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

Optimal. Leaf size=527 \[ \frac{g \sqrt{\sqrt{4 a c+b^2}+b} \left (\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{4 a c+b^2}+b}}\right ),-\frac{2 \sqrt{4 a c+b^2}}{b-\sqrt{4 a c+b^2}}\right )}{\sqrt{2} \sqrt{c} e \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1}{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1}} \sqrt{-a+b x^2+c x^4}}+\frac{\sqrt{\sqrt{4 a c+b^2}-b} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1} (e f-d g) \Pi \left (-\frac{\left (b-\sqrt{b^2+4 a c}\right ) e^2}{2 c d^2};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2+4 a c}-b}}\right )|\frac{b-\sqrt{b^2+4 a c}}{b+\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} d e \sqrt{-a+b x^2+c x^4}}-\frac{(e f-d g) \tanh ^{-1}\left (\frac{-2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt{-a+b x^2+c x^4} \sqrt{-a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt{-a e^4+b d^2 e^2+c d^4}} \]

[Out]

-((e*f - d*g)*ArcTanh[(b*d^2 - 2*a*e^2 + (2*c*d^2 + b*e^2)*x^2)/(2*Sqrt[c*d^4 + b*d^2*e^2 - a*e^4]*Sqrt[-a + b
*x^2 + c*x^4])])/(2*Sqrt[c*d^4 + b*d^2*e^2 - a*e^4]) + (Sqrt[b + Sqrt[b^2 + 4*a*c]]*g*(1 + (2*c*x^2)/(b - Sqrt
[b^2 + 4*a*c]))*EllipticF[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (-2*Sqrt[b^2 + 4*a*c])/(b -
Sqrt[b^2 + 4*a*c])])/(Sqrt[2]*Sqrt[c]*e*Sqrt[(1 + (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c]))/(1 + (2*c*x^2)/(b + Sqrt
[b^2 + 4*a*c]))]*Sqrt[-a + b*x^2 + c*x^4]) + (Sqrt[-b + Sqrt[b^2 + 4*a*c]]*(e*f - d*g)*Sqrt[1 + (2*c*x^2)/(b -
Sqrt[b^2 + 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticPi[-((b - Sqrt[b^2 + 4*a*c])*e^2)/(2*
c*d^2), ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[-b + Sqrt[b^2 + 4*a*c]]], (b - Sqrt[b^2 + 4*a*c])/(b + Sqrt[b^2 + 4*a*
c])])/(Sqrt[2]*Sqrt[c]*d*e*Sqrt[-a + b*x^2 + c*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.717031, antiderivative size = 527, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.323, Rules used = {1741, 12, 1247, 724, 206, 1710, 1104, 418, 1220, 537} \[ \frac{\sqrt{\sqrt{4 a c+b^2}-b} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1} (e f-d g) \Pi \left (-\frac{\left (b-\sqrt{b^2+4 a c}\right ) e^2}{2 c d^2};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2+4 a c}-b}}\right )|\frac{b-\sqrt{b^2+4 a c}}{b+\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} d e \sqrt{-a+b x^2+c x^4}}+\frac{g \sqrt{\sqrt{4 a c+b^2}+b} \left (\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|-\frac{2 \sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} e \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1}{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1}} \sqrt{-a+b x^2+c x^4}}-\frac{(e f-d g) \tanh ^{-1}\left (\frac{-2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt{-a+b x^2+c x^4} \sqrt{-a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt{-a e^4+b d^2 e^2+c d^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((e*f - d*g)*ArcTanh[(b*d^2 - 2*a*e^2 + (2*c*d^2 + b*e^2)*x^2)/(2*Sqrt[c*d^4 + b*d^2*e^2 - a*e^4]*Sqrt[-a + b
*x^2 + c*x^4])])/(2*Sqrt[c*d^4 + b*d^2*e^2 - a*e^4]) + (Sqrt[b + Sqrt[b^2 + 4*a*c]]*g*(1 + (2*c*x^2)/(b - Sqrt
[b^2 + 4*a*c]))*EllipticF[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (-2*Sqrt[b^2 + 4*a*c])/(b -
Sqrt[b^2 + 4*a*c])])/(Sqrt[2]*Sqrt[c]*e*Sqrt[(1 + (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c]))/(1 + (2*c*x^2)/(b + Sqrt
[b^2 + 4*a*c]))]*Sqrt[-a + b*x^2 + c*x^4]) + (Sqrt[-b + Sqrt[b^2 + 4*a*c]]*(e*f - d*g)*Sqrt[1 + (2*c*x^2)/(b -
Sqrt[b^2 + 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticPi[-((b - Sqrt[b^2 + 4*a*c])*e^2)/(2*
c*d^2), ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[-b + Sqrt[b^2 + 4*a*c]]], (b - Sqrt[b^2 + 4*a*c])/(b + Sqrt[b^2 + 4*a*
c])])/(Sqrt[2]*Sqrt[c]*d*e*Sqrt[-a + b*x^2 + c*x^4])

