Integrand size = 22, antiderivative size = 48 \[ \int \frac {\sqrt {2+3 x^2+x^4}}{\left (1+x^2\right )^2} \, dx=\frac {\sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {2+3 x^2+x^4}} \] Output:
2^(1/2)*(x^2+1)*((x^2+2)/(x^2+1))^(1/2)*EllipticE(x/(x^2+1)^(1/2),1/2*2^(1 /2))/(x^4+3*x^2+2)^(1/2)
Result contains complex when optimal does not.
Time = 10.16 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.96 \[ \int \frac {\sqrt {2+3 x^2+x^4}}{\left (1+x^2\right )^2} \, dx=\frac {2 x+x^3+i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{\sqrt {2+3 x^2+x^4}} \] Input:
Integrate[Sqrt[2 + 3*x^2 + x^4]/(1 + x^2)^2,x]
Output:
(2*x + x^3 + I*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - I*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/Sq rt[2 + 3*x^2 + x^4]
Time = 0.17 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1395, 313}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {x^4+3 x^2+2}}{\left (x^2+1\right )^2} \, dx\) |
\(\Big \downarrow \) 1395 |
\(\displaystyle \frac {\sqrt {x^4+3 x^2+2} \int \frac {\sqrt {x^2+2}}{\left (x^2+1\right )^{3/2}}dx}{\sqrt {x^2+1} \sqrt {x^2+2}}\) |
\(\Big \downarrow \) 313 |
\(\displaystyle \frac {\sqrt {2} \sqrt {x^4+3 x^2+2} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}}}\) |
Input:
Int[Sqrt[2 + 3*x^2 + x^4]/(1 + x^2)^2,x]
Output:
(Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]*EllipticE[ArcTan[x], 1/2])/((1 + x^2)*Sqrt[ (2 + x^2)/(1 + x^2)])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Result contains complex when optimal does not.
Time = 0.93 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.67
method | result | size |
risch | \(\frac {x \left (x^{2}+2\right )}{\sqrt {x^{4}+3 x^{2}+2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}\) | \(80\) |
default | \(\frac {\left (x^{2}+2\right ) x}{\sqrt {\left (x^{2}+2\right ) \left (x^{2}+1\right )}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}\) | \(81\) |
elliptic | \(\frac {\left (x^{2}+2\right ) x}{\sqrt {\left (x^{2}+2\right ) \left (x^{2}+1\right )}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}\) | \(81\) |
Input:
int((x^4+3*x^2+2)^(1/2)/(x^2+1)^2,x,method=_RETURNVERBOSE)
Output:
x*(x^2+2)/(x^4+3*x^2+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*x*2^(1/2),2^(1/2))-EllipticE(1/2*I*x* 2^(1/2),2^(1/2)))
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {2+3 x^2+x^4}}{\left (1+x^2\right )^2} \, dx=-\frac {\sqrt {2} \sqrt {-\frac {1}{2}} {\left (x^{2} + 1\right )} E(\arcsin \left (\sqrt {-\frac {1}{2}} x\right )\,|\,2) - \sqrt {2} \sqrt {-\frac {1}{2}} {\left (x^{2} + 1\right )} F(\arcsin \left (\sqrt {-\frac {1}{2}} x\right )\,|\,2) - 2 \, \sqrt {x^{4} + 3 \, x^{2} + 2} x}{2 \, {\left (x^{2} + 1\right )}} \] Input:
integrate((x^4+3*x^2+2)^(1/2)/(x^2+1)^2,x, algorithm="fricas")
Output:
-1/2*(sqrt(2)*sqrt(-1/2)*(x^2 + 1)*elliptic_e(arcsin(sqrt(-1/2)*x), 2) - s qrt(2)*sqrt(-1/2)*(x^2 + 1)*elliptic_f(arcsin(sqrt(-1/2)*x), 2) - 2*sqrt(x ^4 + 3*x^2 + 2)*x)/(x^2 + 1)
\[ \int \frac {\sqrt {2+3 x^2+x^4}}{\left (1+x^2\right )^2} \, dx=\int \frac {\sqrt {\left (x^{2} + 1\right ) \left (x^{2} + 2\right )}}{\left (x^{2} + 1\right )^{2}}\, dx \] Input:
integrate((x**4+3*x**2+2)**(1/2)/(x**2+1)**2,x)
Output:
Integral(sqrt((x**2 + 1)*(x**2 + 2))/(x**2 + 1)**2, x)
\[ \int \frac {\sqrt {2+3 x^2+x^4}}{\left (1+x^2\right )^2} \, dx=\int { \frac {\sqrt {x^{4} + 3 \, x^{2} + 2}}{{\left (x^{2} + 1\right )}^{2}} \,d x } \] Input:
integrate((x^4+3*x^2+2)^(1/2)/(x^2+1)^2,x, algorithm="maxima")
Output:
integrate(sqrt(x^4 + 3*x^2 + 2)/(x^2 + 1)^2, x)
\[ \int \frac {\sqrt {2+3 x^2+x^4}}{\left (1+x^2\right )^2} \, dx=\int { \frac {\sqrt {x^{4} + 3 \, x^{2} + 2}}{{\left (x^{2} + 1\right )}^{2}} \,d x } \] Input:
integrate((x^4+3*x^2+2)^(1/2)/(x^2+1)^2,x, algorithm="giac")
Output:
integrate(sqrt(x^4 + 3*x^2 + 2)/(x^2 + 1)^2, x)
Timed out. \[ \int \frac {\sqrt {2+3 x^2+x^4}}{\left (1+x^2\right )^2} \, dx=\int \frac {\sqrt {x^4+3\,x^2+2}}{{\left (x^2+1\right )}^2} \,d x \] Input:
int((3*x^2 + x^4 + 2)^(1/2)/(x^2 + 1)^2,x)
Output:
int((3*x^2 + x^4 + 2)^(1/2)/(x^2 + 1)^2, x)
\[ \int \frac {\sqrt {2+3 x^2+x^4}}{\left (1+x^2\right )^2} \, dx=\int \frac {\sqrt {x^{4}+3 x^{2}+2}}{x^{4}+2 x^{2}+1}d x \] Input:
int((x^4+3*x^2+2)^(1/2)/(x^2+1)^2,x)
Output:
int(sqrt(x**4 + 3*x**2 + 2)/(x**4 + 2*x**2 + 1),x)