3.398 \(\int \frac{1}{(d+e x^2)^2 \sqrt{2+3 x^2+x^4}} \, dx\)

Optimal. Leaf size=316 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{2 x^2+2}} (2 d-e) \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{2 d \sqrt{x^4+3 x^2+2} (d-e)^2}+\frac{e^2 x \sqrt{x^4+3 x^2+2}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}-\frac{e x \left (x^2+2\right )}{2 d \sqrt{x^4+3 x^2+2} \left (d^2-3 d e+2 e^2\right )}-\frac{e \left (x^2+2\right ) \left (3 d^2-6 d e+2 e^2\right ) \Pi \left (1-\frac{e}{d};\tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{2} d^2 \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2} (d-2 e) (d-e)^2}+\frac{e \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} d \sqrt{x^4+3 x^2+2} (d-2 e) (d-e)} \]

[Out]

-(e*x*(2 + x^2))/(2*d*(d^2 - 3*d*e + 2*e^2)*Sqrt[2 + 3*x^2 + x^4]) + (e^2*x*Sqrt[2 + 3*x^2 + x^4])/(2*d*(d^2 -
3*d*e + 2*e^2)*(d + e*x^2)) + (e*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(Sqrt[2]*d*(d
- 2*e)*(d - e)*Sqrt[2 + 3*x^2 + x^4]) + ((2*d - e)*(1 + x^2)*Sqrt[(2 + x^2)/(2 + 2*x^2)]*EllipticF[ArcTan[x],
1/2])/(2*d*(d - e)^2*Sqrt[2 + 3*x^2 + x^4]) - (e*(3*d^2 - 6*d*e + 2*e^2)*(2 + x^2)*EllipticPi[1 - e/d, ArcTan
[x], 1/2])/(2*Sqrt[2]*d^2*(d - 2*e)*(d - e)^2*Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])

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Rubi [A]  time = 0.326668, antiderivative size = 399, normalized size of antiderivative = 1.26, number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1223, 1716, 1189, 1099, 1135, 1214, 1456, 539} \[ \frac{e^2 x \sqrt{x^4+3 x^2+2}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}-\frac{e x \left (x^2+2\right )}{2 d \sqrt{x^4+3 x^2+2} \left (d^2-3 d e+2 e^2\right )}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \left (3 d^2-6 d e+2 e^2\right ) F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{2} d \sqrt{x^4+3 x^2+2} (d-2 e) (d-e)^2}-\frac{e \left (x^2+2\right ) \left (3 d^2-6 d e+2 e^2\right ) \Pi \left (1-\frac{e}{d};\tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{2} d^2 \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2} (d-2 e) (d-e)^2}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{2} \sqrt{x^4+3 x^2+2} (d-2 e) (d-e)}+\frac{e \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} d \sqrt{x^4+3 x^2+2} (d-2 e) (d-e)} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x^2)^2*Sqrt[2 + 3*x^2 + x^4]),x]

[Out]

-(e*x*(2 + x^2))/(2*d*(d^2 - 3*d*e + 2*e^2)*Sqrt[2 + 3*x^2 + x^4]) + (e^2*x*Sqrt[2 + 3*x^2 + x^4])/(2*d*(d^2 -
3*d*e + 2*e^2)*(d + e*x^2)) + (e*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(Sqrt[2]*d*(d
- 2*e)*(d - e)*Sqrt[2 + 3*x^2 + x^4]) - ((1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(2*Sq
rt[2]*(d - 2*e)*(d - e)*Sqrt[2 + 3*x^2 + x^4]) + ((3*d^2 - 6*d*e + 2*e^2)*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*
EllipticF[ArcTan[x], 1/2])/(2*Sqrt[2]*d*(d - 2*e)*(d - e)^2*Sqrt[2 + 3*x^2 + x^4]) - (e*(3*d^2 - 6*d*e + 2*e^2
)*(2 + x^2)*EllipticPi[1 - e/d, ArcTan[x], 1/2])/(2*Sqrt[2]*d^2*(d - 2*e)*(d - e)^2*Sqrt[(2 + x^2)/(1 + x^2)]*
Sqrt[2 + 3*x^2 + 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 1716

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*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]}, -Dist[(e^2)^(-1), Int[(C*d - B*e - C*e*x^2)/Sqrt[a + b*x^
2 + c*x^4], x], x] + Dist[(C*d^2 - B*d*e + A*e^2)/e^2, Int[1/((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] && Ne
Q[c*d^2 - a*e^2, 0]

