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

Optimal. Leaf size=1125 \[ \text{result too large to display} \]

[Out]

-(Sqrt[c]*(3*A*e*(3*c*d^2 - e*(2*b*d - a*e)) - B*d*(5*c*d^2 - e*(2*b*d + a*e)))*x*Sqrt[a + b*x^2 + c*x^4])/(8*
d^2*(c*d^2 - b*d*e + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (e*(B*d - A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(4*d*(c*d^2
- b*d*e + a*e^2)*(d + e*x^2)^2) + (e*(3*A*e*(3*c*d^2 - e*(2*b*d - a*e)) - B*d*(5*c*d^2 - e*(2*b*d + a*e)))*x*
Sqrt[a + b*x^2 + c*x^4])/(8*d^2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x^2)) - ((B*d*(3*c^2*d^4 - 10*a*c*d^2*e^2 + a
*e^3*(4*b*d - a*e)) - A*e*(15*c^2*d^4 - 2*c*d^2*e*(10*b*d - 3*a*e) + e^2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*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])])/(16*d^(5/2)*Sqrt[e]*(c*d^2
- b*d*e + a*e^2)^(5/2)) + (a^(1/4)*c^(1/4)*(3*A*e*(3*c*d^2 - e*(2*b*d - a*e)) - B*d*(5*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]*EllipticE[2*ArcTan[(c^(1/4)*x)
/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(8*d^2*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) + (c^(1/4)*
(Sqrt[a]*Sqrt[c]*d*(B*d - A*e) + a*e*(B*d + 3*A*e) + 4*A*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)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(
8*a^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) + ((Sqrt[c]*d + Sqrt[a]
*e)*(B*d*(3*c^2*d^4 - 10*a*c*d^2*e^2 + a*e^3*(4*b*d - a*e)) - A*e*(15*c^2*d^4 - 2*c*d^2*e*(10*b*d - 3*a*e) + e
^2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*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/(S
qrt[a]*Sqrt[c]))/4])/(32*a^(1/4)*c^(1/4)*d^3*e*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^
2 + c*x^4])

