3.4.10 \(\int \frac {1}{\sqrt {f x} (d+e x^2) (a+b x^2+c x^4)} \, dx\) [310]

Optimal. Leaf size=866 \[ \frac {c^{3/4} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {c^{3/4} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {e^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {f x}}{\sqrt [4]{d} \sqrt {f}}\right )}{\sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}+\frac {e^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {f x}}{\sqrt [4]{d} \sqrt {f}}\right )}{\sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}+\frac {c^{3/4} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {c^{3/4} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {e^{7/4} \log \left (\sqrt {d} \sqrt {f}+\sqrt {e} \sqrt {f} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {f x}\right )}{2 \sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}+\frac {e^{7/4} \log \left (\sqrt {d} \sqrt {f}+\sqrt {e} \sqrt {f} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {f x}\right )}{2 \sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}} \]

[Out]

-1/2*e^(7/4)*arctan(1-e^(1/4)*2^(1/2)*(f*x)^(1/2)/d^(1/4)/f^(1/2))/d^(3/4)/(a*e^2-b*d*e+c*d^2)*2^(1/2)/f^(1/2)
+1/2*e^(7/4)*arctan(1+e^(1/4)*2^(1/2)*(f*x)^(1/2)/d^(1/4)/f^(1/2))/d^(3/4)/(a*e^2-b*d*e+c*d^2)*2^(1/2)/f^(1/2)
-1/4*e^(7/4)*ln(d^(1/2)*f^(1/2)+x*e^(1/2)*f^(1/2)-d^(1/4)*e^(1/4)*2^(1/2)*(f*x)^(1/2))/d^(3/4)/(a*e^2-b*d*e+c*
d^2)*2^(1/2)/f^(1/2)+1/4*e^(7/4)*ln(d^(1/2)*f^(1/2)+x*e^(1/2)*f^(1/2)+d^(1/4)*e^(1/4)*2^(1/2)*(f*x)^(1/2))/d^(
3/4)/(a*e^2-b*d*e+c*d^2)*2^(1/2)/f^(1/2)+1/2*c^(3/4)*arctan(2^(1/4)*c^(1/4)*(f*x)^(1/2)/(-b-(-4*a*c+b^2)^(1/2)
)^(1/4)/f^(1/2))*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))*2^(3/4)/(a*e^2-b*d*e+c*d^2)/(-b-(-4*a*c+b^2)^(1/2))^(3/4)/(-
4*a*c+b^2)^(1/2)/f^(1/2)+1/2*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*(f*x)^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)/f^(1/2)
)*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))*2^(3/4)/(a*e^2-b*d*e+c*d^2)/(-b-(-4*a*c+b^2)^(1/2))^(3/4)/(-4*a*c+b^2)^(1/2
)/f^(1/2)-1/2*c^(3/4)*arctan(2^(1/4)*c^(1/4)*(f*x)^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)/f^(1/2))*(2*c*d-e*(b+(-
4*a*c+b^2)^(1/2)))*2^(3/4)/(a*e^2-b*d*e+c*d^2)/(-4*a*c+b^2)^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)/f^(1/2)-1/2*c^
(3/4)*arctanh(2^(1/4)*c^(1/4)*(f*x)^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)/f^(1/2))*(2*c*d-e*(b+(-4*a*c+b^2)^(1/2
)))*2^(3/4)/(a*e^2-b*d*e+c*d^2)/(-4*a*c+b^2)^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)/f^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.77, antiderivative size = 866, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 12, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {1283, 1438, 217, 1179, 642, 1176, 631, 210, 1436, 218, 214, 211} \begin {gather*} -\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {f x}}{\sqrt [4]{d} \sqrt {f}}\right ) e^{7/4}}{\sqrt {2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt {f}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {f x}}{\sqrt [4]{d} \sqrt {f}}+1\right ) e^{7/4}}{\sqrt {2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt {f}}-\frac {\log \left (\sqrt {e} \sqrt {f} x+\sqrt {d} \sqrt {f}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {f x}\right ) e^{7/4}}{2 \sqrt {2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt {f}}+\frac {\log \left (\sqrt {e} \sqrt {f} x+\sqrt {d} \sqrt {f}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {f x}\right ) e^{7/4}}{2 \sqrt {2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt {f}}+\frac {c^{3/4} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt {f}}-\frac {c^{3/4} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt {f}}+\frac {c^{3/4} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt {f}}-\frac {c^{3/4} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt {f}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[f*x]*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

