3.83.48 \(\int \frac {4 x+2 x^2+e^{\log (e^x x) \log (2+x)} (-x \log (e^x x)+(-2-3 x-x^2) \log (2+x))}{2 x+x^2} \, dx\)

Optimal. Leaf size=20 \[ -4-e^{\log \left (e^x x\right ) \log (2+x)}+2 x \]

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Rubi [F]  time = 1.58, 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 = {} \begin {gather*} \int \frac {4 x+2 x^2+e^{\log \left (e^x x\right ) \log (2+x)} \left (-x \log \left (e^x x\right )+\left (-2-3 x-x^2\right ) \log (2+x)\right )}{2 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x + 2*x^2 + E^(Log[E^x*x]*Log[2 + x])*(-(x*Log[E^x*x]) + (-2 - 3*x - x^2)*Log[2 + x]))/(2*x + x^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 x+2 x^2+e^{\log \left (e^x x\right ) \log (2+x)} \left (-x \log \left (e^x x\right )+\left (-2-3 x-x^2\right ) \log (2+x)\right )}{x (2+x)} \, dx\\ &=\int \left (2-\frac {\left (e^x x\right )^{\log (2+x)} \left (x \log \left (e^x x\right )+2 \log (2+x)+3 x \log (2+x)+x^2 \log (2+x)\right )}{x (2+x)}\right ) \, dx\\ &=2 x-\int \frac {\left (e^x x\right )^{\log (2+x)} \left (x \log \left (e^x x\right )+2 \log (2+x)+3 x \log (2+x)+x^2 \log (2+x)\right )}{x (2+x)} \, dx\\ &=2 x-\int \frac {\left (e^x x\right )^{\log (2+x)} \left (x \log \left (e^x x\right )+\left (2+3 x+x^2\right ) \log (2+x)\right )}{x (2+x)} \, dx\\ &=2 x-\int \left (\frac {\left (e^x x\right )^{\log (2+x)} \log \left (e^x x\right )}{2+x}+\frac {\left (e^x x\right )^{\log (2+x)} (1+x) \log (2+x)}{x}\right ) \, dx\\ &=2 x-\int \frac {\left (e^x x\right )^{\log (2+x)} \log \left (e^x x\right )}{2+x} \, dx-\int \frac {\left (e^x x\right )^{\log (2+x)} (1+x) \log (2+x)}{x} \, dx\\ &=2 x-\int \frac {\left (e^x x\right )^{\log (2+x)} \log \left (e^x x\right )}{2+x} \, dx-\int \left (\left (e^x x\right )^{\log (2+x)} \log (2+x)+\frac {\left (e^x x\right )^{\log (2+x)} \log (2+x)}{x}\right ) \, dx\\ &=2 x-\int \frac {\left (e^x x\right )^{\log (2+x)} \log \left (e^x x\right )}{2+x} \, dx-\int \left (e^x x\right )^{\log (2+x)} \log (2+x) \, dx-\int \frac {\left (e^x x\right )^{\log (2+x)} \log (2+x)}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 16, normalized size = 0.80 \begin {gather*} 2 x-\left (e^x x\right )^{\log (2+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x + 2*x^2 + E^(Log[E^x*x]*Log[2 + x])*(-(x*Log[E^x*x]) + (-2 - 3*x - x^2)*Log[2 + x]))/(2*x + x^2
),x]

[Out]

2*x - (E^x*x)^Log[2 + x]

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fricas [A]  time = 0.71, size = 17, normalized size = 0.85 \begin {gather*} 2 \, x - e^{\left (\log \left (x e^{x}\right ) \log \left (x + 2\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*log(exp(x)*x)+(-x^2-3*x-2)*log(2+x))*exp(log(2+x)*log(exp(x)*x))+2*x^2+4*x)/(x^2+2*x),x, algori
thm="fricas")

[Out]

2*x - e^(log(x*e^x)*log(x + 2))

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giac [A]  time = 0.29, size = 17, normalized size = 0.85 \begin {gather*} 2 \, x - e^{\left (\log \left (x e^{x}\right ) \log \left (x + 2\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*log(exp(x)*x)+(-x^2-3*x-2)*log(2+x))*exp(log(2+x)*log(exp(x)*x))+2*x^2+4*x)/(x^2+2*x),x, algori
thm="giac")

[Out]

2*x - e^(log(x*e^x)*log(x + 2))

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maple [C]  time = 0.37, size = 69, normalized size = 3.45

method	result	size
risch	\(2 x -\left (2+x \right )^{-\frac {i \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right ) \pi }{2}+\frac {i \pi \,\mathrm {csgn}\left (i x \right )}{2}+\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )}{2}+\ln \relax (x )+\ln \left ({\mathrm e}^{x}\right )}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x*ln(exp(x)*x)+(-x^2-3*x-2)*ln(2+x))*exp(ln(2+x)*ln(exp(x)*x))+2*x^2+4*x)/(x^2+2*x),x,method=_RETURNVER
BOSE)

[Out]

2*x-(2+x)^(-1/2*I*csgn(I*x*exp(x))*Pi+1/2*I*Pi*csgn(I*x)+1/2*I*Pi*csgn(I*exp(x))-1/2*I*Pi*csgn(I*x*exp(x))*csg
n(I*x)*csgn(I*exp(x))+ln(x)+ln(exp(x)))

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maxima [A]  time = 0.46, size = 21, normalized size = 1.05 \begin {gather*} 2 \, x - e^{\left (x \log \left (x + 2\right ) + \log \left (x + 2\right ) \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*log(exp(x)*x)+(-x^2-3*x-2)*log(2+x))*exp(log(2+x)*log(exp(x)*x))+2*x^2+4*x)/(x^2+2*x),x, algori
thm="maxima")

[Out]

2*x - e^(x*log(x + 2) + log(x + 2)*log(x))

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mupad [B]  time = 4.99, size = 17, normalized size = 0.85 \begin {gather*} 2\,x-x^{\ln \left (x+2\right )}\,{\left (x+2\right )}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - exp(log(x*exp(x))*log(x + 2))*(log(x + 2)*(3*x + x^2 + 2) + x*log(x*exp(x))) + 2*x^2)/(2*x + x^2),x
)

[Out]

2*x - x^log(x + 2)*(x + 2)^x

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sympy [A]  time = 60.95, size = 20, normalized size = 1.00 \begin {gather*} 2 x - e^{x \log {\left (x + 2 \right )}} e^{\log {\relax (x )} \log {\left (x + 2 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*ln(exp(x)*x)+(-x**2-3*x-2)*ln(2+x))*exp(ln(2+x)*ln(exp(x)*x))+2*x**2+4*x)/(x**2+2*x),x)

[Out]

2*x - exp(x*log(x + 2))*exp(log(x)*log(x + 2))

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