2 + 48*a^2*b*c*x^2 + 3*b^4*x^4 + 7*a*b^2*c*x^4 + 68*a^2*c^2*x

^4 - b^3*c*x^6 + 28*a*b*c^2*x^6))/(c*(-b^2 + 4*a*c)^2*(a + b*x^2 + c*x^4)^2) + RootSum[a + b*#1^4 + c*#1^8 & ,

Log[Sqrt[x] - #1]/(b*#1^3 + 2*c*#1^7) & ]/(2*c^2) - 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) & ]/(8*a*c^3*(-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[Sqr

t[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) & ])/(64*a*c^3*(-b^2 + 4*a*c)^2)

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

[Out]

Timed out

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

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval

uation time: 192.56Unable to convert to real 1/4 Error: Bad Argument Value

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maple [C] time = 0.04, size = 275, normalized size = 0.44 \begin {gather*} \frac {3 \left (\left (-28 a c +b^{2}\right ) \RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{4} b +12 a^{2} c +a \,b^{2}\right ) \ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )+\sqrt {x}\right )}{64 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c \left (2 \RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{7} c +\RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{3} b \right )}+\frac {-\frac {\left (28 a c -b^{2}\right ) b \,x^{\frac {13}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 \left (8 a c +b^{2}\right ) a b \,x^{\frac {5}{2}}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}-\frac {\left (68 a^{2} c^{2}+7 a \,b^{2} c +3 b^{4}\right ) x^{\frac {9}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}-\frac {3 \left (12 a c +b^{2}\right ) a^{2} \sqrt {x}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[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(-_R+x^(1/2)),_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*} \frac {3 \, {\left (b^{2} c + 12 \, a c^{2}\right )} x^{\frac {17}{2}} + {\left (7 \, b^{3} + 44 \, a b c\right )} x^{\frac {13}{2}} + 24 \, a^{2} b x^{\frac {5}{2}} + {\left (35 \, a b^{2} + 4 \, a^{2} c\right )} x^{\frac {9}{2}}}{16 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{8} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{6} + a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{4} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x^{2}\right )}} - \int \frac {3 \, {\left ({\left (b^{2} + 12 \, a c\right )} x^{\frac {7}{2}} + 40 \, a b x^{\frac {3}{2}}\right )}}{32 \, {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )}}\,{d x} \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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mupad [B] time = 9.35, size = 50970, normalized size = 82.08

result too large to display

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 - 704475299840*a^11*b^18*c^16 + 2113425899520*a^12*b^16*c^17 - 520227913728

0*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^2

1 - 19585050869760*a^17*b^6*c^22 + 13056700579840*a^18*b^4*c^23 - 5497558138880*a^19*b^2*c^24)))^(1/4) - (9*x^

(1/2)*(123201*a^4*b^16 + 483729408*a^12*c^8 - 14619852*a^5*b^14*c + 653342274*a^6*b^12*c^2 - 13105503216*a^7*b

^10*c^3 + 102306071520*a^8*b^8*c^4 - 66486210048*a^9*b^6*c^5 + 9199443456*a^10*b^4*c^6 + 6261608448*a^11*b^2*c

^7))/(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 + 32440320*a^8*b^8*c^9 - 5767

1680*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 + 756253143

8592*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^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^2

0*c^15 - 704475299840*a^11*b^18*c^16 + 2113425899520*a^12*b^16*c^17 - 5202279137280*a^13*b^14*c^18 + 104045582

74560*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 + 13056700579840*a^18*b^4*c^23 - 5497558138880*a^19*b^2*c^24)))^(1/4)*1i - (((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 + 28403

23968*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 + 58

9824*a^8*b^2*c^9)) + ((3*(-(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 - 97320

52992*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 + 98635

4024448*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 + 124

0320*a^4*b^32*c^9 - 15876096*a^5*b^30*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 - 704475299840*a^11*b^18*c^16 + 2113425

899520*a^12*b^16*c^17 - 5202279137280*a^13*b^14*c^18 + 10404558274560*a^14*b^12*c^19 - 16647293239296*a^15*b^1

0*c^20 + 20809116549120*a^16*b^8*c^21 - 19585050869760*a^17*b^6*c^22 + 13056700579840*a^18*b^4*c^23 - 54975581

38880*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 - 17

59218604441600*a^12*b^3*c^14))/(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)) + (9*x^(1/2)*(16777216*a^3*b^25*c^4 - 31243722414882816*a^15*b*c^16 + 23890755584*a^4*b^23*c^5 - 1000190

509056*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 - 73870688712130560*a^11*b^9*c^12 + 130

973825100677120*a^12*b^7*c^13 - 152242778028376064*a^13*b^5*c^14 + 103864266406232064*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 + 32440320*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) - 471104

225280*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 + 1

3151174656*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) - 1

57*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^10 + 158760960*a^6*b^28*c^11 - 127008

7680*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 - 7044

75299840*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 + 130567

00579840*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) - 4

71104225280*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^1

3*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^10 + 158760960*a^6*b^28*c^11 - 1

270087680*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^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 + 1

3056700579840*a^18*b^4*c^23 - 5497558138880*a^19*b^2*c^24)))^(1/4) + (9*x^(1/2)*(123201*a^4*b^16 + 483729408*a

^12*c^8 - 14619852*a^5*b^14*c + 653342274*a^6*b^12*c^2 - 13105503216*a^7*b^10*c^3 + 102306071520*a^8*b^8*c^4 -

66486210048*a^9*b^6*c^5 + 9199443456*a^10*b^4*c^6 + 6261608448*a^11*b^2*c^7))/(4194304*(b^24*c + 16777216*a^1

2*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 + 32440320*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 + 10

509*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^10 + 158760960*a^6*b^28*c^11 - 1270087680*a^7*b^26*c^12 + 82

55569920*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^13*b^14*c^18 + 10404558274560*a^14*b^12*c^19 - 1664729323929

6*a^15*b^10*c^20 + 20809116549120*a^16*b^8*c^21 - 19585050869760*a^17*b^6*c^22 + 13056700579840*a^18*b^4*c^23

- 5497558138880*a^19*b^2*c^24)))^(1/4)*1i)/((((3*(3159*a^3*b^14 - 20155392*a^10*c^7 - 367497*a^4*b^12*c + 1590

0219*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