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Rubi [A]  time = 3.59431, antiderivative size = 1125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1696, 1714, 1195, 1708, 1103, 1706} \[ -\frac{\sqrt{c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) \sqrt{c x^4+b x^2+a} x}{8 d^2 \left (c d^2-b e d+a e^2\right )^2 \left (\sqrt{c} x^2+\sqrt{a}\right )}+\frac{e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) \sqrt{c x^4+b x^2+a} x}{8 d^2 \left (c d^2-b e d+a e^2\right )^2 \left (e x^2+d\right )}-\frac{e (B d-A e) \sqrt{c x^4+b x^2+a} x}{4 d \left (c d^2-b e d+a e^2\right ) \left (e x^2+d\right )^2}-\frac{\left (B \left (3 c^2 d^5-10 a c e^2 d^3+a e^3 (4 b d-a e) d\right )-A e \left (15 c^2 d^4-2 c e (10 b d-3 a e) d^2+e^2 \left (8 b^2 d^2-8 a b e d+3 a^2 e^2\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c d^2-b e d+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+b x^2+a}}\right )}{16 d^{5/2} \sqrt{e} \left (c d^2-b e d+a e^2\right )^{5/2}}+\frac{\sqrt [4]{a} \sqrt [4]{c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\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 )}{8 d^2 \left (c d^2-b e d+a e^2\right )^2 \sqrt{c x^4+b x^2+a}}+\frac{\sqrt [4]{c} \left (\sqrt{a} \sqrt{c} d (B d-A e)+a e (B d+3 A e)+4 A d (c d-b e)\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\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 )}{8 \sqrt [4]{a} d^2 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2-b e d+a e^2\right ) \sqrt{c x^4+b x^2+a}}+\frac{\left (\sqrt{c} d+\sqrt{a} e\right ) \left (B \left (3 c^2 d^5-10 a c e^2 d^3+a e^3 (4 b d-a e) d\right )-A e \left (15 c^2 d^4-2 c e (10 b d-3 a e) d^2+e^2 \left (8 b^2 d^2-8 a b e d+3 a^2 e^2\right )\right )\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\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 )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 e \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt{c x^4+b x^2+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[c]*(3*A*e*(3*c*d^2 - e*(2*b*d - a*e)) - B*(5*c*d^3 - d*e*(2*b*d + a*e)))*x*Sqrt[a + b*x^2 + c*x^4])/(8*
d^2*(c*d^2 - b*d*e + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (e*(B*d - A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(4*d*(c*d^2
- b*d*e + a*e^2)*(d + e*x^2)^2) + (e*(3*A*e*(3*c*d^2 - e*(2*b*d - a*e)) - B*(5*c*d^3 - d*e*(2*b*d + a*e)))*x*
Sqrt[a + b*x^2 + c*x^4])/(8*d^2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x^2)) - ((B*(3*c^2*d^5 - 10*a*c*d^3*e^2 + a*d
*e^3*(4*b*d - a*e)) - A*e*(15*c^2*d^4 - 2*c*d^2*e*(10*b*d - 3*a*e) + e^2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*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])])/(16*d^(5/2)*Sqrt[e]*(c*d^2
- b*d*e + a*e^2)^(5/2)) + (a^(1/4)*c^(1/4)*(3*A*e*(3*c*d^2 - e*(2*b*d - a*e)) - B*(5*c*d^3 - d*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]*EllipticE[2*ArcTan[(c^(1/4)*x)
/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(8*d^2*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) + (c^(1/4)*
(Sqrt[a]*Sqrt[c]*d*(B*d - A*e) + a*e*(B*d + 3*A*e) + 4*A*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)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(
8*a^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) + ((Sqrt[c]*d + Sqrt[a]
*e)*(B*(3*c^2*d^5 - 10*a*c*d^3*e^2 + a*d*e^3*(4*b*d - a*e)) - A*e*(15*c^2*d^4 - 2*c*d^2*e*(10*b*d - 3*a*e) + e
^2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*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/(S
qrt[a]*Sqrt[c]))/4])/(32*a^(1/4)*c^(1/4)*d^3*e*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^
2 + c*x^4])

