\(\int \frac {20+25 e^x+5 x}{e^x+x} \, dx\) [6742]

Optimal result

Integrand size = 18, antiderivative size = 27 \[ \int \frac {20+25 e^x+5 x}{e^x+x} \, dx=x+x \left (4+\frac {5 \left (-3+2 \log \left (\frac {1}{4} \left (e^x+x\right )^2\right )\right )}{x}\right ) \]

[Out]

x*(4+5*(2*ln(1/2*(exp(x)+x)*(1/2*exp(x)+1/2*x))-3)/x)+x

Rubi [F]

\[ \int \frac {20+25 e^x+5 x}{e^x+x} \, dx=\int \frac {20+25 e^x+5 x}{e^x+x} \, dx \]

[In]

Int[(20 + 25*E^x + 5*x)/(E^x + x),x]

[Out]

25*x + 20*Defer[Int][(E^x + x)^(-1), x] - 20*Defer[Int][x/(E^x + x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {5 \left (4+5 e^x+x\right )}{e^x+x} \, dx \\ & = 5 \int \frac {4+5 e^x+x}{e^x+x} \, dx \\ & = 5 \int \left (5-\frac {4 (-1+x)}{e^x+x}\right ) \, dx \\ & = 25 x-20 \int \frac {-1+x}{e^x+x} \, dx \\ & = 25 x-20 \int \left (-\frac {1}{e^x+x}+\frac {x}{e^x+x}\right ) \, dx \\ & = 25 x+20 \int \frac {1}{e^x+x} \, dx-20 \int \frac {x}{e^x+x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.44 \[ \int \frac {20+25 e^x+5 x}{e^x+x} \, dx=5 \left (x+4 \log \left (e^x+x\right )\right ) \]

[In]

Integrate[(20 + 25*E^x + 5*x)/(E^x + x),x]

[Out]

5*(x + 4*Log[E^x + x])

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.44

method	result	size
norman	\(5 x +20 \ln \left ({\mathrm e}^{x}+x \right )\)	\(12\)
risch	\(5 x +20 \ln \left ({\mathrm e}^{x}+x \right )\)	\(12\)
parallelrisch	\(5 x +20 \ln \left ({\mathrm e}^{x}+x \right )\)	\(12\)

[In]

int((25*exp(x)+20+5*x)/(exp(x)+x),x,method=_RETURNVERBOSE)

[Out]

5*x+20*ln(exp(x)+x)

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.41 \[ \int \frac {20+25 e^x+5 x}{e^x+x} \, dx=5 \, x + 20 \, \log \left (x + e^{x}\right ) \]

[In]

integrate((25*exp(x)+20+5*x)/(exp(x)+x),x, algorithm="fricas")

[Out]

5*x + 20*log(x + e^x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.37 \[ \int \frac {20+25 e^x+5 x}{e^x+x} \, dx=5 x + 20 \log {\left (x + e^{x} \right )} \]

[In]

integrate((25*exp(x)+20+5*x)/(exp(x)+x),x)

[Out]

5*x + 20*log(x + exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.41 \[ \int \frac {20+25 e^x+5 x}{e^x+x} \, dx=5 \, x + 20 \, \log \left (x + e^{x}\right ) \]

[In]

integrate((25*exp(x)+20+5*x)/(exp(x)+x),x, algorithm="maxima")

[Out]

5*x + 20*log(x + e^x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.41 \[ \int \frac {20+25 e^x+5 x}{e^x+x} \, dx=5 \, x + 20 \, \log \left (x + e^{x}\right ) \]

[In]

integrate((25*exp(x)+20+5*x)/(exp(x)+x),x, algorithm="giac")

[Out]

5*x + 20*log(x + e^x)

Mupad [B] (verification not implemented)

Time = 17.56 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.41 \[ \int \frac {20+25 e^x+5 x}{e^x+x} \, dx=5\,x+20\,\ln \left (x+{\mathrm {e}}^x\right ) \]

[In]

int((5*x + 25*exp(x) + 20)/(x + exp(x)),x)

[Out]

5*x + 20*log(x + exp(x))