3.13 \(\int \frac{A+B x^2}{(d+e x^2)^2 (a+c x^4)^{3/2}} \, dx\)

Optimal. Leaf size=1494 \[ \text{result too large to display} \]

[Out]

(c*x*(A*c*d^2 + 2*a*B*d*e - a*A*e^2 + (B*c*d^2 - 2*A*c*d*e - a*B*e^2)*x^2))/(2*a*(c*d^2 + a*e^2)^2*Sqrt[a + c*
x^4]) + (Sqrt[c]*e^2*(B*d - A*e)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (Sqrt[c]
*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*x*Sqrt[a + c*x^4])/(2*a*(c*d^2 + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (e^3*(B*
d - A*e)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)^2*(d + e*x^2)) - (e^(3/2)*(B*d - A*e)*(3*c*d^2 + a*e^2)*ArcTa
n[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(4*d^(3/2)*(c*d^2 + a*e^2)^(5/2)) - (e^(3/2)*(B*
c*d^2 - 2*A*c*d*e - a*B*e^2)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(2*Sqrt[d]*(c*
d^2 + a*e^2)^(5/2)) - (a^(1/4)*c^(1/4)*e^2*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqr
t[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*d*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (c^(1/4)
*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE
[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) - (c^(1/4)*e*(B*d - A*e)*(
Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/
2])/(2*a^(1/4)*d*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) - (c^(1/4)*e*(B*c*d^2 - 2*A*c*d*e -
a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^
(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) - (c^(1/4)*(B*c*d^2 - 2*A*
c*d*e - a*B*e^2 - (Sqrt[c]*(A*c*d^2 + 2*a*B*d*e - a*A*e^2))/Sqrt[a])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/
(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(3/4)*(c*d^2 + a*e^2)^2*Sqrt[a
+ c*x^4]) + (e*(Sqrt[c]*d + Sqrt[a]*e)*(B*d - A*e)*(3*c*d^2 + a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/
(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)
/a^(1/4)], 1/2])/(8*a^(1/4)*c^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (a^(3/4)*
e*((Sqrt[c]*d)/Sqrt[a] + e)^2*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a
] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4
)], 1/2])/(4*c^(1/4)*d*(c*d^2 - a*e^2)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4])

________________________________________________________________________________________

Rubi [A]  time = 2.0065, antiderivative size = 1494, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {1721, 1179, 1198, 220, 1196, 1224, 1715, 1709, 1707, 1217} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/((d + e*x^2)^2*(a + c*x^4)^(3/2)),x]

[Out]

(c*x*(A*c*d^2 + 2*a*B*d*e - a*A*e^2 + (B*c*d^2 - 2*A*c*d*e - a*B*e^2)*x^2))/(2*a*(c*d^2 + a*e^2)^2*Sqrt[a + c*
x^4]) + (Sqrt[c]*e^2*(B*d - A*e)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (Sqrt[c]
*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*x*Sqrt[a + c*x^4])/(2*a*(c*d^2 + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (e^3*(B*
d - A*e)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)^2*(d + e*x^2)) - (e^(3/2)*(B*d - A*e)*(3*c*d^2 + a*e^2)*ArcTa
n[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(4*d^(3/2)*(c*d^2 + a*e^2)^(5/2)) - (e^(3/2)*(B*
c*d^2 - 2*A*c*d*e - a*B*e^2)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(2*Sqrt[d]*(c*
d^2 + a*e^2)^(5/2)) - (a^(1/4)*c^(1/4)*e^2*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqr
t[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*d*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (c^(1/4)
*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE
[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) - (c^(1/4)*e*(B*d - A*e)*(
Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/
2])/(2*a^(1/4)*d*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) - (c^(1/4)*e*(B*c*d^2 - 2*A*c*d*e -
a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^
(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) - (c^(1/4)*(B*c*d^2 - 2*A*
c*d*e - a*B*e^2 - (Sqrt[c]*(A*c*d^2 + 2*a*B*d*e - a*A*e^2))/Sqrt[a])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/
(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(3/4)*(c*d^2 + a*e^2)^2*Sqrt[a
+ c*x^4]) + (e*(Sqrt[c]*d + Sqrt[a]*e)*(B*d - A*e)*(3*c*d^2 + a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/
(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)
/a^(1/4)], 1/2])/(8*a^(1/4)*c^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (a^(3/4)*
e*((Sqrt[c]*d)/Sqrt[a] + e)^2*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a
] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4
)], 1/2])/(4*c^(1/4)*d*(c*d^2 - a*e^2)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4])