Rule 1696

Int[((P4x_)*((d_) + (e_.)*(x_)^2)^(q_))/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{A = Coeff
[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -Simp[((C*d^2 - B*d*e + A*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*d*(C*d - B*e) + A*(a*e^2*(2*q + 3) + 2*d*(c*d - b*e)*(q + 1)) - 2*((B*d
- A*e)*(b*e*(q + 2) - c*d*(q + 1)) - C*d*(b*d + a*e*(q + 1)))*x^2 + c*(C*d^2 - B*d*e + A*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] && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4] &
& NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{A+B x^2}{\left (d+e x^2\right )^3 \sqrt{a+b x^2+c x^4}} \, dx &=-\frac{e (B d-A e) x \sqrt{a+b x^2+c x^4}}{4 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^2}-\frac{\int \frac{-4 A c d^2-a B d e+A e (4 b d-3 a e)-2 (B d-A e) (2 c d-b e) x^2+c e (B d-A e) x^4}{\left (d+e x^2\right )^2 \sqrt{a+b x^2+c x^4}} \, dx}{4 d \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e (B d-A e) x \sqrt{a+b x^2+c x^4}}{4 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^2}+\frac{e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x \sqrt{a+b x^2+c x^4}}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}+\frac{\int \frac{a d e (B d-A e) (5 c d-2 b e)+\left (4 A c d^2+a B d e-A e (4 b d-3 a e)\right ) \left (2 c d^2-e (2 b d-a e)\right )-2 c d \left (A e \left (8 c d^2-e (5 b d-2 a e)\right )-B \left (4 c d^3-d e (b d+2 a e)\right )\right ) x^2-c e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x^4}{\left (d+e x^2\right ) \sqrt{a+b x^2+c x^4}} \, dx}{8 d^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{e (B d-A e) x \sqrt{a+b x^2+c x^4}}{4 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^2}+\frac{e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x \sqrt{a+b x^2+c x^4}}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}+\frac{\int \frac{c e \left (a d e (B d-A e) (5 c d-2 b e)+\left (4 A c d^2+a B d e-A e (4 b d-3 a e)\right ) \left (2 c d^2-e (2 b d-a e)\right )\right )-\sqrt{a} c^{3/2} d e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right )+\left (c e \left (c d-\sqrt{a} \sqrt{c} e\right ) \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right )-2 c^2 d e \left (A e \left (8 c d^2-e (5 b d-2 a e)\right )-B \left (4 c d^3-d e (b d+2 a e)\right )\right )\right ) x^2}{\left (d+e x^2\right ) \sqrt{a+b x^2+c x^4}} \, dx}{8 c d^2 e \left (c d^2-b d e+a e^2\right )^2}+\frac{\left (\sqrt{a} \sqrt{c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right )\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+b x^2+c x^4}} \, dx}{8 d^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{\sqrt{c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x \sqrt{a+b x^2+c x^4}}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{e (B d-A e) x \sqrt{a+b x^2+c x^4}}{4 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^2}+\frac{e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x \sqrt{a+b x^2+c x^4}}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}+\frac{\sqrt [4]{a} \sqrt [4]{c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a 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}} 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 )}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt{a+b x^2+c x^4}}+\frac{\left (\sqrt{c} \left (\sqrt{a} \sqrt{c} d (B d-A e)+a e (B d+3 A e)+4 A d (c d-b e)\right )\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{4 d^2 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2-b d e+a e^2\right )}+\frac{\left (\sqrt{a} \left (B \left (3 c^2 d^5-10 a c d^3 e^2+a d e^3 (4 b d-a e)\right )-A e \left (15 c^2 d^4-2 c d^2 e (10 b d-3 a e)+e^2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right )\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}{8 d^2 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{\sqrt{c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x \sqrt{a+b x^2+c x^4}}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{e (B d-A e) x \sqrt{a+b x^2+c