\(\int \frac {27+9 e^x}{89+9 e^x+27 x} \, dx\) [6503]

Optimal result

Integrand size = 20, antiderivative size = 13 \[ \int \frac {27+9 e^x}{89+9 e^x+27 x} \, dx=1+\log \left (\frac {89}{9}+e^x+3 x\right ) \]

[Out]

1+ln(3*x+89/9+exp(x))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6816} \[ \int \frac {27+9 e^x}{89+9 e^x+27 x} \, dx=\log \left (27 x+9 e^x+89\right ) \]

[In]

Int[(27 + 9*E^x)/(89 + 9*E^x + 27*x),x]

[Out]

Log[89 + 9*E^x + 27*x]

Rule 6816

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps \begin{align*} \text {integral}& = \log \left (89+9 e^x+27 x\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {27+9 e^x}{89+9 e^x+27 x} \, dx=\log \left (89+9 e^x+27 x\right ) \]

[In]

Integrate[(27 + 9*E^x)/(89 + 9*E^x + 27*x),x]

[Out]

Log[89 + 9*E^x + 27*x]

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69

method	result	size
risch	\(\ln \left (3 x +\frac {89}{9}+{\mathrm e}^{x}\right )\)	\(9\)
parallelrisch	\(\ln \left (\frac {{\mathrm e}^{x}}{3}+x +\frac {89}{27}\right )\)	\(9\)
derivativedivides	\(\ln \left (9 \,{\mathrm e}^{x}+27 x +89\right )\)	\(11\)
norman	\(\ln \left (9 \,{\mathrm e}^{x}+27 x +89\right )\)	\(11\)

[In]

int((9*exp(x)+27)/(9*exp(x)+27*x+89),x,method=_RETURNVERBOSE)

[Out]

ln(3*x+89/9+exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {27+9 e^x}{89+9 e^x+27 x} \, dx=\log \left (27 \, x + 9 \, e^{x} + 89\right ) \]

[In]

integrate((9*exp(x)+27)/(9*exp(x)+27*x+89),x, algorithm="fricas")

[Out]

log(27*x + 9*e^x + 89)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {27+9 e^x}{89+9 e^x+27 x} \, dx=\log {\left (3 x + e^{x} + \frac {89}{9} \right )} \]

[In]

integrate((9*exp(x)+27)/(9*exp(x)+27*x+89),x)

[Out]

log(3*x + exp(x) + 89/9)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {27+9 e^x}{89+9 e^x+27 x} \, dx=\log \left (27 \, x + 9 \, e^{x} + 89\right ) \]

[In]

integrate((9*exp(x)+27)/(9*exp(x)+27*x+89),x, algorithm="maxima")

[Out]

log(27*x + 9*e^x + 89)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {27+9 e^x}{89+9 e^x+27 x} \, dx=\log \left (27 \, x + 9 \, e^{x} + 89\right ) \]

[In]

integrate((9*exp(x)+27)/(9*exp(x)+27*x+89),x, algorithm="giac")

[Out]

log(27*x + 9*e^x + 89)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {27+9 e^x}{89+9 e^x+27 x} \, dx=\ln \left (27\,x+9\,{\mathrm {e}}^x+89\right ) \]

[In]

int((9*exp(x) + 27)/(27*x + 9*exp(x) + 89),x)

[Out]

log(27*x + 9*exp(x) + 89)