\(\int \frac {e^{.\frac {1}{3}/x} (-4-13 x)+9 x^3}{3 x^3} \, dx\) [8667]

Optimal result

Integrand size = 28, antiderivative size = 26 \[ \int \frac {e^{\left .\frac {1}{3}\right /x} (-4-13 x)+9 x^3}{3 x^3} \, dx=-7+3 x-\frac {x-e^{\left .\frac {1}{3}\right /x} (4+x)}{x} \]

[Out]

3*x-(x-exp(1/3/x)*(4+x))/x-7

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {12, 14, 6874, 2243, 2240} \[ \int \frac {e^{\left .\frac {1}{3}\right /x} (-4-13 x)+9 x^3}{3 x^3} \, dx=3 x+e^{\left .\frac {1}{3}\right /x}+\frac {4 e^{\left .\frac {1}{3}\right /x}}{x} \]

[In]

Int[(E^(1/(3*x))*(-4 - 13*x) + 9*x^3)/(3*x^3),x]

[Out]

E^(1/(3*x)) + (4*E^(1/(3*x)))/x + 3*x

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 2240

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 2243

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

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 \frac {e^{\left .\frac {1}{3}\right /x} (-4-13 x)+9 x^3}{x^3} \, dx \\ & = \frac {1}{3} \int \left (9-\frac {e^{\left .\frac {1}{3}\right /x} (4+13 x)}{x^3}\right ) \, dx \\ & = 3 x-\frac {1}{3} \int \frac {e^{\left .\frac {1}{3}\right /x} (4+13 x)}{x^3} \, dx \\ & = 3 x-\frac {1}{3} \int \left (\frac {4 e^{\left .\frac {1}{3}\right /x}}{x^3}+\frac {13 e^{\left .\frac {1}{3}\right /x}}{x^2}\right ) \, dx \\ & = 3 x-\frac {4}{3} \int \frac {e^{\left .\frac {1}{3}\right /x}}{x^3} \, dx-\frac {13}{3} \int \frac {e^{\left .\frac {1}{3}\right /x}}{x^2} \, dx \\ & = 13 e^{\left .\frac {1}{3}\right /x}+\frac {4 e^{\left .\frac {1}{3}\right /x}}{x}+3 x+4 \int \frac {e^{\left .\frac {1}{3}\right /x}}{x^2} \, dx \\ & = e^{\left .\frac {1}{3}\right /x}+\frac {4 e^{\left .\frac {1}{3}\right /x}}{x}+3 x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\left .\frac {1}{3}\right /x} (-4-13 x)+9 x^3}{3 x^3} \, dx=e^{\left .\frac {1}{3}\right /x}+\frac {4 e^{\left .\frac {1}{3}\right /x}}{x}+3 x \]

[In]

Integrate[(E^(1/(3*x))*(-4 - 13*x) + 9*x^3)/(3*x^3),x]

[Out]

E^(1/(3*x)) + (4*E^(1/(3*x)))/x + 3*x

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69

method	result	size
risch	\(3 x +\frac {\left (4+x \right ) {\mathrm e}^{\frac {1}{3 x}}}{x}\)	\(18\)
derivativedivides	\(3 x +\frac {4 \,{\mathrm e}^{\frac {1}{3 x}}}{x}+{\mathrm e}^{\frac {1}{3 x}}\)	\(22\)
default	\(3 x +\frac {4 \,{\mathrm e}^{\frac {1}{3 x}}}{x}+{\mathrm e}^{\frac {1}{3 x}}\)	\(22\)
parts	\(3 x +\frac {4 \,{\mathrm e}^{\frac {1}{3 x}}}{x}+{\mathrm e}^{\frac {1}{3 x}}\)	\(22\)
parallelrisch	\(\frac {9 x^{2}+3 \,{\mathrm e}^{\frac {1}{3 x}} x +12 \,{\mathrm e}^{\frac {1}{3 x}}}{3 x}\)	\(29\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {1}{3 x}}+3 x^{3}+4 \,{\mathrm e}^{\frac {1}{3 x}} x}{x^{2}}\)	\(30\)

[In]

int(1/3*((-13*x-4)*exp(1/3/x)+9*x^3)/x^3,x,method=_RETURNVERBOSE)

[Out]

3*x+(4+x)/x*exp(1/3/x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {e^{\left .\frac {1}{3}\right /x} (-4-13 x)+9 x^3}{3 x^3} \, dx=\frac {3 \, x^{2} + {\left (x + 4\right )} e^{\left (\frac {1}{3 \, x}\right )}}{x} \]

[In]

integrate(1/3*((-13*x-4)*exp(1/3/x)+9*x^3)/x^3,x, algorithm="fricas")

[Out]

(3*x^2 + (x + 4)*e^(1/3/x))/x

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.54 \[ \int \frac {e^{\left .\frac {1}{3}\right /x} (-4-13 x)+9 x^3}{3 x^3} \, dx=3 x + \frac {\left (x + 4\right ) e^{\frac {1}{3 x}}}{x} \]

[In]

integrate(1/3*((-13*x-4)*exp(1/3/x)+9*x**3)/x**3,x)

[Out]

3*x + (x + 4)*exp(1/(3*x))/x

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {e^{\left .\frac {1}{3}\right /x} (-4-13 x)+9 x^3}{3 x^3} \, dx=3 \, x + 13 \, e^{\left (\frac {1}{3 \, x}\right )} - 12 \, \Gamma \left (2, -\frac {1}{3 \, x}\right ) \]

[In]

integrate(1/3*((-13*x-4)*exp(1/3/x)+9*x^3)/x^3,x, algorithm="maxima")

[Out]

3*x + 13*e^(1/3/x) - 12*gamma(2, -1/3/x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\left .\frac {1}{3}\right /x} (-4-13 x)+9 x^3}{3 x^3} \, dx=x {\left (\frac {e^{\left (\frac {1}{3 \, x}\right )}}{x} + \frac {4 \, e^{\left (\frac {1}{3 \, x}\right )}}{x^{2}} + 3\right )} \]

[In]

integrate(1/3*((-13*x-4)*exp(1/3/x)+9*x^3)/x^3,x, algorithm="giac")

[Out]

x*(e^(1/3/x)/x + 4*e^(1/3/x)/x^2 + 3)

Mupad [B] (verification not implemented)

Time = 12.98 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {e^{\left .\frac {1}{3}\right /x} (-4-13 x)+9 x^3}{3 x^3} \, dx=3\,x+{\mathrm {e}}^{\frac {1}{3\,x}}+\frac {4\,{\mathrm {e}}^{\frac {1}{3\,x}}}{x} \]

[In]

int(-((exp(1/(3*x))*(13*x + 4))/3 - 3*x^3)/x^3,x)

[Out]

3*x + exp(1/(3*x)) + (4*exp(1/(3*x)))/x