3.5.52 \(\int \frac {x^{11/2}}{(a+b x^2) (c+d x^2)^2} \, dx\)

Optimal. Leaf size=570 \[ \frac {a^{9/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)^2}-\frac {a^{9/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)^2}+\frac {a^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{5/4} (b c-a d)^2}-\frac {a^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{5/4} (b c-a d)^2}+\frac {c^{5/4} (5 b c-9 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{9/4} (b c-a d)^2}-\frac {c^{5/4} (5 b c-9 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{9/4} (b c-a d)^2}+\frac {c^{5/4} (5 b c-9 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{9/4} (b c-a d)^2}-\frac {c^{5/4} (5 b c-9 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} d^{9/4} (b c-a d)^2}+\frac {\sqrt {x} (5 b c-4 a d)}{2 b d^2 (b c-a d)}-\frac {c x^{5/2}}{2 d \left (c+d x^2\right ) (b c-a d)} \]

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Rubi [A]  time = 0.85, antiderivative size = 570, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {466, 470, 582, 522, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {a^{9/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)^2}-\frac {a^{9/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)^2}+\frac {a^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{5/4} (b c-a d)^2}-\frac {a^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{5/4} (b c-a d)^2}+\frac {c^{5/4} (5 b c-9 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{9/4} (b c-a d)^2}-\frac {c^{5/4} (5 b c-9 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{9/4} (b c-a d)^2}+\frac {c^{5/4} (5 b c-9 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{9/4} (b c-a d)^2}-\frac {c^{5/4} (5 b c-9 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} d^{9/4} (b c-a d)^2}+\frac {\sqrt {x} (5 b c-4 a d)}{2 b d^2 (b c-a d)}-\frac {c x^{5/2}}{2 d \left (c+d x^2\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(11/2)/((a + b*x^2)*(c + d*x^2)^2),x]

[Out]

((5*b*c - 4*a*d)*Sqrt[x])/(2*b*d^2*(b*c - a*d)) - (c*x^(5/2))/(2*d*(b*c - a*d)*(c + d*x^2)) + (a^(9/4)*ArcTan[
1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(5/4)*(b*c - a*d)^2) - (a^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(5/4)*(b*c - a*d)^2) + (c^(5/4)*(5*b*c - 9*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqr
t[x])/c^(1/4)])/(4*Sqrt[2]*d^(9/4)*(b*c - a*d)^2) - (c^(5/4)*(5*b*c - 9*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[
x])/c^(1/4)])/(4*Sqrt[2]*d^(9/4)*(b*c - a*d)^2) + (a^(9/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqr
t[b]*x])/(2*Sqrt[2]*b^(5/4)*(b*c - a*d)^2) - (a^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*
x])/(2*Sqrt[2]*b^(5/4)*(b*c - a*d)^2) + (c^(5/4)*(5*b*c - 9*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]
+ Sqrt[d]*x])/(8*Sqrt[2]*d^(9/4)*(b*c - a*d)^2) - (c^(5/4)*(5*b*c - 9*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1
/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*d^(9/4)*(b*c - a*d)^2)

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 470

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

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 582

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

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

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

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {x^{11/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^{12}}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {c x^{5/2}}{2 d (b c-a d) \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (5 a c+(5 b c-4 a d) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{2 d (b c-a d)}\\ &=\frac {(5 b c-4 a d) \sqrt {x}}{2 b d^2 (b c-a d)}-\frac {c x^{5/2}}{2 d (b c-a d) \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {a c (5 b c-4 a d)+\left (5 b^2 c^2-4 a b c d-4 a^2 d^2\right ) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{2 b d^2 (b c-a d)}\\ &=\frac {(5 b c-4 a d) \sqrt {x}}{2 b d^2 (b c-a d)}-\frac {c x^{5/2}}{2 d (b c-a d) \left (c+d x^2\right )}-\frac {\left (2 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b (b c-a d)^2}-\frac {\left (c^2 (5 b c-9 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 d^2 (b c-a d)^2}\\ &=\frac {(5 b c-4 a d) \sqrt {x}}{2 b d^2 (b c-a d)}-\frac {c x^{5/2}}{2 d (b c-a d) \left (c+d x^2\right )}-\frac {a^{5/2} \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b (b c-a d)^2}-\frac {a^{5/2} \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b (b c-a d)^2}-\frac {\left (c^{3/2} (5 b c-9 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 d^2 (b c-a d)^2}-\frac {\left (c^{3/2} (5 b c-9 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 d^2 (b c-a d)^2}\\ &=\frac {(5 b c-4 a d) \sqrt {x}}{2 b d^2 (b c-a d)}-\frac {c x^{5/2}}{2 d (b c-a d) \left (c+d x^2\right )}-\frac {a^{5/2} \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 b^{3/2} (b c-a d)^2}-\frac {a^{5/2} \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 b^{3/2} (b c-a d)^2}+\frac {a^{9/4} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} b^{5/4} (b c-a d)^2}+\frac {a^{9/4} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} b^{5/4} (b c-a d)^2}-\frac {\left (c^{3/2} (5 b c-9 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 d^{5/2} (b c-a d)^2}-\frac {\left (c^{3/2} (5 b c-9 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 d^{5/2} (b c-a d)^2}+\frac {\left (c^{5/4} (5 b c-9 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} d^{9/4} (b c-a d)^2}+\frac {\left (c^{5/4} (5 b c-9 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} d^{9/4} (b c-a d)^2}\\ &=\frac {(5 b c-4 a d) \sqrt {x}}{2 b d^2 (b c-a d)}-\frac {c x^{5/2}}{2 d (b c-a d) \left (c+d x^2\right )}+\frac {a^{9/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)^2}-\frac {a^{9/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)^2}+\frac {c^{5/4} (5 b c-9 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{9/4} (b c-a d)^2}-\frac {c^{5/4} (5 b c-9 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{9/4} (b c-a d)^2}-\frac {a^{9/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{5/4} (b c-a d)^2}+\frac {a^{9/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{5/4} (b c-a d)^2}-\frac {\left (c^{5/4} (5 b c-9 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{9/4} (b c-a d)^2}+\frac {\left (c^{5/4} (5 b c-9 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{9/4} (b c-a d)^2}\\ &=\frac {(5 b c-4 a d) \sqrt {x}}{2 b d^2 (b c-a d)}-\frac {c x^{5/2}}{2 d (b c-a d) \left (c+d x^2\right )}+\frac {a^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{5/4} (b c-a d)^2}-\frac {a^{9/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{5/4} (b c-a d)^2}+\frac {c^{5/4} (5 b c-9 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{9/4} (b c-a d)^2}-\frac {c^{5/4} (5 b c-9 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{9/4} (b c-a d)^2}+\frac {a^{9/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)^2}-\frac {a^{9/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)^2}+\frac {c^{5/4} (5 b c-9 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{9/4} (b c-a d)^2}-\frac {c^{5/4} (5 b c-9 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{9/4} (b c-a d)^2}\\ \end {align*}

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Mathematica [A]  time = 0.36, size = 563, normalized size = 0.99 \begin {gather*} \frac {4 \sqrt {2} a^{9/4} d^{9/4} \left (c+d x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-4 \sqrt {2} a^{9/4} d^{9/4} \left (c+d x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+8 \sqrt {2} a^{9/4} d^{9/4} \left (c+d x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )-8 \sqrt {2} a^{9/4} d^{9/4} \left (c+d x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )+\sqrt {2} b^{5/4} c^{5/4} \left (c+d x^2\right ) (5 b c-9 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )-\sqrt {2} b^{5/4} c^{5/4} \left (c+d x^2\right ) (5 b c-9 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )+2 \sqrt {2} b^{5/4} c^{5/4} \left (c+d x^2\right ) (5 b c-9 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )-2 \sqrt {2} b^{5/4} c^{5/4} \left (c+d x^2\right ) (5 b c-9 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )+8 b^{5/4} c^2 \sqrt [4]{d} \sqrt {x} (b c-a d)+32 \sqrt [4]{b} \sqrt [4]{d} \sqrt {x} \left (c+d x^2\right ) (b c-a d)^2}{16 b^{5/4} d^{9/4} \left (c+d x^2\right ) (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(11/2)/((a + b*x^2)*(c + d*x^2)^2),x]

[Out]