x^4}}{4 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^2}+\frac{e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x \sqrt{a+b x^2+c x^4}}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}-\frac{\left (B \left (3 c^2 d^5-10 a c d^3 e^2+a d e^3 (4 b d-a e)\right )-A e \left (15 c^2 d^4-2 c d^2 e (10 b d-3 a e)+e^2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right )\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 )}{16 d^{5/2} \sqrt{e} \left (c d^2-b d e+a e^2\right )^{5/2}}+\frac{\sqrt [4]{a} \sqrt [4]{c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a 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}} 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 )}{8 d^2 \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 (\sqrt{a} \sqrt{c} d (B d-A e)+a e (B d+3 A e)+4 A d (c d-b 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}} 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 )}{8 \sqrt [4]{a} d^2 \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{\left (\sqrt{c} d+\sqrt{a} e\right ) \left (B \left (3 c^2 d^5-10 a c d^3 e^2+a d e^3 (4 b d-a e)\right )-A e \left (15 c^2 d^4-2 c d^2 e (10 b d-3 a e)+e^2 \left (8 b^2 d^2-8 a b d e+3 a^2 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}} \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 )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 e \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 = 6.1701, size = 781, normalized size = 0.69 \[ \frac{-\frac{4 d e^2 x \left (a+b x^2+c x^4\right ) \left (\left (d+e x^2\right ) \left (B \left (5 c d^3-d e (a e+2 b d)\right )-3 A e \left (e (a e-2 b d)+3 c d^2\right )\right )+2 d (B d-A e) \left (e (a e-b d)+c d^2\right )\right )}{\left (d+e x^2\right )^2}-\frac{i \sqrt{2} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (d \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right ) \left (B d \left (-e^2 \left (\sqrt{b^2-4 a c}-b\right ) (a e+2 b d)+c d e \left (5 d \sqrt{b^2-4 a c}-6 a e-5 b d\right )+6 c^2 d^3\right )-A e \left (-3 e^2 \left (\sqrt{b^2-4 a c}-b\right ) (2 b d-a e)+c d e \left (9 d \sqrt{b^2-4 a c}+2 a e-17 b d\right )+14 c^2 d^3\right )\right )+2 \left (A e \left (e^2 \left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right )+2 c d^2 e (3 a e-10 b d)+15 c^2 d^4\right )+B \left (a d e^3 (a e-4 b d)+10 a c d^3 e^2-3 c^2 d^5\right )\right ) \Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )+d e \left (\sqrt{b^2-4 a c}-b\right ) E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right ) \left (3 A e \left (e (a e-2 b d)+3 c d^2\right )+B \left (d e (a e+2 b d)-5 c d^3\right )\right )\right )}{\sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}}}{32 d^3 e \sqrt{a+b x^2+c x^4} \left (e (a e-b d)+c d^2\right )^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-4*d*e^2*x*(a + b*x^2 + c*x^4)*(2*d*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e)) + (-3*A*e*(3*c*d^2 + e*(-2*b*d +
a*e)) + B*(5*c*d^3 - d*e*(2*b*d + a*e)))*(d + e*x^2)))/(d + e*x^2)^2 - (I*Sqrt[2]*Sqrt[(b + Sqrt[b^2 - 4*a*c]
+ 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*((-b + Sqrt[b^2 - 4*a*c])*d*e*
(3*A*e*(3*c*d^2 + e*(-2*b*d + a*e)) + B*(-5*c*d^3 + d*e*(2*b*d + a*e)))*EllipticE[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])] + d*(B*d*(6*c^2*d^3 + c*d*e*(-5*b*d
+ 5*Sqrt[b^2 - 4*a*c]*d - 6*a*e) - (-b + Sqrt[b^2 - 4*a*c])*e^2*(2*b*d + a*e)) - A*e*(14*c^2*d^3 - 3*(-b + Sq
rt[b^2 - 4*a*c])*e^2*(2*b*d - a*e) + c*d*e*(-17*b*d + 9*Sqrt[b^2 - 4*a*c]*d + 2*a*e)))*EllipticF[I*ArcSinh[Sqr
t[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])] + 2*(B*(-3*c^2*d^5 +
10*a*c*d^3*e^2 + a*d*e^3*(-4*b*d + a*e)) + A*e*(15*c^2*d^4 + 2*c*d^2*e*(-10*b*d + 3*a*e) + e^2*(8*b^2*d^2 - 8
*a*b*d*e + 3*a^2*e^2)))*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), 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])]))/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])])/(32*d^3*e*
(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + b*x^2 + c*x^4])

