Integrand size = 86, antiderivative size = 23 \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx=\left (4+e^{\frac {2 e^2}{3 x}}\right ) \left (1+x+x^2\right )^2 \]
Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx=\left (4+e^{\frac {2 e^2}{3 x}}\right ) \left (1+x+x^2\right )^2 \]
Integrate[(24*x^2 + 72*x^3 + 72*x^4 + 48*x^5 + E^((2*E^2)/(3*x))*(6*x^2 + 18*x^3 + 18*x^4 + 12*x^5 + E^2*(-2 - 4*x - 6*x^2 - 4*x^3 - 2*x^4)))/(3*x^2 ),x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.68 (sec) , antiderivative size = 301, normalized size of antiderivative = 13.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {27, 27, 2010, 2009}
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.
| \(\displaystyle \int \frac {48 x^5+72 x^4+72 x^3+24 x^2+e^{\frac {2 e^2}{3 x}} \left (12 x^5+18 x^4+18 x^3+6 x^2+e^2 \left (-2 x^4-4 x^3-6 x^2-4 x-2\right )\right )}{3 x^2} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {2 \left (24 x^5+36 x^4+36 x^3+12 x^2+e^{\frac {2 e^2}{3 x}} \left (6 x^5+9 x^4+9 x^3+3 x^2-e^2 \left (x^4+2 x^3+3 x^2+2 x+1\right )\right )\right )}{x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} \int \frac {24 x^5+36 x^4+36 x^3+12 x^2+e^{\frac {2 e^2}{3 x}} \left (6 x^5+9 x^4+9 x^3+3 x^2-e^2 \left (x^4+2 x^3+3 x^2+2 x+1\right )\right )}{x^2}dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {2}{3} \int \left (12 \left (2 x^3+3 x^2+3 x+1\right )+\frac {e^{\frac {2 e^2}{3 x}} \left (x^2+x+1\right ) \left (6 x^3+\left (3-e^2\right ) x^2-e^2 x-e^2\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} \left (-\frac {4}{81} e^6 \left (9-e^2\right ) \operatorname {ExpIntegralEi}\left (\frac {2 e^2}{3 x}\right )-2 e^2 \left (1-e^2\right ) \operatorname {ExpIntegralEi}\left (\frac {2 e^2}{3 x}\right )-\frac {2}{9} e^4 \left (9-2 e^2\right ) \operatorname {ExpIntegralEi}\left (\frac {2 e^2}{3 x}\right )+2 e^2 \operatorname {ExpIntegralEi}\left (\frac {2 e^2}{3 x}\right )+6 x^4+\frac {1}{3} \left (9-e^2\right ) e^{\frac {2 e^2}{3 x}} x^3+12 x^3+\frac {1}{9} \left (9-e^2\right ) e^{\frac {2 e^2}{3 x}+2} x^2+\frac {1}{2} \left (9-2 e^2\right ) e^{\frac {2 e^2}{3 x}} x^2+18 x^2+\frac {2}{27} \left (9-e^2\right ) e^{\frac {2 e^2}{3 x}+4} x+3 \left (1-e^2\right ) e^{\frac {2 e^2}{3 x}} x+\frac {1}{3} \left (9-2 e^2\right ) e^{\frac {2 e^2}{3 x}+2} x+12 x+\frac {3}{2} e^{\frac {2 e^2}{3 x}}+\frac {32}{27} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right )\right )\) |
Int[(24*x^2 + 72*x^3 + 72*x^4 + 48*x^5 + E^((2*E^2)/(3*x))*(6*x^2 + 18*x^3 + 18*x^4 + 12*x^5 + E^2*(-2 - 4*x - 6*x^2 - 4*x^3 - 2*x^4)))/(3*x^2),x]
(2*((3*E^((2*E^2)/(3*x)))/2 + 12*x + (E^(2 + (2*E^2)/(3*x))*(9 - 2*E^2)*x) /3 + 3*E^((2*E^2)/(3*x))*(1 - E^2)*x + (2*E^(4 + (2*E^2)/(3*x))*(9 - E^2)* x)/27 + 18*x^2 + (E^((2*E^2)/(3*x))*(9 - 2*E^2)*x^2)/2 + (E^(2 + (2*E^2)/( 3*x))*(9 - E^2)*x^2)/9 + 12*x^3 + (E^((2*E^2)/(3*x))*(9 - E^2)*x^3)/3 + 6* x^4 + 2*E^2*ExpIntegralEi[(2*E^2)/(3*x)] - (2*E^4*(9 - 2*E^2)*ExpIntegralE i[(2*E^2)/(3*x)])/9 - 2*E^2*(1 - E^2)*ExpIntegralEi[(2*E^2)/(3*x)] - (4*E^ 6*(9 - E^2)*ExpIntegralEi[(2*E^2)/(3*x)])/81 + (32*E^8*Gamma[-4, (-2*E^2)/ (3*x)])/27))/3
3.22.71.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(19)=38\).