(8*b^(5/4)*c^2*d^(1/4)*(b*c - a*d)*Sqrt[x] + 32*b^(1/4)*d^(1/4)*(b*c - a*d)^2*Sqrt[x]*(c + d*x^2) + 8*Sqrt[2]*
a^(9/4)*d^(9/4)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 8*Sqrt[2]*a^(9/4)*d^(9/4)*(c + d*x
^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 2*Sqrt[2]*b^(5/4)*c^(5/4)*(5*b*c - 9*a*d)*(c + d*x^2)*ArcT
an[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 2*Sqrt[2]*b^(5/4)*c^(5/4)*(5*b*c - 9*a*d)*(c + d*x^2)*ArcTan[1 + (
Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 4*Sqrt[2]*a^(9/4)*d^(9/4)*(c + d*x^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x] - 4*Sqrt[2]*a^(9/4)*d^(9/4)*(c + d*x^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x] + Sqrt[2]*b^(5/4)*c^(5/4)*(5*b*c - 9*a*d)*(c + d*x^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x
] + Sqrt[d]*x] - Sqrt[2]*b^(5/4)*c^(5/4)*(5*b*c - 9*a*d)*(c + d*x^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqr
t[x] + Sqrt[d]*x])/(16*b^(5/4)*d^(9/4)*(b*c - a*d)^2*(c + d*x^2))

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IntegrateAlgebraic [A]  time = 1.20, size = 353, normalized size = 0.62 \begin {gather*} \frac {a^{9/4} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{a}}{\sqrt {2} \sqrt [4]{b}}-\frac {\sqrt [4]{b} x}{\sqrt {2} \sqrt [4]{a}}}{\sqrt {x}}\right )}{\sqrt {2} b^{5/4} (b c-a d)^2}-\frac {a^{9/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} b^{5/4} (b c-a d)^2}+\frac {\left (5 b c^{9/4}-9 a c^{5/4} d\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{4 \sqrt {2} d^{9/4} (a d-b c)^2}-\frac {\left (5 b c^{9/4}-9 a c^{5/4} d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{4 \sqrt {2} d^{9/4} (a d-b c)^2}+\frac {\sqrt {x} \left (-4 a c d-4 a d^2 x^2+5 b c^2+4 b c d x^2\right )}{2 b d^2 \left (c+d x^2\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(11/2)/((a + b*x^2)*(c + d*x^2)^2),x]

[Out]

(Sqrt[x]*(5*b*c^2 - 4*a*c*d + 4*b*c*d*x^2 - 4*a*d^2*x^2))/(2*b*d^2*(b*c - a*d)*(c + d*x^2)) + (a^(9/4)*ArcTan[
(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*b^(5/4)*(b*c - a*d)^2) + ((5*b*
c^(9/4) - 9*a*c^(5/4)*d)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(4*Sqrt[2]*d^(9/4)*(
-(b*c) + a*d)^2) - (a^(9/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*b^(5/4)
*(b*c - a*d)^2) - ((5*b*c^(9/4) - 9*a*c^(5/4)*d)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*
x)])/(4*Sqrt[2]*d^(9/4)*(-(b*c) + a*d)^2)

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fricas [B]  time = 111.54, size = 3393, normalized size = 5.95

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(11/2)/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

1/8*(4*(b^2*c^2*d^2 - a*b*c*d^3 + (b^2*c*d^3 - a*b*d^4)*x^2)*(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2
*c^7*d^2 - 14580*a^3*b*c^6*d^3 + 6561*a^4*c^5*d^4)/(b^8*c^8*d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*
a^3*b^5*c^5*d^12 + 70*a^4*b^4*c^4*d^13 - 56*a^5*b^3*c^3*d^14 + 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17
))^(1/4)*arctan(((b^6*c^6*d^7 - 6*a*b^5*c^5*d^8 + 15*a^2*b^4*c^4*d^9 - 20*a^3*b^3*c^3*d^10 + 15*a^4*b^2*c^2*d^
11 - 6*a^5*b*c*d^12 + a^6*d^13)*sqrt((25*b^2*c^4 - 90*a*b*c^3*d + 81*a^2*c^2*d^2)*x + (b^4*c^4*d^4 - 4*a*b^3*c
^3*d^5 + 6*a^2*b^2*c^2*d^6 - 4*a^3*b*c*d^7 + a^4*d^8)*sqrt(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2*c^
7*d^2 - 14580*a^3*b*c^6*d^3 + 6561*a^4*c^5*d^4)/(b^8*c^8*d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*a^3
*b^5*c^5*d^12 + 70*a^4*b^4*c^4*d^13 - 56*a^5*b^3*c^3*d^14 + 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17)))
*(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2*c^7*d^2 - 14580*a^3*b*c^6*d^3 + 6561*a^4*c^5*d^4)/(b^8*c^8*
d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*a^3*b^5*c^5*d^12 + 70*a^4*b^4*c^4*d^13 - 56*a^5*b^3*c^3*d^14
+ 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17))^(3/4) + (5*b^7*c^8*d^7 - 39*a*b^6*c^7*d^8 + 129*a^2*b^5*c
^6*d^9 - 235*a^3*b^4*c^5*d^10 + 255*a^4*b^3*c^4*d^11 - 165*a^5*b^2*c^3*d^12 + 59*a^6*b*c^2*d^13 - 9*a^7*c*d^14
)*sqrt(x)*(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2*c^7*d^2 - 14580*a^3*b*c^6*d^3 + 6561*a^4*c^5*d^4)/
(b^8*c^8*d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*a^3*b^5*c^5*d^12 + 70*a^4*b^4*c^4*d^13 - 56*a^5*b^3
*c^3*d^14 + 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17))^(3/4))/(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a
^2*b^2*c^7*d^2 - 14580*a^3*b*c^6*d^3 + 6561*a^4*c^5*d^4)) - 16*(-a^9/(b^13*c^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*
c^6*d^2 - 56*a^3*b^10*c^5*d^3 + 70*a^4*b^9*c^4*d^4 - 56*a^5*b^8*c^3*d^5 + 28*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7
+ a^8*b^5*d^8))^(1/4)*(b^2*c^2*d^2 - a*b*c*d^3 + (b^2*c*d^3 - a*b*d^4)*x^2)*arctan(((b^10*c^6 - 6*a*b^9*c^5*d
+ 15*a^2*b^8*c^4*d^2 - 20*a^3*b^7*c^3*d^3 + 15*a^4*b^6*c^2*d^4 - 6*a^5*b^5*c*d^5 + a^6*b^4*d^6)*(-a^9/(b^13*c
^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*c^6*d^2 - 56*a^3*b^10*c^5*d^3 + 70*a^4*b^9*c^4*d^4 - 56*a^5*b^8*c^3*d^5 + 28
*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7 + a^8*b^5*d^8))^(3/4)*sqrt(a^4*x + (b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2
*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*sqrt(-a^9/(b^13*c^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*c^6*d^2 - 56*a^3*b^10
*c^5*d^3 + 70*a^4*b^9*c^4*d^4 - 56*a^5*b^8*c^3*d^5 + 28*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7 + a^8*b^5*d^8))) - (
a^2*b^10*c^6 - 6*a^3*b^9*c^5*d + 15*a^4*b^8*c^4*d^2 - 20*a^5*b^7*c^3*d^3 + 15*a^6*b^6*c^2*d^4 - 6*a^7*b^5*c*d^
5 + a^8*b^4*d^6)*(-a^9/(b^13*c^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*c^6*d^2 - 56*a^3*b^10*c^5*d^3 + 70*a^4*b^9*c^4
*d^4 - 56*a^5*b^8*c^3*d^5 + 28*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7 + a^8*b^5*d^8))^(3/4)*sqrt(x))/a^9) - 4*(-a^9
/(b^13*c^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*c^6*d^2 - 56*a^3*b^10*c^5*d^3 + 70*a^4*b^9*c^4*d^4 - 56*a^5*b^8*c^3*
d^5 + 28*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7 + a^8*b^5*d^8))^(1/4)*(b^2*c^2*d^2 - a*b*c*d^3 + (b^2*c*d^3 - a*b*d
^4)*x^2)*log(a^2*sqrt(x) + (-a^9/(b^13*c^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*c^6*d^2 - 56*a^3*b^10*c^5*d^3 + 70*a
^4*b^9*c^4*d^4 - 56*a^5*b^8*c^3*d^5 + 28*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7 + a^8*b^5*d^8))^(1/4)*(b^3*c^2 - 2*
a*b^2*c*d + a^2*b*d^2)) + 4*(-a^9/(b^13*c^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*c^6*d^2 - 56*a^3*b^10*c^5*d^3 + 70*
a^4*b^9*c^4*d^4 - 56*a^5*b^8*c^3*d^5 + 28*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7 + a^8*b^5*d^8))^(1/4)*(b^2*c^2*d^2
- a*b*c*d^3 + (b^2*c*d^3 - a*b*d^4)*x^2)*log(a^2*sqrt(x) - (-a^9/(b^13*c^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*c^6
*d^2 - 56*a^3*b^10*c^5*d^3 + 70*a^4*b^9*c^4*d^4 - 56*a^5*b^8*c^3*d^5 + 28*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7 +
a^8*b^5*d^8))^(1/4)*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)) + (b^2*c^2*d^2 - a*b*c*d^3 + (b^2*c*d^3 - a*b*d^4)*x^
2)*(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2*c^7*d^2 - 14580*a^3*b*c^6*d^3 + 6561*a^4*c^5*d^4)/(b^8*c^
8*d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*a^3*b^5*c^5*d^12 + 70*a^4*b^4*c^4*d^13 - 56*a^5*b^3*c^3*d^
14 + 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17))^(1/4)*log(-(5*b*c^2 - 9*a*c*d)*sqrt(x) + (b^2*c^2*d^2 -
2*a*b*c*d^3 + a^2*d^4)*(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2*c^7*d^2 - 14580*a^3*b*c^6*d^3 + 6561
*a^4*c^5*d^4)/(b^8*c^8*d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*a^3*b^5*c^5*d^12 + 70*a^4*b^4*c^4*d^1
3 - 56*a^5*b^3*c^3*d^14 + 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17))^(1/4)) - (b^2*c^2*d^2 - a*b*c*d^3
+ (b^2*c*d^3 - a*b*d^4)*x^2)*(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2*c^7*d^2 - 14580*a^3*b*c^6*d^3 +
6561*a^4*c^5*d^4)/(b^8*c^8*d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*a^3*b^5*c^5*d^12 + 70*a^4*b^4*c^
4*d^13 - 56*a^5*b^3*c^3*d^14 + 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17))^(1/4)*log(-(5*b*c^2 - 9*a*c*d
)*sqrt(x) - (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2*c^7*d^2 -
14580*a^3*b*c^6*d^3 + 6561*a^4*c^5*d^4)/(b^8*c^8*d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*a^3*b^5*c^5
*d^12 + 70*a^4*b^4*c^4*d^13 - 56*a^5*b^3*c^3*d^14 + 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17))^(1/4)) +
4*(5*b*c^2 - 4*a*c*d + 4*(b*c*d - a*d^2)*x^2)*sqrt(x))/(b^2*c^2*d^2 - a*b*c*d^3 + (b^2*c*d^3 - a*b*d^4)*x^2)

