\(\int \frac {e^{\frac {x}{2+x^2}} (2+2 x+3 x^2-x^3+2 x^4)}{2 x+x^3} \, dx\) [3]

Optimal result

Integrand size = 41, antiderivative size = 28 \[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx=e^{\frac {x}{2+x^2}} \left (2+x^2\right )+\operatorname {ExpIntegralEi}\left (\frac {x}{2+x^2}\right ) \] Output:

exp(x/(x^2+2))*(x^2+2)+Ei(x/(x^2+2))
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx=2 e^{\frac {x}{2+x^2}}+e^{\frac {x}{2+x^2}} x^2+\operatorname {ExpIntegralEi}\left (\frac {x}{2+x^2}\right ) \] Input:

Integrate[(E^(x/(2 + x^2))*(2 + 2*x + 3*x^2 - x^3 + 2*x^4))/(2*x + x^3),x]
 

Output:

2*E^(x/(2 + x^2)) + E^(x/(2 + x^2))*x^2 + ExpIntegralEi[x/(2 + x^2)]
 

Rubi [F]

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.

\(\displaystyle \int \frac {e^{\frac {x}{x^2+2}} \left (2 x^4-x^3+3 x^2+2 x+2\right )}{x^3+2 x} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {e^{\frac {x}{x^2+2}} \left (2 x^4-x^3+3 x^2+2 x+2\right )}{x \left (x^2+2\right )}dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (-\frac {2 e^{\frac {x}{x^2+2}} (x-2)}{x^2+2}-e^{\frac {x}{x^2+2}}+2 e^{\frac {x}{x^2+2}} x+\frac {e^{\frac {x}{x^2+2}}}{x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\int e^{\frac {x}{x^2+2}}dx+\left (1+i \sqrt {2}\right ) \int \frac {e^{\frac {x}{x^2+2}}}{i \sqrt {2}-x}dx+\int \frac {e^{\frac {x}{x^2+2}}}{x}dx+2 \int e^{\frac {x}{x^2+2}} xdx-\left (1-i \sqrt {2}\right ) \int \frac {e^{\frac {x}{x^2+2}}}{x+i \sqrt {2}}dx\)

Input:

Int[(E^(x/(2 + x^2))*(2 + 2*x + 3*x^2 - x^3 + 2*x^4))/(2*x + x^3),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [F]

\[\int \frac {\left (2 x^{4}-x^{3}+3 x^{2}+2 x +2\right ) {\mathrm e}^{\frac {x}{x^{2}+2}}}{x^{3}+2 x}d x\]

Input:

int((2*x^4-x^3+3*x^2+2*x+2)*exp(x/(x^2+2))/(x^3+2*x),x)
 

Output:

int((2*x^4-x^3+3*x^2+2*x+2)*exp(x/(x^2+2))/(x^3+2*x),x)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx={\left (x^{2} + 2\right )} e^{\left (\frac {x}{x^{2} + 2}\right )} + {\rm Ei}\left (\frac {x}{x^{2} + 2}\right ) \] Input:

integrate((2*x^4-x^3+3*x^2+2*x+2)*exp(x/(x^2+2))/(x^3+2*x),x, algorithm="f 
ricas")
 

Output:

(x^2 + 2)*e^(x/(x^2 + 2)) + Ei(x/(x^2 + 2))
 

Sympy [F]

\[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx=\int \frac {\left (2 x^{4} - x^{3} + 3 x^{2} + 2 x + 2\right ) e^{\frac {x}{x^{2} + 2}}}{x \left (x^{2} + 2\right )}\, dx \] Input:

integrate((2*x**4-x**3+3*x**2+2*x+2)*exp(x/(x**2+2))/(x**3+2*x),x)
 

Output:

Integral((2*x**4 - x**3 + 3*x**2 + 2*x + 2)*exp(x/(x**2 + 2))/(x*(x**2 + 2 
)), x)
 

Maxima [F]

\[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx=\int { \frac {{\left (2 \, x^{4} - x^{3} + 3 \, x^{2} + 2 \, x + 2\right )} e^{\left (\frac {x}{x^{2} + 2}\right )}}{x^{3} + 2 \, x} \,d x } \] Input:

integrate((2*x^4-x^3+3*x^2+2*x+2)*exp(x/(x^2+2))/(x^3+2*x),x, algorithm="m 
axima")
 