(c^(3/4)*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[f*x])/((-b - Sqrt[b^2 - 4*a*c])^(1/4
)*Sqrt[f])])/(2^(1/4)*Sqrt[b^2 - 4*a*c]*(-b - Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sqrt[f]) - (c^(
3/4)*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[f*x])/((-b + Sqrt[b^2 - 4*a*c])^(1/4)*Sq
rt[f])])/(2^(1/4)*Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sqrt[f]) - (e^(7/4)
*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[f*x])/(d^(1/4)*Sqrt[f])])/(Sqrt[2]*d^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sqrt[f])
+ (e^(7/4)*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[f*x])/(d^(1/4)*Sqrt[f])])/(Sqrt[2]*d^(3/4)*(c*d^2 - b*d*e + a*e^2)
*Sqrt[f]) + (c^(3/4)*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[f*x])/((-b - Sqrt[b^2 -
4*a*c])^(1/4)*Sqrt[f])])/(2^(1/4)*Sqrt[b^2 - 4*a*c]*(-b - Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sq
rt[f]) - (c^(3/4)*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[f*x])/((-b + Sqrt[b^2 - 4*
a*c])^(1/4)*Sqrt[f])])/(2^(1/4)*Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sqrt[
f]) - (e^(7/4)*Log[Sqrt[d]*Sqrt[f] + Sqrt[e]*Sqrt[f]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[f*x]])/(2*Sqrt[2]*d^(3/4
)*(c*d^2 - b*d*e + a*e^2)*Sqrt[f]) + (e^(7/4)*Log[Sqrt[d]*Sqrt[f] + Sqrt[e]*Sqrt[f]*x + Sqrt[2]*d^(1/4)*e^(1/4
)*Sqrt[f*x]])/(2*Sqrt[2]*d^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sqrt[f])

Rule 210

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

Rule 211

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

Rule 214

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

Rule 217

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 218

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

Rule 631

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 642

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 1176

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 1179

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]