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giac [A]  time = 1.11, size = 718, normalized size = 1.26 \begin {gather*} -\frac {\left (a b^{3}\right )^{\frac {1}{4}} a^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b^{4} c^{2} - 2 \, \sqrt {2} a b^{3} c d + \sqrt {2} a^{2} b^{2} d^{2}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} a^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b^{4} c^{2} - 2 \, \sqrt {2} a b^{3} c d + \sqrt {2} a^{2} b^{2} d^{2}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} a^{2} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} b^{4} c^{2} - 2 \, \sqrt {2} a b^{3} c d + \sqrt {2} a^{2} b^{2} d^{2}\right )}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} a^{2} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} b^{4} c^{2} - 2 \, \sqrt {2} a b^{3} c d + \sqrt {2} a^{2} b^{2} d^{2}\right )}} - \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b c^{2} - 9 \, \left (c d^{3}\right )^{\frac {1}{4}} a c d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{2} c^{2} d^{3} - 2 \, \sqrt {2} a b c d^{4} + \sqrt {2} a^{2} d^{5}\right )}} - \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b c^{2} - 9 \, \left (c d^{3}\right )^{\frac {1}{4}} a c d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{2} c^{2} d^{3} - 2 \, \sqrt {2} a b c d^{4} + \sqrt {2} a^{2} d^{5}\right )}} - \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b c^{2} - 9 \, \left (c d^{3}\right )^{\frac {1}{4}} a c d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{2} d^{3} - 2 \, \sqrt {2} a b c d^{4} + \sqrt {2} a^{2} d^{5}\right )}} + \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b c^{2} - 9 \, \left (c d^{3}\right )^{\frac {1}{4}} a c d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{2} d^{3} - 2 \, \sqrt {2} a b c d^{4} + \sqrt {2} a^{2} d^{5}\right )}} + \frac {c^{2} \sqrt {x}}{2 \, {\left (b c d^{2} - a d^{3}\right )} {\left (d x^{2} + c\right )}} + \frac {2 \, \sqrt {x}}{b d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(11/2)/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="giac")

[Out]

-(a*b^3)^(1/4)*a^2*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^4*c^2 - 2*sqrt
(2)*a*b^3*c*d + sqrt(2)*a^2*b^2*d^2) - (a*b^3)^(1/4)*a^2*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))
/(a/b)^(1/4))/(sqrt(2)*b^4*c^2 - 2*sqrt(2)*a*b^3*c*d + sqrt(2)*a^2*b^2*d^2) - 1/2*(a*b^3)^(1/4)*a^2*log(sqrt(2
)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^4*c^2 - 2*sqrt(2)*a*b^3*c*d + sqrt(2)*a^2*b^2*d^2) + 1/2*(a*
b^3)^(1/4)*a^2*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^4*c^2 - 2*sqrt(2)*a*b^3*c*d + sqrt
(2)*a^2*b^2*d^2) - 1/4*(5*(c*d^3)^(1/4)*b*c^2 - 9*(c*d^3)^(1/4)*a*c*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4)
+ 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^2*d^3 - 2*sqrt(2)*a*b*c*d^4 + sqrt(2)*a^2*d^5) - 1/4*(5*(c*d^3)^(1/4
)*b*c^2 - 9*(c*d^3)^(1/4)*a*c*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b
^2*c^2*d^3 - 2*sqrt(2)*a*b*c*d^4 + sqrt(2)*a^2*d^5) - 1/8*(5*(c*d^3)^(1/4)*b*c^2 - 9*(c*d^3)^(1/4)*a*c*d)*log(
sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^2*d^3 - 2*sqrt(2)*a*b*c*d^4 + sqrt(2)*a^2*d^5) + 1
/8*(5*(c*d^3)^(1/4)*b*c^2 - 9*(c*d^3)^(1/4)*a*c*d)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*
b^2*c^2*d^3 - 2*sqrt(2)*a*b*c*d^4 + sqrt(2)*a^2*d^5) + 1/2*c^2*sqrt(x)/((b*c*d^2 - a*d^3)*(d*x^2 + c)) + 2*sqr
t(x)/(b*d^2)

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maple [A]  time = 0.02, size = 582, normalized size = 1.02 \begin {gather*} -\frac {a \,c^{2} \sqrt {x}}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right ) d}+\frac {b \,c^{3} \sqrt {x}}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right ) d^{2}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (a d -b c \right )^{2} b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (a d -b c \right )^{2} b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 \left (a d -b c \right )^{2} b}+\frac {9 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{2} d}+\frac {9 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{2} d}+\frac {9 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a c \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 \left (a d -b c \right )^{2} d}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b \,c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{2} d^{2}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b \,c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{2} d^{2}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b \,c^{2} \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 \left (a d -b c \right )^{2} d^{2}}+\frac {2 \sqrt {x}}{b \,d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(11/2)/(b*x^2+a)/(d*x^2+c)^2,x)

[Out]

