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

Optimal. Leaf size=718 \[ \frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 \sqrt [4]{a} d \sqrt{a+b x^2+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )}+\frac{e^2 x \sqrt{a+b x^2+c x^4}}{2 d \left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} e x \sqrt{a+b x^2+c x^4}}{2 d \left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{e} \left (3 c d^2-e (2 b d-a e)\right ) \tan ^{-1}\left (\frac{x \sqrt{a e^2-b d e+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+b x^2+c x^4}}\right )}{4 d^{3/2} \left (a e^2-b d e+c d^2\right )^{3/2}}+\frac{\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 d \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (3 c d^2-e (2 b d-a e)\right ) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \sqrt{a+b x^2+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2-b d e+c d^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.08066, antiderivative size = 718, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1223, 1714, 1195, 1708, 1103, 1706} \[ \frac{e^2 x \sqrt{a+b x^2+c x^4}}{2 d \left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} e x \sqrt{a+b x^2+c x^4}}{2 d \left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{e} \left (3 c d^2-e (2 b d-a e)\right ) \tan ^{-1}\left (\frac{x \sqrt{a e^2-b d e+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+b x^2+c x^4}}\right )}{4 d^{3/2} \left (a e^2-b d e+c d^2\right )^{3/2}}+\frac{\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 d \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (3 c d^2-e (2 b d-a e)\right ) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \sqrt{a+b x^2+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 \sqrt [4]{a} d \sqrt{a+b x^2+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 1.88822, size = 1069, normalized size = 1.49 \[ \frac{2 i \sqrt{2} c \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (e x^2+d\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right ),\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right ) d^2-6 i \sqrt{2} c \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (e x^2+d\right ) \Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right ) d^2+4 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} e^2 x \left (c x^4+b x^2+a\right ) d+i \sqrt{2} \left (b-\sqrt{b^2-4 a c}\right ) e \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (e x^2+d\right ) \left (E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right ),\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right ) d+4 i \sqrt{2} b e \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (e x^2+d\right ) \Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right ) d-2 i \sqrt{2} a e^2 \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (e x^2+d\right ) \Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{8 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d \left (c d^3+e (a e-b d) d\right ) \left (e x^2+d\right ) \sqrt{c x^4+b x^2+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e^2*x*(a + b*x^2 + c*x^4) + I*Sqrt[2]*(b - Sqrt[b^2 - 4*a*c])*d*e*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])]*(d + e*x^
2)*(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])] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2
- 4*a*c])]) + (2*I)*Sqrt[2]*c*d^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])]*(d + e*x^2)*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (
b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - (6*I)*Sqrt[2]*c*d^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])]*(d + e*x^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])] + (4*I)*Sqrt[2]*b*d*e*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])]*(d + e*x^2)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*S
qrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - (2*I)*Sqrt[2]*a*e^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])]*(d + e
*x^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])])/(8*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*(c*d^3 + d*e*(-(b*d) + a*
e))*(d + e*x^2)*Sqrt[a + b*x^2 + c*x^4])

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Maple [A]  time = 0.027, size = 1279, normalized size = 1.8 \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^2+d)^2/(c*x^4+b*x^2+a)^(1/2),x)

[Out]

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

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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^{2} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((d + e*x**2)**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^{2} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+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^2 + d)^2), x)