3.52.71 \(\int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} (6 x^2+18 x^3+18 x^4+12 x^5+e^2 (-2-4 x-6 x^2-4 x^3-2 x^4))}{3 x^2} \, dx\)

Optimal. Leaf size=23 \[ \left (4+e^{\frac {2 e^2}{3 x}}\right ) \left (1+x+x^2\right )^2 \]

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Rubi [C]  time = 0.56, antiderivative size = 297, normalized size of antiderivative = 12.91, number of steps used = 18, number of rules used = 8, integrand size = 86, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {12, 14, 6742, 2206, 2210, 2209, 2214, 2218} \begin {gather*} -\frac {8}{243} e^6 \left (9-e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{3} e^2 \left (1-e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{27} e^4 \left (9-2 e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )+\frac {4}{3} e^2 \text {Ei}\left (\frac {2 e^2}{3 x}\right )+4 x^4+\frac {2}{9} \left (9-e^2\right ) e^{\frac {2 e^2}{3 x}} x^3+8 x^3+\frac {2}{27} \left (9-e^2\right ) e^{\frac {2 e^2}{3 x}+2} x^2+\frac {1}{3} \left (9-2 e^2\right ) e^{\frac {2 e^2}{3 x}} x^2+12 x^2+\frac {4}{81} \left (9-e^2\right ) e^{\frac {2 e^2}{3 x}+4} x+2 \left (1-e^2\right ) e^{\frac {2 e^2}{3 x}} x+\frac {2}{9} \left (9-2 e^2\right ) e^{\frac {2 e^2}{3 x}+2} x+8 x+e^{\frac {2 e^2}{3 x}}+\frac {64}{81} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

E^((2*E^2)/(3*x)) + 8*x + (2*E^(2 + (2*E^2)/(3*x))*(9 - 2*E^2)*x)/9 + 2*E^((2*E^2)/(3*x))*(1 - E^2)*x + (4*E^(
4 + (2*E^2)/(3*x))*(9 - E^2)*x)/81 + 12*x^2 + (E^((2*E^2)/(3*x))*(9 - 2*E^2)*x^2)/3 + (2*E^(2 + (2*E^2)/(3*x))
*(9 - E^2)*x^2)/27 + 8*x^3 + (2*E^((2*E^2)/(3*x))*(9 - E^2)*x^3)/9 + 4*x^4 + (4*E^2*ExpIntegralEi[(2*E^2)/(3*x
)])/3 - (4*E^4*(9 - 2*E^2)*ExpIntegralEi[(2*E^2)/(3*x)])/27 - (4*E^2*(1 - E^2)*ExpIntegralEi[(2*E^2)/(3*x)])/3
- (8*E^6*(9 - E^2)*ExpIntegralEi[(2*E^2)/(3*x)])/243 + (64*E^8*Gamma[-4, (-2*E^2)/(3*x)])/81

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

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]]