2/b/d^2*x^(1/2)-1/4/b*a^2/(a*d-b*c)^2*(a/b)^(1/4)*2^(1/2)*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a
/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))-1/2/b*a^2/(a*d-b*c)^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x
^(1/2)+1)-1/2/b*a^2/(a*d-b*c)^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-1/2*c^2/d/(a*d-b*c)^
2*x^(1/2)/(d*x^2+c)*a+1/2*c^3/d^2/(a*d-b*c)^2*x^(1/2)/(d*x^2+c)*b+9/8*c/d/(a*d-b*c)^2*(c/d)^(1/4)*2^(1/2)*arct
an(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a-5/8*c^2/d^2/(a*d-b*c)^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(
1/2)+1)*b+9/8*c/d/(a*d-b*c)^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a-5/8*c^2/d^2/(a*d-b*c
)^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b+9/16*c/d/(a*d-b*c)^2*(c/d)^(1/4)*2^(1/2)*ln((x
+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*a-5/16*c^2/d^2/(a*d-b*c
)^2*(c/d)^(1/4)*2^(1/2)*ln((x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1
/2)))*b

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maxima [A]  time = 2.02, size = 494, normalized size = 0.87 \begin {gather*} -\frac {{\left (\frac {2 \, \sqrt {2} {\left (5 \, b c - 9 \, a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (5 \, b c - 9 \, a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (5 \, b c - 9 \, a d\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, b c - 9 \, a d\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}\right )} c^{2}}{16 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )}} + \frac {c^{2} \sqrt {x}}{2 \, {\left (b c^{2} d^{2} - a c d^{3} + {\left (b c d^{3} - a d^{4}\right )} x^{2}\right )}} - \frac {\frac {2 \, \sqrt {2} a^{\frac {5}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} a^{\frac {5}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} a^{\frac {9}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} a^{\frac {9}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}}}{4 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}} + \frac {2 \, \sqrt {x}}{b d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(11/2)/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

-1/16*(2*sqrt(2)*(5*b*c - 9*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)
*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(5*b*c - 9*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^
(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(5*b*c - 9*a*d)*lo
g(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(5*b*c - 9*a*d)*log(-sqrt
(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))*c^2/(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4
) + 1/2*c^2*sqrt(x)/(b*c^2*d^2 - a*c*d^3 + (b*c*d^3 - a*d^4)*x^2) - 1/4*(2*sqrt(2)*a^(5/2)*arctan(1/2*sqrt(2)*
(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/sqrt(sqrt(a)*sqrt(b)) + 2*sqrt(2)*a^(5/2)
*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/sqrt(sqrt(a)*sqrt(b)
) + sqrt(2)*a^(9/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/b^(1/4) - sqrt(2)*a^(9/4)*log(-
sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/b^(1/4))/(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2) + 2*sqrt(x
)/(b*d^2)

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mupad [B]  time = 3.25, size = 22978, normalized size = 40.31

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(11/2)/((a + b*x^2)*(c + d*x^2)^2),x)

[Out]

