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]

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

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Rubi [F]  time = 0.63, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.17, size = 13, normalized size = 1.00 \[ \log (x)-\frac {2}{\sqrt {x+\log (x)}} \]

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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fricas [B]  time = 0.43, size = 24, normalized size = 1.85 \[ \frac {x \log \relax (x) + \log \relax (x)^{2} - 2 \, \sqrt {x + \log \relax (x)}}{x + \log \relax (x)} \]

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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giac [A]  time = 1.08, size = 11, normalized size = 0.85 \[ -\frac {2}{\sqrt {x + \log \relax (x)}} + \log \relax (x) \]

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]

-2/sqrt(x + log(x)) + log(x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}+2 x \ln \relax (x )+\ln \relax (x )^{2}+\left (x +1\right ) \sqrt {x +\ln \relax (x )}}{x^{3}+2 x^{2} \ln \relax (x )+x \ln \relax (x )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x + \log \relax (x)} {\left (x + 1\right )}}{x^{3} + 2 \, x^{2} \log \relax (x) + x \log \relax (x)^{2}}\,{d x} + \log \relax (x) \]

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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mupad [B]  time = 0.20, size = 11, normalized size = 0.85 \[ \ln \relax (x)-\frac {2}{\sqrt {x+\ln \relax (x)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} + x \sqrt {x + \log {\relax (x )}} + 2 x \log {\relax (x )} + \sqrt {x + \log {\relax (x )}} + \log {\relax (x )}^{2}}{x \left (x + \log {\relax (x )}\right )^{2}}\, dx \]

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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