\(\int \frac {-3+7 x+3 e^x x}{3 x} \, dx\) [4370]

Optimal result

Integrand size = 18, antiderivative size = 16 \[ \int \frac {-3+7 x+3 e^x x}{3 x} \, dx=-5+e^x+\frac {7 x}{3}+\log \left (\frac {12}{x}\right ) \]

[Out]

7/3*x+ln(12/x)-5+exp(x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {12, 14, 2225, 45} \[ \int \frac {-3+7 x+3 e^x x}{3 x} \, dx=\frac {7 x}{3}+e^x-\log (x) \]

[In]

Int[(-3 + 7*x + 3*E^x*x)/(3*x),x]

[Out]

E^x + (7*x)/3 - Log[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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {-3+7 x+3 e^x x}{x} \, dx \\ & = \frac {1}{3} \int \left (3 e^x+\frac {-3+7 x}{x}\right ) \, dx \\ & = \frac {1}{3} \int \frac {-3+7 x}{x} \, dx+\int e^x \, dx \\ & = e^x+\frac {1}{3} \int \left (7-\frac {3}{x}\right ) \, dx \\ & = e^x+\frac {7 x}{3}-\log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {-3+7 x+3 e^x x}{3 x} \, dx=e^x+\frac {7 x}{3}-\log (x) \]

[In]

Integrate[(-3 + 7*x + 3*E^x*x)/(3*x),x]

[Out]

E^x + (7*x)/3 - Log[x]

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69

method	result	size
default	\(\frac {7 x}{3}-\ln \left (x \right )+{\mathrm e}^{x}\)	\(11\)
norman	\(\frac {7 x}{3}-\ln \left (x \right )+{\mathrm e}^{x}\)	\(11\)
risch	\(\frac {7 x}{3}-\ln \left (x \right )+{\mathrm e}^{x}\)	\(11\)
parallelrisch	\(\frac {7 x}{3}-\ln \left (x \right )+{\mathrm e}^{x}\)	\(11\)
parts	\(\frac {7 x}{3}-\ln \left (x \right )+{\mathrm e}^{x}\)	\(11\)

[In]

int(1/3*(3*exp(x)*x+7*x-3)/x,x,method=_RETURNVERBOSE)

[Out]

7/3*x-ln(x)+exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \frac {-3+7 x+3 e^x x}{3 x} \, dx=\frac {7}{3} \, x + e^{x} - \log \left (x\right ) \]

[In]

integrate(1/3*(3*exp(x)*x+7*x-3)/x,x, algorithm="fricas")

[Out]

7/3*x + e^x - log(x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \frac {-3+7 x+3 e^x x}{3 x} \, dx=\frac {7 x}{3} + e^{x} - \log {\left (x \right )} \]

[In]

integrate(1/3*(3*exp(x)*x+7*x-3)/x,x)

[Out]

7*x/3 + exp(x) - log(x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \frac {-3+7 x+3 e^x x}{3 x} \, dx=\frac {7}{3} \, x + e^{x} - \log \left (x\right ) \]

[In]

integrate(1/3*(3*exp(x)*x+7*x-3)/x,x, algorithm="maxima")

[Out]

7/3*x + e^x - log(x)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \frac {-3+7 x+3 e^x x}{3 x} \, dx=\frac {7}{3} \, x + e^{x} - \log \left (x\right ) \]

[In]

integrate(1/3*(3*exp(x)*x+7*x-3)/x,x, algorithm="giac")

[Out]

7/3*x + e^x - log(x)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \frac {-3+7 x+3 e^x x}{3 x} \, dx=\frac {7\,x}{3}+{\mathrm {e}}^x-\ln \left (x\right ) \]

[In]

int(((7*x)/3 + x*exp(x) - 1)/x,x)

[Out]

(7*x)/3 + exp(x) - log(x)