Rule 1721

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a +
c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x^2] && NeQ[c*d^
2 + a*e^2, 0] && IntegerQ[p + 1/2] && IntegerQ[q]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)*(a + c*x^4)^(p + 1))/(
4*a*(p + 1)), x] + Dist[1/(4*a*(p + 1)), Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x], x
] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1198

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

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1196

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

Rule 1224

Int[((d_) + (e_.)*(x_)^2)^(q_)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> -Simp[(e^2*x*(d + e*x^2)^(q + 1)*Sqrt[a
+ c*x^4])/(2*d*(q + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*d*(q + 1)*(c*d^2 + a*e^2)), Int[((d + e*x^2)^(q + 1)*
Simp[a*e^2*(2*q + 3) + 2*c*d^2*(q + 1) - 2*e*c*d*(q + 1)*x^2 + c*e^2*(2*q + 5)*x^4, x])/Sqrt[a + c*x^4], x], x
] /; FreeQ[{a, c, d, e}, x] && ILtQ[q, -1]

Rule 1715

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2], A = Coeff[P4x
, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -Dist[C/(e*q), Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] +
Dist[1/(c*e), Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] /;
FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]

Rule 1709

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

Rule 1707

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]
}, -Simp[((B*d - A*e)*ArcTan[(Rt[(c*d)/e + (a*e)/d, 2]*x)/Sqrt[a + c*x^4]])/(2*d*e*Rt[(c*d)/e + (a*e)/d, 2]),
x] + Simp[((B*d + A*e)*(A + B*x^2)*Sqrt[(A^2*(a + c*x^4))/(a*(A + B*x^2)^2)]*EllipticPi[Cancel[-((B*d - A*e)^2
/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2])/(4*d*e*A*q*Sqrt[a + c*x^4]), x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 1217

