\(\int \frac {51+15 e^{2+x}-3 x+(3 x-15 e^{2+x} x) \log (x)}{4624 x+400 e^{4+2 x} x-544 x^2+16 x^3+e^{2+x} (2720 x-160 x^2)} \, dx\) [8004]

Optimal result

Integrand size = 69, antiderivative size = 22 \[ \int \frac {51+15 e^{2+x}-3 x+\left (3 x-15 e^{2+x} x\right ) \log (x)}{4624 x+400 e^{4+2 x} x-544 x^2+16 x^3+e^{2+x} \left (2720 x-160 x^2\right )} \, dx=\frac {3 \log (x)}{16 \left (2+5 \left (3+e^{2+x}\right )-x\right )} \]

[Out]

3/16*ln(x)/(17+5*exp(2+x)-x)

Rubi [F]

\[ \int \frac {51+15 e^{2+x}-3 x+\left (3 x-15 e^{2+x} x\right ) \log (x)}{4624 x+400 e^{4+2 x} x-544 x^2+16 x^3+e^{2+x} \left (2720 x-160 x^2\right )} \, dx=\int \frac {51+15 e^{2+x}-3 x+\left (3 x-15 e^{2+x} x\right ) \log (x)}{4624 x+400 e^{4+2 x} x-544 x^2+16 x^3+e^{2+x} \left (2720 x-160 x^2\right )} \, dx \]

[In]

Int[(51 + 15*E^(2 + x) - 3*x + (3*x - 15*E^(2 + x)*x)*Log[x])/(4624*x + 400*E^(4 + 2*x)*x - 544*x^2 + 16*x^3 +
E^(2 + x)*(2720*x - 160*x^2)),x]

[Out]

(27*Log[x]*Defer[Int][(17 + 5*E^(2 + x) - x)^(-2), x])/8 - (3*Log[x]*Defer[Int][(17 + 5*E^(2 + x) - x)^(-1), x
])/16 + (3*Defer[Int][1/((17 + 5*E^(2 + x) - x)*x), x])/16 - (3*Log[x]*Defer[Int][x/(17 + 5*E^(2 + x) - x)^2,
x])/16 + (3*Defer[Int][Defer[Int][(17 + 5*E^(2 + x) - x)^(-1), x]/x, x])/16 - (27*Defer[Int][Defer[Int][(-17 -
5*E^(2 + x) + x)^(-2), x]/x, x])/8 + (3*Defer[Int][Defer[Int][x/(-17 - 5*E^(2 + x) + x)^2, x]/x, x])/16

Rubi steps \begin{align*} \text {integral}& = \int \frac {3 \left (17+5 e^{2+x}-x-\left (-1+5 e^{2+x}\right ) x \log (x)\right )}{16 \left (17+5 e^{2+x}-x\right )^2 x} \, dx \\ & = \frac {3}{16} \int \frac {17+5 e^{2+x}-x-\left (-1+5 e^{2+x}\right ) x \log (x)}{\left (17+5 e^{2+x}-x\right )^2 x} \, dx \\ & = \frac {3}{16} \int \left (-\frac {(-18+x) \log (x)}{\left (17+5 e^{2+x}-x\right )^2}-\frac {-1+x \log (x)}{\left (17+5 e^{2+x}-x\right ) x}\right ) \, dx \\ & = -\left (\frac {3}{16} \int \frac {(-18+x) \log (x)}{\left (17+5 e^{2+x}-x\right )^2} \, dx\right )-\frac {3}{16} \int \frac {-1+x \log (x)}{\left (17+5 e^{2+x}-x\right ) x} \, dx \\ & = -\left (\frac {3}{16} \int \left (-\frac {1}{\left (17+5 e^{2+x}-x\right ) x}+\frac {\log (x)}{17+5 e^{2+x}-x}\right ) \, dx\right )+\frac {3}{16} \int \frac {-18 \int \frac {1}{\left (-17-5 e^{2+x}+x\right )^2} \, dx+\int \frac {x}{\left (-17-5 e^{2+x}+x\right )^2} \, dx}{x} \, dx-\frac {1}{16} (3 \log (x)) \int \frac {x}{\left (17+5 e^{2+x}-x\right )^2} \, dx+\frac {1}{8} (27 \log (x)) \int \frac {1}{\left (17+5 e^{2+x}-x\right )^2} \, dx \\ & = \frac {3}{16} \int \frac {1}{\left (17+5 e^{2+x}-x\right ) x} \, dx-\frac {3}{16} \int \frac {\log (x)}{17+5 e^{2+x}-x} \, dx+\frac {3}{16} \int \left (-\frac {18 \int \frac {1}{\left (-17-5 e^{2+x}+x\right )^2} \, dx}{x}+\frac {\int \frac {x}{\left (-17-5 e^{2+x}+x\right )^2} \, dx}{x}\right ) \, dx-\frac {1}{16} (3 \log (x)) \int \frac {x}{\left (17+5 e^{2+x}-x\right )^2} \, dx+\frac {1}{8} (27 \log (x)) \int \frac {1}{\left (17+5 e^{2+x}-x\right )^2} \, dx \\ & = \frac {3}{16} \int \frac {1}{\left (17+5 e^{2+x}-x\right ) x} \, dx+\frac {3}{16} \int \frac {\int \frac {1}{17+5 e^{2+x}-x} \, dx}{x} \, dx+\frac {3}{16} \int \frac {\int \frac {x}{\left (-17-5 e^{2+x}+x\right )^2} \, dx}{x} \, dx-\frac {27}{8} \int \frac {\int \frac {1}{\left (-17-5 e^{2+x}+x\right )^2} \, dx}{x} \, dx-\frac {1}{16} (3 \log (x)) \int \frac {1}{17+5 e^{2+x}-x} \, dx-\frac {1}{16} (3 \log (x)) \int \frac {x}{\left (17+5 e^{2+x}-x\right )^2} \, dx+\frac {1}{8} (27 \log (x)) \int \frac {1}{\left (17+5 e^{2+x}-x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {51+15 e^{2+x}-3 x+\left (3 x-15 e^{2+x} x\right ) \log (x)}{4624 x+400 e^{4+2 x} x-544 x^2+16 x^3+e^{2+x} \left (2720 x-160 x^2\right )} \, dx=\frac {3 \log (x)}{16 \left (17+5 e^{2+x}-x\right )} \]

