3.53.96 \(\int \frac {11+12 x^2}{4 x^2} \, dx\)

Optimal. Leaf size=16 \[ -\frac {11}{4 x}+3 x+\log ^2(\log (5)) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 0.69, 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 {11}{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(11 + 12*x^2)/(4*x^2),x]

[Out]

-11/(4*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}{4} \int \frac {11+12 x^2}{x^2} \, dx\\ &=\frac {1}{4} \int \left (12+\frac {11}{x^2}\right ) \, dx\\ &=-\frac {11}{4 x}+3 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.69 \begin {gather*} -\frac {11}{4 x}+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(11 + 12*x^2)/(4*x^2),x]

[Out]

-11/(4*x) + 3*x

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fricas [A]  time = 0.52, size = 12, normalized size = 0.75 \begin {gather*} \frac {12 \, x^{2} - 11}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(12*x^2+11)/x^2,x, algorithm="fricas")

[Out]

1/4*(12*x^2 - 11)/x

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giac [A]  time = 0.13, size = 9, normalized size = 0.56 \begin {gather*} 3 \, x - \frac {11}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(12*x^2+11)/x^2,x, algorithm="giac")

[Out]

3*x - 11/4/x

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maple [A]  time = 0.03, size = 10, normalized size = 0.62

method	result	size
default	\(3 x -\frac {11}{4 x}\)	\(10\)
risch	\(3 x -\frac {11}{4 x}\)	\(10\)
norman	\(\frac {-\frac {11}{4}+3 x^{2}}{x}\)	\(12\)
gosper	\(\frac {12 x^{2}-11}{4 x}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(12*x^2+11)/x^2,x,method=_RETURNVERBOSE)

[Out]

3*x-11/4/x

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maxima [A]  time = 0.38, size = 9, normalized size = 0.56 \begin {gather*} 3 \, x - \frac {11}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(12*x^2+11)/x^2,x, algorithm="maxima")

[Out]

3*x - 11/4/x

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mupad [B]  time = 0.02, size = 9, normalized size = 0.56 \begin {gather*} 3\,x-\frac {11}{4\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2 + 11/4)/x^2,x)

[Out]

3*x - 11/(4*x)

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sympy [A]  time = 0.06, size = 7, normalized size = 0.44 \begin {gather*} 3 x - \frac {11}{4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(12*x**2+11)/x**2,x)

[Out]

3*x - 11/(4*x)

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