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

Rubi steps

\begin{align*} \int \frac{A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx &=\int \left (\frac{c \left (A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^4\right )^{3/2}}+\frac{e (-B d+A e)}{\left (c d^2+a e^2\right ) \left (d+e x^2\right )^2 \sqrt{a+c x^4}}+\frac{e \left (-B c d^2+2 A c d e+a B e^2\right )}{\left (c d^2+a e^2\right )^2 \left (d+e x^2\right ) \sqrt{a+c x^4}}\right ) \, dx\\ &=\frac{c \int \frac{A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2}{\left (a+c x^4\right )^{3/2}} \, dx}{\left (c d^2+a e^2\right )^2}-\frac{(e (B d-A e)) \int \frac{1}{\left (d+e x^2\right )^2 \sqrt{a+c x^4}} \, dx}{c d^2+a e^2}-\frac{\left (e \left (B c d^2-2 A c d e-a B e^2\right )\right ) \int \frac{1}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx}{\left (c d^2+a e^2\right )^2}\\ &=\frac{c x \left (A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}-\frac{e^3 (B d-A e) x \sqrt{a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac{c \int \frac{-A c d^2-2 a B d e+a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2}{\sqrt{a+c x^4}} \, dx}{2 a \left (c d^2+a e^2\right )^2}+\frac{(e (B d-A e)) \int \frac{-2 c d^2-a e^2+2 c d e x^2+c e^2 x^4}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx}{2 d \left (c d^2+a e^2\right )^2}-\frac{\left (\sqrt{c} e \left (B c d^2-2 A c d e-a B e^2\right )\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{\left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )^2}+\frac{\left (\sqrt{a} e^2 \left (B c d^2-2 A c d e-a B e^2\right )\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx}{\left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )^2}\\ &=\frac{c x \left (A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}-\frac{e^3 (B d-A e) x \sqrt{a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac{e^{3/2} \left (B c d^2-2 A c d e-a B e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{2 \sqrt{d} \left (c d^2+a e^2\right )^{5/2}}-\frac{\sqrt [4]{c} e \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}+\frac{e \left (\sqrt{c} d+\sqrt{a} e\right ) \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}+\frac{(B d-A e) \int \frac{\sqrt{a} c^{3/2} d e^2+c e \left (-2 c d^2-a e^2\right )+\left (2 c^2 d e^2-c e^2 \left (c d-\sqrt{a} \sqrt{c} e\right )\right ) x^2}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx}{2 c d \left (c d^2+a e^2\right )^2}-\frac{\left (\sqrt{a} \sqrt{c} e^2 (B d-A e)\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{2 d \left (c d^2+a e^2\right )^2}+\frac{\left (\sqrt{c} \left (B c d^2-2 A c d e-a B e^2\right )\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{2 \sqrt{a} \left (c d^2+a e^2\right )^2}-\frac{\left (\sqrt{c} \left (B c d^2-2 A c d e-a B e^2-\frac{\sqrt{c} \left (A c d^2+2 a B d e-a A e^2\right )}{\sqrt{a}}\right )\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{2 \sqrt{a} \left (c d^2+a e^2\right )^2}\\ &=\frac{c x \left (A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}+\frac{\sqrt{c} e^2 (B d-A e) x \sqrt{a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt{c} \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt{a+c x^4}}{2 a \left (c d^2+a e^2\right )^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{e^3 (B d-A e) x \sqrt{a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac{e^{3/2} \left (B c d^2-2 A c d e-a B e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{2 \sqrt{d} \left (c d^2+a e^2\right )^{5/2}}-\frac{\sqrt [4]{a} \sqrt [4]{c} e^2 (B d-A e) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 d \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}+\frac{\sqrt [4]{c} \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}-\frac{\sqrt [4]{c} e \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}-\frac{\sqrt [4]{c} \left (B c d^2-2 A c d e-a B e^2-\frac{\sqrt{c} \left (A c d^2+2 a B d e-a A e^2\right )}{\sqrt{a}}\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}+\frac{e \left (\sqrt{c} d+\sqrt{a} e\right ) \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}-\frac{\left (\sqrt{c} e (B d-A e)\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{d \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )}+\frac{\left (\sqrt{a} e^2 (B d-A e) \left (3 c d^2+a e^2\right )\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx}{2 d \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )^2}\\ &=\frac{c x \left (A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}+\frac{\sqrt{c} e^2 (B d-A e) x \sqrt{a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt{c} \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt{a+c x^4}}{2 a \left (c d^2+a e^2\right )^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{e^3 (B d-A e) x \sqrt{a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac{e^{3/2} (B d-A e) \left (3 c d^2+a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{4 d^{3/2} \left (c d^2+a e^2\right )^{5/2}}-\frac{e^{3/2} \left (B c d^2-2 A c d e-a B e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{2 \sqrt{d} \left (c d^2+a e^2\right )^{5/2}}-\frac{\sqrt [4]{a} \sqrt [4]{c} e^2 (B d-A e) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 d \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}+\frac{\sqrt [4]{c} \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}-\frac{\sqrt [4]{c} e (B d-A e) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} d \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right ) \sqrt{a+c x^4}}-\frac{\sqrt [4]{c} e \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}-\frac{\sqrt [4]{c} \left (B c d^2-2 A c d e-a B e^2-\frac{\sqrt{c} \left (A c d^2+2 a B d e-a A e^2\right )}{\sqrt{a}}\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}+\frac{e \left (\sqrt{c} d+\sqrt{a} e\right ) (B d-A e) \left (3 c d^2+a e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}+\frac{e \left (\sqrt{c} d+\sqrt{a} e\right ) \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}\\ \end{align*}

Mathematica [C]  time = 1.62582, size = 427, normalized size = 0.29 \[ \frac{d \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left (c d x \left (d+e x^2\right ) \left (-a A e^2+a B e \left (2 d-e x^2\right )+A c d \left (d-2 e x^2\right )+B c d^2 x^2\right )+a e^3 x \left (a+c x^4\right ) (A e-B d)\right )-\sqrt{\frac{c x^4}{a}+1} \left (d+e x^2\right ) \left (i \left (\sqrt{c} d \left (\sqrt{c} d-i \sqrt{a} e\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\right ),-1\right ) \left (i \sqrt{a} \sqrt{c} d (B d-A e)+a e (2 B d-A e)+A c d^2\right )+a e \left (a A e^3+a B d e^2+7 A c d^2 e-5 B c d^3\right ) \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )\right )-\sqrt{a} \sqrt{c} d E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right ) \left (-a A e^3+2 a B d e^2+2 A c d^2 e-B c d^3\right )\right )}{2 a \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} \left (d+e x^2\right ) \left (a d e^2+c d^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/((d + e*x^2)^2*(a + c*x^4)^(3/2)),x]

