3.4 \(\int \frac{1}{(d+e x)^2 \sqrt{a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=822 \[ -\frac{\sqrt{c x^4+b x^2+a} e^3}{\left (c d^4+b e^2 d^2+a e^4\right ) (d+e x)}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right ) e^2}{\left (c d^4+b e^2 d^2+a e^4\right ) \sqrt{c x^4+b x^2+a}}+\frac{\sqrt{c} x \sqrt{c x^4+b x^2+a} e^2}{\left (c d^4+b e^2 d^2+a e^4\right ) \left (\sqrt{c} x^2+\sqrt{a}\right )}-\frac{d \left (2 c d^2+b e^2\right ) \tan ^{-1}\left (\frac{\sqrt{-c d^4-b e^2 d^2-a e^4} x}{d e \sqrt{c x^4+b x^2+a}}\right ) e}{2 \left (-c d^4-b e^2 d^2-a e^4\right )^{3/2}}-\frac{d \left (2 c d^2+b e^2\right ) \tanh ^{-1}\left (\frac{b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt{c d^4+b e^2 d^2+a e^4} \sqrt{c x^4+b x^2+a}}\right ) e}{2 \left (c d^4+b e^2 d^2+a e^4\right )^{3/2}}+\frac{\sqrt [4]{c} \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 \sqrt [4]{a} \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \sqrt{c x^4+b x^2+a}}-\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (2 c d^2+b e^2\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \sqrt{c x^4+b x^2+a}} \]

[Out]

-((e^3*Sqrt[a + b*x^2 + c*x^4])/((c*d^4 + b*d^2*e^2 + a*e^4)*(d + e*x))) + (Sqrt[c]*e^2*x*Sqrt[a + b*x^2 + c*x
^4])/((c*d^4 + b*d^2*e^2 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)) - (d*e*(2*c*d^2 + b*e^2)*ArcTan[(Sqrt[-(c*d^4) - b*
d^2*e^2 - a*e^4]*x)/(d*e*Sqrt[a + b*x^2 + c*x^4])])/(2*(-(c*d^4) - b*d^2*e^2 - a*e^4)^(3/2)) - (d*e*(2*c*d^2 +
b*e^2)*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*(c*d^4 + b*d^2*e^2 + a*e^4)^(3/2)) - (a^(1/4)*c^(1/4)*e^2*(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])/((
c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[a + b*x^2 + c*x^4]) + (c^(1/4)*(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^2 + Sqrt[a]*e^2)*Sqrt[a + b*x^2 + c*x^4]) - ((Sqrt[c]*d^2 - Sqrt[a]*e^2)*(2*c*d^2 + b*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^2 + Sqrt[a]*e^2)^2/
(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*a^(1/4)*c^(1/4)*(
Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[a + b*x^2 + c*x^4])

________________________________________________________________________________________

Rubi [A]  time = 1.2564, antiderivative size = 822, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {1726, 1741, 12, 1247, 724, 206, 1714, 1195, 1708, 1103, 1706} \[ -\frac{\sqrt{c x^4+b x^2+a} e^3}{\left (c d^4+b e^2 d^2+a e^4\right ) (d+e x)}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right ) e^2}{\left (c d^4+b e^2 d^2+a e^4\right ) \sqrt{c x^4+b x^2+a}}+\frac{\sqrt{c} x \sqrt{c x^4+b x^2+a} e^2}{\left (c d^4+b e^2 d^2+a e^4\right ) \left (\sqrt{c} x^2+\sqrt{a}\right )}-\frac{d \left (2 c d^2+b e^2\right ) \tan ^{-1}\left (\frac{\sqrt{-c d^4-b e^2 d^2-a e^4} x}{d e \sqrt{c x^4+b x^2+a}}\right ) e}{2 \left (-c d^4-b e^2 d^2-a e^4\right )^{3/2}}-\frac{d \left (2 c d^2+b e^2\right ) \tanh ^{-1}\left (\frac{b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt{c d^4+b e^2 d^2+a e^4} \sqrt{c x^4+b x^2+a}}\right ) e}{2 \left (c d^4+b e^2 d^2+a e^4\right )^{3/2}}+\frac{\sqrt [4]{c} \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 \sqrt [4]{a} \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \sqrt{c x^4+b x^2+a}}-\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (2 c d^2+b e^2\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \sqrt{c x^4+b x^2+a}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^2*Sqrt[a + b*x^2 + c*x^4]),x]

[Out]