atan((((-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 +
1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*((-a^9/(16*b^13*c
^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 -
896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(3/4)*((x^(1/2)*(6400*a^3*b^15*c^14*d^6 - 7424
0*a^4*b^14*c^13*d^7 + 384256*a^5*b^13*c^12*d^8 - 1165312*a^6*b^12*c^11*d^9 + 2286080*a^7*b^11*c^10*d^10 - 3017
728*a^8*b^10*c^9*d^11 + 2691584*a^9*b^9*c^8*d^12 - 1570816*a^10*b^8*c^7*d^13 + 541952*a^11*b^7*c^6*d^14 - 7424
0*a^12*b^6*c^5*d^15 - 12032*a^13*b^5*c^4*d^16 + 4096*a^14*b^4*c^3*d^17))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c
^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9) - (2*(-a^9/(16*b^13*
c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4
- 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*(5120*a^3*b^13*c^11*d^8 - 40960*a^4*b^1
2*c^10*d^9 + 143360*a^5*b^11*c^9*d^10 - 286720*a^6*b^10*c^8*d^11 + 358400*a^7*b^9*c^7*d^12 - 286720*a^8*b^8*c^
6*d^13 + 143360*a^9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d^15 + 5120*a^11*b^5*c^3*d^16))/(a^3*b*d^8 - b^4*c^3*d^5
+ 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7)) - (2*(625*a^4*b^8*c^11 + 576*a^12*c^3*d^8 - 3875*a^5*b^7*c^10*d + 256*a
^11*b*c^4*d^7 + 8275*a^6*b^6*c^9*d^2 - 6305*a^7*b^5*c^8*d^3 + 256*a^8*b^4*c^7*d^4 + 256*a^9*b^3*c^6*d^5 + 256*
a^10*b^2*c^5*d^6))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7)) + (x^(1/2)*(625*a^6*b^8*c^12
+ 1296*a^14*c^4*d^8 - 4500*a^7*b^7*c^11*d - 1440*a^13*b*c^5*d^7 + 12150*a^8*b^6*c^10*d^2 - 14580*a^9*b^5*c^9*
d^3 + 6561*a^10*b^4*c^8*d^4 + 400*a^12*b^2*c^6*d^6))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c
*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9))*(-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 1
28*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 +
448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*1i + ((-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 +
448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^
6 - 128*a*b^12*c^7*d))^(1/4)*((-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 -
896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^
(3/4)*((x^(1/2)*(6400*a^3*b^15*c^14*d^6 - 74240*a^4*b^14*c^13*d^7 + 384256*a^5*b^13*c^12*d^8 - 1165312*a^6*b^1
2*c^11*d^9 + 2286080*a^7*b^11*c^10*d^10 - 3017728*a^8*b^10*c^9*d^11 + 2691584*a^9*b^9*c^8*d^12 - 1570816*a^10*
b^8*c^7*d^13 + 541952*a^11*b^7*c^6*d^14 - 74240*a^12*b^6*c^5*d^15 - 12032*a^13*b^5*c^4*d^16 + 4096*a^14*b^4*c^
3*d^17))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*
d^8 + 15*a^4*b^3*c^2*d^9) + (2*(-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2
- 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))
^(1/4)*(5120*a^3*b^13*c^11*d^8 - 40960*a^4*b^12*c^10*d^9 + 143360*a^5*b^11*c^9*d^10 - 286720*a^6*b^10*c^8*d^11
+ 358400*a^7*b^9*c^7*d^12 - 286720*a^8*b^8*c^6*d^13 + 143360*a^9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d^15 + 512
0*a^11*b^5*c^3*d^16))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7)) + (2*(625*a^4*b^8*c^11 +
576*a^12*c^3*d^8 - 3875*a^5*b^7*c^10*d + 256*a^11*b*c^4*d^7 + 8275*a^6*b^6*c^9*d^2 - 6305*a^7*b^5*c^8*d^3 + 25
6*a^8*b^4*c^7*d^4 + 256*a^9*b^3*c^6*d^5 + 256*a^10*b^2*c^5*d^6))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 -
3*a^2*b^2*c*d^7)) + (x^(1/2)*(625*a^6*b^8*c^12 + 1296*a^14*c^4*d^8 - 4500*a^7*b^7*c^11*d - 1440*a^13*b*c^5*d^7
+ 12150*a^8*b^6*c^10*d^2 - 14580*a^9*b^5*c^9*d^3 + 6561*a^10*b^4*c^8*d^4 + 400*a^12*b^2*c^6*d^6))/(a^6*b*d^11
+ b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2
*d^9))*(-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 +
1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*1i)/(((-a^9/(16*b
^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*
d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*((-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^
8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d
^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(3/4)*((x^(1/2)*(6400*a^3*b^15*c^14*d^6 - 74240*a^4*b^14*c^13*d^
7 + 384256*a^5*b^13*c^12*d^8 - 1165312*a^6*b^12*c^11*d^9 + 2286080*a^7*b^11*c^10*d^10 - 3017728*a^8*b^10*c^9*d
^11 + 2691584*a^9*b^9*c^8*d^12 - 1570816*a^10*b^8*c^7*d^13 + 541952*a^11*b^7*c^6*d^14 - 74240*a^12*b^6*c^5*d^1
5 - 12032*a^13*b^5*c^4*d^16 + 4096*a^14*b^4*c^3*d^17))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2
*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9) - (2*(-a^9/(16*b^13*c^8 + 16*a^8*b^5*d
^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*
d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*(5120*a^3*b^13*c^11*d^8 - 40960*a^4*b^12*c^10*d^9 + 14336
0*a^5*b^11*c^9*d^10 - 286720*a^6*b^10*c^8*d^11 + 358400*a^7*b^9*c^7*d^12 - 286720*a^8*b^8*c^6*d^13 + 143360*a^
9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d^15 + 5120*a^11*b^5*c^3*d^16))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6
- 3*a^2*b^2*c*d^7)) - (2*(625*a^4*b^8*c^11 + 576*a^12*c^3*d^8 - 3875*a^5*b^7*c^10*d + 256*a^11*b*c^4*d^7 + 82
75*a^6*b^6*c^9*d^2 - 6305*a^7*b^5*c^8*d^3 + 256*a^8*b^4*c^7*d^4 + 256*a^9*b^3*c^6*d^5 + 256*a^10*b^2*c^5*d^6))
/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7)) + (x^(1/2)*(625*a^6*b^8*c^12 + 1296*a^14*c^4*d
^8 - 4500*a^7*b^7*c^11*d - 1440*a^13*b*c^5*d^7 + 12150*a^8*b^6*c^10*d^2 - 14580*a^9*b^5*c^9*d^3 + 6561*a^10*b^
4*c^8*d^4 + 400*a^12*b^2*c^6*d^6))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5
*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9))*(-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 +
448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^
6 - 128*a*b^12*c^7*d))^(1/4) - ((-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2
- 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d)
)^(1/4)*((-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3
+ 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(3/4)*((x^(1/2)*(6400
*a^3*b^15*c^14*d^6 - 74240*a^4*b^14*c^13*d^7 + 384256*a^5*b^13*c^12*d^8 - 1165312*a^6*b^12*c^11*d^9 + 2286080*
a^7*b^11*c^10*d^10 - 3017728*a^8*b^10*c^9*d^11 + 2691584*a^9*b^9*c^8*d^12 - 1570816*a^10*b^8*c^7*d^13 + 541952
*a^11*b^7*c^6*d^14 - 74240*a^12*b^6*c^5*d^15 - 12032*a^13*b^5*c^4*d^16 + 4096*a^14*b^4*c^3*d^17))/(a^6*b*d^11
+ b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*
d^9) + (2*(-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^
3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*(5120*a^3*b^13
*c^11*d^8 - 40960*a^4*b^12*c^10*d^9 + 143360*a^5*b^11*c^9*d^10 - 286720*a^6*b^10*c^8*d^11 + 358400*a^7*b^9*c^7
*d^12 - 286720*a^8*b^8*c^6*d^13 + 143360*a^9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d^15 + 5120*a^11*b^5*c^3*d^16))
/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7)) + (2*(625*a^4*b^8*c^11 + 576*a^12*c^3*d^8 - 38
75*a^5*b^7*c^10*d + 256*a^11*b*c^4*d^7 + 8275*a^6*b^6*c^9*d^2 - 6305*a^7*b^5*c^8*d^3 + 256*a^8*b^4*c^7*d^4 + 2
56*a^9*b^3*c^6*d^5 + 256*a^10*b^2*c^5*d^6))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7)) + (
x^(1/2)*(625*a^6*b^8*c^12 + 1296*a^14*c^4*d^8 - 4500*a^7*b^7*c^11*d - 1440*a^13*b*c^5*d^7 + 12150*a^8*b^6*c^10
*d^2 - 14580*a^9*b^5*c^9*d^3 + 6561*a^10*b^4*c^8*d^4 + 400*a^12*b^2*c^6*d^6))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*
b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9))*(-a^9/(16*b^13
*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4
- 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)))*(-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8
- 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5
+ 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*2i + 2*atan((((-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*
b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6
*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*((-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^1
1*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^
12*c^7*d))^(3/4)*((x^(1/2)*(6400*a^3*b^15*c^14*d^6 - 74240*a^4*b^14*c^13*d^7 + 384256*a^5*b^13*c^12*d^8 - 1165
312*a^6*b^12*c^11*d^9 + 2286080*a^7*b^11*c^10*d^10 - 3017728*a^8*b^10*c^9*d^11 + 2691584*a^9*b^9*c^8*d^12 - 15
70816*a^10*b^8*c^7*d^13 + 541952*a^11*b^7*c^6*d^14 - 74240*a^12*b^6*c^5*d^15 - 12032*a^13*b^5*c^4*d^16 + 4096*
