\(\int \frac {3+3 x+2 x^2}{x} \, dx\) [9487]

Optimal result

Integrand size = 14, antiderivative size = 13 \[ \int \frac {3+3 x+2 x^2}{x} \, dx=x^2+3 (-2+x+\log (2 x)) \]

[Out]

x^2+3*x+3*ln(2*x)-6

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {14} \[ \int \frac {3+3 x+2 x^2}{x} \, dx=x^2+3 x+3 \log (x) \]

[In]

Int[(3 + 3*x + 2*x^2)/x,x]

[Out]

3*x + x^2 + 3*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {3+3 x+2 x^2}{x} \, dx=3 x+x^2+3 \log (x) \]

[In]

Integrate[(3 + 3*x + 2*x^2)/x,x]

[Out]

3*x + x^2 + 3*Log[x]

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

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

[In]

int((2*x^2+3*x+3)/x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {3+3 x+2 x^2}{x} \, dx=x^{2} + 3 \, x + 3 \, \log \left (x\right ) \]

[In]

integrate((2*x^2+3*x+3)/x,x, algorithm="fricas")

[Out]

x^2 + 3*x + 3*log(x)

Sympy [A] (verification not implemented)

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

[In]

integrate((2*x**2+3*x+3)/x,x)

[Out]

x**2 + 3*x + 3*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {3+3 x+2 x^2}{x} \, dx=x^{2} + 3 \, x + 3 \, \log \left (x\right ) \]

[In]

integrate((2*x^2+3*x+3)/x,x, algorithm="maxima")

[Out]

x^2 + 3*x + 3*log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {3+3 x+2 x^2}{x} \, dx=x^{2} + 3 \, x + 3 \, \log \left ({\left | x \right |}\right ) \]

[In]

integrate((2*x^2+3*x+3)/x,x, algorithm="giac")

[Out]

x^2 + 3*x + 3*log(abs(x))

Mupad [B] (verification not implemented)

Time = 13.96 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {3+3 x+2 x^2}{x} \, dx=3\,x+3\,\ln \left (x\right )+x^2 \]

[In]

int((3*x + 2*x^2 + 3)/x,x)

[Out]

3*x + 3*log(x) + x^2