-((e^3*Sqrt[a + b*x^2 + c*x^4])/((c*d^4 + b*d^2*e^2 + a*e^4)*(d + e*x))) + (Sqrt[c]*e^2*x*Sqrt[a + b*x^2 + c*x
^4])/((c*d^4 + b*d^2*e^2 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)) - (d*e*(2*c*d^2 + b*e^2)*ArcTan[(Sqrt[-(c*d^4) - b*
d^2*e^2 - a*e^4]*x)/(d*e*Sqrt[a + b*x^2 + c*x^4])])/(2*(-(c*d^4) - b*d^2*e^2 - a*e^4)^(3/2)) - (d*e*(2*c*d^2 +
b*e^2)*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*(c*d^4 + b*d^2*e^2 + a*e^4)^(3/2)) - (a^(1/4)*c^(1/4)*e^2*(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])/((
c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[a + b*x^2 + c*x^4]) + (c^(1/4)*(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^2 + Sqrt[a]*e^2)*Sqrt[a + b*x^2 + c*x^4]) - ((Sqrt[c]*d^2 - Sqrt[a]*e^2)*(2*c*d^2 + b*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^2 + Sqrt[a]*e^2)^2/
(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*a^(1/4)*c^(1/4)*(
Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[a + b*x^2 + c*x^4])

Rule 1726

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

Rule 1741

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

Rule 12

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

Rule 1247

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

Rule 724

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

Rule 206

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

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

Mathematica [C]  time = 7.79877, size = 4019, normalized size = 4.89 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((d + e*x)^2*Sqrt[a + b*x^2 + c*x^4]),x]

[Out]

-((e^3*Sqrt[a + b*x^2 + c*x^4])/((c*d^4 + b*d^2*e^2 + a*e^4)*(d + e*x))) + (((I/2)*(-b + Sqrt[b^2 - 4*a*c])*e^
2*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])]*(EllipticE[I*ArcSi
nh[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])] - Ellipt
icF[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[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) + (I*c*d^2*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 - S
qrt[b^2 - 4*a*c]))]*x], (-b - Sqrt[b^2 - 4*a*c])/(-b + Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 -
4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) + (4*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])*d^3*(-(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 - S
qrt[b^2 - 4*a*c])/c] - 2*x))]*((-d + (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*EllipticF[ArcSin[Sqrt[((S
qrt[(-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*Elliptic
Pi[((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 + 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[(-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])*(-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]) + (2*b*(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*e^2*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sq
rt[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))/((Sq
rt[-(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 + S
qrt[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)/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) + Sqr
t[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)/S
qrt[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 + 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[(-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])*(-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]))/(c*d^4 + b*d^2*e^2 + a*e^4)

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 771, 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(1/(e*x+d)^2/(c*x^4+b*x^2+a)^(1/2),x)

[Out]

-e^3*(c*x^4+b*x^2+a)^(1/2)/(a*e^4+b*d^2*e^2+c*d^4)/(e*x+d)-1/4*c*d^2/(a*e^4+b*d^2*e^2+c*d^4)*2^(1/2)/(((-4*a*c
+b^2)^(1/2)-b)/a)^(1/2)*(4-2*((-4*a*c+b^2)^(1/2)-b)/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x
^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(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*c*e^2/(a*e^4+b*d^2*e^2+c*d^4)*a*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)*(4-2*((-4*a*c+b^2)^(1
/2)-b)/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(Ell
ipticF(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))-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)))+d*(b*e^2+2*c*d
^2)/(a*e^4+b*d^2*e^2+c*d^4)/e*(-1/2/(c*d^4/e^4+b*d^2/e^2+a)^(1/2)*arctanh(1/2*(2*c*x^2*d^2/e^2+b*x^2+b*d^2/e^2
+2*a)/(c*d^4/e^4+b*d^2/e^2+a)^(1/2)/(c*x^4+b*x^2+a)^(1/2))+2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)/d*e*(1-1/2
*((-4*a*c+b^2)^(1/2)-b)/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)*Elliptic
Pi(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/d^2*e^2,(-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)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{4} + b x^{2} + a}{\left (e x + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(c*x^4 + b*x^2 + a)*(e*x + 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(1/(e*x+d)^2/(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{1}{\left (d + e x\right )^{2} \sqrt{a + b x^{2} + c x^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((d + e*x)**2*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{1}{\sqrt{c x^{4} + b x^{2} + a}{\left (e x + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(c*x^4 + b*x^2 + a)*(e*x + d)^2), x)