a^14*b^4*c^3*d^17))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a
^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9) - ((-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11
*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^1
2*c^7*d))^(1/4)*(5120*a^3*b^13*c^11*d^8 - 40960*a^4*b^12*c^10*d^9 + 143360*a^5*b^11*c^9*d^10 - 286720*a^6*b^10
*c^8*d^11 + 358400*a^7*b^9*c^7*d^12 - 286720*a^8*b^8*c^6*d^13 + 143360*a^9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d
^15 + 5120*a^11*b^5*c^3*d^16)*2i)/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7))*1i + (2*(625*
a^4*b^8*c^11 + 576*a^12*c^3*d^8 - 3875*a^5*b^7*c^10*d + 256*a^11*b*c^4*d^7 + 8275*a^6*b^6*c^9*d^2 - 6305*a^7*b
^5*c^8*d^3 + 256*a^8*b^4*c^7*d^4 + 256*a^9*b^3*c^6*d^5 + 256*a^10*b^2*c^5*d^6))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a
*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7))*1i - (x^(1/2)*(625*a^6*b^8*c^12 + 1296*a^14*c^4*d^8 - 4500*a^7*b^7*c^11*d - 1
440*a^13*b*c^5*d^7 + 12150*a^8*b^6*c^10*d^2 - 14580*a^9*b^5*c^9*d^3 + 6561*a^10*b^4*c^8*d^4 + 400*a^12*b^2*c^6
*d^6))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^
8 + 15*a^4*b^3*c^2*d^9))*(-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*
a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)
+ ((-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 11
20*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*((-a^9/(16*b^13*c^8
+ 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 89
6*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(3/4)*((x^(1/2)*(6400*a^3*b^15*c^14*d^6 - 74240*a
^4*b^14*c^13*d^7 + 384256*a^5*b^13*c^12*d^8 - 1165312*a^6*b^12*c^11*d^9 + 2286080*a^7*b^11*c^10*d^10 - 3017728
*a^8*b^10*c^9*d^11 + 2691584*a^9*b^9*c^8*d^12 - 1570816*a^10*b^8*c^7*d^13 + 541952*a^11*b^7*c^6*d^14 - 74240*a
^12*b^6*c^5*d^15 - 12032*a^13*b^5*c^4*d^16 + 4096*a^14*b^4*c^3*d^17))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*
d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9) + ((-a^9/(16*b^13*c^8 +
16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896
*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*(5120*a^3*b^13*c^11*d^8 - 40960*a^4*b^12*c^1
0*d^9 + 143360*a^5*b^11*c^9*d^10 - 286720*a^6*b^10*c^8*d^11 + 358400*a^7*b^9*c^7*d^12 - 286720*a^8*b^8*c^6*d^1
3 + 143360*a^9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d^15 + 5120*a^11*b^5*c^3*d^16)*2i)/(a^3*b*d^8 - b^4*c^3*d^5 +
3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7))*1i - (2*(625*a^4*b^8*c^11 + 576*a^12*c^3*d^8 - 3875*a^5*b^7*c^10*d + 256*
a^11*b*c^4*d^7 + 8275*a^6*b^6*c^9*d^2 - 6305*a^7*b^5*c^8*d^3 + 256*a^8*b^4*c^7*d^4 + 256*a^9*b^3*c^6*d^5 + 256
*a^10*b^2*c^5*d^6))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7))*1i - (x^(1/2)*(625*a^6*b^8*
c^12 + 1296*a^14*c^4*d^8 - 4500*a^7*b^7*c^11*d - 1440*a^13*b*c^5*d^7 + 12150*a^8*b^6*c^10*d^2 - 14580*a^9*b^5*
c^9*d^3 + 6561*a^10*b^4*c^8*d^4 + 400*a^12*b^2*c^6*d^6))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b
^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9))*(-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8
- 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^
5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4))/(((-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7
+ 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d
^6 - 128*a*b^12*c^7*d))^(1/4)*((-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2
- 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))
^(3/4)*((x^(1/2)*(6400*a^3*b^15*c^14*d^6 - 74240*a^4*b^14*c^13*d^7 + 384256*a^5*b^13*c^12*d^8 - 1165312*a^6*b^
12*c^11*d^9 + 2286080*a^7*b^11*c^10*d^10 - 3017728*a^8*b^10*c^9*d^11 + 2691584*a^9*b^9*c^8*d^12 - 1570816*a^10
*b^8*c^7*d^13 + 541952*a^11*b^7*c^6*d^14 - 74240*a^12*b^6*c^5*d^15 - 12032*a^13*b^5*c^4*d^16 + 4096*a^14*b^4*c
^3*d^17))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3
*d^8 + 15*a^4*b^3*c^2*d^9) - ((-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 -
896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^
(1/4)*(5120*a^3*b^13*c^11*d^8 - 40960*a^4*b^12*c^10*d^9 + 143360*a^5*b^11*c^9*d^10 - 286720*a^6*b^10*c^8*d^11
+ 358400*a^7*b^9*c^7*d^12 - 286720*a^8*b^8*c^6*d^13 + 143360*a^9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d^15 + 5120
*a^11*b^5*c^3*d^16)*2i)/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7))*1i + (2*(625*a^4*b^8*c^
11 + 576*a^12*c^3*d^8 - 3875*a^5*b^7*c^10*d + 256*a^11*b*c^4*d^7 + 8275*a^6*b^6*c^9*d^2 - 6305*a^7*b^5*c^8*d^3
+ 256*a^8*b^4*c^7*d^4 + 256*a^9*b^3*c^6*d^5 + 256*a^10*b^2*c^5*d^6))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d
^6 - 3*a^2*b^2*c*d^7))*1i - (x^(1/2)*(625*a^6*b^8*c^12 + 1296*a^14*c^4*d^8 - 4500*a^7*b^7*c^11*d - 1440*a^13*b
*c^5*d^7 + 12150*a^8*b^6*c^10*d^2 - 14580*a^9*b^5*c^9*d^3 + 6561*a^10*b^4*c^8*d^4 + 400*a^12*b^2*c^6*d^6))/(a^
6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4
*b^3*c^2*d^9))*(-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c
^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*1i - ((-a
^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*
b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*((-a^9/(16*b^13*c^8 + 16*a^
8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b
^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(3/4)*((x^(1/2)*(6400*a^3*b^15*c^14*d^6 - 74240*a^4*b^14
*c^13*d^7 + 384256*a^5*b^13*c^12*d^8 - 1165312*a^6*b^12*c^11*d^9 + 2286080*a^7*b^11*c^10*d^10 - 3017728*a^8*b^
10*c^9*d^11 + 2691584*a^9*b^9*c^8*d^12 - 1570816*a^10*b^8*c^7*d^13 + 541952*a^11*b^7*c^6*d^14 - 74240*a^12*b^6
*c^5*d^15 - 12032*a^13*b^5*c^4*d^16 + 4096*a^14*b^4*c^3*d^17))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6
*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9) + ((-a^9/(16*b^13*c^8 + 16*a^8
*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^
8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*(5120*a^3*b^13*c^11*d^8 - 40960*a^4*b^12*c^10*d^9 +
143360*a^5*b^11*c^9*d^10 - 286720*a^6*b^10*c^8*d^11 + 358400*a^7*b^9*c^7*d^12 - 286720*a^8*b^8*c^6*d^13 + 143
360*a^9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d^15 + 5120*a^11*b^5*c^3*d^16)*2i)/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^
3*c^2*d^6 - 3*a^2*b^2*c*d^7))*1i - (2*(625*a^4*b^8*c^11 + 576*a^12*c^3*d^8 - 3875*a^5*b^7*c^10*d + 256*a^11*b*
c^4*d^7 + 8275*a^6*b^6*c^9*d^2 - 6305*a^7*b^5*c^8*d^3 + 256*a^8*b^4*c^7*d^4 + 256*a^9*b^3*c^6*d^5 + 256*a^10*b
^2*c^5*d^6))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7))*1i - (x^(1/2)*(625*a^6*b^8*c^12 +
1296*a^14*c^4*d^8 - 4500*a^7*b^7*c^11*d - 1440*a^13*b*c^5*d^7 + 12150*a^8*b^6*c^10*d^2 - 14580*a^9*b^5*c^9*d^3
+ 6561*a^10*b^4*c^8*d^4 + 400*a^12*b^2*c^6*d^6))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^
10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9))*(-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*
a^7*b^6*c*d^7 + 448*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448
*a^6*b^7*c^2*d^6 - 128*a*b^12*c^7*d))^(1/4)*1i))*(-a^9/(16*b^13*c^8 + 16*a^8*b^5*d^8 - 128*a^7*b^6*c*d^7 + 448
*a^2*b^11*c^6*d^2 - 896*a^3*b^10*c^5*d^3 + 1120*a^4*b^9*c^4*d^4 - 896*a^5*b^8*c^3*d^5 + 448*a^6*b^7*c^2*d^6 -
128*a*b^12*c^7*d))^(1/4) + atan(((-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d
^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 22
9376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^
7*b*c*d^16))^(1/4)*((-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b
^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*
c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^
(1/4)*((-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(40
96*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 28
6720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(3/4)*((x^(1/
2)*(6400*a^3*b^15*c^14*d^6 - 74240*a^4*b^14*c^13*d^7 + 384256*a^5*b^13*c^12*d^8 - 1165312*a^6*b^12*c^11*d^9 +
2286080*a^7*b^11*c^10*d^10 - 3017728*a^8*b^10*c^9*d^11 + 2691584*a^9*b^9*c^8*d^12 - 1570816*a^10*b^8*c^7*d^13
+ 541952*a^11*b^7*c^6*d^14 - 74240*a^12*b^6*c^5*d^15 - 12032*a^13*b^5*c^4*d^16 + 4096*a^14*b^4*c^3*d^17))/(a^6
*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*
b^3*c^2*d^9) - (2*(-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3
*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^
5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1
