\(\int \frac {e^{-x/9} (4 e^{x/9} x^2+e^{\frac {e^{-x/9} (e^{x/9} (-81-x)+45 x)}{x}} (324 e^{x/9}-20 x^2))}{x^2} \, dx\) [9061]

Optimal result

Integrand size = 70, antiderivative size = 25 \[ \int \frac {e^{-x/9} \left (4 e^{x/9} x^2+e^{\frac {e^{-x/9} \left (e^{x/9} (-81-x)+45 x\right )}{x}} \left (324 e^{x/9}-20 x^2\right )\right )}{x^2} \, dx=4 \left (e^{\frac {-81-x+45 e^{-x/9} x}{x}}+x\right ) \]

[Out]

4*exp((45*x/exp(1/9*x)-81-x)/x)+4*x

Rubi [F]

\[ \int \frac {e^{-x/9} \left (4 e^{x/9} x^2+e^{\frac {e^{-x/9} \left (e^{x/9} (-81-x)+45 x\right )}{x}} \left (324 e^{x/9}-20 x^2\right )\right )}{x^2} \, dx=\int \frac {e^{-x/9} \left (4 e^{x/9} x^2+e^{\frac {e^{-x/9} \left (e^{x/9} (-81-x)+45 x\right )}{x}} \left (324 e^{x/9}-20 x^2\right )\right )}{x^2} \, dx \]

[In]

Int[(4*E^(x/9)*x^2 + E^((E^(x/9)*(-81 - x) + 45*x)/(E^(x/9)*x))*(324*E^(x/9) - 20*x^2))/(E^(x/9)*x^2),x]

[Out]

4*x - 20*Defer[Int][E^(-1 + 45/E^(x/9) - 81/x - x/9), x] + 324*Defer[Int][E^(-1 + 45/E^(x/9) - 81/x)/x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (4-20 e^{-1+45 e^{-x/9}-\frac {81}{x}-\frac {x}{9}}+\frac {324 e^{-1+45 e^{-x/9}-\frac {81}{x}}}{x^2}\right ) \, dx \\ & = 4 x-20 \int e^{-1+45 e^{-x/9}-\frac {81}{x}-\frac {x}{9}} \, dx+324 \int \frac {e^{-1+45 e^{-x/9}-\frac {81}{x}}}{x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-x/9} \left (4 e^{x/9} x^2+e^{\frac {e^{-x/9} \left (e^{x/9} (-81-x)+45 x\right )}{x}} \left (324 e^{x/9}-20 x^2\right )\right )}{x^2} \, dx=4 e^{-1+45 e^{-x/9}-\frac {81}{x}}+4 x \]

[In]

Integrate[(4*E^(x/9)*x^2 + E^((E^(x/9)*(-81 - x) + 45*x)/(E^(x/9)*x))*(324*E^(x/9) - 20*x^2))/(E^(x/9)*x^2),x]

[Out]

4*E^(-1 + 45/E^(x/9) - 81/x) + 4*x

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28

method	result	size
parallelrisch	\(4 x +4 \,{\mathrm e}^{\frac {\left (\left (-x -81\right ) {\mathrm e}^{\frac {x}{9}}+45 x \right ) {\mathrm e}^{-\frac {x}{9}}}{x}}\)	\(32\)
parts	\(4 x +4 \,{\mathrm e}^{\frac {\left (\left (-x -81\right ) {\mathrm e}^{\frac {x}{9}}+45 x \right ) {\mathrm e}^{-\frac {x}{9}}}{x}}\)	\(32\)
risch	\(4 x +4 \,{\mathrm e}^{-\frac {\left (x \,{\mathrm e}^{\frac {x}{9}}+81 \,{\mathrm e}^{\frac {x}{9}}-45 x \right ) {\mathrm e}^{-\frac {x}{9}}}{x}}\)	\(33\)
norman	\(\frac {\left (4 x^{2} {\mathrm e}^{\frac {x}{9}}+4 x \,{\mathrm e}^{\frac {x}{9}} {\mathrm e}^{\frac {\left (\left (-x -81\right ) {\mathrm e}^{\frac {x}{9}}+45 x \right ) {\mathrm e}^{-\frac {x}{9}}}{x}}\right ) {\mathrm e}^{-\frac {x}{9}}}{x}\)	\(53\)