Rule 1283

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

Rule 1436

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

Rule 1438

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Int[ExpandIntegran
d[(d + e*x^n)^q/(a + b*x^n + c*x^(2*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] && IntegerQ[q]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {f x} \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{\left (d+\frac {e x^4}{f^2}\right ) \left (a+\frac {b x^4}{f^2}+\frac {c x^8}{f^4}\right )} \, dx,x,\sqrt {f x}\right )}{f}\\ &=\frac {2 \text {Subst}\left (\int \left (\frac {e^2 f^2}{\left (c d^2-b d e+a e^2\right ) \left (d f^2+e x^4\right )}+\frac {c d f^4-b e f^4-c e f^2 x^4}{\left (c d^2-b d e+a e^2\right ) \left (a f^4+b f^2 x^4+c x^8\right )}\right ) \, dx,x,\sqrt {f x}\right )}{f}\\ &=\frac {2 \text {Subst}\left (\int \frac {c d f^4-b e f^4-c e f^2 x^4}{a f^4+b f^2 x^4+c x^8} \, dx,x,\sqrt {f x}\right )}{\left (c d^2-b d e+a e^2\right ) f}+\frac {\left (2 e^2 f\right ) \text {Subst}\left (\int \frac {1}{d f^2+e x^4} \, dx,x,\sqrt {f x}\right )}{c d^2-b d e+a e^2}\\ &=\frac {e^2 \text {Subst}\left (\int \frac {\sqrt {d} f-\sqrt {e} x^2}{d f^2+e x^4} \, dx,x,\sqrt {f x}\right )}{\sqrt {d} \left (c d^2-b d e+a e^2\right )}+\frac {e^2 \text {Subst}\left (\int \frac {\sqrt {d} f+\sqrt {e} x^2}{d f^2+e x^4} \, dx,x,\sqrt {f x}\right )}{\sqrt {d} \left (c d^2-b d e+a e^2\right )}-\frac {\left (c \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) f\right ) \text {Subst}\left (\int \frac {1}{\frac {b f^2}{2}+\frac {1}{2} \sqrt {b^2-4 a c} f^2+c x^4} \, dx,x,\sqrt {f x}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}+\frac {\left (c \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) f\right ) \text {Subst}\left (\int \frac {1}{\frac {b f^2}{2}-\frac {1}{2} \sqrt {b^2-4 a c} f^2+c x^4} \, dx,x,\sqrt {f x}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}\\ &=\frac {e^{3/2} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {d} f}{\sqrt {e}}-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {f} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt {f x}\right )}{2 \sqrt {d} \left (c d^2-b d e+a e^2\right )}+\frac {e^{3/2} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {d} f}{\sqrt {e}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {f} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt {f x}\right )}{2 \sqrt {d} \left (c d^2-b d e+a e^2\right )}+\frac {\left (c \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}} f-\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {f x}\right )}{\sqrt {b^2-4 a c} \sqrt {-b-\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}+\frac {\left (c \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}} f+\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {f x}\right )}{\sqrt {b^2-4 a c} \sqrt {-b-\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}-\frac {\left (c \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b+\sqrt {b^2-4 a c}} f-\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {f x}\right )}{\sqrt {b^2-4 a c} \sqrt {-b+\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}-\frac {\left (c \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b+\sqrt {b^2-4 a c}} f+\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {f x}\right )}{\sqrt {b^2-4 a c} \sqrt {-b+\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}-\frac {e^{7/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{d} \sqrt {f}}{\sqrt [4]{e}}+2 x}{-\frac {\sqrt {d} f}{\sqrt {e}}-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {f} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt {f x}\right )}{2 \sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {e^{7/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{d} \sqrt {f}}{\sqrt [4]{e}}-2 x}{-\frac {\sqrt {d} f}{\sqrt {e}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {f} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt {f x}\right )}{2 \sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}\\ &=\frac {c^{3/4} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {c^{3/4} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}+\frac {c^{3/4} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {c^{3/4} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {e^{7/4} \log \left (\sqrt {d} \sqrt {f}+\sqrt {e} \sqrt {f} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {f x}\right )}{2 \sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}+\frac {e^{7/4} \log \left (\sqrt {d} \sqrt {f}+\sqrt {e} \sqrt {f} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {f x}\right )}{2 \sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}+\frac {e^{7/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {f x}}{\sqrt [4]{d} \sqrt {f}}\right )}{\sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {e^{7/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {f x}}{\sqrt [4]{d} \sqrt {f}}\right )}{\sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}\\ &=\frac {c^{3/4} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {c^{3/4} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {e^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {f x}}{\sqrt [4]{d} \sqrt {f}}\right )}{\sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}+\frac {e^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {f x}}{\sqrt [4]{d} \sqrt {f}}\right )}{\sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}+\frac {c^{3/4} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {c^{3/4} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {e^{7/4} \log \left (\sqrt {d} \sqrt {f}+\sqrt {e} \sqrt {f} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {f x}\right )}{2 \sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}+\frac {e^{7/4} \log \left (\sqrt {d} \sqrt {f}+\sqrt {e} \sqrt {f} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {f x}\right )}{2 \sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.25, size = 215, normalized size = 0.25 \begin {gather*} -\frac {\sqrt {x} \left (\sqrt {2} e^{7/4} \left (\tan ^{-1}\left (\frac {\sqrt {d}-\sqrt {e} x}{\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}}\right )-\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}}{\sqrt {d}+\sqrt {e} x}\right )\right )+d^{3/4} \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-c d \log \left (\sqrt {x}-\text {$\#$1}\right )+b e \log \left (\sqrt {x}-\text {$\#$1}\right )+c e \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]\right )}{2 d^{3/4} \left (c d^2+e (-b d+a e)\right ) \sqrt {f x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[f*x]*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

