\(\int \frac {-450-1350 x+e^x (-30+120 x)+(-450 x+30 e^x x) \log (x)}{225 x+e^{2 x} x-450 x^2+225 x^3+e^x (30 x-30 x^2)} \, dx\) [5140]

Optimal result

Integrand size = 64, antiderivative size = 24 \[ \int \frac {-450-1350 x+e^x (-30+120 x)+\left (-450 x+30 e^x x\right ) \log (x)}{225 x+e^{2 x} x-450 x^2+225 x^3+e^x \left (30 x-30 x^2\right )} \, dx=\frac {4+\log (x)}{-\frac {e^x}{30}+\frac {1}{4} (-2+2 x)} \]

[Out]

(ln(x)+4)/(1/2*x-1/2-1/30*exp(x))

Rubi [F]

\[ \int \frac {-450-1350 x+e^x (-30+120 x)+\left (-450 x+30 e^x x\right ) \log (x)}{225 x+e^{2 x} x-450 x^2+225 x^3+e^x \left (30 x-30 x^2\right )} \, dx=\int \frac {-450-1350 x+e^x (-30+120 x)+\left (-450 x+30 e^x x\right ) \log (x)}{225 x+e^{2 x} x-450 x^2+225 x^3+e^x \left (30 x-30 x^2\right )} \, dx \]

[In]

Int[(-450 - 1350*x + E^x*(-30 + 120*x) + (-450*x + 30*E^x*x)*Log[x])/(225*x + E^(2*x)*x - 450*x^2 + 225*x^3 +
E^x*(30*x - 30*x^2)),x]

[Out]

