\(\int \frac {-3+6 x^2}{2 x^2} \, dx\) [3110]

Optimal result

Integrand size = 14, antiderivative size = 20 \[ \int \frac {-3+6 x^2}{2 x^2} \, dx=2-3 \left (3-\frac {1}{2 x}-x+\log ^2(4)\right ) \]

[Out]

-7-12*ln(2)^2+3*x+3/2/x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.55, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 14} \[ \int \frac {-3+6 x^2}{2 x^2} \, dx=3 x+\frac {3}{2 x} \]

[In]

Int[(-3 + 6*x^2)/(2*x^2),x]

[Out]

3/(2*x) + 3*x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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}& = \frac {1}{2} \int \frac {-3+6 x^2}{x^2} \, dx \\ & = \frac {1}{2} \int \left (6-\frac {3}{x^2}\right ) \, dx \\ & = \frac {3}{2 x}+3 x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.55 \[ \int \frac {-3+6 x^2}{2 x^2} \, dx=\frac {3}{2 x}+3 x \]

[In]

Integrate[(-3 + 6*x^2)/(2*x^2),x]

[Out]

3/(2*x) + 3*x

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.50

method	result	size
default	\(\frac {3}{2 x}+3 x\)	\(10\)
risch	\(\frac {3}{2 x}+3 x\)	\(10\)
norman	\(\frac {\frac {3}{2}+3 x^{2}}{x}\)	\(12\)
gosper	\(\frac {\frac {3}{2}+3 x^{2}}{x}\)	\(13\)
parallelrisch	\(\frac {6 x^{2}+3}{2 x}\)	\(13\)

[In]

int(1/2*(6*x^2-3)/x^2,x,method=_RETURNVERBOSE)

[Out]

3/2/x+3*x

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.60 \[ \int \frac {-3+6 x^2}{2 x^2} \, dx=\frac {3 \, {\left (2 \, x^{2} + 1\right )}}{2 \, x} \]

[In]

integrate(1/2*(6*x^2-3)/x^2,x, algorithm="fricas")

[Out]

3/2*(2*x^2 + 1)/x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.35 \[ \int \frac {-3+6 x^2}{2 x^2} \, dx=3 x + \frac {3}{2 x} \]

[In]

integrate(1/2*(6*x**2-3)/x**2,x)

[Out]

3*x + 3/(2*x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.45 \[ \int \frac {-3+6 x^2}{2 x^2} \, dx=3 \, x + \frac {3}{2 \, x} \]

[In]

integrate(1/2*(6*x^2-3)/x^2,x, algorithm="maxima")

[Out]

3*x + 3/2/x

Giac [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.45 \[ \int \frac {-3+6 x^2}{2 x^2} \, dx=3 \, x + \frac {3}{2 \, x} \]

[In]

integrate(1/2*(6*x^2-3)/x^2,x, algorithm="giac")

[Out]

3*x + 3/2/x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.45 \[ \int \frac {-3+6 x^2}{2 x^2} \, dx=3\,x+\frac {3}{2\,x} \]

[In]

int((3*x^2 - 3/2)/x^2,x)

[Out]

3*x + 3/(2*x)