Rule 2206

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[((c + d*x)*F^(a + b*(c + d*x)^n))/d, x]
- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]
&& ILtQ[n, 0]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \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 )}{x^2} \, dx\\ &=\frac {1}{3} \int \left (24 \left (1+3 x+3 x^2+2 x^3\right )+\frac {2 e^{\frac {2 e^2}{3 x}} \left (1+x+x^2\right ) \left (-e^2-e^2 x+\left (3-e^2\right ) x^2+6 x^3\right )}{x^2}\right ) \, dx\\ &=\frac {2}{3} \int \frac {e^{\frac {2 e^2}{3 x}} \left (1+x+x^2\right ) \left (-e^2-e^2 x+\left (3-e^2\right ) x^2+6 x^3\right )}{x^2} \, dx+8 \int \left (1+3 x+3 x^2+2 x^3\right ) \, dx\\ &=8 x+12 x^2+8 x^3+4 x^4+\frac {2}{3} \int \left (-3 e^{\frac {2 e^2}{3 x}} \left (-1+e^2\right )-\frac {e^{2+\frac {2 e^2}{3 x}}}{x^2}-\frac {2 e^{2+\frac {2 e^2}{3 x}}}{x}-e^{\frac {2 e^2}{3 x}} \left (-9+2 e^2\right ) x-e^{\frac {2 e^2}{3 x}} \left (-9+e^2\right ) x^2+6 e^{\frac {2 e^2}{3 x}} x^3\right ) \, dx\\ &=8 x+12 x^2+8 x^3+4 x^4-\frac {2}{3} \int \frac {e^{2+\frac {2 e^2}{3 x}}}{x^2} \, dx-\frac {4}{3} \int \frac {e^{2+\frac {2 e^2}{3 x}}}{x} \, dx+4 \int e^{\frac {2 e^2}{3 x}} x^3 \, dx+\frac {1}{3} \left (2 \left (9-2 e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} x \, dx+\left (2 \left (1-e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} \, dx+\frac {1}{3} \left (2 \left (9-e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} x^2 \, dx\\ &=e^{\frac {2 e^2}{3 x}}+8 x+2 e^{\frac {2 e^2}{3 x}} \left (1-e^2\right ) x+12 x^2+\frac {1}{3} e^{\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x^2+8 x^3+\frac {2}{9} e^{\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^3+4 x^4+\frac {4}{3} e^2 \text {Ei}\left (\frac {2 e^2}{3 x}\right )+\frac {64}{81} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right )+\frac {1}{9} \left (2 e^2 \left (9-2 e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} \, dx+\frac {1}{3} \left (4 e^2 \left (1-e^2\right )\right ) \int \frac {e^{\frac {2 e^2}{3 x}}}{x} \, dx+\frac {1}{27} \left (4 e^2 \left (9-e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} x \, dx\\ &=e^{\frac {2 e^2}{3 x}}+8 x+\frac {2}{9} e^{2+\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x+2 e^{\frac {2 e^2}{3 x}} \left (1-e^2\right ) x+12 x^2+\frac {1}{3} e^{\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x^2+\frac {2}{27} e^{2+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^2+8 x^3+\frac {2}{9} e^{\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^3+4 x^4+\frac {4}{3} e^2 \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{3} e^2 \left (1-e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )+\frac {64}{81} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right )+\frac {1}{27} \left (4 e^4 \left (9-2 e^2\right )\right ) \int \frac {e^{\frac {2 e^2}{3 x}}}{x} \, dx+\frac {1}{81} \left (4 e^4 \left (9-e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} \, dx\\ &=e^{\frac {2 e^2}{3 x}}+8 x+\frac {2}{9} e^{2+\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x+2 e^{\frac {2 e^2}{3 x}} \left (1-e^2\right ) x+\frac {4}{81} e^{4+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x+12 x^2+\frac {1}{3} e^{\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x^2+\frac {2}{27} e^{2+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^2+8 x^3+\frac {2}{9} e^{\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^3+4 x^4+\frac {4}{3} e^2 \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{27} e^4 \left (9-2 e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{3} e^2 \left (1-e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )+\frac {64}{81} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right )+\frac {1}{243} \left (8 e^6 \left (9-e^2\right )\right ) \int \frac {e^{\frac {2 e^2}{3 x}}}{x} \, dx\\ &=e^{\frac {2 e^2}{3 x}}+8 x+\frac {2}{9} e^{2+\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x+2 e^{\frac {2 e^2}{3 x}} \left (1-e^2\right ) x+\frac {4}{81} e^{4+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x+12 x^2+\frac {1}{3} e^{\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x^2+\frac {2}{27} e^{2+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^2+8 x^3+\frac {2}{9} e^{\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^3+4 x^4+\frac {4}{3} e^2 \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{27} e^4 \left (9-2 e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{3} e^2 \left (1-e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {8}{243} e^6 \left (9-e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )+\frac {64}{81} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 40, normalized size = 1.74 \begin {gather*} 8 x+12 x^2+8 x^3+4 x^4+e^{\frac {2 e^2}{3 x}} \left (1+x+x^2\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

8*x + 12*x^2 + 8*x^3 + 4*x^4 + E^((2*E^2)/(3*x))*(1 + x + x^2)^2

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fricas [B]  time = 0.50, size = 46, normalized size = 2.00 \begin {gather*} 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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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="fricas")

