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

Optimal. Leaf size=1301 \[ \text{result too large to display} \]

[Out]

(x*(a*b*c*(A*e*(2*c*d - b*e) - B*(c*d^2 - a*e^2)) + (b^2 - 2*a*c)*(a*B*e*(2*c*d - b*e) + A*(c^2*d^2 + b^2*e^2
- c*e*(2*b*d + a*e))) - c*(a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) + A*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*
e^2 - b*c*(c*d^2 - 3*a*e^2)))*x^2))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) + (Sqr
t[c]*(a*B*d*(-4*c^2*d^2 - 3*b^2*e^2 + 4*c*e*(b*d + 2*a*e)) + A*(2*b^3*d*e^2 + 2*b*c*d*(c*d^2 - 3*a*e^2) - 4*a*
c*e*(-2*c*d^2 + a*e^2) + b^2*(-4*c*d^2*e + a*e^3)))*x*Sqrt[a + b*x^2 + c*x^4])/(2*a*(-b^2 + 4*a*c)*d*(c*d^2 +
e*(-(b*d) + a*e))^2*(Sqrt[a] + Sqrt[c]*x^2)) - (e^3*(B*d - A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(2*d*(c*d^2 - b*d*e
+ a*e^2)^2*(d + e*x^2)) + (e^(3/2)*(A*e*(7*c*d^2 - e*(4*b*d - a*e)) - B*d*(5*c*d^2 - e*(2*b*d + a*e)))*ArcTan
[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(4*d^(3/2)*(c*d^2 - b*d*e + a*e^2
)^(5/2)) - (c^(1/4)*(a*B*d*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e)) - A*(2*b^3*d*e^2 + 2*b*c*d*(c*d^2 - 3
*a*e^2) - 4*a*c*e*(-2*c*d^2 + a*e^2) + b^2*(-4*c*d^2*e + a*e^3)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*
x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2 - b/(4*Sqrt[a]*Sqrt[c])])/(2*a^(3
/4)*(b^2 - 4*a*c)*d*(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + b*x^2 + c*x^4]) + (c^(1/4)*(a*Sqrt[c]*e*(B*d - 2*A*e
) + Sqrt[a]*(B*d - A*e)*(c*d - b*e) + A*Sqrt[c]*d*(-(c*d) + b*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*
x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2 - b/(4*Sqrt[a]*Sqrt[c])])/(2*a^(3
/4)*(b - 2*Sqrt[a]*Sqrt[c])*d*(-(Sqrt[c]*d) + Sqrt[a]*e)*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + b*x^2 + c*x^4]) -
(e*(Sqrt[c]*d + Sqrt[a]*e)*(A*e*(7*c*d^2 - e*(4*b*d - a*e)) - B*d*(5*c*d^2 - e*(2*b*d + a*e)))*(Sqrt[a] + Sqr
t[c]*x^2)*Sqrt[(a + b*x^2 + 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)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(8*a^(1/4)*c^(1/4)*d^2*(Sqrt[c]*d
- Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4])

________________________________________________________________________________________

Rubi [A]  time = 3.38621, antiderivative size = 2112, normalized size of antiderivative = 1.62, number of steps used = 15, number of rules used = 10, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.303, Rules used = {1720, 1178, 1197, 1103, 1195, 1223, 1714, 1708, 1706, 1216} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a*b*c*(A*e*(2*c*d - b*e) - B*(c*d^2 - a*e^2)) + (b^2 - 2*a*c)*(a*B*e*(2*c*d - b*e) + A*(c^2*d^2 + b^2*e^2
- c*e*(2*b*d + a*e))) - c*(a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) + A*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*
e^2 - b*c*(c*d^2 - 3*a*e^2)))*x^2))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) + (Sqr
t[c]*e^2*(B*d - A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(2*d*(c*d^2 - b*d*e + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) + (Sqr
t[c]*(a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) + A*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*
a*e^2)))*x*Sqrt[a + b*x^2 + c*x^4])/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (e^3
*(B*d - A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(2*d*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x^2)) - (e^(3/2)*(B*d - A*e)*(3*
c*d^2 - e*(2*b*d - a*e))*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(4
*d^(3/2)*(c*d^2 - b*d*e + a*e^2)^(5/2)) + (e^(3/2)*(A*e*(2*c*d - b*e) - B*(c*d^2 - a*e^2))*ArcTan[(Sqrt[c*d^2
- b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[d]*(c*d^2 - b*d*e + 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 + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ell
ipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*d*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*
x^2 + c*x^4]) - (c^(1/4)*(a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) + A*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e
^2 - b*c*(c*d^2 - 3*a*e^2)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellip
ticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(a^(3/4)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^
2)^2*Sqrt[a + b*x^2 + c*x^4]) - (c^(1/4)*e*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[
a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*d*(Sqrt
[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) + (c^(1/4)*e*(A*e*(2*c*d - b*e) - B*(c*d^2
- a*e^2))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(
1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)^2*S
qrt[a + b*x^2 + c*x^4]) - (c^(1/4)*(a^(3/2)*B*Sqrt[c]*e^2 + A*(c*d - b*e)^2 + a*e*(2*B*c*d - b*B*e - A*c*e) -
Sqrt[a]*Sqrt[c]*(B*c*d^2 - A*e*(2*c*d - b*e)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqr
t[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(3/4)*(b - 2*Sqrt[a]*
Sqrt[c])*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) + (e*(Sqrt[c]*d + Sqrt[a]*e)*(B*d - A*e)*(3*c*d^2
- e*(2*b*d - a*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sq
rt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(
8*a^(1/4)*c^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) - (a^(3/4)*e*
((Sqrt[c]*d)/Sqrt[a] + e)^2*(A*e*(2*c*d - b*e) - B*(c*d^2 - a*e^2))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 +
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)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*c^(1/4)*d*(c*d^2 - a*e^2)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a
+ b*x^2 + c*x^4])

Rule 1720

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

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1197

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

Rule 1103

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

Rule 1195

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

Rule 1223

Int[((d_) + (e_.)*(x_)^2)^(q_)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> -Simp[(e^2*x*(d + e*x^2)
^(q + 1)*Sqrt[a + b*x^2 + c*x^4])/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(2*d*(q + 1)*(c*d^2 - b*d
*e + a*e^2)), Int[((d + e*x^2)^(q + 1)*Simp[a*e^2*(2*q + 3) + 2*d*(c*d - b*e)*(q + 1) - 2*e*(c*d*(q + 1) - b*e
*(q + 2))*x^2 + c*e^2*(2*q + 5)*x^4, x])/Sqrt[a + b*x^2 + c*x^4], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b
^2 - 4*a*c, 0] && ILtQ[q, -1]

Rule 1714

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (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 + b
*x^2 + 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 + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && 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] && PosQ[c/a] &&  !GtQ[b^2 - 4*a*c, 0]

Rule 1708

Int[((A_.) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (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 + b*x^2 + 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 + 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] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]

Rule 1706

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

Rule 1216

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

Rubi steps

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

Mathematica [C]  time = 8.00736, size = 8031, normalized size = 6.17 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.041, size = 8276, normalized size = 6.4 \begin{align*} \text{output 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+b*x^2+a)^(3/2),x)

[Out]

result too large to display

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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} + b x^{2} + 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+b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

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

[Out]

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