\(\int \frac {1}{3} e^{\frac {1}{3} (3+3 x-\log (x))} (3+(2+3 x) \log (x)) \, dx\) [1947]

Optimal result

Integrand size = 29, antiderivative size = 18 \[ \int \frac {1}{3} e^{\frac {1}{3} (3+3 x-\log (x))} (3+(2+3 x) \log (x)) \, dx=e^{x+\frac {1}{3} (3-\log (x))} x \log (x) \]

[Out]

x*exp(-1/3*ln(x)+x+1)*ln(x)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.32 (sec) , antiderivative size = 175, normalized size of antiderivative = 9.72, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {12, 2306, 6820, 6874, 2250, 2258, 2634, 15, 14, 6696} \[ \int \frac {1}{3} e^{\frac {1}{3} (3+3 x-\log (x))} (3+(2+3 x) \log (x)) \, dx=-\frac {9}{25} e x^{5/3} \, _2F_2\left (\frac {5}{3},\frac {5}{3};\frac {8}{3},\frac {8}{3};x\right )-\frac {3}{2} e x^{2/3} \, _2F_2\left (\frac {2}{3},\frac {2}{3};\frac {5}{3},\frac {5}{3};x\right )-\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {3 e x^{2/3} \operatorname {Gamma}\left (\frac {5}{3}\right ) \log \left (\sqrt [3]{x}\right )}{(-x)^{2/3}}+\frac {2 e x^{2/3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \log \left (\sqrt [3]{x}\right )}{(-x)^{2/3}}-\frac {e x^{5/3} \log (x) \Gamma \left (\frac {5}{3},-x\right )}{(-x)^{5/3}}-\frac {2 e x^{2/3} \log (x) \Gamma \left (\frac {2}{3},-x\right )}{3 (-x)^{2/3}} \]

[In]

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

[Out]

-((E*x^(2/3)*Gamma[2/3, -x])/(-x)^(2/3)) - (3*E*x^(2/3)*HypergeometricPFQ[{2/3, 2/3}, {5/3, 5/3}, x])/2 - (9*E
*x^(5/3)*HypergeometricPFQ[{5/3, 5/3}, {8/3, 8/3}, x])/25 + (2*E*x^(2/3)*Gamma[2/3]*Log[x^(1/3)])/(-x)^(2/3) -
(3*E*x^(2/3)*Gamma[5/3]*Log[x^(1/3)])/(-x)^(2/3) - (2*E*x^(2/3)*Gamma[2/3, -x]*Log[x])/(3*(-x)^(2/3)) - (E*x^
(5/3)*Gamma[5/3, -x]*Log[x])/(-x)^(5/3)

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 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 2250

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

Rule 2258

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rule 2306

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rule 2634

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]
/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6696

Int[Gamma[n_, (b_.)*(x_)]/(x_), x_Symbol] :> Simp[Gamma[n]*Log[x], x] - Simp[((b*x)^n/n^2)*HypergeometricPFQ[{
n, n}, {1 + n, 1 + n}, (-b)*x], x] /; FreeQ[{b, n}, x] &&  !IntegerQ[n]

Rule 6820

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int e^{\frac {1}{3} (3+3 x-\log (x))} (3+(2+3 x) \log (x)) \, dx \\ & = \frac {1}{3} \int \frac {e^{\frac {1}{3} (3+3 x)} (3+(2+3 x) \log (x))}{\sqrt [3]{x}} \, dx \\ & = \frac {1}{3} \int \frac {e^{1+x} (3+(2+3 x) \log (x))}{\sqrt [3]{x}} \, dx \\ & = \text {Subst}\left (\int e^{1+x^3} x \left (3+\left (2+3 x^3\right ) \log \left (x^3\right )\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \text {Subst}\left (\int \left (3 e^{1+x^3} x+e^{1+x^3} x \left (2+3 x^3\right ) \log \left (x^3\right )\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int e^{1+x^3} x \, dx,x,\sqrt [3]{x}\right )+\text {Subst}\left (\int e^{1+x^3} x \left (2+3 x^3\right ) \log \left (x^3\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3},-x\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{5/3} \Gamma \left (\frac {5}{3},-x\right ) \log (x)}{(-x)^{5/3}}-\text {Subst}\left (\int \frac {e x \left (-2 \Gamma \left (\frac {2}{3},-x^3\right )+3 \Gamma \left (\frac {5}{3},-x^3\right )\right )}{\left (-x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3},-x\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{5/3} \Gamma \left (\frac {5}{3},-x\right ) \log (x)}{(-x)^{5/3}}-e \text {Subst}\left (\int \frac {x \left (-2 \Gamma \left (\frac {2}{3},-x^3\right )+3 \Gamma \left (\frac {5}{3},-x^3\right )\right )}{\left (-x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3},-x\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{5/3} \Gamma \left (\frac {5}{3},-x\right ) \log (x)}{(-x)^{5/3}}-\frac {\left (e x^{2/3}\right ) \text {Subst}\left (\int \frac {-2 \Gamma \left (\frac {2}{3},-x^3\right )+3 \Gamma \left (\frac {5}{3},-x^3\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-x)^{2/3}} \\ & = -\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3},-x\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{5/3} \Gamma \left (\frac {5}{3},-x\right ) \log (x)}{(-x)^{5/3}}-\frac {\left (e x^{2/3}\right ) \text {Subst}\left (\int \left (-\frac {2 \Gamma \left (\frac {2}{3},-x^3\right )}{x}+\frac {3 \Gamma \left (\frac {5}{3},-x^3\right )}{x}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-x)^{2/3}} \\ & = -\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3},-x\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{5/3} \Gamma \left (\frac {5}{3},-x\right ) \log (x)}{(-x)^{5/3}}+\frac {\left (2 e x^{2/3}\right ) \text {Subst}\left (\int \frac {\Gamma \left (\frac {2}{3},-x^3\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-x)^{2/3}}-\frac {\left (3 e x^{2/3}\right ) \text {Subst}\left (\int \frac {\Gamma \left (\frac {5}{3},-x^3\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-x)^{2/3}} \\ & = -\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3},-x\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{5/3} \Gamma \left (\frac {5}{3},-x\right ) \log (x)}{(-x)^{5/3}}+\frac {\left (2 e x^{2/3}\right ) \text {Subst}\left (\int \frac {\Gamma \left (\frac {2}{3},-x\right )}{x} \, dx,x,x\right )}{3 (-x)^{2/3}}-\frac {\left (e x^{2/3}\right ) \text {Subst}\left (\int \frac {\Gamma \left (\frac {5}{3},-x\right )}{x} \, dx,x,x\right )}{(-x)^{2/3}} \\ & = -\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {3}{2} e x^{2/3} \, _2F_2\left (\frac {2}{3},\frac {2}{3};\frac {5}{3},\frac {5}{3};x\right )-\frac {9}{25} e x^{5/3} \, _2F_2\left (\frac {5}{3},\frac {5}{3};\frac {8}{3},\frac {8}{3};x\right )+\frac {2 e x^{2/3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{2/3} \operatorname {Gamma}\left (\frac {5}{3}\right ) \log (x)}{(-x)^{2/3}}-\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3},-x\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{5/3} \Gamma \left (\frac {5}{3},-x\right ) \log (x)}{(-x)^{5/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \frac {1}{3} e^{\frac {1}{3} (3+3 x-\log (x))} (3+(2+3 x) \log (x)) \, dx=e^{1+x} x^{2/3} \log (x) \]