/4)*(5120*a^3*b^13*c^11*d^8 - 40960*a^4*b^12*c^10*d^9 + 143360*a^5*b^11*c^9*d^10 - 286720*a^6*b^10*c^8*d^11 +
358400*a^7*b^9*c^7*d^12 - 286720*a^8*b^8*c^6*d^13 + 143360*a^9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d^15 + 5120*a
^11*b^5*c^3*d^16))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7)) - (2*(625*a^4*b^8*c^11 + 576
*a^12*c^3*d^8 - 3875*a^5*b^7*c^10*d + 256*a^11*b*c^4*d^7 + 8275*a^6*b^6*c^9*d^2 - 6305*a^7*b^5*c^8*d^3 + 256*a
^8*b^4*c^7*d^4 + 256*a^9*b^3*c^6*d^5 + 256*a^10*b^2*c^5*d^6))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a
^2*b^2*c*d^7)) + (x^(1/2)*(625*a^6*b^8*c^12 + 1296*a^14*c^4*d^8 - 4500*a^7*b^7*c^11*d - 1440*a^13*b*c^5*d^7 +
12150*a^8*b^6*c^10*d^2 - 14580*a^9*b^5*c^9*d^3 + 6561*a^10*b^4*c^8*d^4 + 400*a^12*b^2*c^6*d^6))/(a^6*b*d^11 +
b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^
9))*1i + (-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(
4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 +
286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*((-(6
25*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17
+ 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^
4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*((-(625*b^4*c^9 +
6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c
^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 -
229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(3/4)*((x^(1/2)*(6400*a^3*b^15*c^14*d
^6 - 74240*a^4*b^14*c^13*d^7 + 384256*a^5*b^13*c^12*d^8 - 1165312*a^6*b^12*c^11*d^9 + 2286080*a^7*b^11*c^10*d^
10 - 3017728*a^8*b^10*c^9*d^11 + 2691584*a^9*b^9*c^8*d^12 - 1570816*a^10*b^8*c^7*d^13 + 541952*a^11*b^7*c^6*d^
14 - 74240*a^12*b^6*c^5*d^15 - 12032*a^13*b^5*c^4*d^16 + 4096*a^14*b^4*c^3*d^17))/(a^6*b*d^11 + b^7*c^6*d^5 -
6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9) + (2*(-(625
*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 +
4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*
c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*(5120*a^3*b^13*c^11*
d^8 - 40960*a^4*b^12*c^10*d^9 + 143360*a^5*b^11*c^9*d^10 - 286720*a^6*b^10*c^8*d^11 + 358400*a^7*b^9*c^7*d^12
- 286720*a^8*b^8*c^6*d^13 + 143360*a^9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d^15 + 5120*a^11*b^5*c^3*d^16))/(a^3*
b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7)) + (2*(625*a^4*b^8*c^11 + 576*a^12*c^3*d^8 - 3875*a^5
*b^7*c^10*d + 256*a^11*b*c^4*d^7 + 8275*a^6*b^6*c^9*d^2 - 6305*a^7*b^5*c^8*d^3 + 256*a^8*b^4*c^7*d^4 + 256*a^9
*b^3*c^6*d^5 + 256*a^10*b^2*c^5*d^6))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7)) + (x^(1/2
)*(625*a^6*b^8*c^12 + 1296*a^14*c^4*d^8 - 4500*a^7*b^7*c^11*d - 1440*a^13*b*c^5*d^7 + 12150*a^8*b^6*c^10*d^2 -
14580*a^9*b^5*c^9*d^3 + 6561*a^10*b^4*c^8*d^4 + 400*a^12*b^2*c^6*d^6))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^
5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9))*1i)/((-(625*b^4*c^9
+ 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8
*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13
- 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*((-(625*b^4*c^9 + 6561*a^4*c^
5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32
768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*
b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*((-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580
*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7
*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14
+ 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(3/4)*((x^(1/2)*(6400*a^3*b^15*c^14*d^6 - 74240*a^4*b^14*c^13
*d^7 + 384256*a^5*b^13*c^12*d^8 - 1165312*a^6*b^12*c^11*d^9 + 2286080*a^7*b^11*c^10*d^10 - 3017728*a^8*b^10*c^
9*d^11 + 2691584*a^9*b^9*c^8*d^12 - 1570816*a^10*b^8*c^7*d^13 + 541952*a^11*b^7*c^6*d^14 - 74240*a^12*b^6*c^5*
d^15 - 12032*a^13*b^5*c^4*d^16 + 4096*a^14*b^4*c^3*d^17))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*
b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9) - (2*(-(625*b^4*c^9 + 6561*a^4*c^5*
d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 3276
8*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^
3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*(5120*a^3*b^13*c^11*d^8 - 40960*a^4*b^12*c^1
0*d^9 + 143360*a^5*b^11*c^9*d^10 - 286720*a^6*b^10*c^8*d^11 + 358400*a^7*b^9*c^7*d^12 - 286720*a^8*b^8*c^6*d^1
3 + 143360*a^9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d^15 + 5120*a^11*b^5*c^3*d^16))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*
a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7)) - (2*(625*a^4*b^8*c^11 + 576*a^12*c^3*d^8 - 3875*a^5*b^7*c^10*d + 256*a^11*b
*c^4*d^7 + 8275*a^6*b^6*c^9*d^2 - 6305*a^7*b^5*c^8*d^3 + 256*a^8*b^4*c^7*d^4 + 256*a^9*b^3*c^6*d^5 + 256*a^10*
b^2*c^5*d^6))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7)) + (x^(1/2)*(625*a^6*b^8*c^12 + 12
96*a^14*c^4*d^8 - 4500*a^7*b^7*c^11*d - 1440*a^13*b*c^5*d^7 + 12150*a^8*b^6*c^10*d^2 - 14580*a^9*b^5*c^9*d^3 +
6561*a^10*b^4*c^8*d^4 + 400*a^12*b^2*c^6*d^6))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10
+ 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9)) - (-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*
a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*
d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 +
114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*((-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3
+ 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688
*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b
^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*((-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*
b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*
d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 -
32768*a^7*b*c*d^16))^(3/4)*((x^(1/2)*(6400*a^3*b^15*c^14*d^6 - 74240*a^4*b^14*c^13*d^7 + 384256*a^5*b^13*c^12
*d^8 - 1165312*a^6*b^12*c^11*d^9 + 2286080*a^7*b^11*c^10*d^10 - 3017728*a^8*b^10*c^9*d^11 + 2691584*a^9*b^9*c^
8*d^12 - 1570816*a^10*b^8*c^7*d^13 + 541952*a^11*b^7*c^6*d^14 - 74240*a^12*b^6*c^5*d^15 - 12032*a^13*b^5*c^4*d
^16 + 4096*a^14*b^4*c^3*d^17))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4
*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9) + (2*(-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 +
12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a
^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2
*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*(5120*a^3*b^13*c^11*d^8 - 40960*a^4*b^12*c^10*d^9 + 143360*a^5*b^11*c^9
*d^10 - 286720*a^6*b^10*c^8*d^11 + 358400*a^7*b^9*c^7*d^12 - 286720*a^8*b^8*c^6*d^13 + 143360*a^9*b^7*c^5*d^14
- 40960*a^10*b^6*c^4*d^15 + 5120*a^11*b^5*c^3*d^16))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c
*d^7)) + (2*(625*a^4*b^8*c^11 + 576*a^12*c^3*d^8 - 3875*a^5*b^7*c^10*d + 256*a^11*b*c^4*d^7 + 8275*a^6*b^6*c^9
*d^2 - 6305*a^7*b^5*c^8*d^3 + 256*a^8*b^4*c^7*d^4 + 256*a^9*b^3*c^6*d^5 + 256*a^10*b^2*c^5*d^6))/(a^3*b*d^8 -
b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7)) + (x^(1/2)*(625*a^6*b^8*c^12 + 1296*a^14*c^4*d^8 - 4500*a^7*
b^7*c^11*d - 1440*a^13*b*c^5*d^7 + 12150*a^8*b^6*c^10*d^2 - 14580*a^9*b^5*c^9*d^3 + 6561*a^10*b^4*c^8*d^4 + 40
0*a^12*b^2*c^6*d^6))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*
a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9))))*(-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b
^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d
^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 -
32768*a^7*b*c*d^16))^(1/4)*2i + 2*atan(((-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^
2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^
11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 3
2768*a^7*b*c*d^16))^(1/4)*((-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4
500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a
^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*
d^16))^(1/4)*((-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8
*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^
12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(3/4)*
((x^(1/2)*(6400*a^3*b^15*c^14*d^6 - 74240*a^4*b^14*c^13*d^7 + 384256*a^5*b^13*c^12*d^8 - 1165312*a^6*b^12*c^11