[Out]

4*x^4 + 8*x^3 + 12*x^2 + (x^4 + 2*x^3 + 3*x^2 + 2*x + 1)*e^(2/3*e^2/x) + 8*x

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giac [B]  time = 0.19, size = 101, normalized size = 4.39 \begin {gather*} 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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="giac")

[Out]

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)

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maple [B]  time = 0.32, size = 50, normalized size = 2.17

method	result	size
risch	\(4 x^{4}+8 x^{3}+12 x^{2}+8 x +\frac {\left (3 x^{4}+6 x^{3}+9 x^{2}+6 x +3\right ) {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{3}\)	\(50\)
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 \,{\mathrm e}^{-8} x^{4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{64}-\frac {9 \,{\mathrm e}^{-6} x^{3} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {3 \,{\mathrm e}^{-4} x^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{16}-\frac {\expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{24}\right )+288 \,{\mathrm e}^{8} \left (-\frac {9 \,{\mathrm e}^{-6} x^{3} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 \,{\mathrm e}^{-4} x^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )-32 \,{\mathrm e}^{10} \left (-\frac {9 \,{\mathrm e}^{-6} x^{3} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 \,{\mathrm e}^{-4} x^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )+432 \,{\mathrm e}^{6} \left (-\frac {9 \,{\mathrm e}^{-4} x^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 \,{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )-96 \,{\mathrm e}^{8} \left (-\frac {9 \,{\mathrm e}^{-4} x^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 \,{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )+216 \,{\mathrm e}^{4} \left (-\frac {3 \,{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )-216 \,{\mathrm e}^{6} \left (-\frac {3 \,{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )+216 \,{\mathrm e}^{4} \expIntegralEi \left (1, -\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 \,{\mathrm e}^{-8} x^{4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{64}-\frac {9 \,{\mathrm e}^{-6} x^{3} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {3 \,{\mathrm e}^{-4} x^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{16}-\frac {\expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{24}\right )+288 \,{\mathrm e}^{8} \left (-\frac {9 \,{\mathrm e}^{-6} x^{3} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 \,{\mathrm e}^{-4} x^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )-32 \,{\mathrm e}^{10} \left (-\frac {9 \,{\mathrm e}^{-6} x^{3} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 \,{\mathrm e}^{-4} x^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )+432 \,{\mathrm e}^{6} \left (-\frac {9 \,{\mathrm e}^{-4} x^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 \,{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )-96 \,{\mathrm e}^{8} \left (-\frac {9 \,{\mathrm e}^{-4} x^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 \,{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )+216 \,{\mathrm e}^{4} \left (-\frac {3 \,{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )-216 \,{\mathrm e}^{6} \left (-\frac {3 \,{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )+216 \,{\mathrm e}^{4} \expIntegralEi \left (1, -\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )}{162}\)	\(438\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)*exp(2/3*exp(2)/x)+48*x^5+72*x^4+72*x^3
+24*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

4*x^4+8*x^3+12*x^2+8*x+1/3*(3*x^4+6*x^3+9*x^2+6*x+3)*exp(2/3*exp(2)/x)

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maxima [C]  time = 0.42, size = 130, normalized size = 5.65 \begin {gather*} 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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="maxima")

[Out]

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)

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mupad [B]  time = 3.29, size = 69, normalized size = 3.00 \begin {gather*} {\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

exp((2*exp(2))/(3*x)) + x*(2*exp((2*exp(2))/(3*x)) + 8) + x^4*(exp((2*exp(2))/(3*x)) + 4) + x^3*(2*exp((2*exp(
2))/(3*x)) + 8) + x^2*(3*exp((2*exp(2))/(3*x)) + 12)

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sympy [B]  time = 0.27, size = 46, normalized size = 2.00 \begin {gather*} 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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

4*x**4 + 8*x**3 + 12*x**2 + 8*x + (x**4 + 2*x**3 + 3*x**2 + 2*x + 1)*exp(2*exp(2)/(3*x))

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