-1/2*(Sqrt[x]*(Sqrt[2]*e^(7/4)*(ArcTan[(Sqrt[d] - Sqrt[e]*x)/(Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x])] - ArcTanh[(Sqr
t[2]*d^(1/4)*e^(1/4)*Sqrt[x])/(Sqrt[d] + Sqrt[e]*x)]) + d^(3/4)*RootSum[a + b*#1^4 + c*#1^8 & , (-(c*d*Log[Sqr
t[x] - #1]) + b*e*Log[Sqrt[x] - #1] + c*e*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ]))/(d^(3/4)*(c*d^2 +
e*(-(b*d) + a*e))*Sqrt[f*x])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.17, size = 264, normalized size = 0.30

method	result	size
derivativedivides	\(2 f^{5} \left (\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+b \,f^{2} \textit {\_Z}^{4}+a \,f^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{4} c e -b e \,f^{2}+c d \,f^{2}\right ) \ln \left (\sqrt {f x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b \,f^{2}}}{4 f^{4} \left (a \,e^{2}-d e b +c \,d^{2}\right )}+\frac {e^{2} \left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {f x +\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} \sqrt {f x}\, \sqrt {2}+\sqrt {\frac {d \,f^{2}}{e}}}{f x -\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} \sqrt {f x}\, \sqrt {2}+\sqrt {\frac {d \,f^{2}}{e}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {f x}}{\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {f x}}{\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}}}-1\right )\right )}{8 f^{6} \left (a \,e^{2}-d e b +c \,d^{2}\right ) d}\right )\)	\(264\)
default	\(2 f^{5} \left (\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+b \,f^{2} \textit {\_Z}^{4}+a \,f^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{4} c e -b e \,f^{2}+c d \,f^{2}\right ) \ln \left (\sqrt {f x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b \,f^{2}}}{4 f^{4} \left (a \,e^{2}-d e b +c \,d^{2}\right )}+\frac {e^{2} \left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {f x +\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} \sqrt {f x}\, \sqrt {2}+\sqrt {\frac {d \,f^{2}}{e}}}{f x -\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} \sqrt {f x}\, \sqrt {2}+\sqrt {\frac {d \,f^{2}}{e}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {f x}}{\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {f x}}{\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}}}-1\right )\right )}{8 f^{6} \left (a \,e^{2}-d e b +c \,d^{2}\right ) d}\right )\)	\(264\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^2+d)/(c*x^4+b*x^2+a)/(f*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*f^5*(1/4/f^4/(a*e^2-b*d*e+c*d^2)*sum((-_R^4*c*e-b*e*f^2+c*d*f^2)/(2*_R^7*c+_R^3*b*f^2)*ln((f*x)^(1/2)-_R),_R
=RootOf(_Z^8*c+_Z^4*b*f^2+a*f^4))+1/8*e^2/f^6/(a*e^2-b*d*e+c*d^2)*(d*f^2/e)^(1/4)/d*2^(1/2)*(ln((f*x+(d*f^2/e)
^(1/4)*(f*x)^(1/2)*2^(1/2)+(d*f^2/e)^(1/2))/(f*x-(d*f^2/e)^(1/4)*(f*x)^(1/2)*2^(1/2)+(d*f^2/e)^(1/2)))+2*arcta
n(2^(1/2)/(d*f^2/e)^(1/4)*(f*x)^(1/2)+1)+2*arctan(2^(1/2)/(d*f^2/e)^(1/4)*(f*x)^(1/2)-1)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)/(c*x^4+b*x^2+a)/(f*x)^(1/2),x, algorithm="maxima")

[Out]