Rule 1741

Int[(Px_)/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[Px, x,
0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^
2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4)/((d^2 - e^2*x^2)*Sqrt[a + b*
x^2 + c*x^4]), x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + b*d^2*e
^2 + a*e^4, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

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

Rule 1710

Int[((A_.) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Dist
[B/e, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[(e*A - d*B)/e, Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),
x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^
2 - a*e^2, 0] && NegQ[c/a]

Rule 1104

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[1 + (2*
c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)])/Sqrt[a + b*x^2 + c*x^4], Int[1/(Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[
1 + (2*c*x^2)/(b + q)]), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 1220

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c,
2]}, Dist[(Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)])/Sqrt[a + b*x^2 + c*x^4], Int[1/((d + e*x^
2)*Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 -
4*a*c, 0] && NegQ[c/a]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rubi steps

\begin{align*} \int \frac{f+g x}{(d+e x) \sqrt{-a+b x^2+c x^4}} \, dx &=\int \frac{(-e f+d g) x}{\left (d^2-e^2 x^2\right ) \sqrt{-a+b x^2+c x^4}} \, dx+\int \frac{d f-e g x^2}{\left (d^2-e^2 x^2\right ) \sqrt{-a+b x^2+c x^4}} \, dx\\ &=\frac{g \int \frac{1}{\sqrt{-a+b x^2+c x^4}} \, dx}{e}+\frac{(d (e f-d g)) \int \frac{1}{\left (d^2-e^2 x^2\right ) \sqrt{-a+b x^2+c x^4}} \, dx}{e}+(-e f+d g) \int \frac{x}{\left (d^2-e^2 x^2\right ) \sqrt{-a+b x^2+c x^4}} \, dx\\ &=\frac{1}{2} (-e f+d g) \operatorname{Subst}\left (\int \frac{1}{\left (d^2-e^2 x\right ) \sqrt{-a+b x+c x^2}} \, dx,x,x^2\right )+\frac{\left (g \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}\right ) \int \frac{1}{\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}} \, dx}{e \sqrt{-a+b x^2+c x^4}}+\frac{\left (d (e f-d g) \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}\right ) \int \frac{1}{\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}} \left (d^2-e^2 x^2\right )} \, dx}{e \sqrt{-a+b x^2+c x^4}}\\ &=\frac{\sqrt{b+\sqrt{b^2+4 a c}} g \left (1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|-\frac{2 \sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} e \sqrt{\frac{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}}{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}} \sqrt{-a+b x^2+c x^4}}+\frac{\sqrt{-b+\sqrt{b^2+4 a c}} (e f-d g) \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}} \Pi \left (-\frac{\left (b-\sqrt{b^2+4 a c}\right ) e^2}{2 c d^2};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{-b+\sqrt{b^2+4 a c}}}\right )|\frac{b-\sqrt{b^2+4 a c}}{b+\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} d e \sqrt{-a+b x^2+c x^4}}+(e f-d g) \operatorname{Subst}\left (\int \frac{1}{4 c d^4+4 b d^2 e^2-4 a e^4-x^2} \, dx,x,\frac{-b d^2+2 a e^2-\left (2 c d^2+b e^2\right ) x^2}{\sqrt{-a+b x^2+c x^4}}\right )\\ &=-\frac{(e f-d g) \tanh ^{-1}\left (\frac{b d^2-2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt{c d^4+b d^2 e^2-a e^4} \sqrt{-a+b x^2+c x^4}}\right )}{2 \sqrt{c d^4+b d^2 e^2-a e^4}}+\frac{\sqrt{b+\sqrt{b^2+4 a c}} g \left (1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|-\frac{2 \sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} e \sqrt{\frac{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}}{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}} \sqrt{-a+b x^2+c x^4}}+\frac{\sqrt{-b+\sqrt{b^2+4 a c}} (e f-d g) \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}} \Pi \left (-\frac{\left (b-\sqrt{b^2+4 a c}\right ) e^2}{2 c d^2};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{-b+\sqrt{b^2+4 a c}}}\right )|\frac{b-\sqrt{b^2+4 a c}}{b+\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} d e \sqrt{-a+b x^2+c x^4}}\\ \end{align*}

