\(\int \sqrt {2+3 x^2+x^4} \, dx\) [289]

Optimal result

Integrand size = 14, antiderivative size = 141 \[ \int \sqrt {2+3 x^2+x^4} \, dx=\frac {x \left (2+x^2\right )}{\sqrt {2+3 x^2+x^4}}+\frac {1}{3} x \sqrt {2+3 x^2+x^4}-\frac {\sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {2+3 x^2+x^4}}+\frac {2 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{3 \sqrt {2+3 x^2+x^4}} \]

[Out]

x*(x^2+2)/(x^4+3*x^2+2)^(1/2)-(x^2+1)^(3/2)*(1/(x^2+1))^(1/2)*EllipticE(x/(x^2+1)^(1/2),1/2*2^(1/2))*2^(1/2)*(
(x^2+2)/(x^2+1))^(1/2)/(x^4+3*x^2+2)^(1/2)+2/3*(x^2+1)^(3/2)*(1/(x^2+1))^(1/2)*EllipticF(x/(x^2+1)^(1/2),1/2*2
^(1/2))*2^(1/2)*((x^2+2)/(x^2+1))^(1/2)/(x^4+3*x^2+2)^(1/2)+1/3*x*(x^4+3*x^2+2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1105, 1203, 1113, 1149} \[ \int \sqrt {2+3 x^2+x^4} \, dx=\frac {2 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{3 \sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}+\frac {1}{3} \sqrt {x^4+3 x^2+2} x+\frac {\left (x^2+2\right ) x}{\sqrt {x^4+3 x^2+2}} \]

[In]

Int[Sqrt[2 + 3*x^2 + x^4],x]

[Out]

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

Rule 1105

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*((a + b*x^2 + c*x^4)^p/(4*p + 1)), x] + Dis
t[2*(p/(4*p + 1)), Int[(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4
*a*c, 0] && GtQ[p, 0] && IntegerQ[2*p]

Rule 1113

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)]/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*Elli
pticF[ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], 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 1149

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)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*(q
/(b + q))], 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 1203

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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x \sqrt {2+3 x^2+x^4}+\frac {1}{3} \int \frac {4+3 x^2}{\sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {1}{3} x \sqrt {2+3 x^2+x^4}+\frac {4}{3} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx+\int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {x \left (2+x^2\right )}{\sqrt {2+3 x^2+x^4}}+\frac {1}{3} x \sqrt {2+3 x^2+x^4}-\frac {\sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2+3 x^2+x^4}}+\frac {2 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {2+3 x^2+x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.65 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.72 \[ \int \sqrt {2+3 x^2+x^4} \, dx=\frac {2 x+3 x^3+x^5-3 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{3 \sqrt {2+3 x^2+x^4}} \]

[In]

Integrate[Sqrt[2 + 3*x^2 + x^4],x]

[Out]

(2*x + 3*x^3 + x^5 - (3*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - I*Sqrt[1 + x^2]*Sq
rt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(3*Sqrt[2 + 3*x^2 + x^4])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.61 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {x \sqrt {x^{4}+3 x^{2}+2}}{3}-\frac {2 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{3 \sqrt {x^{4}+3 x^{2}+2}}+\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}\)	\(121\)
risch	\(\frac {x \sqrt {x^{4}+3 x^{2}+2}}{3}-\frac {2 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{3 \sqrt {x^{4}+3 x^{2}+2}}+\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}\)	\(121\)
elliptic	\(\frac {x \sqrt {x^{4}+3 x^{2}+2}}{3}-\frac {2 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{3 \sqrt {x^{4}+3 x^{2}+2}}+\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}\)	\(121\)

[In]

int((x^4+3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*x*(x^4+3*x^2+2)^(1/2)-2/3*I*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*2^(1
/2)*x,2^(1/2))+1/2*I*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*2^(1/2)*x,2^(1
/2))-EllipticE(1/2*I*2^(1/2)*x,2^(1/2)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.33 \[ \int \sqrt {2+3 x^2+x^4} \, dx=\frac {-3 i \, x E(\arcsin \left (\frac {i}{x}\right )\,|\,2) + 7 i \, x F(\arcsin \left (\frac {i}{x}\right )\,|\,2) + \sqrt {x^{4} + 3 \, x^{2} + 2} {\left (x^{2} + 3\right )}}{3 \, x} \]

[In]

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

[Out]

1/3*(-3*I*x*elliptic_e(arcsin(I/x), 2) + 7*I*x*elliptic_f(arcsin(I/x), 2) + sqrt(x^4 + 3*x^2 + 2)*(x^2 + 3))/x

Sympy [F]

\[ \int \sqrt {2+3 x^2+x^4} \, dx=\int \sqrt {x^{4} + 3 x^{2} + 2}\, dx \]

[In]

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

[Out]

Integral(sqrt(x**4 + 3*x**2 + 2), x)

Maxima [F]

\[ \int \sqrt {2+3 x^2+x^4} \, dx=\int { \sqrt {x^{4} + 3 \, x^{2} + 2} \,d x } \]

[In]

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

[Out]

integrate(sqrt(x^4 + 3*x^2 + 2), x)

Giac [F]

\[ \int \sqrt {2+3 x^2+x^4} \, dx=\int { \sqrt {x^{4} + 3 \, x^{2} + 2} \,d x } \]

[In]

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

[Out]

integrate(sqrt(x^4 + 3*x^2 + 2), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {2+3 x^2+x^4} \, dx=\int \sqrt {x^4+3\,x^2+2} \,d x \]

[In]

int((3*x^2 + x^4 + 2)^(1/2),x)

[Out]

int((3*x^2 + x^4 + 2)^(1/2), x)