Rule 1189

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

Rule 1099

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

Rule 1135

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(x*(b +
q + 2*c*x^2))/(2*c*Sqrt[a + b*x^2 + c*x^4]), x] - Simp[(Rt[(b + q)/(2*a), 2]*(2*a + (b + q)*x^2)*Sqrt[(2*a + (
b - q)*x^2)/(2*a + (b + q)*x^2)]*EllipticE[ArcTan[Rt[(b + q)/(2*a), 2]*x], (2*q)/(b + q)])/(2*c*Sqrt[a + b*x^2
+ c*x^4]), x] /; PosQ[(b + q)/a] &&  !(PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; Fre
eQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]

Rule 1214

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

Rule 1456

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^
(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPar
t[p]), Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c*x^n)/e)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p,
q, r}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p]

Rule 539

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(c*Sqrt[e +
f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[Rt[d/c, 2]*x], 1 - (c*f)/(d*e)])/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sq
rt[(c*(e + f*x^2))/(e*(c + d*x^2))]), x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]

Rubi steps

\begin{align*} \int \frac{1}{\left (d+e x^2\right )^2 \sqrt{2+3 x^2+x^4}} \, dx &=\frac{e^2 x \sqrt{2+3 x^2+x^4}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}-\frac{\int \frac{-2 \left (d^2-3 d e+e^2\right )+2 d e x^2+e^2 x^4}{\left (d+e x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{2 d (d-2 e) (d-e)}\\ &=\frac{e^2 x \sqrt{2+3 x^2+x^4}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}+\frac{\int \frac{-d e^2-e^3 x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{2 d (d-2 e) (d-e) e^2}+\frac{\left (3 d^2-6 d e+2 e^2\right ) \int \frac{1}{\left (d+e x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{2 d (d-2 e) (d-e)}\\ &=\frac{e^2 x \sqrt{2+3 x^2+x^4}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}-\frac{\int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{2 (d-2 e) (d-e)}+\frac{\left (3 d^2-6 d e+2 e^2\right ) \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{2 d (d-2 e) (d-e)^2}-\frac{\left (e \left (3 d^2-6 d e+2 e^2\right )\right ) \int \frac{2+2 x^2}{\left (d+e x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{4 d (d-2 e) (d-e)^2}-\frac{e \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{2 d \left (d^2-3 d e+2 e^2\right )}\\ &=-\frac{e x \left (2+x^2\right )}{2 d \left (d^2-3 d e+2 e^2\right ) \sqrt{2+3 x^2+x^4}}+\frac{e^2 x \sqrt{2+3 x^2+x^4}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}+\frac{e \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} d (d-2 e) (d-e) \sqrt{2+3 x^2+x^4}}-\frac{\left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{2} (d-2 e) (d-e) \sqrt{2+3 x^2+x^4}}+\frac{\left (3 d^2-6 d e+2 e^2\right ) \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{2} d (d-2 e) (d-e)^2 \sqrt{2+3 x^2+x^4}}-\frac{\left (e \left (3 d^2-6 d e+2 e^2\right ) \sqrt{1+\frac{x^2}{2}} \sqrt{2+2 x^2}\right ) \int \frac{\sqrt{2+2 x^2}}{\sqrt{1+\frac{x^2}{2}} \left (d+e x^2\right )} \, dx}{4 d (d-2 e) (d-e)^2 \sqrt{2+3 x^2+x^4}}\\ &=-\frac{e x \left (2+x^2\right )}{2 d \left (d^2-3 d e+2 e^2\right ) \sqrt{2+3 x^2+x^4}}+\frac{e^2 x \sqrt{2+3 x^2+x^4}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}+\frac{e \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} d (d-2 e) (d-e) \sqrt{2+3 x^2+x^4}}-\frac{\left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{2} (d-2 e) (d-e) \sqrt{2+3 x^2+x^4}}+\frac{\left (3 d^2-6 d e+2 e^2\right ) \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{2} d (d-2 e) (d-e)^2 \sqrt{2+3 x^2+x^4}}-\frac{e \left (3 d^2-6 d e+2 e^2\right ) \left (2+x^2\right ) \Pi \left (1-\frac{e}{d};\tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{2} d^2 (d-2 e) (d-e)^2 \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}\\ \end{align*}

Mathematica [C]  time = 0.583131, size = 175, normalized size = 0.55 \[ \frac{\frac{e^2 x \left (x^4+3 x^2+2\right )}{\left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}+\frac{i \sqrt{x^2+1} \sqrt{x^2+2} \left (d (d-e) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ),2\right )+\left (-3 d^2+6 d e-2 e^2\right ) \Pi \left (\frac{2 e}{d};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+d e E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )\right )}{d (d-2 e) (d-e)}}{2 d \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x^2)^2*Sqrt[2 + 3*x^2 + x^4]),x]