1/4*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*d^(1/4)*e^(1/4) + 2*e^(1/2*log(x) + 1/2))*e^(-1/4)/d^(1/4))*e^(7/4)
/d^(3/4) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*d^(1/4)*e^(1/4) - 2*e^(1/2*log(x) + 1/2))*e^(-1/4)/d^(1/4))*
e^(7/4)/d^(3/4) + sqrt(2)*e^(7/4)*log(sqrt(2)*d^(1/4)*e^(1/2*log(x) + 1/4) + sqrt(d) + e^(log(x) + 1/2))/d^(3/
4) - sqrt(2)*e^(7/4)*log(-sqrt(2)*d^(1/4)*e^(1/2*log(x) + 1/4) + sqrt(d) + e^(log(x) + 1/2))/d^(3/4))/(c*d^2*s
qrt(f) - b*d*sqrt(f)*e + a*sqrt(f)*e^2) - 2*e^(1/2*log(x) + 2)/(c*d^3*sqrt(f) - b*d^2*sqrt(f)*e + a*d*sqrt(f)*
e^2) + 2*sqrt(x)/(a*d*sqrt(f)) + integrate(-((c^2*d - b*c*e)*x^(7/2) + (b*c*d - (b^2 - a*c)*e)*x^(3/2))/(a^2*c
*d^2*sqrt(f) - a^2*b*d*sqrt(f)*e + (a*c^2*d^2*sqrt(f) - a*b*c*d*sqrt(f)*e + a^2*c*sqrt(f)*e^2)*x^4 + a^3*sqrt(
f)*e^2 + (a*b*c*d^2*sqrt(f) - a*b^2*d*sqrt(f)*e + a^2*b*sqrt(f)*e^2)*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)/(c*x^4+b*x^2+a)/(f*x)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {f x} \left (d + e x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)/(c*x**4+b*x**2+a)/(f*x)**(1/2),x)

[Out]

Integral(1/(sqrt(f*x)*(d + e*x**2)*(a + b*x**2 + c*x**4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)/(c*x^4+b*x^2+a)/(f*x)^(1/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 6.84, size = 2500, normalized size = 2.89 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((f*x)^(1/2)*(d + e*x^2)*(a + b*x^2 + c*x^4)),x)

[Out]

