3.36.57 \(\int \frac {2+5 x-2 e^x x}{x} \, dx\)

Optimal. Leaf size=20 \[ x-2 \left (5+e^x-2 x-\log (4)-\log (x)\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 0.65, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {14, 2194, 43} \begin {gather*} 5 x-2 e^x+2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 5*x - 2*E^x*x)/x,x]

[Out]

-2*E^x + 5*x + 2*Log[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 43

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 2194

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 {gather*} \begin {aligned} \text {integral} &=\int \left (-2 e^x+\frac {2+5 x}{x}\right ) \, dx\\ &=-\left (2 \int e^x \, dx\right )+\int \frac {2+5 x}{x} \, dx\\ &=-2 e^x+\int \left (5+\frac {2}{x}\right ) \, dx\\ &=-2 e^x+5 x+2 \log (x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 13, normalized size = 0.65 \begin {gather*} -2 e^x+5 x+2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 5*x - 2*E^x*x)/x,x]

[Out]

-2*E^x + 5*x + 2*Log[x]

________________________________________________________________________________________

fricas [A]  time = 0.74, size = 12, normalized size = 0.60 \begin {gather*} 5 \, x - 2 \, e^{x} + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(x)*x+5*x+2)/x,x, algorithm="fricas")

[Out]

5*x - 2*e^x + 2*log(x)

________________________________________________________________________________________

giac [A]  time = 0.14, size = 12, normalized size = 0.60 \begin {gather*} 5 \, x - 2 \, e^{x} + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(x)*x+5*x+2)/x,x, algorithm="giac")

[Out]

5*x - 2*e^x + 2*log(x)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 13, normalized size = 0.65

method	result	size
default	\(5 x +2 \ln \relax (x )-2 \,{\mathrm e}^{x}\)	\(13\)
norman	\(5 x +2 \ln \relax (x )-2 \,{\mathrm e}^{x}\)	\(13\)
risch	\(5 x +2 \ln \relax (x )-2 \,{\mathrm e}^{x}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*exp(x)*x+5*x+2)/x,x,method=_RETURNVERBOSE)

[Out]

5*x+2*ln(x)-2*exp(x)

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 12, normalized size = 0.60 \begin {gather*} 5 \, x - 2 \, e^{x} + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(x)*x+5*x+2)/x,x, algorithm="maxima")

[Out]

5*x - 2*e^x + 2*log(x)

________________________________________________________________________________________

mupad [B]  time = 2.07, size = 12, normalized size = 0.60 \begin {gather*} 5\,x-2\,{\mathrm {e}}^x+2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x - 2*x*exp(x) + 2)/x,x)

[Out]

5*x - 2*exp(x) + 2*log(x)

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 12, normalized size = 0.60 \begin {gather*} 5 x - 2 e^{x} + 2 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(x)*x+5*x+2)/x,x)

[Out]

5*x - 2*exp(x) + 2*log(x)

________________________________________________________________________________________