\(\int \frac {x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt {x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx\) [12]

Optimal result

Integrand size = 45, antiderivative size = 13 \[ \int \frac {x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt {x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx=\log (x)-\frac {2}{\sqrt {x+\log (x)}} \]

[Out]

ln(x)-2/(x+ln(x))^(1/2)

Rubi [F]

\[ \int \frac {x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt {x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx=\int \frac {x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt {x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx \]

[In]

Int[(x^2 + 2*x*Log[x] + Log[x]^2 + (1 + x)*Sqrt[x + Log[x]])/(x^3 + 2*x^2*Log[x] + x*Log[x]^2),x]

[Out]

Log[x] + Defer[Int][(x + Log[x])^(-3/2), x] - Defer[Int][1/(Log[x]*(x + Log[x])^(3/2)), x] - Defer[Int][1/(Log
[x]^2*Sqrt[x + Log[x]]), x] + Defer[Int][Sqrt[x + Log[x]]/(x*Log[x]^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt {x+\log (x)}}{x (x+\log (x))^2} \, dx \\ & = \int \left (\frac {1}{x}+\frac {1}{(x+\log (x))^{3/2}}-\frac {1}{\log (x) (x+\log (x))^{3/2}}-\frac {1}{\log ^2(x) \sqrt {x+\log (x)}}+\frac {\sqrt {x+\log (x)}}{x \log ^2(x)}\right ) \, dx \\ & = \log (x)+\int \frac {1}{(x+\log (x))^{3/2}} \, dx-\int \frac {1}{\log (x) (x+\log (x))^{3/2}} \, dx-\int \frac {1}{\log ^2(x) \sqrt {x+\log (x)}} \, dx+\int \frac {\sqrt {x+\log (x)}}{x \log ^2(x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt {x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx=\log (x)-\frac {2}{\sqrt {x+\log (x)}} \]

[In]

Integrate[(x^2 + 2*x*Log[x] + Log[x]^2 + (1 + x)*Sqrt[x + Log[x]])/(x^3 + 2*x^2*Log[x] + x*Log[x]^2),x]

[Out]

Log[x] - 2/Sqrt[x + Log[x]]

Maple [F]

\[\int \frac {x^{2}+2 x \ln \left (x \right )+\ln \left (x \right )^{2}+\left (1+x \right ) \sqrt {x +\ln \left (x \right )}}{x^{3}+2 x^{2} \ln \left (x \right )+x \ln \left (x \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (11) = 22\).

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.85 \[ \int \frac {x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt {x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx=\frac {x \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, \sqrt {x + \log \left (x\right )}}{x + \log \left (x\right )} \]

[In]

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

[Out]

(x*log(x) + log(x)^2 - 2*sqrt(x + log(x)))/(x + log(x))

Sympy [F]

\[ \int \frac {x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt {x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx=\int \frac {x^{2} + x \sqrt {x + \log {\left (x \right )}} + 2 x \log {\left (x \right )} + \sqrt {x + \log {\left (x \right )}} + \log {\left (x \right )}^{2}}{x \left (x + \log {\left (x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

Integral((x**2 + x*sqrt(x + log(x)) + 2*x*log(x) + sqrt(x + log(x)) + log(x)**2)/(x*(x + log(x))**2), x)

Maxima [F]

\[ \int \frac {x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt {x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx=\int { \frac {x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2} + \sqrt {x + \log \left (x\right )} {\left (x + 1\right )}}{x^{3} + 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(x + log(x))*(x + 1)/(x^3 + 2*x^2*log(x) + x*log(x)^2), x) + log(x)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt {x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx=-\frac {2}{\sqrt {x + \log \left (x\right )}} + \log \left (x\right ) \]

[In]

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

[Out]

-2/sqrt(x + log(x)) + log(x)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt {x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx=\ln \left (x\right )-\frac {2}{\sqrt {x+\ln \left (x\right )}} \]

[In]

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

[Out]

log(x) - 2/(x + log(x))^(1/2)