symsum(log(-root(8388608*a^7*b*c^11*d^18*e*f^6*h^12 - 513802240*a^10*b^2*c^7*d^11*e^8*f^6*h^12 - 381681664*a^1
1*b^2*c^6*d^9*e^10*f^6*h^12 - 381681664*a^9*b^2*c^8*d^13*e^6*f^6*h^12 - 300941312*a^9*b^5*c^5*d^10*e^9*f^6*h^1
2 - 300941312*a^8*b^5*c^6*d^12*e^7*f^6*h^12 + 293601280*a^10*b^3*c^6*d^10*e^9*f^6*h^12 + 293601280*a^9*b^3*c^7
*d^12*e^7*f^6*h^12 - 168820736*a^10*b^5*c^4*d^8*e^11*f^6*h^12 - 168820736*a^7*b^5*c^7*d^14*e^5*f^6*h^12 + 1660
68224*a^8*b^6*c^5*d^11*e^8*f^6*h^12 - 146800640*a^12*b^2*c^5*d^7*e^12*f^6*h^12 - 146800640*a^8*b^2*c^9*d^15*e^
4*f^6*h^12 + 124780544*a^10*b^4*c^5*d^9*e^10*f^6*h^12 + 124780544*a^8*b^4*c^7*d^13*e^6*f^6*h^12 + 119275520*a^
9*b^4*c^6*d^11*e^8*f^6*h^12 + 117440512*a^11*b^3*c^5*d^8*e^11*f^6*h^12 + 117440512*a^8*b^3*c^8*d^14*e^5*f^6*h^
12 + 102760448*a^9*b^6*c^4*d^9*e^10*f^6*h^12 + 102760448*a^7*b^6*c^6*d^13*e^6*f^6*h^12 + 91750400*a^11*b^4*c^4
*d^7*e^12*f^6*h^12 + 91750400*a^7*b^4*c^8*d^15*e^4*f^6*h^12 - 71065600*a^7*b^8*c^4*d^11*e^8*f^6*h^12 - 5344460
8*a^8*b^8*c^3*d^9*e^10*f^6*h^12 - 53444608*a^6*b^8*c^5*d^13*e^6*f^6*h^12 + 40370176*a^9*b^7*c^3*d^8*e^11*f^6*h
^12 + 40370176*a^6*b^7*c^6*d^14*e^5*f^6*h^12 - 36700160*a^11*b^5*c^3*d^6*e^13*f^6*h^12 - 36700160*a^6*b^5*c^8*
d^16*e^3*f^6*h^12 + 34078720*a^8*b^7*c^4*d^10*e^9*f^6*h^12 + 34078720*a^7*b^7*c^5*d^12*e^7*f^6*h^12 + 26214400
*a^12*b^4*c^3*d^5*e^14*f^6*h^12 + 26214400*a^6*b^4*c^9*d^17*e^2*f^6*h^12 + 22118400*a^7*b^9*c^3*d^10*e^9*f^6*h
^12 + 22118400*a^6*b^9*c^4*d^12*e^7*f^6*h^12 - 20971520*a^13*b^2*c^4*d^5*e^14*f^6*h^12 - 20971520*a^7*b^2*c^10
*d^17*e^2*f^6*h^12 + 18350080*a^10*b^7*c^2*d^6*e^13*f^6*h^12 + 18350080*a^5*b^7*c^7*d^16*e^3*f^6*h^12 - 166297
60*a^9*b^8*c^2*d^7*e^12*f^6*h^12 - 16629760*a^5*b^8*c^6*d^15*e^4*f^6*h^12 - 10485760*a^11*b^6*c^2*d^5*e^14*f^6
*h^12 - 10485760*a^5*b^6*c^8*d^17*e^2*f^6*h^12 + 9175040*a^10*b^6*c^3*d^7*e^12*f^6*h^12 + 9175040*a^6*b^6*c^7*
d^15*e^4*f^6*h^12 - 8388608*a^13*b^3*c^3*d^4*e^15*f^6*h^12 + 5619712*a^7*b^10*c^2*d^9*e^10*f^6*h^12 + 5619712*
a^5*b^10*c^4*d^13*e^6*f^6*h^12 - 5570560*a^6*b^11*c^2*d^10*e^9*f^6*h^12 - 5570560*a^5*b^11*c^3*d^12*e^7*f^6*h^
12 + 4358144*a^8*b^9*c^2*d^8*e^11*f^6*h^12 + 4358144*a^5*b^9*c^5*d^14*e^5*f^6*h^12 + 4259840*a^6*b^10*c^3*d^11
*e^8*f^6*h^12 + 3899392*a^4*b^10*c^5*d^15*e^4*f^6*h^12 - 3440640*a^4*b^9*c^6*d^16*e^3*f^6*h^12 + 3145728*a^12*
b^5*c^2*d^4*e^15*f^6*h^12 - 2523136*a^4*b^11*c^4*d^14*e^5*f^6*h^12 + 1802240*a^4*b^8*c^7*d^17*e^2*f^6*h^12 + 1
556480*a^5*b^12*c^2*d^11*e^8*f^6*h^12 + 1048576*a^14*b^2*c^3*d^3*e^16*f^6*h^12 + 688128*a^4*b^12*c^3*d^13*e^6*