Time = 0.47 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22
| method | result | size |
| risch | \(4 x^{4}+8 x^{3}+12 x^{2}+8 x +4+\frac {\left (3 x^{4}+6 x^{3}+9 x^{2}+6 x +3\right ) {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{3}\) | \(51\) |
| parallelrisch | \({\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}} x^{4}+4 x^{4}+2 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}} x^{3}+8 x^{3}+3 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}} x^{2}+12 x^{2}+2 x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}+8 x +{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}\) | \(77\) |
| norman | \(\frac {x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}+{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}} x^{5}+8 x^{2}+12 x^{3}+8 x^{4}+4 x^{5}+2 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}} x^{2}+3 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}} x^{3}+2 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}} x^{4}}{x}\) | \(87\) |
| derivativedivides | \(4 x^{4}+8 x^{3}+12 x^{2}+8 x -\frac {{\mathrm e}^{-2} \left (-162 \,{\mathrm e}^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}+128 \,{\mathrm e}^{10} \left (-\frac {81 x^{4} {\mathrm e}^{-8} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{64}-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{16}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{24}\right )+288 \,{\mathrm e}^{8} \left (-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )-32 \,{\mathrm e}^{10} \left (-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )+432 \,{\mathrm e}^{6} \left (-\frac {9 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )-96 \,{\mathrm e}^{8} \left (-\frac {9 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )+216 \,{\mathrm e}^{4} \left (-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )-216 \,{\mathrm e}^{6} \left (-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )+216 \,{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )}{162}\) | \(438\) |
| default | \(4 x^{4}+8 x^{3}+12 x^{2}+8 x -\frac {{\mathrm e}^{-2} \left (-162 \,{\mathrm e}^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}+128 \,{\mathrm e}^{10} \left (-\frac {81 x^{4} {\mathrm e}^{-8} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{64}-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{16}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{24}\right )+288 \,{\mathrm e}^{8} \left (-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )-32 \,{\mathrm e}^{10} \left (-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )+432 \,{\mathrm e}^{6} \left (-\frac {9 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )-96 \,{\mathrm e}^{8} \left (-\frac {9 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )+216 \,{\mathrm e}^{4} \left (-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )-216 \,{\mathrm e}^{6} \left (-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )+216 \,{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )}{162}\) | \(438\) |
| parts | \(4 x^{4}+8 x^{3}+12 x^{2}+8 x +\frac {{\mathrm e}^{-2} \left (81 \,{\mathrm e}^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}-64 \,{\mathrm e}^{10} \left (-\frac {81 x^{4} {\mathrm e}^{-8} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{64}-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{16}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{24}\right )-144 \,{\mathrm e}^{8} \left (-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )+16 \,{\mathrm e}^{10} \left (-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )-216 \,{\mathrm e}^{6} \left (-\frac {9 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )+48 \,{\mathrm e}^{8} \left (-\frac {9 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )-108 \,{\mathrm e}^{4} \left (-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )-108 \,{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )+108 \,{\mathrm e}^{6} \left (-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )\right )}{81}\) | \(438\) |
int(1/3*(((-2*x^4-4*x^3-6*x^2-4*x-2)*exp(2)+12*x^5+18*x^4+18*x^3+6*x^2)*ex p(2/3*exp(2)/x)+48*x^5+72*x^4+72*x^3+24*x^2)/x^2,x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx=4 \, x^{4} + 8 \, x^{3} + 12 \, x^{2} + {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )} e^{\left (\frac {2 \, e^{2}}{3 \, x}\right )} + 8 \, x \]
integrate(1/3*(((-2*x^4-4*x^3-6*x^2-4*x-2)*exp(2)+12*x^5+18*x^4+18*x^3+6*x ^2)*exp(2/3*exp(2)/x)+48*x^5+72*x^4+72*x^3+24*x^2)/x^2,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).
Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx=4 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + \left (x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1\right ) e^{\frac {2 e^{2}}{3 x}} \]
integrate(1/3*(((-2*x**4-4*x**3-6*x**2-4*x-2)*exp(2)+12*x**5+18*x**4+18*x* *3+6*x**2)*exp(2/3*exp(2)/x)+48*x**5+72*x**4+72*x**3+24*x**2)/x**2,x)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 5.65 \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx=4 \, x^{4} + 8 \, x^{3} + 12 \, x^{2} + \frac {4}{3} \, {\rm Ei}\left (\frac {2 \, e^{2}}{3 \, x}\right ) e^{2} + \frac {4}{3} \, e^{4} \Gamma \left (-1, -\frac {2 \, e^{2}}{3 \, x}\right ) - \frac {4}{3} \, e^{2} \Gamma \left (-1, -\frac {2 \, e^{2}}{3 \, x}\right ) - \frac {16}{27} \, e^{6} \Gamma \left (-2, -\frac {2 \, e^{2}}{3 \, x}\right ) + \frac {8}{3} \, e^{4} \Gamma \left (-2, -\frac {2 \, e^{2}}{3 \, x}\right ) + \frac {16}{81} \, e^{8} \Gamma \left (-3, -\frac {2 \, e^{2}}{3 \, x}\right ) - \frac {16}{9} \, e^{6} \Gamma \left (-3, -\frac {2 \, e^{2}}{3 \, x}\right ) + \frac {64}{81} \, e^{8} \Gamma \left (-4, -\frac {2 \, e^{2}}{3 \, x}\right ) + 8 \, x + e^{\left (\frac {2 \, e^{2}}{3 \, x}\right )} \]
integrate(1/3*(((-2*x^4-4*x^3-6*x^2-4*x-2)*exp(2)+12*x^5+18*x^4+18*x^3+6*x ^2)*exp(2/3*exp(2)/x)+48*x^5+72*x^4+72*x^3+24*x^2)/x^2,x, algorithm=\
4*x^4 + 8*x^3 + 12*x^2 + 4/3*Ei(2/3*e^2/x)*e^2 + 4/3*e^4*gamma(-1, -2/3*e^ 2/x) - 4/3*e^2*gamma(-1, -2/3*e^2/x) - 16/27*e^6*gamma(-2, -2/3*e^2/x) + 8 /3*e^4*gamma(-2, -2/3*e^2/x) + 16/81*e^8*gamma(-3, -2/3*e^2/x) - 16/9*e^6* gamma(-3, -2/3*e^2/x) + 64/81*e^8*gamma(-4, -2/3*e^2/x) + 8*x + e^(2/3*e^2 /x)
Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 4.39 \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx=x^{4} {\left (\frac {8 \, e^{10}}{x} + \frac {2 \, e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}}{x} + \frac {12 \, e^{10}}{x^{2}} + \frac {3 \, e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}}{x^{2}} + \frac {8 \, e^{10}}{x^{3}} + \frac {2 \, e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}}{x^{3}} + \frac {e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}}{x^{4}} + 4 \, e^{10} + e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}\right )} e^{\left (-10\right )} \]
integrate(1/3*(((-2*x^4-4*x^3-6*x^2-4*x-2)*exp(2)+12*x^5+18*x^4+18*x^3+6*x ^2)*exp(2/3*exp(2)/x)+48*x^5+72*x^4+72*x^3+24*x^2)/x^2,x, algorithm=\
x^4*(8*e^10/x + 2*e^(2/3*e^2/x + 10)/x + 12*e^10/x^2 + 3*e^(2/3*e^2/x + 10 )/x^2 + 8*e^10/x^3 + 2*e^(2/3*e^2/x + 10)/x^3 + e^(2/3*e^2/x + 10)/x^4 + 4 *e^10 + e^(2/3*e^2/x + 10))*e^(-10)
Time = 11.32 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.00 \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx={\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+x\,\left (2\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+8\right )+x^4\,\left ({\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+4\right )+x^3\,\left (2\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+8\right )+x^2\,\left (3\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+12\right ) \]
int(((exp((2*exp(2))/(3*x))*(6*x^2 - exp(2)*(4*x + 6*x^2 + 4*x^3 + 2*x^4 + 2) + 18*x^3 + 18*x^4 + 12*x^5))/3 + 8*x^2 + 24*x^3 + 24*x^4 + 16*x^5)/x^2 ,x)