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

Optimal. Leaf size=13 \[ \log (x)-\frac{2}{\sqrt{x+\log (x)}} \]

[Out]

Log[x] - 2/Sqrt[x + Log[x]]

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Rubi [F]  time = 0.625373, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt{x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx \]

Verification is Not applicable to the result.

[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*} \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 (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]  time = 0.150842, size = 13, normalized size = 1. \[ \log (x)-\frac{2}{\sqrt{x+\log (x)}} \]

Antiderivative was successfully verified.

[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]]

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Maple [F]  time = 0.012, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}+2\,{x}^{2}\ln \left ( x \right ) +x \left ( \ln \left ( x \right ) \right ) ^{2}} \left ({x}^{2}+2\,x\ln \left ( x \right ) + \left ( \ln \left ( x \right ) \right ) ^{2}+ \left ( 1+x \right ) \sqrt{x+\ln \left ( x \right ) } \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\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} + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 2.20676, size = 77, normalized size = 5.92 \begin{align*} \frac{x \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, \sqrt{x + \log \left (x\right )}}{x + \log \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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