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 = 1.00954, 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. \[ \text{Int}\left (\frac{x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt{x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)},x\right ) \]

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]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

Timed out

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Mathematica [A]  time = 0.0334075, 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 = 0.036, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}+2\,{x}^{2}\ln \left ( x \right ) +x \left ( \ln \left ( x \right ) \right ) ^{2}} \left ({x}^{2}+2\,x\ln \left ( x \right ) + \left ( \ln \left ( x \right ) \right ) ^{2}+ \left ( 1+x \right ) \sqrt{x+\ln \left ( x \right ) } \right ) }\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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 [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

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]

Exception raised: RecursionError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2} + \sqrt{x + \log \left (x\right )}{\left (x + 1\right )}}{x^{3} + 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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)