[In]

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

[Out]

E^(1 + x)*x^(2/3)*Log[x]

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61

method	result	size
default	\(x^{\frac {2}{3}} {\mathrm e}^{1+x} \ln \left (x \right )\)	\(11\)
risch	\(x^{\frac {2}{3}} {\mathrm e}^{1+x} \ln \left (x \right )\)	\(11\)
norman	\(x \,{\mathrm e}^{-\frac {\ln \left (x \right )}{3}+x +1} \ln \left (x \right )\)	\(13\)
parallelrisch	\(x \,{\mathrm e}^{-\frac {\ln \left (x \right )}{3}+x +1} \ln \left (x \right )\)	\(13\)

[In]

int(1/3*((2+3*x)*ln(x)+3)*exp(-1/3*ln(x)+x+1),x,method=_RETURNVERBOSE)

[Out]

x^(2/3)*exp(1+x)*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {1}{3} e^{\frac {1}{3} (3+3 x-\log (x))} (3+(2+3 x) \log (x)) \, dx=x e^{\left (x - \frac {1}{3} \, \log \left (x\right ) + 1\right )} \log \left (x\right ) \]

[In]

integrate(1/3*((2+3*x)*log(x)+3)*exp(-1/3*log(x)+x+1),x, algorithm="fricas")

[Out]

x*e^(x - 1/3*log(x) + 1)*log(x)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{3} e^{\frac {1}{3} (3+3 x-\log (x))} (3+(2+3 x) \log (x)) \, dx=\text {Timed out} \]

[In]

integrate(1/3*((2+3*x)*ln(x)+3)*exp(-1/3*ln(x)+x+1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{3} e^{\frac {1}{3} (3+3 x-\log (x))} (3+(2+3 x) \log (x)) \, dx=\int { \frac {1}{3} \, {\left ({\left (3 \, x + 2\right )} \log \left (x\right ) + 3\right )} e^{\left (x - \frac {1}{3} \, \log \left (x\right ) + 1\right )} \,d x } \]

[In]

integrate(1/3*((2+3*x)*log(x)+3)*exp(-1/3*log(x)+x+1),x, algorithm="maxima")

[Out]

1/3*integrate(((3*x + 2)*log(x) + 3)*e^(x - 1/3*log(x) + 1), x)

Giac [F]

\[ \int \frac {1}{3} e^{\frac {1}{3} (3+3 x-\log (x))} (3+(2+3 x) \log (x)) \, dx=\int { \frac {1}{3} \, {\left ({\left (3 \, x + 2\right )} \log \left (x\right ) + 3\right )} e^{\left (x - \frac {1}{3} \, \log \left (x\right ) + 1\right )} \,d x } \]

[In]

integrate(1/3*((2+3*x)*log(x)+3)*exp(-1/3*log(x)+x+1),x, algorithm="giac")

[Out]

integrate(1/3*((3*x + 2)*log(x) + 3)*e^(x - 1/3*log(x) + 1), x)

Mupad [B] (verification not implemented)

Time = 8.46 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.56 \[ \int \frac {1}{3} e^{\frac {1}{3} (3+3 x-\log (x))} (3+(2+3 x) \log (x)) \, dx=x^{2/3}\,\mathrm {e}\,{\mathrm {e}}^x\,\ln \left (x\right ) \]

[In]

int((exp(x - log(x)/3 + 1)*(log(x)*(3*x + 2) + 3))/3,x)

[Out]

x^(2/3)*exp(1)*exp(x)*log(x)