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

Optimal. Leaf size=13 \[ 1+\log \left (\frac {89}{9}+e^x+3 x\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 11, normalized size of antiderivative = 0.85, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6684} \begin {gather*} \log \left (27 x+9 e^x+89\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[89 + 9*E^x + 27*x]

Rule 6684

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

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 11, normalized size = 0.85 \begin {gather*} \log \left (89+9 e^x+27 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[89 + 9*E^x + 27*x]

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fricas [A]  time = 0.72, size = 10, normalized size = 0.77 \begin {gather*} \log \left (27 \, x + 9 \, e^{x} + 89\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(27*x + 9*e^x + 89)

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giac [A]  time = 0.14, size = 10, normalized size = 0.77 \begin {gather*} \log \left (27 \, x + 9 \, e^{x} + 89\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(27*x + 9*e^x + 89)

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maple [A]  time = 0.04, size = 9, normalized size = 0.69

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 10, normalized size = 0.77 \begin {gather*} \log \left (27 \, x + 9 \, e^{x} + 89\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(27*x + 9*e^x + 89)

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mupad [B]  time = 4.01, size = 10, normalized size = 0.77 \begin {gather*} \ln \left (27\,x+9\,{\mathrm {e}}^x+89\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(27*x + 9*exp(x) + 89)

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sympy [A]  time = 0.10, size = 10, normalized size = 0.77 \begin {gather*} \log {\left (3 x + e^{x} + \frac {89}{9} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(3*x + exp(x) + 89/9)

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