Output:

integrate((2*x^4 - x^3 + 3*x^2 + 2*x + 2)*e^(x/(x^2 + 2))/(x^3 + 2*x), x)
 

Giac [F]

\[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx=\int { \frac {{\left (2 \, x^{4} - x^{3} + 3 \, x^{2} + 2 \, x + 2\right )} e^{\left (\frac {x}{x^{2} + 2}\right )}}{x^{3} + 2 \, x} \,d x } \] Input:

integrate((2*x^4-x^3+3*x^2+2*x+2)*exp(x/(x^2+2))/(x^3+2*x),x, algorithm="g 
iac")
 

Output:

integrate((2*x^4 - x^3 + 3*x^2 + 2*x + 2)*e^(x/(x^2 + 2))/(x^3 + 2*x), x)
 

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx=\mathrm {ei}\left (\frac {x}{x^2+2}\right )+2\,{\mathrm {e}}^{\frac {x}{x^2+2}}+x^2\,{\mathrm {e}}^{\frac {x}{x^2+2}} \] Input:

int((exp(x/(x^2 + 2))*(2*x + 3*x^2 - x^3 + 2*x^4 + 2))/(2*x + x^3),x)
 

Output:

ei(x/(x^2 + 2)) + 2*exp(x/(x^2 + 2)) + x^2*exp(x/(x^2 + 2))
 

Reduce [F]

\[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx=\frac {167 e^{\frac {x}{x^{2}+2}} x^{8}+42 e^{\frac {x}{x^{2}+2}} x^{6}+1294 e^{\frac {x}{x^{2}+2}} x^{5}-4236 e^{\frac {x}{x^{2}+2}} x^{4}+5336 e^{\frac {x}{x^{2}+2}} x^{3}-11184 e^{\frac {x}{x^{2}+2}} x^{2}+5184 e^{\frac {x}{x^{2}+2}} x -7824 e^{\frac {x}{x^{2}+2}}+5344 \left (\int \frac {e^{\frac {x}{x^{2}+2}}}{x^{11}+10 x^{9}+40 x^{7}+80 x^{5}+80 x^{3}+32 x}d x \right ) x^{6}+32064 \left (\int \frac {e^{\frac {x}{x^{2}+2}}}{x^{11}+10 x^{9}+40 x^{7}+80 x^{5}+80 x^{3}+32 x}d x \right ) x^{4}+64128 \left (\int \frac {e^{\frac {x}{x^{2}+2}}}{x^{11}+10 x^{9}+40 x^{7}+80 x^{5}+80 x^{3}+32 x}d x \right ) x^{2}+42752 \left (\int \frac {e^{\frac {x}{x^{2}+2}}}{x^{11}+10 x^{9}+40 x^{7}+80 x^{5}+80 x^{3}+32 x}d x \right )+256 \left (\int \frac {e^{\frac {x}{x^{2}+2}}}{x^{10}+10 x^{8}+40 x^{6}+80 x^{4}+80 x^{2}+32}d x \right ) x^{6}+1536 \left (\int \frac {e^{\frac {x}{x^{2}+2}}}{x^{10}+10 x^{8}+40 x^{6}+80 x^{4}+80 x^{2}+32}d x \right ) x^{4}+3072 \left (\int \frac {e^{\frac {x}{x^{2}+2}}}{x^{10}+10 x^{8}+40 x^{6}+80 x^{4}+80 x^{2}+32}d x \right ) x^{2}+2048 \left (\int \frac {e^{\frac {x}{x^{2}+2}}}{x^{10}+10 x^{8}+40 x^{6}+80 x^{4}+80 x^{2}+32}d x \right )-167 \left (\int \frac {e^{\frac {x}{x^{2}+2}} x^{5}}{x^{6}+6 x^{4}+12 x^{2}+8}d x \right ) x^{6}-1002 \left (\int \frac {e^{\frac {x}{x^{2}+2}} x^{5}}{x^{6}+6 x^{4}+12 x^{2}+8}d x \right ) x^{4}-2004 \left (\int \frac {e^{\frac {x}{x^{2}+2}} x^{5}}{x^{6}+6 x^{4}+12 x^{2}+8}d x \right ) x^{2}-1336 \left (\int \frac {e^{\frac {x}{x^{2}+2}} x^{5}}{x^{6}+6 x^{4}+12 x^{2}+8}d x \right )+3920 \left (\int \frac {e^{\frac {x}{x^{2}+2}} x}{x^{10}+10 x^{8}+40 x^{6}+80 x^{4}+80 x^{2}+32}d x \right ) x^{6}+23520 \left (\int \frac {e^{\frac {x}{x^{2}+2}} x}{x^{10}+10 x^{8}+40 x^{6}+80 x^{4}+80 x^{2}+32}d x \right ) x^{4}+47040 \left (\int \frac {e^{\frac {x}{x^{2}+2}} x}{x^{10}+10 x^{8}+40 x^{6}+80 x^{4}+80 x^{2}+32}d x \right ) x^{2}+31360 \left (\int \frac {e^{\frac {x}{x^{2}+2}} x}{x^{10}+10 x^{8}+40 x^{6}+80 x^{4}+80 x^{2}+32}d x \right )}{167 x^{6}+1002 x^{4}+2004 x^{2}+1336} \] Input:

int((2*x^4-x^3+3*x^2+2*x+2)*exp(x/(x^2+2))/(x^3+2*x),x)
 

Output:

(167*e**(x/(x**2 + 2))*x**8 + 42*e**(x/(x**2 + 2))*x**6 + 1294*e**(x/(x**2 
 + 2))*x**5 - 4236*e**(x/(x**2 + 2))*x**4 + 5336*e**(x/(x**2 + 2))*x**3 - 
11184*e**(x/(x**2 + 2))*x**2 + 5184*e**(x/(x**2 + 2))*x - 7824*e**(x/(x**2 
 + 2)) + 5344*int(e**(x/(x**2 + 2))/(x**11 + 10*x**9 + 40*x**7 + 80*x**5 + 
 80*x**3 + 32*x),x)*x**6 + 32064*int(e**(x/(x**2 + 2))/(x**11 + 10*x**9 + 
40*x**7 + 80*x**5 + 80*x**3 + 32*x),x)*x**4 + 64128*int(e**(x/(x**2 + 2))/ 
(x**11 + 10*x**9 + 40*x**7 + 80*x**5 + 80*x**3 + 32*x),x)*x**2 + 42752*int 
(e**(x/(x**2 + 2))/(x**11 + 10*x**9 + 40*x**7 + 80*x**5 + 80*x**3 + 32*x), 
x) + 256*int(e**(x/(x**2 + 2))/(x**10 + 10*x**8 + 40*x**6 + 80*x**4 + 80*x 
**2 + 32),x)*x**6 + 1536*int(e**(x/(x**2 + 2))/(x**10 + 10*x**8 + 40*x**6 
+ 80*x**4 + 80*x**2 + 32),x)*x**4 + 3072*int(e**(x/(x**2 + 2))/(x**10 + 10 
*x**8 + 40*x**6 + 80*x**4 + 80*x**2 + 32),x)*x**2 + 2048*int(e**(x/(x**2 + 
 2))/(x**10 + 10*x**8 + 40*x**6 + 80*x**4 + 80*x**2 + 32),x) - 167*int((e* 
*(x/(x**2 + 2))*x**5)/(x**6 + 6*x**4 + 12*x**2 + 8),x)*x**6 - 1002*int((e* 
*(x/(x**2 + 2))*x**5)/(x**6 + 6*x**4 + 12*x**2 + 8),x)*x**4 - 2004*int((e* 
*(x/(x**2 + 2))*x**5)/(x**6 + 6*x**4 + 12*x**2 + 8),x)*x**2 - 1336*int((e* 
*(x/(x**2 + 2))*x**5)/(x**6 + 6*x**4 + 12*x**2 + 8),x) + 3920*int((e**(x/( 
x**2 + 2))*x)/(x**10 + 10*x**8 + 40*x**6 + 80*x**4 + 80*x**2 + 32),x)*x**6 
 + 23520*int((e**(x/(x**2 + 2))*x)/(x**10 + 10*x**8 + 40*x**6 + 80*x**4 + 
80*x**2 + 32),x)*x**4 + 47040*int((e**(x/(x**2 + 2))*x)/(x**10 + 10*x**...