________________________________________________________________________________________

Maple [B]  time = 0.031, size = 4476, normalized size = 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)^3/(c*x^4+b*x^2+a)^(1/2),x)

[Out]

B/e*(1/2*e^2/(a*e^2-b*d*e+c*d^2)/d*x*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)-1/8*c/(a*e^2-b*d*e+c*d^2)*2^(1/2)/(1/a*(-
4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(4-2/a*x^2*(-4*a*c+b^2)^(1/2)+2/a*b*x^2)^(1/2)*(4+2/a*b*x^2+2/a*x^2*(-4*a*c+b^2)
^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-
4*a*c+b^2)^(1/2))/a/c)^(1/2))+1/4*c*e/(a*e^2-b*d*e+c*d^2)/d*a*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(4-
2/a*x^2*(-4*a*c+b^2)^(1/2)+2/a*b*x^2)^(1/2)*(4+2/a*b*x^2+2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/
2)/(b+(-4*a*c+b^2)^(1/2))*EllipticF(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)
^(1/2))/a/c)^(1/2))-1/4*c*e/(a*e^2-b*d*e+c*d^2)/d*a*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(4-2/a*x^2*(-
4*a*c+b^2)^(1/2)+2/a*b*x^2)^(1/2)*(4+2/a*b*x^2+2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*
a*c+b^2)^(1/2))*EllipticE(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/
c)^(1/2))+1/2/(a*e^2-b*d*e+c*d^2)/d^2*e^2*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(1-1/2/a*x^2*(-4*a*c+b^
2)^(1/2)+1/2/a*b*x^2)^(1/2)*(1+1/2/a*b*x^2+1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticP
i(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),-2/((-4*a*c+b^2)^(1/2)-b)*a*e/d,(-1/2*(b+(-4*a*c+b^2)^(1/2))/
a)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2))*a-1/(a*e^2-b*d*e+c*d^2)*e/d*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)
-1/a*b)^(1/2)*(1-1/2/a*x^2*(-4*a*c+b^2)^(1/2)+1/2/a*b*x^2)^(1/2)*(1+1/2/a*b*x^2+1/2/a*x^2*(-4*a*c+b^2)^(1/2))^
(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),-2/((-4*a*c+b^2)^(1/2)-b
)*a*e/d,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2))*b+3/2/(a*e^2-b*d*e+c*d
^2)*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(1-1/2/a*x^2*(-4*a*c+b^2)^(1/2)+1/2/a*b*x^2)^(1/2)*(1+1/2/a*b
*x^2+1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b
)/a)^(1/2),-2/((-4*a*c+b^2)^(1/2)-b)*a*e/d,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-
b)/a)^(1/2))*c)+(A*e-B*d)/e*(1/4*e^2/(a*e^2-b*d*e+c*d^2)/d*x*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^2+3/8*e^2*(a*e^2-
2*b*d*e+3*c*d^2)/(a*e^2-b*d*e+c*d^2)^2/d^2*x*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)+3/16*e^3*c/(a*e^2-b*d*e+c*d^2)^2/
d^2*a^2*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(4-2/a*x^2*(-4*a*c+b^2)^(1/2)+2/a*b*x^2)^(1/2)*(4+2/a*b*x
^2+2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*EllipticF(1/2*x*2^(1/2)*(((-
4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))+9/16*e*c^2/(a*e^2-b*d*e+c*d^2)^2*a
*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(4-2/a*x^2*(-4*a*c+b^2)^(1/2)+2/a*b*x^2)^(1/2)*(4+2/a*b*x^2+2/a*
x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*EllipticF(1/2*x*2^(1/2)*(((-4*a*c+b
^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-3/8*e^2*c/(a*e^2-b*d*e+c*d^2)^2/d*a*2^(1/
2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(4-2/a*x^2*(-4*a*c+b^2)^(1/2)+2/a*b*x^2)^(1/2)*(4+2/a*b*x^2+2/a*x^2*(-
4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*EllipticF(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1
/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))*b+3/8*e^2*c/(a*e^2-b*d*e+c*d^2)^2/d*a*2^(1/2)/(
1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(4-2/a*x^2*(-4*a*c+b^2)^(1/2)+2/a*b*x^2)^(1/2)*(4+2/a*b*x^2+2/a*x^2*(-4*a*
c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*EllipticE(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-
b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))*b-7/32*c^2/(a*e^2-b*d*e+c*d^2)^2*d*2^(1/2)/(1/a*(-4
*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(4-2/a*x^2*(-4*a*c+b^2)^(1/2)+2/a*b*x^2)^(1/2)*(4+2/a*b*x^2+2/a*x^2*(-4*a*c+b^2)^
(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4