[In]

int(((324*exp(1/9*x)-20*x^2)*exp(((-x-81)*exp(1/9*x)+45*x)/x/exp(1/9*x))+4*x^2*exp(1/9*x))/x^2/exp(1/9*x),x,me
thod=_RETURNVERBOSE)

[Out]

4*x+4*exp(((-x-81)*exp(1/9*x)+45*x)/x/exp(1/9*x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-x/9} \left (4 e^{x/9} x^2+e^{\frac {e^{-x/9} \left (e^{x/9} (-81-x)+45 x\right )}{x}} \left (324 e^{x/9}-20 x^2\right )\right )}{x^2} \, dx=4 \, x + 4 \, e^{\left (-\frac {{\left ({\left (x + 81\right )} e^{\left (\frac {1}{9} \, x\right )} - 45 \, x\right )} e^{\left (-\frac {1}{9} \, x\right )}}{x}\right )} \]

[In]

integrate(((324*exp(1/9*x)-20*x^2)*exp(((-x-81)*exp(1/9*x)+45*x)/x/exp(1/9*x))+4*x^2*exp(1/9*x))/x^2/exp(1/9*x
),x, algorithm="fricas")

[Out]

4*x + 4*e^(-((x + 81)*e^(1/9*x) - 45*x)*e^(-1/9*x)/x)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-x/9} \left (4 e^{x/9} x^2+e^{\frac {e^{-x/9} \left (e^{x/9} (-81-x)+45 x\right )}{x}} \left (324 e^{x/9}-20 x^2\right )\right )}{x^2} \, dx=4 x + 4 e^{\frac {\left (45 x + \left (- x - 81\right ) e^{\frac {x}{9}}\right ) e^{- \frac {x}{9}}}{x}} \]

[In]

integrate(((324*exp(1/9*x)-20*x**2)*exp(((-x-81)*exp(1/9*x)+45*x)/x/exp(1/9*x))+4*x**2*exp(1/9*x))/x**2/exp(1/
9*x),x)

[Out]

4*x + 4*exp((45*x + (-x - 81)*exp(x/9))*exp(-x/9)/x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {e^{-x/9} \left (4 e^{x/9} x^2+e^{\frac {e^{-x/9} \left (e^{x/9} (-81-x)+45 x\right )}{x}} \left (324 e^{x/9}-20 x^2\right )\right )}{x^2} \, dx=4 \, x + 4 \, e^{\left (-\frac {81}{x} + 45 \, e^{\left (-\frac {1}{9} \, x\right )} - 1\right )} \]

[In]

integrate(((324*exp(1/9*x)-20*x^2)*exp(((-x-81)*exp(1/9*x)+45*x)/x/exp(1/9*x))+4*x^2*exp(1/9*x))/x^2/exp(1/9*x
),x, algorithm="maxima")

[Out]

4*x + 4*e^(-81/x + 45*e^(-1/9*x) - 1)

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {e^{-x/9} \left (4 e^{x/9} x^2+e^{\frac {e^{-x/9} \left (e^{x/9} (-81-x)+45 x\right )}{x}} \left (324 e^{x/9}-20 x^2\right )\right )}{x^2} \, dx=4 \, x + 4 \, e^{\left (-\frac {81}{x} + 45 \, e^{\left (-\frac {1}{9} \, x\right )} - 1\right )} \]

[In]

integrate(((324*exp(1/9*x)-20*x^2)*exp(((-x-81)*exp(1/9*x)+45*x)/x/exp(1/9*x))+4*x^2*exp(1/9*x))/x^2/exp(1/9*x
),x, algorithm="giac")

[Out]

4*x + 4*e^(-81/x + 45*e^(-1/9*x) - 1)

Mupad [B] (verification not implemented)

Time = 13.42 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-x/9} \left (4 e^{x/9} x^2+e^{\frac {e^{-x/9} \left (e^{x/9} (-81-x)+45 x\right )}{x}} \left (324 e^{x/9}-20 x^2\right )\right )}{x^2} \, dx=4\,x+4\,{\mathrm {e}}^{\frac {45}{{\left ({\mathrm {e}}^x\right )}^{1/9}}}\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{-\frac {81}{x}} \]

[In]

int((exp(-x/9)*(exp((exp(-x/9)*(45*x - exp(x/9)*(x + 81)))/x)*(324*exp(x/9) - 20*x^2) + 4*x^2*exp(x/9)))/x^2,x
)

[Out]

4*x + 4*exp(45/exp(x)^(1/9))*exp(-1)*exp(-81/x)