-3600*Defer[Int][(15 + E^x - 15*x)^(-2), x] - 900*Log[x]*Defer[Int][(15 + E^x - 15*x)^(-2), x] + 120*Defer[Int
][(15 + E^x - 15*x)^(-1), x] + 30*Log[x]*Defer[Int][(15 + E^x - 15*x)^(-1), x] - 30*Defer[Int][1/((15 + E^x -
15*x)*x), x] + 1800*Defer[Int][x/(15 + E^x - 15*x)^2, x] + 450*Log[x]*Defer[Int][x/(15 + E^x - 15*x)^2, x] + 9
00*Defer[Int][Defer[Int][(15 + E^x - 15*x)^(-2), x]/x, x] - 30*Defer[Int][Defer[Int][(15 + E^x - 15*x)^(-1), x
]/x, x] - 450*Defer[Int][Defer[Int][x/(15 + E^x - 15*x)^2, x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {30 \left (-15 (1+3 x)+e^x (-1+4 x)+\left (-15+e^x\right ) x \log (x)\right )}{\left (15+e^x-15 x\right )^2 x} \, dx \\ & = 30 \int \frac {-15 (1+3 x)+e^x (-1+4 x)+\left (-15+e^x\right ) x \log (x)}{\left (15+e^x-15 x\right )^2 x} \, dx \\ & = 30 \int \left (\frac {15 (-2+x) (4+\log (x))}{\left (15+e^x-15 x\right )^2}+\frac {-1+4 x+x \log (x)}{\left (15+e^x-15 x\right ) x}\right ) \, dx \\ & = 30 \int \frac {-1+4 x+x \log (x)}{\left (15+e^x-15 x\right ) x} \, dx+450 \int \frac {(-2+x) (4+\log (x))}{\left (15+e^x-15 x\right )^2} \, dx \\ & = 30 \int \left (\frac {4}{15+e^x-15 x}-\frac {1}{\left (15+e^x-15 x\right ) x}+\frac {\log (x)}{15+e^x-15 x}\right ) \, dx+450 \int \left (-\frac {2 (4+\log (x))}{\left (15+e^x-15 x\right )^2}+\frac {x (4+\log (x))}{\left (15+e^x-15 x\right )^2}\right ) \, dx \\ & = -\left (30 \int \frac {1}{\left (15+e^x-15 x\right ) x} \, dx\right )+30 \int \frac {\log (x)}{15+e^x-15 x} \, dx+120 \int \frac {1}{15+e^x-15 x} \, dx+450 \int \frac {x (4+\log (x))}{\left (15+e^x-15 x\right )^2} \, dx-900 \int \frac {4+\log (x)}{\left (15+e^x-15 x\right )^2} \, dx \\ & = -\left (30 \int \frac {1}{\left (15+e^x-15 x\right ) x} \, dx\right )-30 \int \frac {\int \frac {1}{15+e^x-15 x} \, dx}{x} \, dx+120 \int \frac {1}{15+e^x-15 x} \, dx+450 \int \left (\frac {4 x}{\left (15+e^x-15 x\right )^2}+\frac {x \log (x)}{\left (15+e^x-15 x\right )^2}\right ) \, dx-900 \int \left (\frac {4}{\left (15+e^x-15 x\right )^2}+\frac {\log (x)}{\left (15+e^x-15 x\right )^2}\right ) \, dx+(30 \log (x)) \int \frac {1}{15+e^x-15 x} \, dx \\ & = -\left (30 \int \frac {1}{\left (15+e^x-15 x\right ) x} \, dx\right )-30 \int \frac {\int \frac {1}{15+e^x-15 x} \, dx}{x} \, dx+120 \int \frac {1}{15+e^x-15 x} \, dx+450 \int \frac {x \log (x)}{\left (15+e^x-15 x\right )^2} \, dx-900 \int \frac {\log (x)}{\left (15+e^x-15 x\right )^2} \, dx+1800 \int \frac {x}{\left (15+e^x-15 x\right )^2} \, dx-3600 \int \frac {1}{\left (15+e^x-15 x\right )^2} \, dx+(30 \log (x)) \int \frac {1}{15+e^x-15 x} \, dx \\ & = -\left (30 \int \frac {1}{\left (15+e^x-15 x\right ) x} \, dx\right )-30 \int \frac {\int \frac {1}{15+e^x-15 x} \, dx}{x} \, dx+120 \int \frac {1}{15+e^x-15 x} \, dx-450 \int \frac {\int \frac {x}{\left (15+e^x-15 x\right )^2} \, dx}{x} \, dx+900 \int \frac {\int \frac {1}{\left (15+e^x-15 x\right )^2} \, dx}{x} \, dx+1800 \int \frac {x}{\left (15+e^x-15 x\right )^2} \, dx-3600 \int \frac {1}{\left (15+e^x-15 x\right )^2} \, dx+(30 \log (x)) \int \frac {1}{15+e^x-15 x} \, dx+(450 \log (x)) \int \frac {x}{\left (15+e^x-15 x\right )^2} \, dx-(900 \log (x)) \int \frac {1}{\left (15+e^x-15 x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {-450-1350 x+e^x (-30+120 x)+\left (-450 x+30 e^x x\right ) \log (x)}{225 x+e^{2 x} x-450 x^2+225 x^3+e^x \left (30 x-30 x^2\right )} \, dx=-\frac {30 (4+\log (x))}{15+e^x-15 x} \]

[In]

Integrate[(-450 - 1350*x + E^x*(-30 + 120*x) + (-450*x + 30*E^x*x)*Log[x])/(225*x + E^(2*x)*x - 450*x^2 + 225*
x^3 + E^x*(30*x - 30*x^2)),x]

[Out]

(-30*(4 + Log[x]))/(15 + E^x - 15*x)

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79

method	result	size
parallelrisch	\(\frac {120+30 \ln \left (x \right )}{15 x -{\mathrm e}^{x}-15}\)	\(19\)
risch	\(\frac {30 \ln \left (x \right )}{15 x -{\mathrm e}^{x}-15}+\frac {120}{15 x -{\mathrm e}^{x}-15}\)	\(30\)

[In]

int(((30*exp(x)*x-450*x)*ln(x)+(120*x-30)*exp(x)-1350*x-450)/(x*exp(x)^2+(-30*x^2+30*x)*exp(x)+225*x^3-450*x^2
+225*x),x,method=_RETURNVERBOSE)

[Out]