*d^9 + 2286080*a^7*b^11*c^10*d^10 - 3017728*a^8*b^10*c^9*d^11 + 2691584*a^9*b^9*c^8*d^12 - 1570816*a^10*b^8*c^
7*d^13 + 541952*a^11*b^7*c^6*d^14 - 74240*a^12*b^6*c^5*d^15 - 12032*a^13*b^5*c^4*d^16 + 4096*a^14*b^4*c^3*d^17
))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 +
15*a^4*b^3*c^2*d^9) - ((-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*
a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b
^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16
))^(1/4)*(5120*a^3*b^13*c^11*d^8 - 40960*a^4*b^12*c^10*d^9 + 143360*a^5*b^11*c^9*d^10 - 286720*a^6*b^10*c^8*d^
11 + 358400*a^7*b^9*c^7*d^12 - 286720*a^8*b^8*c^6*d^13 + 143360*a^9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d^15 + 5
120*a^11*b^5*c^3*d^16)*2i)/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7))*1i + (2*(625*a^4*b^8
*c^11 + 576*a^12*c^3*d^8 - 3875*a^5*b^7*c^10*d + 256*a^11*b*c^4*d^7 + 8275*a^6*b^6*c^9*d^2 - 6305*a^7*b^5*c^8*
d^3 + 256*a^8*b^4*c^7*d^4 + 256*a^9*b^3*c^6*d^5 + 256*a^10*b^2*c^5*d^6))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^
2*d^6 - 3*a^2*b^2*c*d^7))*1i - (x^(1/2)*(625*a^6*b^8*c^12 + 1296*a^14*c^4*d^8 - 4500*a^7*b^7*c^11*d - 1440*a^1
3*b*c^5*d^7 + 12150*a^8*b^6*c^10*d^2 - 14580*a^9*b^5*c^9*d^3 + 6561*a^10*b^4*c^8*d^4 + 400*a^12*b^2*c^6*d^6))/
(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*
a^4*b^3*c^2*d^9)) + (-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b
^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*
c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^
(1/4)*((-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(40
96*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 28
6720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*((-(625
*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 +
4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*
c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(3/4)*((x^(1/2)*(6400*a^3*
b^15*c^14*d^6 - 74240*a^4*b^14*c^13*d^7 + 384256*a^5*b^13*c^12*d^8 - 1165312*a^6*b^12*c^11*d^9 + 2286080*a^7*b
^11*c^10*d^10 - 3017728*a^8*b^10*c^9*d^11 + 2691584*a^9*b^9*c^8*d^12 - 1570816*a^10*b^8*c^7*d^13 + 541952*a^11
*b^7*c^6*d^14 - 74240*a^12*b^6*c^5*d^15 - 12032*a^13*b^5*c^4*d^16 + 4096*a^14*b^4*c^3*d^17))/(a^6*b*d^11 + b^7
*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9)
+ ((-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a
^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720
*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*(5120*a^3*b
^13*c^11*d^8 - 40960*a^4*b^12*c^10*d^9 + 143360*a^5*b^11*c^9*d^10 - 286720*a^6*b^10*c^8*d^11 + 358400*a^7*b^9*
c^7*d^12 - 286720*a^8*b^8*c^6*d^13 + 143360*a^9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d^15 + 5120*a^11*b^5*c^3*d^1
6)*2i)/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7))*1i - (2*(625*a^4*b^8*c^11 + 576*a^12*c^3
*d^8 - 3875*a^5*b^7*c^10*d + 256*a^11*b*c^4*d^7 + 8275*a^6*b^6*c^9*d^2 - 6305*a^7*b^5*c^8*d^3 + 256*a^8*b^4*c^
7*d^4 + 256*a^9*b^3*c^6*d^5 + 256*a^10*b^2*c^5*d^6))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*
d^7))*1i - (x^(1/2)*(625*a^6*b^8*c^12 + 1296*a^14*c^4*d^8 - 4500*a^7*b^7*c^11*d - 1440*a^13*b*c^5*d^7 + 12150*
a^8*b^6*c^10*d^2 - 14580*a^9*b^5*c^9*d^3 + 6561*a^10*b^4*c^8*d^4 + 400*a^12*b^2*c^6*d^6))/(a^6*b*d^11 + b^7*c^
6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9)))/(
(-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*
d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^
4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*((-(625*b^4*c^
9 + 6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b
^8*c^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^1
3 - 229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*((-(625*b^4*c^9 + 6561*a^4*
c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 -
32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^
5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(3/4)*((x^(1/2)*(6400*a^3*b^15*c^14*d^6 - 7424
0*a^4*b^14*c^13*d^7 + 384256*a^5*b^13*c^12*d^8 - 1165312*a^6*b^12*c^11*d^9 + 2286080*a^7*b^11*c^10*d^10 - 3017
728*a^8*b^10*c^9*d^11 + 2691584*a^9*b^9*c^8*d^12 - 1570816*a^10*b^8*c^7*d^13 + 541952*a^11*b^7*c^6*d^14 - 7424
0*a^12*b^6*c^5*d^15 - 12032*a^13*b^5*c^4*d^16 + 4096*a^14*b^4*c^3*d^17))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c
^5*d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9) - ((-(625*b^4*c^9 +
6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c
^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 -
229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*(5120*a^3*b^13*c^11*d^8 - 40960
*a^4*b^12*c^10*d^9 + 143360*a^5*b^11*c^9*d^10 - 286720*a^6*b^10*c^8*d^11 + 358400*a^7*b^9*c^7*d^12 - 286720*a^
8*b^8*c^6*d^13 + 143360*a^9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d^15 + 5120*a^11*b^5*c^3*d^16)*2i)/(a^3*b*d^8 -
b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7))*1i + (2*(625*a^4*b^8*c^11 + 576*a^12*c^3*d^8 - 3875*a^5*b^7*
c^10*d + 256*a^11*b*c^4*d^7 + 8275*a^6*b^6*c^9*d^2 - 6305*a^7*b^5*c^8*d^3 + 256*a^8*b^4*c^7*d^4 + 256*a^9*b^3*
c^6*d^5 + 256*a^10*b^2*c^5*d^6))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7))*1i - (x^(1/2)*
(625*a^6*b^8*c^12 + 1296*a^14*c^4*d^8 - 4500*a^7*b^7*c^11*d - 1440*a^13*b*c^5*d^7 + 12150*a^8*b^6*c^10*d^2 - 1
4580*a^9*b^5*c^9*d^3 + 6561*a^10*b^4*c^8*d^4 + 400*a^12*b^2*c^6*d^6))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*
d^6 - 6*a^5*b^2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9))*1i - (-(625*b^4*c^9 +
6561*a^4*c^5*d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c
^8*d^9 - 32768*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 -
229376*a^5*b^3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*((-(625*b^4*c^9 + 6561*a^4*c^5*
d^4 - 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 3276
8*a*b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^
3*c^3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*((-(625*b^4*c^9 + 6561*a^4*c^5*d^4 - 14580*a
^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*b^7*c^7*d
^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^3*d^14 +
114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(3/4)*((x^(1/2)*(6400*a^3*b^15*c^14*d^6 - 74240*a^4*b^14*c^13*d
^7 + 384256*a^5*b^13*c^12*d^8 - 1165312*a^6*b^12*c^11*d^9 + 2286080*a^7*b^11*c^10*d^10 - 3017728*a^8*b^10*c^9*
d^11 + 2691584*a^9*b^9*c^8*d^12 - 1570816*a^10*b^8*c^7*d^13 + 541952*a^11*b^7*c^6*d^14 - 74240*a^12*b^6*c^5*d^
15 - 12032*a^13*b^5*c^4*d^16 + 4096*a^14*b^4*c^3*d^17))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^
2*c*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9) + ((-(625*b^4*c^9 + 6561*a^4*c^5*d^4
- 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*
b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^
3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4)*(5120*a^3*b^13*c^11*d^8 - 40960*a^4*b^12*c^10*d^
9 + 143360*a^5*b^11*c^9*d^10 - 286720*a^6*b^10*c^8*d^11 + 358400*a^7*b^9*c^7*d^12 - 286720*a^8*b^8*c^6*d^13 +
143360*a^9*b^7*c^5*d^14 - 40960*a^10*b^6*c^4*d^15 + 5120*a^11*b^5*c^3*d^16)*2i)/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a
*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7))*1i - (2*(625*a^4*b^8*c^11 + 576*a^12*c^3*d^8 - 3875*a^5*b^7*c^10*d + 256*a^11
*b*c^4*d^7 + 8275*a^6*b^6*c^9*d^2 - 6305*a^7*b^5*c^8*d^3 + 256*a^8*b^4*c^7*d^4 + 256*a^9*b^3*c^6*d^5 + 256*a^1
0*b^2*c^5*d^6))/(a^3*b*d^8 - b^4*c^3*d^5 + 3*a*b^3*c^2*d^6 - 3*a^2*b^2*c*d^7))*1i - (x^(1/2)*(625*a^6*b^8*c^12
+ 1296*a^14*c^4*d^8 - 4500*a^7*b^7*c^11*d - 1440*a^13*b*c^5*d^7 + 12150*a^8*b^6*c^10*d^2 - 14580*a^9*b^5*c^9*
d^3 + 6561*a^10*b^4*c^8*d^4 + 400*a^12*b^2*c^6*d^6))/(a^6*b*d^11 + b^7*c^6*d^5 - 6*a*b^6*c^5*d^6 - 6*a^5*b^2*c
*d^10 + 15*a^2*b^5*c^4*d^7 - 20*a^3*b^4*c^3*d^8 + 15*a^4*b^3*c^2*d^9))*1i))*(-(625*b^4*c^9 + 6561*a^4*c^5*d^4
- 14580*a^3*b*c^6*d^3 + 12150*a^2*b^2*c^7*d^2 - 4500*a*b^3*c^8*d)/(4096*a^8*d^17 + 4096*b^8*c^8*d^9 - 32768*a*
b^7*c^7*d^10 + 114688*a^2*b^6*c^6*d^11 - 229376*a^3*b^5*c^5*d^12 + 286720*a^4*b^4*c^4*d^13 - 229376*a^5*b^3*c^
3*d^14 + 114688*a^6*b^2*c^2*d^15 - 32768*a^7*b*c*d^16))^(1/4) + (2*x^(1/2))/(b*d^2) - (b*c^2*x^(1/2))/(2*(b*d^
3*x^2 + b*c*d^2)*(a*d - b*c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(11/2)/(b*x**2+a)/(d*x**2+c)**2,x)

[Out]

Timed out

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