3.10.73 \(\int \frac {2+11 x}{2 x} \, dx\)

Optimal. Leaf size=13 \[ -\frac {x}{2}+6 (3+x)+\log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 0.62, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 43} \begin {gather*} \frac {11 x}{2}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 11*x)/(2*x),x]

[Out]

(11*x)/2 + Log[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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {2+11 x}{x} \, dx\\ &=\frac {1}{2} \int \left (11+\frac {2}{x}\right ) \, dx\\ &=\frac {11 x}{2}+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 0.62 \begin {gather*} \frac {11 x}{2}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 11*x)/(2*x),x]

[Out]

(11*x)/2 + Log[x]

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fricas [A]  time = 0.63, size = 6, normalized size = 0.46 \begin {gather*} \frac {11}{2} \, x + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/2*x + log(x)

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giac [A]  time = 0.33, size = 7, normalized size = 0.54 \begin {gather*} \frac {11}{2} \, x + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/2*x + log(abs(x))

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maple [A]  time = 0.01, size = 7, normalized size = 0.54

method	result	size
default	\(\frac {11 x}{2}+\ln \relax (x )\)	\(7\)
norman	\(\frac {11 x}{2}+\ln \relax (x )\)	\(7\)
risch	\(\frac {11 x}{2}+\ln \relax (x )\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/2*x+ln(x)

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maxima [A]  time = 0.54, size = 6, normalized size = 0.46 \begin {gather*} \frac {11}{2} \, x + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/2*x + log(x)

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mupad [B]  time = 0.02, size = 6, normalized size = 0.46 \begin {gather*} \frac {11\,x}{2}+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((11*x)/2 + 1)/x,x)

[Out]

(11*x)/2 + log(x)

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sympy [A]  time = 0.07, size = 7, normalized size = 0.54 \begin {gather*} \frac {11 x}{2} + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11*x/2 + log(x)

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