[In]

Integrate[(51 + 15*E^(2 + x) - 3*x + (3*x - 15*E^(2 + x)*x)*Log[x])/(4624*x + 400*E^(4 + 2*x)*x - 544*x^2 + 16
*x^3 + E^(2 + x)*(2720*x - 160*x^2)),x]

[Out]

(3*Log[x])/(16*(17 + 5*E^(2 + x) - x))

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {3 \ln \left (x \right )}{16 \left (x -5 \,{\mathrm e}^{2+x}-17\right )}\)	\(16\)
parallelrisch	\(-\frac {3 \ln \left (x \right )}{16 \left (x -5 \,{\mathrm e}^{2+x}-17\right )}\)	\(16\)

[In]

int(((-15*x*exp(2+x)+3*x)*ln(x)+15*exp(2+x)-3*x+51)/(400*x*exp(2+x)^2+(-160*x^2+2720*x)*exp(2+x)+16*x^3-544*x^
2+4624*x),x,method=_RETURNVERBOSE)

[Out]

-3/16/(x-5*exp(2+x)-17)*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {51+15 e^{2+x}-3 x+\left (3 x-15 e^{2+x} x\right ) \log (x)}{4624 x+400 e^{4+2 x} x-544 x^2+16 x^3+e^{2+x} \left (2720 x-160 x^2\right )} \, dx=-\frac {3 \, \log \left (x\right )}{16 \, {\left (x - 5 \, e^{\left (x + 2\right )} - 17\right )}} \]

[In]

integrate(((-15*x*exp(2+x)+3*x)*log(x)+15*exp(2+x)-3*x+51)/(400*x*exp(2+x)^2+(-160*x^2+2720*x)*exp(2+x)+16*x^3
-544*x^2+4624*x),x, algorithm="fricas")

[Out]

-3/16*log(x)/(x - 5*e^(x + 2) - 17)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {51+15 e^{2+x}-3 x+\left (3 x-15 e^{2+x} x\right ) \log (x)}{4624 x+400 e^{4+2 x} x-544 x^2+16 x^3+e^{2+x} \left (2720 x-160 x^2\right )} \, dx=\frac {3 \log {\left (x \right )}}{- 16 x + 80 e^{x + 2} + 272} \]

[In]

integrate(((-15*x*exp(2+x)+3*x)*ln(x)+15*exp(2+x)-3*x+51)/(400*x*exp(2+x)**2+(-160*x**2+2720*x)*exp(2+x)+16*x*
*3-544*x**2+4624*x),x)

[Out]

3*log(x)/(-16*x + 80*exp(x + 2) + 272)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {51+15 e^{2+x}-3 x+\left (3 x-15 e^{2+x} x\right ) \log (x)}{4624 x+400 e^{4+2 x} x-544 x^2+16 x^3+e^{2+x} \left (2720 x-160 x^2\right )} \, dx=-\frac {3 \, \log \left (x\right )}{16 \, {\left (x - 5 \, e^{\left (x + 2\right )} - 17\right )}} \]

[In]

integrate(((-15*x*exp(2+x)+3*x)*log(x)+15*exp(2+x)-3*x+51)/(400*x*exp(2+x)^2+(-160*x^2+2720*x)*exp(2+x)+16*x^3
-544*x^2+4624*x),x, algorithm="maxima")

[Out]

-3/16*log(x)/(x - 5*e^(x + 2) - 17)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {51+15 e^{2+x}-3 x+\left (3 x-15 e^{2+x} x\right ) \log (x)}{4624 x+400 e^{4+2 x} x-544 x^2+16 x^3+e^{2+x} \left (2720 x-160 x^2\right )} \, dx=-\frac {3 \, \log \left (x\right )}{16 \, {\left (x - 5 \, e^{\left (x + 2\right )} - 17\right )}} \]

[In]

integrate(((-15*x*exp(2+x)+3*x)*log(x)+15*exp(2+x)-3*x+51)/(400*x*exp(2+x)^2+(-160*x^2+2720*x)*exp(2+x)+16*x^3
-544*x^2+4624*x),x, algorithm="giac")

[Out]

-3/16*log(x)/(x - 5*e^(x + 2) - 17)

Mupad [B] (verification not implemented)

Time = 13.57 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {51+15 e^{2+x}-3 x+\left (3 x-15 e^{2+x} x\right ) \log (x)}{4624 x+400 e^{4+2 x} x-544 x^2+16 x^3+e^{2+x} \left (2720 x-160 x^2\right )} \, dx=\frac {3\,\ln \left (x\right )}{16\,\left (5\,{\mathrm {e}}^{x+2}-x+17\right )} \]

[In]

int((15*exp(x + 2) - 3*x + log(x)*(3*x - 15*x*exp(x + 2)) + 51)/(4624*x + exp(x + 2)*(2720*x - 160*x^2) + 400*
x*exp(2*x + 4) - 544*x^2 + 16*x^3),x)

[Out]

(3*log(x))/(16*(5*exp(x + 2) - x + 17))