3.413 \(\int \frac {f+g x}{(d+e x) \sqrt {-a+b x^2+c x^4}} \, dx\)
Optimal. Leaf size=527 \[ \frac {\sqrt {\sqrt {4 a c+b^2}-b} \sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1} \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1} (e f-d g) \Pi \left (-\frac {\left (b-\sqrt {b^2+4 a c}\right ) e^2}{2 c d^2};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2+4 a c}-b}}\right )|\frac {b-\sqrt {b^2+4 a c}}{b+\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {c} d e \sqrt {-a+b x^2+c x^4}}+\frac {g \sqrt {\sqrt {4 a c+b^2}+b} \left (\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|-\frac {2 \sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {c} e \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1}{\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1}} \sqrt {-a+b x^2+c x^4}}-\frac {(e f-d g) \tanh ^{-1}\left (\frac {-2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {-a+b x^2+c x^4} \sqrt {-a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt {-a e^4+b d^2 e^2+c d^4}} \]
[Out]
-1/2*(-d*g+e*f)*arctanh(1/2*(b*d^2-2*a*e^2+(b*e^2+2*c*d^2)*x^2)/(-a*e^4+b*d^2*e^2+c*d^4)^(1/2)/(c*x^4+b*x^2-a)
^(1/2))/(-a*e^4+b*d^2*e^2+c*d^4)^(1/2)+1/2*g*(1/(1+2*c*x^2/(b+(4*a*c+b^2)^(1/2))))^(1/2)*(1+2*c*x^2/(b+(4*a*c+
b^2)^(1/2)))^(1/2)*EllipticF(x*2^(1/2)*c^(1/2)/(b+(4*a*c+b^2)^(1/2))^(1/2)/(1+2*c*x^2/(b+(4*a*c+b^2)^(1/2)))^(
1/2),(-2*(4*a*c+b^2)^(1/2)/(b-(4*a*c+b^2)^(1/2)))^(1/2))*(1+2*c*x^2/(b-(4*a*c+b^2)^(1/2)))*(b+(4*a*c+b^2)^(1/2
))^(1/2)/e*2^(1/2)/c^(1/2)/(c*x^4+b*x^2-a)^(1/2)/((1+2*c*x^2/(b-(4*a*c+b^2)^(1/2)))/(1+2*c*x^2/(b+(4*a*c+b^2)^
(1/2))))^(1/2)+1/2*(-d*g+e*f)*EllipticPi(x*2^(1/2)*c^(1/2)/(-b+(4*a*c+b^2)^(1/2))^(1/2),-1/2*e^2*(b-(4*a*c+b^2
)^(1/2))/c/d^2,((b-(4*a*c+b^2)^(1/2))/(b+(4*a*c+b^2)^(1/2)))^(1/2))*(1+2*c*x^2/(b-(4*a*c+b^2)^(1/2)))^(1/2)*(-
b+(4*a*c+b^2)^(1/2))^(1/2)*(1+2*c*x^2/(b+(4*a*c+b^2)^(1/2)))^(1/2)/d/e*2^(1/2)/c^(1/2)/(c*x^4+b*x^2-a)^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.72, antiderivative size = 527, normalized size of antiderivative = 1.00,
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 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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 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 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)
])
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 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 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 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 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 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]
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 = 7.87, 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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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {g x + f}{\sqrt {c x^{4} + b x^{2} - a} {\left (e x + d\right )}}\,{d x} \]
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)
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maple [A] time = 0.02, size = 439, normalized size = 0.83 \[ \frac {\sqrt {\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \sqrt {-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}+4}\, g \EllipticF \left (\frac {\sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}\, x}{2}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )}{2 \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}-a}\, e}+\frac {\left (-d g +e f \right ) \left (\frac {\sqrt {\frac {\left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{2 a}+1}\, \sqrt {-\frac {\left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{2 a}+1}\, e \EllipticPi \left (\sqrt {-\frac {-b +\sqrt {4 a c +b^{2}}}{2 a}}\, x , -\frac {2 a \,e^{2}}{\left (-b +\sqrt {4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}}{2 \sqrt {-\frac {-b +\sqrt {4 a c +b^{2}}}{2 a}}}\right )}{\sqrt {-\frac {-b +\sqrt {4 a c +b^{2}}}{2 a}}\, \sqrt {c \,x^{4}+b \,x^{2}-a}\, d}-\frac {\arctanh \left (\frac {b \,x^{2}+\frac {2 c \,d^{2} x^{2}}{e^{2}}-2 a +\frac {b \,d^{2}}{e^{2}}}{2 \sqrt {-a +\frac {b \,d^{2}}{e^{2}}+\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}\right )}{2 \sqrt {-a +\frac {b \,d^{2}}{e^{2}}+\frac {c \,d^{4}}{e^{4}}}}\right )}{e^{2}} \]
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)*(2*(-b+(4*a*c+b^2)^(1/2))/a*x^2+4)^(1/2)*(-2*(b+(4*a*c+b^2)^(1/2))
/a*x^2+4)^(1/2)/(c*x^4+b*x^2-a)^(1/2)*EllipticF(1/2*(-2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2)*x,1/2*(-2*(b+(4*a*c+b^
2)^(1/2))/a*b/c-4)^(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*d^2/e^2*x^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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {g x + f}{\sqrt {c x^{4} + b x^{2} - a} {\left (e x + d\right )}}\,{d x} \]
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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {f+g\,x}{\left (d+e\,x\right )\,\sqrt {c\,x^4+b\,x^2-a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f + g*x)/((d + e*x)*(b*x^2 - a + c*x^4)^(1/2)),x)
[Out]
int((f + g*x)/((d + e*x)*(b*x^2 - a + c*x^4)^(1/2)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f + g x}{\left (d + e x\right ) \sqrt {- a + b x^{2} + c x^{4}}}\, dx \]
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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