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

Optimal. Leaf size=17 \[ 3 \left (4-e+x^2+\frac {2 \log (x)}{x}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 13, normalized size of antiderivative = 0.76, number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {14, 2304} \begin {gather*} 3 x^2+\frac {6 \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 + 6*x^3 - 6*Log[x])/x^2,x]

[Out]

3*x^2 + (6*Log[x])/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]]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {6 \left (1+x^3\right )}{x^2}-\frac {6 \log (x)}{x^2}\right ) \, dx\\ &=6 \int \frac {1+x^3}{x^2} \, dx-6 \int \frac {\log (x)}{x^2} \, dx\\ &=\frac {6}{x}+\frac {6 \log (x)}{x}+6 \int \left (\frac {1}{x^2}+x\right ) \, dx\\ &=3 x^2+\frac {6 \log (x)}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 0.76 \begin {gather*} 3 x^2+\frac {6 \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 + 6*x^3 - 6*Log[x])/x^2,x]

[Out]

3*x^2 + (6*Log[x])/x

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fricas [A]  time = 0.71, size = 13, normalized size = 0.76 \begin {gather*} \frac {3 \, {\left (x^{3} + 2 \, \log \relax (x)\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(x^3 + 2*log(x))/x

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giac [A]  time = 0.14, size = 13, normalized size = 0.76 \begin {gather*} 3 \, x^{2} + \frac {6 \, \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^2 + 6*log(x)/x

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maple [A]  time = 0.02, size = 14, normalized size = 0.82

method	result	size
default	\(3 x^{2}+\frac {6 \ln \relax (x )}{x}\)	\(14\)
risch	\(3 x^{2}+\frac {6 \ln \relax (x )}{x}\)	\(14\)
norman	\(\frac {3 x^{3}+6 \ln \relax (x )}{x}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 13, normalized size = 0.76 \begin {gather*} 3 \, x^{2} + \frac {6 \, \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^2 + 6*log(x)/x

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mupad [B]  time = 7.20, size = 13, normalized size = 0.76 \begin {gather*} \frac {6\,\ln \relax (x)}{x}+3\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x^3 - 6*log(x) + 6)/x^2,x)

[Out]

(6*log(x))/x + 3*x^2

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sympy [A]  time = 0.09, size = 10, normalized size = 0.59 \begin {gather*} 3 x^{2} + \frac {6 \log {\relax (x )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*ln(x)+6*x**3+6)/x**2,x)

[Out]

3*x**2 + 6*log(x)/x

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