\(\int \frac {1+3 x+x^2-x \log (x)}{-4 x+5 e^x x-x^2+x \log (x)} \, dx\) [1344]

Optimal result

Integrand size = 35, antiderivative size = 17 \[ \int \frac {1+3 x+x^2-x \log (x)}{-4 x+5 e^x x-x^2+x \log (x)} \, dx=-x+\log \left (-4+5 e^x-x+\log (x)\right ) \]

[Out]

ln(ln(x)+5*exp(x)-4-x)-x

Rubi [F]

\[ \int \frac {1+3 x+x^2-x \log (x)}{-4 x+5 e^x x-x^2+x \log (x)} \, dx=\int \frac {1+3 x+x^2-x \log (x)}{-4 x+5 e^x x-x^2+x \log (x)} \, dx \]

[In]

Int[(1 + 3*x + x^2 - x*Log[x])/(-4*x + 5*E^x*x - x^2 + x*Log[x]),x]

[Out]

-Defer[Int][1/(x*(4 - 5*E^x + x - Log[x])), x] - Defer[Int][x/(4 - 5*E^x + x - Log[x]), x] + 3*Defer[Int][(-4
+ 5*E^x - x + Log[x])^(-1), x] - Defer[Int][Log[x]/(-4 + 5*E^x - x + Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{x \left (4-5 e^x+x-\log (x)\right )}-\frac {x}{4-5 e^x+x-\log (x)}+\frac {3}{-4+5 e^x-x+\log (x)}-\frac {\log (x)}{-4+5 e^x-x+\log (x)}\right ) \, dx \\ & = 3 \int \frac {1}{-4+5 e^x-x+\log (x)} \, dx-\int \frac {1}{x \left (4-5 e^x+x-\log (x)\right )} \, dx-\int \frac {x}{4-5 e^x+x-\log (x)} \, dx-\int \frac {\log (x)}{-4+5 e^x-x+\log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1+3 x+x^2-x \log (x)}{-4 x+5 e^x x-x^2+x \log (x)} \, dx=-x+\log \left (4-5 e^x+x-\log (x)\right ) \]

[In]

Integrate[(1 + 3*x + x^2 - x*Log[x])/(-4*x + 5*E^x*x - x^2 + x*Log[x]),x]

[Out]

-x + Log[4 - 5*E^x + x - Log[x]]

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00

method	result	size
norman	\(-x +\ln \left (x -5 \,{\mathrm e}^{x}-\ln \left (x \right )+4\right )\)	\(17\)
risch	\(\ln \left (\ln \left (x \right )+5 \,{\mathrm e}^{x}-4-x \right )-x\)	\(17\)
parallelrisch	\(-x +\ln \left (x -5 \,{\mathrm e}^{x}-\ln \left (x \right )+4\right )\)	\(17\)

[In]

int((-x*ln(x)+x^2+3*x+1)/(x*ln(x)+5*exp(x)*x-x^2-4*x),x,method=_RETURNVERBOSE)

[Out]

-x+ln(x-5*exp(x)-ln(x)+4)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {1+3 x+x^2-x \log (x)}{-4 x+5 e^x x-x^2+x \log (x)} \, dx=-x + \log \left (-x + 5 \, e^{x} + \log \left (x\right ) - 4\right ) \]

[In]

integrate((-x*log(x)+x^2+3*x+1)/(x*log(x)+5*exp(x)*x-x^2-4*x),x, algorithm="fricas")

[Out]

-x + log(-x + 5*e^x + log(x) - 4)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1+3 x+x^2-x \log (x)}{-4 x+5 e^x x-x^2+x \log (x)} \, dx=- x + \log {\left (- \frac {x}{5} + e^{x} + \frac {\log {\left (x \right )}}{5} - \frac {4}{5} \right )} \]

[In]

integrate((-x*ln(x)+x**2+3*x+1)/(x*ln(x)+5*exp(x)*x-x**2-4*x),x)

[Out]

-x + log(-x/5 + exp(x) + log(x)/5 - 4/5)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {1+3 x+x^2-x \log (x)}{-4 x+5 e^x x-x^2+x \log (x)} \, dx=-x + \log \left (-\frac {1}{5} \, x + e^{x} + \frac {1}{5} \, \log \left (x\right ) - \frac {4}{5}\right ) \]

[In]

integrate((-x*log(x)+x^2+3*x+1)/(x*log(x)+5*exp(x)*x-x^2-4*x),x, algorithm="maxima")

[Out]

-x + log(-1/5*x + e^x + 1/5*log(x) - 4/5)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {1+3 x+x^2-x \log (x)}{-4 x+5 e^x x-x^2+x \log (x)} \, dx=-x + \log \left (x - 5 \, e^{x} - \log \left (x\right ) + 4\right ) \]

[In]

integrate((-x*log(x)+x^2+3*x+1)/(x*log(x)+5*exp(x)*x-x^2-4*x),x, algorithm="giac")

[Out]

-x + log(x - 5*e^x - log(x) + 4)

Mupad [B] (verification not implemented)

Time = 8.97 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {1+3 x+x^2-x \log (x)}{-4 x+5 e^x x-x^2+x \log (x)} \, dx=\ln \left (x-5\,{\mathrm {e}}^x-\ln \left (x\right )+4\right )-x \]

[In]

int(-(3*x - x*log(x) + x^2 + 1)/(4*x - 5*x*exp(x) - x*log(x) + x^2),x)

[Out]

log(x - 5*exp(x) - log(x) + 4) - x