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

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 2.50858, antiderivative size = 866, normalized size of antiderivative = 1., 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 = {1269, 1424, 211, 1165, 628, 1162, 617, 204, 1422, 212, 208, 205} \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{f x}}{\sqrt [4]{d} \sqrt{f}}\right ) e^{7/4}}{\sqrt{2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{f x}}{\sqrt [4]{d} \sqrt{f}}+1\right ) e^{7/4}}{\sqrt{2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}-\frac{\log \left (\sqrt{e} \sqrt{f} x+\sqrt{d} \sqrt{f}-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{f x}\right ) e^{7/4}}{2 \sqrt{2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}+\frac{\log \left (\sqrt{e} \sqrt{f} x+\sqrt{d} \sqrt{f}+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{f x}\right ) e^{7/4}}{2 \sqrt{2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}+\frac{c^{3/4} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{f x}}{\sqrt [4]{-b-\sqrt{b^2-4 a c}} \sqrt{f}}\right )}{\sqrt [4]{2} \sqrt{b^2-4 a c} \left (-b-\sqrt{b^2-4 a c}\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}-\frac{c^{3/4} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{f x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b} \sqrt{f}}\right )}{\sqrt [4]{2} \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}-b\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}+\frac{c^{3/4} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{f x}}{\sqrt [4]{-b-\sqrt{b^2-4 a c}} \sqrt{f}}\right )}{\sqrt [4]{2} \sqrt{b^2-4 a c} \left (-b-\sqrt{b^2-4 a c}\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}-\frac{c^{3/4} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{f x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b} \sqrt{f}}\right )}{\sqrt [4]{2} \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}-b\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}} \]

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 1269

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 1424

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]

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 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]

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 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 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 1422

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

Rule 212

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

Rule 208

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

Rule 205

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

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 \operatorname{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 \operatorname{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 \operatorname{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 ) \operatorname{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 \operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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} \operatorname{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} \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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*}

Mathematica [C]  time = 0.374288, size = 267, normalized size = 0.31 \[ \frac{\sqrt{x} \left (\sqrt{2} e^{7/4} \left (-\log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )+\log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}+1\right )\right )-2 d^{3/4} \text{RootSum}\left [\text{$\#$1}^4 b+\text{$\#$1}^8 c+a\& ,\frac{\text{$\#$1}^4 c e \log \left (\sqrt{x}-\text{$\#$1}\right )+b e \log \left (\sqrt{x}-\text{$\#$1}\right )-c d \log \left (\sqrt{x}-\text{$\#$1}\right )}{\text{$\#$1}^3 b+2 \text{$\#$1}^7 c}\& \right ]\right )}{4 d^{3/4} \sqrt{f x} \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.092, size = 336, normalized size = 0.4 \begin{align*}{\frac{{e}^{2}\sqrt{2}}{4\,f \left ( a{e}^{2}-deb+c{d}^{2} \right ) d}\sqrt [4]{{\frac{d{f}^{2}}{e}}}\ln \left ({ \left ( fx+\sqrt [4]{{\frac{d{f}^{2}}{e}}}\sqrt{fx}\sqrt{2}+\sqrt{{\frac{d{f}^{2}}{e}}} \right ) \left ( fx-\sqrt [4]{{\frac{d{f}^{2}}{e}}}\sqrt{fx}\sqrt{2}+\sqrt{{\frac{d{f}^{2}}{e}}} \right ) ^{-1}} \right ) }+{\frac{{e}^{2}\sqrt{2}}{2\,f \left ( a{e}^{2}-deb+c{d}^{2} \right ) d}\sqrt [4]{{\frac{d{f}^{2}}{e}}}\arctan \left ({\sqrt{2}\sqrt{fx}{\frac{1}{\sqrt [4]{{\frac{d{f}^{2}}{e}}}}}}+1 \right ) }+{\frac{{e}^{2}\sqrt{2}}{2\,f \left ( a{e}^{2}-deb+c{d}^{2} \right ) d}\sqrt [4]{{\frac{d{f}^{2}}{e}}}\arctan \left ({\sqrt{2}\sqrt{fx}{\frac{1}{\sqrt [4]{{\frac{d{f}^{2}}{e}}}}}}-1 \right ) }+{\frac{f}{2\,a{e}^{2}-2\,deb+2\,c{d}^{2}}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+b{f}^{2}{{\it \_Z}}^{4}+a{f}^{4} \right ) }{\frac{-{{\it \_R}}^{4}ce-be{f}^{2}+cd{f}^{2}}{2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b{f}^{2}}\ln \left ( \sqrt{fx}-{\it \_R} \right ) }} \end{align*}

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)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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

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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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]

Exception raised: TypeError