(120+30*ln(x))/(15*x-exp(x)-15)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {-450-1350 x+e^x (-30+120 x)+\left (-450 x+30 e^x x\right ) \log (x)}{225 x+e^{2 x} x-450 x^2+225 x^3+e^x \left (30 x-30 x^2\right )} \, dx=\frac {30 \, {\left (\log \left (x\right ) + 4\right )}}{15 \, x - e^{x} - 15} \]

[In]

integrate(((30*exp(x)*x-450*x)*log(x)+(120*x-30)*exp(x)-1350*x-450)/(x*exp(x)^2+(-30*x^2+30*x)*exp(x)+225*x^3-
450*x^2+225*x),x, algorithm="fricas")

[Out]

30*(log(x) + 4)/(15*x - e^x - 15)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {-450-1350 x+e^x (-30+120 x)+\left (-450 x+30 e^x x\right ) \log (x)}{225 x+e^{2 x} x-450 x^2+225 x^3+e^x \left (30 x-30 x^2\right )} \, dx=\frac {- 30 \log {\left (x \right )} - 120}{- 15 x + e^{x} + 15} \]

[In]

integrate(((30*exp(x)*x-450*x)*ln(x)+(120*x-30)*exp(x)-1350*x-450)/(x*exp(x)**2+(-30*x**2+30*x)*exp(x)+225*x**
3-450*x**2+225*x),x)

[Out]

(-30*log(x) - 120)/(-15*x + exp(x) + 15)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {-450-1350 x+e^x (-30+120 x)+\left (-450 x+30 e^x x\right ) \log (x)}{225 x+e^{2 x} x-450 x^2+225 x^3+e^x \left (30 x-30 x^2\right )} \, dx=\frac {30 \, {\left (\log \left (x\right ) + 4\right )}}{15 \, x - e^{x} - 15} \]

[In]

integrate(((30*exp(x)*x-450*x)*log(x)+(120*x-30)*exp(x)-1350*x-450)/(x*exp(x)^2+(-30*x^2+30*x)*exp(x)+225*x^3-
450*x^2+225*x),x, algorithm="maxima")

[Out]

30*(log(x) + 4)/(15*x - e^x - 15)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {-450-1350 x+e^x (-30+120 x)+\left (-450 x+30 e^x x\right ) \log (x)}{225 x+e^{2 x} x-450 x^2+225 x^3+e^x \left (30 x-30 x^2\right )} \, dx=\frac {30 \, {\left (\log \left (x\right ) + 4\right )}}{15 \, x - e^{x} - 15} \]

[In]

integrate(((30*exp(x)*x-450*x)*log(x)+(120*x-30)*exp(x)-1350*x-450)/(x*exp(x)^2+(-30*x^2+30*x)*exp(x)+225*x^3-
450*x^2+225*x),x, algorithm="giac")

[Out]

30*(log(x) + 4)/(15*x - e^x - 15)

Mupad [F(-1)]

Timed out. \[ \int \frac {-450-1350 x+e^x (-30+120 x)+\left (-450 x+30 e^x x\right ) \log (x)}{225 x+e^{2 x} x-450 x^2+225 x^3+e^x \left (30 x-30 x^2\right )} \, dx=\int -\frac {1350\,x+\ln \left (x\right )\,\left (450\,x-30\,x\,{\mathrm {e}}^x\right )-{\mathrm {e}}^x\,\left (120\,x-30\right )+450}{225\,x+x\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (30\,x-30\,x^2\right )-450\,x^2+225\,x^3} \,d x \]

[In]

int(-(1350*x + log(x)*(450*x - 30*x*exp(x)) - exp(x)*(120*x - 30) + 450)/(225*x + x*exp(2*x) + exp(x)*(30*x -
30*x^2) - 450*x^2 + 225*x^3),x)

[Out]

int(-(1350*x + log(x)*(450*x - 30*x*exp(x)) - exp(x)*(120*x - 30) + 450)/(225*x + x*exp(2*x) + exp(x)*(30*x -
30*x^2) - 450*x^2 + 225*x^3), x)