f^6*h^12 - 393216*a^13*b^4*c^2*d^3*e^16*f^6*h^12 - 286720*a^3*b^12*c^4*d^15*e^4*f^6*h^12 + 229376*a^3*b^13*c^3
*d^14*e^5*f^6*h^12 + 229376*a^3*b^11*c^5*d^16*e^3*f^6*h^12 + 163840*a^4*b^13*c^2*d^12*e^7*f^6*h^12 - 114688*a^
3*b^14*c^2*d^13*e^6*f^6*h^12 - 114688*a^3*b^10*c^6*d^17*e^2*f^6*h^12 + 293601280*a^11*b*c^7*d^10*e^9*f^6*h^12
+ 293601280*a^10*b*c^8*d^12*e^7*f^6*h^12 + 176160768*a^12*b*c^6*d^8*e^11*f^6*h^12 + 176160768*a^9*b*c^9*d^14*e
^5*f^6*h^12 + 58720256*a^13*b*c^5*d^6*e^13*f^6*h^12 + 58720256*a^8*b*c^10*d^16*e^3*f^6*h^12 + 8388608*a^14*b*c
^4*d^4*e^15*f^6*h^12 - 8388608*a^6*b^3*c^10*d^18*e*f^6*h^12 + 3899392*a^8*b^10*c*d^7*e^12*f^6*h^12 - 3440640*a
^9*b^9*c*d^6*e^13*f^6*h^12 + 3145728*a^5*b^5*c^9*d^18*e*f^6*h^12 - 2523136*a^7*b^11*c*d^8*e^11*f^6*h^12 + 1802
240*a^10*b^8*c*d^5*e^14*f^6*h^12 + 688128*a^6*b^12*c*d^9*e^10*f^6*h^12 - 524288*a^11*b^7*c*d^4*e^15*f^6*h^12 -
524288*a^4*b^7*c^8*d^18*e*f^6*h^12 + 163840*a^5*b^13*c*d^10*e^9*f^6*h^12 - 163840*a^4*b^14*c*d^11*e^8*f^6*h^1
2 + 65536*a^12*b^6*c*d^3*e^16*f^6*h^12 + 32768*a^3*b^15*c*d^12*e^7*f^6*h^12 + 32768*a^3*b^9*c^7*d^18*e*f^6*h^1
2 - 73400320*a^11*c^8*d^11*e^8*f^6*h^12 - 58720256*a^12*c^7*d^9*e^10*f^6*h^12 - 58720256*a^10*c^9*d^13*e^6*f^6
*h^12 - 29360128*a^13*c^6*d^7*e^12*f^6*h^12 - 29360128*a^9*c^10*d^15*e^4*f^6*h^12 - 8388608*a^14*c^5*d^5*e^14*
f^6*h^12 - 8388608*a^8*c^11*d^17*e^2*f^6*h^12 - 1048576*a^15*c^4*d^3*e^16*f^6*h^12 - 286720*a^7*b^12*d^7*e^12*
f^6*h^12 + 229376*a^8*b^11*d^6*e^13*f^6*h^12 + 229376*a^6*b^13*d^8*e^11*f^6*h^12 - 114688*a^9*b^10*d^5*e^14*f^
6*h^12 - 114688*a^5*b^14*d^9*e^10*f^6*h^12 + 32768*a^10*b^9*d^4*e^15*f^6*h^12 + 32768*a^4*b^15*d^10*e^9*f^6*h^
12 - 4096*a^11*b^8*d^3*e^16*f^6*h^12 - 4096*a^3*b^16*d^11*e^8*f^6*h^12 + 1048576*a^6*b^2*c^11*d^19*f^6*h^12 -
393216*a^5*b^4*c^10*d^19*f^6*h^12 + 65536*a^4*b^6*c^9*d^19*f^6*h^12 - 4096*a^3*b^8*c^8*d^19*f^6*h^12 - 1048576
*a^7*c^12*d^19*f^6*h^12 + 262144*a^10*b*c^4*d*e^14*f^4*h^8 - 23552*a*b^6*c^8*d^14*e*f^4*h^8 - 16384*a^7*b^7*c*
d*e^14*f^4*h^8 - 3328*a*b^13*c*d^7*e^8*f^4*h^8 + 2429952*a^4*b^5*c^6*d^9*e^6*f^4*h^8 - 1865728*a^6*b^3*c^6*d^7
*e^8*f^4*h^8 - 1716224*a^4*b^4*c^7*d^10*e^5*f^4*h^8 + 1605632*a^6*b^2*c^7*d^8*e^7*f^4*h^8 + 1584384*a^5*b^5*c^
5*d^7*e^8*f^4*h^8 + 1572864*a^5*b^2*c^8*d^10*e^5*f^4*h^8 - 1433600*a^5*b^3*c^7*d^9*e^6*f^4*h^8 - 1261568*a^4*b
^6*c^5*d^8*e^7*f^4*h^8 - 1124352*a^3*b^4*c^8*d^12*e^3*f^4*h^8 - 1110016*a^7*b^3*c^5*d^5*e^10*f^4*h^8 + 1106176
*a^3*b^5*c^7*d^11*e^4*f^4*h^8 - 936960*a^5*b^6*...

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