\(\int (\frac {1}{x}+2 x+x^2) \, dx\) [1909]

Optimal result

Integrand size = 10, antiderivative size = 13 \[ \int \left (\frac {1}{x}+2 x+x^2\right ) \, dx=x^2+\frac {x^3}{3}+\log (x) \]

[Out]

x^2+1/3*x^3+ln(x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (\frac {1}{x}+2 x+x^2\right ) \, dx=\frac {x^3}{3}+x^2+\log (x) \]

[In]

Int[x^(-1) + 2*x + x^2,x]

[Out]

x^2 + x^3/3 + Log[x]

Rubi steps \begin{align*} \text {integral}& = x^2+\frac {x^3}{3}+\log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \left (\frac {1}{x}+2 x+x^2\right ) \, dx=x^2+\frac {x^3}{3}+\log (x) \]

[In]

Integrate[x^(-1) + 2*x + x^2,x]

[Out]

x^2 + x^3/3 + Log[x]

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
default	\(x^{2}+\frac {x^{3}}{3}+\ln \left (x \right )\)	\(12\)
norman	\(x^{2}+\frac {x^{3}}{3}+\ln \left (x \right )\)	\(12\)
risch	\(x^{2}+\frac {x^{3}}{3}+\ln \left (x \right )\)	\(12\)
parallelrisch	\(x^{2}+\frac {x^{3}}{3}+\ln \left (x \right )\)	\(12\)

[In]

int(1/x+2*x+x^2,x,method=_RETURNVERBOSE)

[Out]

x^2+1/3*x^3+ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \left (\frac {1}{x}+2 x+x^2\right ) \, dx=\frac {1}{3} \, x^{3} + x^{2} + \log \left (x\right ) \]

[In]

integrate(1/x+2*x+x^2,x, algorithm="fricas")

[Out]

1/3*x^3 + x^2 + log(x)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \left (\frac {1}{x}+2 x+x^2\right ) \, dx=\frac {x^{3}}{3} + x^{2} + \log {\left (x \right )} \]

[In]

integrate(1/x+2*x+x**2,x)

[Out]

x**3/3 + x**2 + log(x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \left (\frac {1}{x}+2 x+x^2\right ) \, dx=\frac {1}{3} \, x^{3} + x^{2} + \log \left (x\right ) \]

[In]

integrate(1/x+2*x+x^2,x, algorithm="maxima")

[Out]

1/3*x^3 + x^2 + log(x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \left (\frac {1}{x}+2 x+x^2\right ) \, dx=\frac {1}{3} \, x^{3} + x^{2} + \log \left ({\left | x \right |}\right ) \]

[In]

integrate(1/x+2*x+x^2,x, algorithm="giac")

[Out]

1/3*x^3 + x^2 + log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \left (\frac {1}{x}+2 x+x^2\right ) \, dx=\ln \left (x\right )+x^2+\frac {x^3}{3} \]

[In]

int(2*x + 1/x + x^2,x)

[Out]

log(x) + x^2 + x^3/3