\(\int \frac {150 x^2+e^{\frac {e^x}{x^2}} (50 x^2+e^x (-50+25 x))}{2 x} \, dx\) [4305]

Optimal result

Integrand size = 38, antiderivative size = 20 \[ \int \frac {150 x^2+e^{\frac {e^x}{x^2}} \left (50 x^2+e^x (-50+25 x)\right )}{2 x} \, dx=1+\frac {25}{2} \left (3+e^{\frac {e^x}{x^2}}\right ) x^2 \]

[Out]

5/2*x^2*(15+5*exp(exp(x)/x^2))+1

Rubi [F]

\[ \int \frac {150 x^2+e^{\frac {e^x}{x^2}} \left (50 x^2+e^x (-50+25 x)\right )}{2 x} \, dx=\int \frac {150 x^2+e^{\frac {e^x}{x^2}} \left (50 x^2+e^x (-50+25 x)\right )}{2 x} \, dx \]

[In]

Int[(150*x^2 + E^(E^x/x^2)*(50*x^2 + E^x*(-50 + 25*x)))/(2*x),x]

[Out]

(75*x^2)/2 + (25*Defer[Int][E^(E^x/x^2 + x), x])/2 - 25*Defer[Int][E^(E^x/x^2 + x)/x, x] + 25*Defer[Int][E^(E^
x/x^2)*x, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {150 x^2+e^{\frac {e^x}{x^2}} \left (50 x^2+e^x (-50+25 x)\right )}{x} \, dx \\ & = \frac {1}{2} \int \left (\frac {25 e^{\frac {e^x}{x^2}+x} (-2+x)}{x}+50 \left (3+e^{\frac {e^x}{x^2}}\right ) x\right ) \, dx \\ & = \frac {25}{2} \int \frac {e^{\frac {e^x}{x^2}+x} (-2+x)}{x} \, dx+25 \int \left (3+e^{\frac {e^x}{x^2}}\right ) x \, dx \\ & = \frac {25}{2} \int \left (e^{\frac {e^x}{x^2}+x}-\frac {2 e^{\frac {e^x}{x^2}+x}}{x}\right ) \, dx+25 \int \left (3 x+e^{\frac {e^x}{x^2}} x\right ) \, dx \\ & = \frac {75 x^2}{2}+\frac {25}{2} \int e^{\frac {e^x}{x^2}+x} \, dx-25 \int \frac {e^{\frac {e^x}{x^2}+x}}{x} \, dx+25 \int e^{\frac {e^x}{x^2}} x \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {150 x^2+e^{\frac {e^x}{x^2}} \left (50 x^2+e^x (-50+25 x)\right )}{2 x} \, dx=\frac {25}{2} \left (3+e^{\frac {e^x}{x^2}}\right ) x^2 \]

[In]

Integrate[(150*x^2 + E^(E^x/x^2)*(50*x^2 + E^x*(-50 + 25*x)))/(2*x),x]

[Out]

(25*(3 + E^(E^x/x^2))*x^2)/2

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95

method	result	size
norman	\(\frac {75 x^{2}}{2}+\frac {25 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{x^{2}}} x^{2}}{2}\)	\(19\)
risch	\(\frac {75 x^{2}}{2}+\frac {25 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{x^{2}}} x^{2}}{2}\)	\(19\)
parallelrisch	\(\frac {75 x^{2}}{2}+\frac {25 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{x^{2}}} x^{2}}{2}\)	\(19\)

[In]

int(1/2*(((25*x-50)*exp(x)+50*x^2)*exp(exp(x)/x^2)+150*x^2)/x,x,method=_RETURNVERBOSE)

[Out]

75/2*x^2+25/2*exp(exp(x)/x^2)*x^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {150 x^2+e^{\frac {e^x}{x^2}} \left (50 x^2+e^x (-50+25 x)\right )}{2 x} \, dx=\frac {25}{2} \, x^{2} e^{\left (\frac {e^{x}}{x^{2}}\right )} + \frac {75}{2} \, x^{2} \]

[In]

integrate(1/2*(((25*x-50)*exp(x)+50*x^2)*exp(exp(x)/x^2)+150*x^2)/x,x, algorithm="fricas")

[Out]

25/2*x^2*e^(e^x/x^2) + 75/2*x^2

Sympy [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {150 x^2+e^{\frac {e^x}{x^2}} \left (50 x^2+e^x (-50+25 x)\right )}{2 x} \, dx=\frac {25 x^{2} e^{\frac {e^{x}}{x^{2}}}}{2} + \frac {75 x^{2}}{2} \]

[In]

integrate(1/2*(((25*x-50)*exp(x)+50*x**2)*exp(exp(x)/x**2)+150*x**2)/x,x)

[Out]

25*x**2*exp(exp(x)/x**2)/2 + 75*x**2/2

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {150 x^2+e^{\frac {e^x}{x^2}} \left (50 x^2+e^x (-50+25 x)\right )}{2 x} \, dx=\frac {25}{2} \, x^{2} e^{\left (\frac {e^{x}}{x^{2}}\right )} + \frac {75}{2} \, x^{2} \]

[In]

integrate(1/2*(((25*x-50)*exp(x)+50*x^2)*exp(exp(x)/x^2)+150*x^2)/x,x, algorithm="maxima")

[Out]

25/2*x^2*e^(e^x/x^2) + 75/2*x^2

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {150 x^2+e^{\frac {e^x}{x^2}} \left (50 x^2+e^x (-50+25 x)\right )}{2 x} \, dx=\frac {25}{2} \, {\left (3 \, x^{2} e^{x} + x^{2} e^{\left (\frac {x^{3} + e^{x}}{x^{2}}\right )}\right )} e^{\left (-x\right )} \]

[In]

integrate(1/2*(((25*x-50)*exp(x)+50*x^2)*exp(exp(x)/x^2)+150*x^2)/x,x, algorithm="giac")

[Out]

25/2*(3*x^2*e^x + x^2*e^((x^3 + e^x)/x^2))*e^(-x)

Mupad [B] (verification not implemented)

Time = 11.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {150 x^2+e^{\frac {e^x}{x^2}} \left (50 x^2+e^x (-50+25 x)\right )}{2 x} \, dx=\frac {25\,x^2\,\left ({\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x^2}}+3\right )}{2} \]

[In]

int((75*x^2 + (exp(exp(x)/x^2)*(exp(x)*(25*x - 50) + 50*x^2))/2)/x,x)

[Out]

(25*x^2*(exp(exp(x)/x^2) + 3))/2