[Out]

(Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*(a*e^3*(-(B*d) + A*e)*x*(a + c*x^4) + c*d*x*(d + e*x^2)*(-(a*A*e^2) + B*c*d^2*x^2
+ A*c*d*(d - 2*e*x^2) + a*B*e*(2*d - e*x^2))) - (d + e*x^2)*Sqrt[1 + (c*x^4)/a]*(-(Sqrt[a]*Sqrt[c]*d*(-(B*c*d
^3) + 2*A*c*d^2*e + 2*a*B*d*e^2 - a*A*e^3)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]) + I*(Sqrt[c]
*d*(Sqrt[c]*d - I*Sqrt[a]*e)*(A*c*d^2 + I*Sqrt[a]*Sqrt[c]*d*(B*d - A*e) + a*e*(2*B*d - A*e))*EllipticF[I*ArcSi
nh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + a*e*(-5*B*c*d^3 + 7*A*c*d^2*e + a*B*d*e^2 + a*A*e^3)*EllipticPi[((-I)*S
qrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])))/(2*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*(c*d^3 +
a*d*e^2)^2*(d + e*x^2)*Sqrt[a + c*x^4])

________________________________________________________________________________________

Maple [C]  time = 0.033, size = 1384, normalized size = 0.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/(e*x^2+d)^2/(c*x^4+a)^(3/2),x)

[Out]

(A*e-B*d)/e*(1/2*e^4/(a*e^2+c*d^2)^2/d*x*(c*x^4+a)^(1/2)/(e*x^2+d)-2*c*(1/2/a*c*d*e/(a*e^2+c*d^2)^2*x^3+1/4/a*
(a*e^2-c*d^2)/(a*e^2+c*d^2)^2*x)/((x^4+a/c)*c)^(1/2)-1/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/
2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)*e^2*c/(a*e^2+c*d^2
)^2+1/2/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1
/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)/a*c^2/(a*e^2+c*d^2)^2*d^2-1/2*I*a^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)
*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*c^(1/2)*e^3/(a*e^2+c*d^2)^2/d
*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+1/2*I*a^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1
/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*c^(1/2)*e^3/(a*e^2+c*d^2)^2/d*EllipticE(x*(I/a^(1/2)*c^(1/
2))^(1/2),I)+I/a^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/
2)/(c*x^4+a)^(1/2)*c^(3/2)*d*e/(a*e^2+c*d^2)^2*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-I/a^(1/2)/(I/a^(1/2)*c
^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*c^(3/2)*d*e/(a*e
^2+c*d^2)^2*EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+1/2*e^4/(a*e^2+c*d^2)^2/d^2/(I/a^(1/2)*c^(1/2))^(1/2)*(1-
I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*c^(1/2))^
(1/2),I*a^(1/2)/c^(1/2)*e/d,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2))*a+7/2*e^2/(a*e^2+c*d^2)^2/(I
/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*Ellipt
icPi(x*(I/a^(1/2)*c^(1/2))^(1/2),I*a^(1/2)/c^(1/2)*e/d,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2))*c
)+B/e*(-2*c*(1/4/a*e/(a*e^2+c*d^2)*x^3-1/4/a*d/(a*e^2+c*d^2)*x)/((x^4+a/c)*c)^(1/2)+1/2/a*c*d/(a*e^2+c*d^2)/(I
/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*Ellipt
icF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+1/2*I/a^(1/2)*c^(1/2)*e/(a*e^2+c*d^2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2
)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-
1/2*I/a^(1/2)*c^(1/2)*e/(a*e^2+c*d^2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c
^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+1/(a*e^2+c*d^2)*e^2/d/(I/a^(1/2)*c^
(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticPi(x*(I/a
^(1/2)*c^(1/2))^(1/2),I*a^(1/2)/c^(1/2)*e/d,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x^{2} + A}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/(e*x^2+d)^2/(c*x^4+a)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)/((c*x^4 + a)^(3/2)*(e*x^2 + d)^2), 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((B*x^2+A)/(e*x^2+d)^2/(c*x^4+a)^(3/2),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((B*x**2+A)/(e*x**2+d)**2/(c*x**4+a)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x^{2} + A}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/(e*x^2+d)^2/(c*x^4+a)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)/((c*x^4 + a)^(3/2)*(e*x^2 + d)^2), x)