3.48.56 \(\int \frac {36+21 x+2 x^2+e (12+3 x)}{9 x+3 e x+2 x^2} \, dx\)

Optimal. Leaf size=18 \[ 7+x+\log \left (\left (3+e+\frac {2 x}{3}\right )^2 x^4\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6, 1593, 1820} \begin {gather*} x+4 \log (x)+2 \log (2 x+3 (3+e)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(36 + 21*x + 2*x^2 + E*(12 + 3*x))/(9*x + 3*E*x + 2*x^2),x]

[Out]

x + 4*Log[x] + 2*Log[3*(3 + E) + 2*x]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1820

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {36+21 x+2 x^2+e (12+3 x)}{(9+3 e) x+2 x^2} \, dx\\ &=\int \frac {36+21 x+2 x^2+e (12+3 x)}{x (9+3 e+2 x)} \, dx\\ &=\int \left (1+\frac {4}{x}+\frac {4}{9+3 e+2 x}\right ) \, dx\\ &=x+4 \log (x)+2 \log (3 (3+e)+2 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.94 \begin {gather*} x+4 \log (x)+2 \log (9+3 e+2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36 + 21*x + 2*x^2 + E*(12 + 3*x))/(9*x + 3*E*x + 2*x^2),x]

[Out]

x + 4*Log[x] + 2*Log[9 + 3*E + 2*x]

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fricas [A]  time = 0.68, size = 18, normalized size = 1.00 \begin {gather*} x + 2 \, \log \left (2 \, x + 3 \, e + 9\right ) + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x+12)*exp(1)+2*x^2+21*x+36)/(3*x*exp(1)+2*x^2+9*x),x, algorithm="fricas")

[Out]

x + 2*log(2*x + 3*e + 9) + 4*log(x)

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giac [A]  time = 0.13, size = 20, normalized size = 1.11 \begin {gather*} x + 2 \, \log \left ({\left | 2 \, x + 3 \, e + 9 \right |}\right ) + 4 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x+12)*exp(1)+2*x^2+21*x+36)/(3*x*exp(1)+2*x^2+9*x),x, algorithm="giac")

[Out]

x + 2*log(abs(2*x + 3*e + 9)) + 4*log(abs(x))

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maple [A]  time = 0.10, size = 19, normalized size = 1.06

method	result	size
default	\(x +2 \ln \left (3 \,{\mathrm e}+2 x +9\right )+4 \ln \relax (x )\)	\(19\)
norman	\(x +2 \ln \left (3 \,{\mathrm e}+2 x +9\right )+4 \ln \relax (x )\)	\(19\)
risch	\(x +2 \ln \left (3 \,{\mathrm e}+2 x +9\right )+4 \ln \relax (x )\)	\(19\)
meijerg	\(\frac {\left (3 \,{\mathrm e}+21\right ) \left (\frac {3 \,{\mathrm e}}{2}+\frac {9}{2}\right ) \ln \left (1+\frac {2 x}{3 \left (3+{\mathrm e}\right )}\right )}{3 \,{\mathrm e}+9}+\frac {2 \left (\frac {3 \,{\mathrm e}}{2}+\frac {9}{2}\right )^{2} \left (\frac {2 x}{3 \left (3+{\mathrm e}\right )}-\ln \left (1+\frac {2 x}{3 \left (3+{\mathrm e}\right )}\right )\right )}{3 \,{\mathrm e}+9}+\frac {8 \,{\mathrm e} \left (\frac {3 \,{\mathrm e}}{2}+\frac {9}{2}\right ) \left (-\ln \left (1+\frac {2 x}{3 \left (3+{\mathrm e}\right )}\right )+\ln \relax (x )+\ln \relax (2)-\ln \relax (3)-\ln \left (3+{\mathrm e}\right )\right )}{\left (3 \,{\mathrm e}+9\right ) \left (3+{\mathrm e}\right )}+\frac {24 \left (\frac {3 \,{\mathrm e}}{2}+\frac {9}{2}\right ) \left (-\ln \left (1+\frac {2 x}{3 \left (3+{\mathrm e}\right )}\right )+\ln \relax (x )+\ln \relax (2)-\ln \relax (3)-\ln \left (3+{\mathrm e}\right )\right )}{\left (3 \,{\mathrm e}+9\right ) \left (3+{\mathrm e}\right )}\)	\(183\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x+12)*exp(1)+2*x^2+21*x+36)/(3*x*exp(1)+2*x^2+9*x),x,method=_RETURNVERBOSE)

[Out]

x+2*ln(3*exp(1)+2*x+9)+4*ln(x)

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maxima [A]  time = 0.36, size = 18, normalized size = 1.00 \begin {gather*} x + 2 \, \log \left (2 \, x + 3 \, e + 9\right ) + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x+12)*exp(1)+2*x^2+21*x+36)/(3*x*exp(1)+2*x^2+9*x),x, algorithm="maxima")

[Out]

x + 2*log(2*x + 3*e + 9) + 4*log(x)

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mupad [B]  time = 0.12, size = 16, normalized size = 0.89 \begin {gather*} x+2\,\ln \left (x+\frac {3\,\mathrm {e}}{2}+\frac {9}{2}\right )+4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((21*x + 2*x^2 + exp(1)*(3*x + 12) + 36)/(9*x + 3*x*exp(1) + 2*x^2),x)

[Out]

x + 2*log(x + (3*exp(1))/2 + 9/2) + 4*log(x)

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sympy [A]  time = 0.52, size = 20, normalized size = 1.11 \begin {gather*} x + 4 \log {\relax (x )} + 2 \log {\left (x + \frac {3 e}{2} + \frac {9}{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x+12)*exp(1)+2*x**2+21*x+36)/(3*x*exp(1)+2*x**2+9*x),x)

[Out]

x + 4*log(x) + 2*log(x + 3*E/2 + 9/2)

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