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

Optimal. Leaf size=20 \[ 2-3 \left (3-\frac {1}{2 x}-x+\log ^2(4)\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 0.55, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 14} \begin {gather*} 3 x+\frac {3}{2 x} \end {gather*}

Antiderivative was successfully verified.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.55 \begin {gather*} \frac {3}{2 x}+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 6*x^2)/(2*x^2),x]

[Out]

3/(2*x) + 3*x

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fricas [A]  time = 0.53, size = 12, normalized size = 0.60 \begin {gather*} \frac {3 \, {\left (2 \, x^{2} + 1\right )}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(2*x^2 + 1)/x

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giac [A]  time = 0.18, size = 9, normalized size = 0.45 \begin {gather*} 3 \, x + \frac {3}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x + 3/2/x

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maple [A]  time = 0.03, size = 10, normalized size = 0.50

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x+3/2/x

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maxima [A]  time = 0.40, size = 9, normalized size = 0.45 \begin {gather*} 3 \, x + \frac {3}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x + 3/2/x

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mupad [B]  time = 0.02, size = 9, normalized size = 0.45 \begin {gather*} 3\,x+\frac {3}{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2 - 3/2)/x^2,x)

[Out]

3*x + 3/(2*x)

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sympy [A]  time = 0.06, size = 7, normalized size = 0.35 \begin {gather*} 3 x + \frac {3}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x + 3/(2*x)

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