[Out]

((e^2*x*(2 + 3*x^2 + x^4))/((d^2 - 3*d*e + 2*e^2)*(d + e*x^2)) + (I*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(d*e*EllipticE
[I*ArcSinh[x/Sqrt[2]], 2] + d*(d - e)*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] + (-3*d^2 + 6*d*e - 2*e^2)*EllipticPi
[(2*e)/d, I*ArcSinh[x/Sqrt[2]], 2]))/(d*(d - 2*e)*(d - e)))/(2*d*Sqrt[2 + 3*x^2 + x^4])

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Maple [C]  time = 0.026, size = 443, normalized size = 1.4 \begin{align*}{\frac{{e}^{2}x}{2\, \left ({d}^{2}-3\,de+2\,{e}^{2} \right ) d \left ( e{x}^{2}+d \right ) }\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{{\frac{i}{4}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) }{{d}^{2}-3\,de+2\,{e}^{2}}\sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{\frac{{\frac{i}{4}}e\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) }{ \left ({d}^{2}-3\,de+2\,{e}^{2} \right ) d}\sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{\frac{{\frac{i}{4}}e\sqrt{2}{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) }{ \left ({d}^{2}-3\,de+2\,{e}^{2} \right ) d}\sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{\frac{{\frac{3\,i}{2}}\sqrt{2}}{{d}^{2}-3\,de+2\,{e}^{2}}\sqrt{1+{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{e}{d}},\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{\frac{3\,ie\sqrt{2}}{ \left ({d}^{2}-3\,de+2\,{e}^{2} \right ) d}\sqrt{1+{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{e}{d}},\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{\frac{i{e}^{2}\sqrt{2}}{ \left ({d}^{2}-3\,de+2\,{e}^{2} \right ){d}^{2}}\sqrt{1+{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{e}{d}},\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^2+d)^2/(x^4+3*x^2+2)^(1/2),x)

[Out]

1/2*e^2*x*(x^4+3*x^2+2)^(1/2)/d/(d^2-3*d*e+2*e^2)/(e*x^2+d)+1/4*I/(d^2-3*d*e+2*e^2)*2^(1/2)*(2*x^2+4)^(1/2)*(x
^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticF(1/2*I*x*2^(1/2),2^(1/2))-1/4*I*e/(d^2-3*d*e+2*e^2)/d*2^(1/2)*(2*x^2+
4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticF(1/2*I*x*2^(1/2),2^(1/2))+1/4*I*e/(d^2-3*d*e+2*e^2)/d*2^(1
/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticE(1/2*I*x*2^(1/2),2^(1/2))-3/2*I/(d^2-3*d*e+2*e^
2)*2^(1/2)*(1+1/2*x^2)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticPi(1/2*I*x*2^(1/2),2*e/d,2^(1/2))+3*I/(
d^2-3*d*e+2*e^2)/d*e*2^(1/2)*(1+1/2*x^2)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticPi(1/2*I*x*2^(1/2),2*
e/d,2^(1/2))-I/(d^2-3*d*e+2*e^2)/d^2*e^2*2^(1/2)*(1+1/2*x^2)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticP
i(1/2*I*x*2^(1/2),2*e/d,2^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{4} + 3 \, x^{2} + 2}{\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/(x^4+3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^4 + 3*x^2 + 2)*(e*x^2 + d)^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{4} + 3 \, x^{2} + 2}}{e^{2} x^{8} +{\left (2 \, d e + 3 \, e^{2}\right )} x^{6} +{\left (d^{2} + 6 \, d e + 2 \, e^{2}\right )} x^{4} +{\left (3 \, d^{2} + 4 \, d e\right )} x^{2} + 2 \, d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^2/(x^4+3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(x^4 + 3*x^2 + 2)/(e^2*x^8 + (2*d*e + 3*e^2)*x^6 + (d^2 + 6*d*e + 2*e^2)*x^4 + (3*d^2 + 4*d*e)*x^
2 + 2*d^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)**2/(x**4+3*x**2+2)**(1/2),x)

[Out]

Integral(1/(sqrt((x**2 + 1)*(x**2 + 2))*(d + e*x**2)**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{4} + 3 \, x^{2} + 2}{\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/(x^4+3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^4 + 3*x^2 + 2)*(e*x^2 + d)^2), x)