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

Optimal. Leaf size=27 \[ \log \left (\frac {e^{\frac {2}{e \left (4+\log ^2(5)\right )}} x^2}{e^5+x}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 0.48, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {631} \begin {gather*} 2 \log (x)-\log \left (x+e^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*E^5 + x)/(E^5*x + x^2),x]

[Out]

2*Log[x] - Log[E^5 + x]

Rule 631

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{-e^5-x}+\frac {2}{x}\right ) \, dx\\ &=2 \log (x)-\log \left (e^5+x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 0.48 \begin {gather*} 2 \log (x)-\log \left (e^5+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*E^5 + x)/(E^5*x + x^2),x]

[Out]

2*Log[x] - Log[E^5 + x]

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fricas [A]  time = 1.02, size = 12, normalized size = 0.44 \begin {gather*} -\log \left (x + e^{5}\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + e^5) + 2*log(x)

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giac [A]  time = 0.20, size = 14, normalized size = 0.52 \begin {gather*} -\log \left ({\left | x + e^{5} \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(x + e^5)) + 2*log(abs(x))

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maple [A]  time = 0.13, size = 13, normalized size = 0.48

method	result	size
default	\(2 \ln \relax (x )-\ln \left ({\mathrm e}^{5}+x \right )\)	\(13\)
norman	\(2 \ln \relax (x )-\ln \left ({\mathrm e}^{5}+x \right )\)	\(13\)
risch	\(2 \ln \relax (x )-\ln \left ({\mathrm e}^{5}+x \right )\)	\(13\)
meijerg	\(2 \ln \relax (x )-10-\ln \left (1+x \,{\mathrm e}^{-5}\right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.35, size = 12, normalized size = 0.44 \begin {gather*} -\log \left (x + e^{5}\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + e^5) + 2*log(x)

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mupad [B]  time = 0.09, size = 12, normalized size = 0.44 \begin {gather*} 2\,\ln \relax (x)-\ln \left (x+{\mathrm {e}}^5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 2*exp(5))/(x*exp(5) + x^2),x)

[Out]

2*log(x) - log(x + exp(5))

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sympy [A]  time = 0.22, size = 10, normalized size = 0.37 \begin {gather*} 2 \log {\relax (x )} - \log {\left (x + e^{5} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x) - log(x + exp(5))

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