3.38.11 \(\int \frac {3+2 x+x^2+e (2 x+2 x^2)+(2 x+2 x^2) \log (x)}{e (3 x+2 x^2+x^3)+(3 x+2 x^2+x^3) \log (x)} \, dx\)

Optimal. Leaf size=14 \[ \log \left (\left (3+2 x+x^2\right ) (e+\log (x))\right ) \]

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Rubi [A]  time = 0.42, antiderivative size = 15, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {6688, 6728, 628, 2302, 29} \begin {gather*} \log \left (x^2+2 x+3\right )+\log (\log (x)+e) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 2*x + x^2 + E*(2*x + 2*x^2) + (2*x + 2*x^2)*Log[x])/(E*(3*x + 2*x^2 + x^3) + (3*x + 2*x^2 + x^3)*Log[
x]),x]

[Out]

Log[3 + 2*x + x^2] + Log[E + Log[x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 15, normalized size = 1.07 \begin {gather*} \log \left (3+2 x+x^2\right )+\log (e+\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x + x^2 + E*(2*x + 2*x^2) + (2*x + 2*x^2)*Log[x])/(E*(3*x + 2*x^2 + x^3) + (3*x + 2*x^2 + x^3
)*Log[x]),x]

[Out]

Log[3 + 2*x + x^2] + Log[E + Log[x]]

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fricas [A]  time = 0.64, size = 16, normalized size = 1.14 \begin {gather*} \log \left (x^{2} + 2 \, x + 3\right ) + \log \left (e + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*log(x)+(2*x^2+2*x)*exp(1)+x^2+2*x+3)/((x^3+2*x^2+3*x)*log(x)+(x^3+2*x^2+3*x)*exp(1)),x,
algorithm="fricas")

[Out]

log(x^2 + 2*x + 3) + log(e + log(x))

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giac [A]  time = 0.21, size = 16, normalized size = 1.14 \begin {gather*} \log \left (x^{2} + 2 \, x + 3\right ) + \log \left (e + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*log(x)+(2*x^2+2*x)*exp(1)+x^2+2*x+3)/((x^3+2*x^2+3*x)*log(x)+(x^3+2*x^2+3*x)*exp(1)),x,
algorithm="giac")

[Out]

log(x^2 + 2*x + 3) + log(e + log(x))

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maple [A]  time = 0.06, size = 17, normalized size = 1.21

method	result	size
norman	\(\ln \left ({\mathrm e}+\ln \relax (x )\right )+\ln \left (x^{2}+2 x +3\right )\)	\(17\)
risch	\(\ln \left ({\mathrm e}+\ln \relax (x )\right )+\ln \left (x^{2}+2 x +3\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2+2*x)*ln(x)+(2*x^2+2*x)*exp(1)+x^2+2*x+3)/((x^3+2*x^2+3*x)*ln(x)+(x^3+2*x^2+3*x)*exp(1)),x,method=_
RETURNVERBOSE)

[Out]

ln(exp(1)+ln(x))+ln(x^2+2*x+3)

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maxima [A]  time = 0.42, size = 16, normalized size = 1.14 \begin {gather*} \log \left (x^{2} + 2 \, x + 3\right ) + \log \left (e + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*log(x)+(2*x^2+2*x)*exp(1)+x^2+2*x+3)/((x^3+2*x^2+3*x)*log(x)+(x^3+2*x^2+3*x)*exp(1)),x,
algorithm="maxima")

[Out]

log(x^2 + 2*x + 3) + log(e + log(x))

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mupad [B]  time = 2.27, size = 16, normalized size = 1.14 \begin {gather*} \ln \left (\mathrm {e}+\ln \relax (x)\right )+\ln \left (x^2+2\,x+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + exp(1)*(2*x + 2*x^2) + log(x)*(2*x + 2*x^2) + x^2 + 3)/(exp(1)*(3*x + 2*x^2 + x^3) + log(x)*(3*x +
2*x^2 + x^3)),x)

[Out]

log(exp(1) + log(x)) + log(2*x + x^2 + 3)

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sympy [A]  time = 0.15, size = 17, normalized size = 1.21 \begin {gather*} \log {\left (\log {\relax (x )} + e \right )} + \log {\left (x^{2} + 2 x + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2+2*x)*ln(x)+(2*x**2+2*x)*exp(1)+x**2+2*x+3)/((x**3+2*x**2+3*x)*ln(x)+(x**3+2*x**2+3*x)*exp(1
)),x)

[Out]

log(log(x) + E) + log(x**2 + 2*x + 3)

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