*a*c+b^2)^(1/2))/a/c)^(1/2))+1/8*c/(a*e^2-b*d*e+c*d^2)^2*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(4-2/a*x
^2*(-4*a*c+b^2)^(1/2)+2/a*b*x^2)^(1/2)*(4+2/a*b*x^2+2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*El
lipticF(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))*b*e-1/(a
*e^2-b*d*e+c*d^2)^2/d^2*e^3*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(1-1/2/a*x^2*(-4*a*c+b^2)^(1/2)+1/2/a
*b*x^2)^(1/2)*(1+1/2/a*b*x^2+1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2
)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),-2/((-4*a*c+b^2)^(1/2)-b)*a*e/d,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/
2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2))*a*b+3/4/(a*e^2-b*d*e+c*d^2)^2*e^2/d*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b
)^(1/2)*(1-1/2/a*x^2*(-4*a*c+b^2)^(1/2)+1/2/a*b*x^2)^(1/2)*(1+1/2/a*b*x^2+1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/
(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),-2/((-4*a*c+b^2)^(1/2)-b)*a*e/
d,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2))*a*c-1/32*c/(a*e^2-b*d*e+c*d^
2)^2/d*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(4-2/a*x^2*(-4*a*c+b^2)^(1/2)+2/a*b*x^2)^(1/2)*(4+2/a*b*x^
2+2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(
1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))*a*e^2+3/8/(a*e^2-b*d*e+c*d^2)^2/d^3*e^4*2^(1/2)/(1/a*(-4*a
*c+b^2)^(1/2)-1/a*b)^(1/2)*(1-1/2/a*x^2*(-4*a*c+b^2)^(1/2)+1/2/a*b*x^2)^(1/2)*(1+1/2/a*b*x^2+1/2/a*x^2*(-4*a*c
+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),-2/((-4*a*c
+b^2)^(1/2)-b)*a*e/d,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2))*a^2+1/(a*
e^2-b*d*e+c*d^2)^2*e^2/d*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(1-1/2/a*x^2*(-4*a*c+b^2)^(1/2)+1/2/a*b*
x^2)^(1/2)*(1+1/2/a*b*x^2+1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*(
((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),-2/((-4*a*c+b^2)^(1/2)-b)*a*e/d,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/
(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2))*b^2-5/2/(a*e^2-b*d*e+c*d^2)^2*e*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)
*(1-1/2/a*x^2*(-4*a*c+b^2)^(1/2)+1/2/a*b*x^2)^(1/2)*(1+1/2/a*b*x^2+1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+
b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),-2/((-4*a*c+b^2)^(1/2)-b)*a*e/d,(-1/2
*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2))*b*c+15/8/(a*e^2-b*d*e+c*d^2)^2*d*2^
(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(1-1/2/a*x^2*(-4*a*c+b^2)^(1/2)+1/2/a*b*x^2)^(1/2)*(1+1/2/a*b*x^2+1
/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(
1/2),-2/((-4*a*c+b^2)^(1/2)-b)*a*e/d,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^
(1/2))*c^2-3/16*e^3*c/(a*e^2-b*d*e+c*d^2)^2/d^2*a^2*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(4-2/a*x^2*(-
4*a*c+b^2)^(1/2)+2/a*b*x^2)^(1/2)*(4+2/a*b*x^2+2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*
a*c+b^2)^(1/2))*EllipticE(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/
c)^(1/2))-9/16*e*c^2/(a*e^2-b*d*e+c*d^2)^2*a*2^(1/2)/(1/a*(-4*a*c+b^2)^(1/2)-1/a*b)^(1/2)*(4-2/a*x^2*(-4*a*c+b
^2)^(1/2)+2/a*b*x^2)^(1/2)*(4+2/a*b*x^2+2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2
)^(1/2))*EllipticE(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2
)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/(e*x^2+d)^3/(c*x^4+b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)^3), 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)^3/(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{A + B x^{2}}{\left (d + e x^{2}\right )^{3} \sqrt{a + b x^{2} + c x^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/(e*x**2+d)**3/(c*x**4+b*x**2+a)**(1/2),x)

[Out]

Integral((A + B*x**2)/((d + e*x**2)**3*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{B x^{2} + A}{\sqrt{c x^{4} + b x^{2} + a}{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/(e*x^2+d)^3/(c*x^4+b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

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