Mathematica [C]  time = 7.86307, size = 3658, normalized size = 6.94 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*g*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*EllipticF[I*
ArcSinh[Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 + 4*a*c]))]*x], (-b - Sqrt[b^2 + 4*a*c])/(-b + Sqrt[b^2 + 4*a*c])])/(S
qrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 + 4*a*c]))]*e*Sqrt[-a + b*x^2 + c*x^4]) + (2*(Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c
]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2])*f*(-(Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2]) + x)^
2*Sqrt[(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c]*(-(Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2]) + x))/((Sqrt[-(b/c) -
Sqrt[b^2 + 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/
c]/Sqrt[2]) + x))]*Sqrt[(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c]*(Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2] + x))/((
Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt
[b^2 + 4*a*c]/c]/Sqrt[2]) + x))]*Sqrt[((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(
Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*
a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - 2*x))]*((-d + (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqr
t[2])*EllipticF[ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sq
rt[(-b - Sqrt[b^2 + 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*
(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 +
4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])^2] - Sqrt[2]*Sqrt[(-b - S
qrt[b^2 + 4*a*c])/c]*e*EllipticPi[((Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 + 4*a*
c]/c]/Sqrt[2])*(d + (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2]))/((-(Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sq
rt[2]) + Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2])*(d - (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2])), Ar
cSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2
+ 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b
- Sqrt[b^2 + 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])^2/(Sq
rt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])^2]))/(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c]*(Sqr
t[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2])*(-d - (Sqrt[-(b/c) - Sqr
t[b^2 + 4*a*c]/c]*e)/Sqrt[2])*(d - (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2])*Sqrt[-a + b*x^2 + c*x^4]) -
(2*(Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2])*d*g*(-(Sqrt[-(b/
c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2]) + x)^2*Sqrt[(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c]*(-(Sqrt[-(b/c) + Sqrt[b^2 + 4
*a*c]/c]/Sqrt[2]) + x))/((Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt
[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2]) + x))]*Sqrt[(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c]*(Sqrt[-(b/c)
+ Sqrt[b^2 + 4*a*c]/c]/Sqrt[2] + x))/((Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 +
4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2]) + x))]*Sqrt[((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/
c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2
+ 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - 2*x))]*((-d + (S
qrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2])*EllipticF[ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[
(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 + 4*a*c])
/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[
b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2
+ 4*a*c])/c])^2] - Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c]*e*EllipticPi[((Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]
/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2])*(d + (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2]))/(
(-(Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2])*(d - (Sqrt[-(b/c)
- Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2])), ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4
*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + S
qrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] +
Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])^2])
)/(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c]*(Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 + 4*a
*c]/c]/Sqrt[2])*e*(-d - (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2])*(d - (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/
c]*e)/Sqrt[2])*Sqrt[-a + b*x^2 + c*x^4])

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Maple [A]  time = 0.021, size = 439, normalized size = 0.8 \begin{align*}{\frac{g}{2\,e}\sqrt{4+2\,{\frac{ \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4-2\,{\frac{ \left ( b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{x}{2}\sqrt{-2\,{\frac{-b+\sqrt{4\,ac+{b}^{2}}}{a}}}},{\frac{1}{2}\sqrt{-4-2\,{\frac{b \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{-2\,{\frac{-b+\sqrt{4\,ac+{b}^{2}}}{a}}}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}-a}}}}+{\frac{-dg+ef}{{e}^{2}} \left ( -{\frac{1}{2}{\it Artanh} \left ({\frac{1}{2} \left ( 2\,{\frac{c{x}^{2}{d}^{2}}{{e}^{2}}}+b{x}^{2}+{\frac{b{d}^{2}}{{e}^{2}}}-2\,a \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+{\frac{b{d}^{2}}{{e}^{2}}}-a}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}-a}}}} \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+{\frac{b{d}^{2}}{{e}^{2}}}-a}}}}+{\frac{e}{d}\sqrt{1+{\frac{{x}^{2}}{2\,a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}\sqrt{1-{\frac{{x}^{2}}{2\,a} \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }}{\it EllipticPi} \left ( \sqrt{-{\frac{1}{2\,a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}x,-2\,{\frac{a{e}^{2}}{ \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ){d}^{2}}},{\frac{\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{-{\frac{1}{2\,a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{-{\frac{1}{2\,a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}-a}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*g/e/(-2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2)*(4+2*(-b+(4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4-2*(b+(4*a*c+b^2)^(1/2)
)/a*x^2)^(1/2)/(c*x^4+b*x^2-a)^(1/2)*EllipticF(1/2*x*(-2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4-2*b*(b+(4*a*c
+b^2)^(1/2))/a/c)^(1/2))+(-d*g+e*f)/e^2*(-1/2/(c*d^4/e^4+b*d^2/e^2-a)^(1/2)*arctanh(1/2*(2*c*x^2*d^2/e^2+b*x^2
+b*d^2/e^2-2*a)/(c*d^4/e^4+b*d^2/e^2-a)^(1/2)/(c*x^4+b*x^2-a)^(1/2))+1/(-1/2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2)/d
*e*(1+1/2*(-b+(4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(1-1/2*(b+(4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2-a)^(1/2)*
EllipticPi((-1/2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2)*x,-2/(-b+(4*a*c+b^2)^(1/2))*a/d^2*e^2,1/2*2^(1/2)*((b+(4*a*c+
b^2)^(1/2))/a)^(1/2)/(-1/2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{g x + f}{\sqrt{c x^{4} + b x^{2} - a}{\left (e x + d\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((g*x + f)/(sqrt(c*x^4 + b*x^2 - a)*(e*x + d)), x)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{g x + f}{\sqrt{c x^{4} + b x^{2} - a}{\left (e x + d\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((g*x + f)/(sqrt(c*x^4 + b*x^2 - a)*(e*x + d)), x)