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

Optimal. Leaf size=31 \[ \frac {3}{2} \left (x^2-\log \left (\frac {6-e^2}{x}\right )\right )+\frac {\log \left (x^3\right )}{x} \]

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Rubi [A]  time = 0.03, antiderivative size = 22, normalized size of antiderivative = 0.71, number of steps used = 6, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {12, 14, 2304} \begin {gather*} \frac {\log \left (x^3\right )}{x}+\frac {3 x^2}{2}+\frac {3 \log (x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x^2)/2 + (3*Log[x])/2 + Log[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]]

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

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Mathematica [A]  time = 0.00, size = 22, normalized size = 0.71 \begin {gather*} \frac {3 x^2}{2}+\frac {3 \log (x)}{2}+\frac {\log \left (x^3\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x^2)/2 + (3*Log[x])/2 + Log[x^3]/x

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fricas [A]  time = 0.60, size = 19, normalized size = 0.61 \begin {gather*} \frac {3 \, x^{3} + {\left (x + 2\right )} \log \left (x^{3}\right )}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*x^3 + (x + 2)*log(x^3))/x

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giac [A]  time = 0.46, size = 18, normalized size = 0.58 \begin {gather*} \frac {3}{2} \, x^{2} + \frac {\log \left (x^{3}\right )}{x} + \frac {3}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*x^2 + log(x^3)/x + 3/2*log(x)

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maple [A]  time = 0.02, size = 19, normalized size = 0.61

method	result	size
default	\(\frac {\ln \left (x^{3}\right )}{x}+\frac {3 x^{2}}{2}+\frac {3 \ln \relax (x )}{2}\)	\(19\)
risch	\(\frac {\ln \left (x^{3}\right )}{x}+\frac {3 x^{2}}{2}+\frac {3 \ln \relax (x )}{2}\)	\(19\)
norman	\(\frac {\frac {3 x^{3}}{2}+\ln \left (x^{3}\right )}{x}+\frac {3 \ln \relax (x )}{2}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 18, normalized size = 0.58 \begin {gather*} \frac {3}{2} \, x^{2} + \frac {\log \left (x^{3}\right )}{x} + \frac {3}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*x^2 + log(x^3)/x + 3/2*log(x)

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mupad [B]  time = 4.18, size = 20, normalized size = 0.65 \begin {gather*} \frac {\ln \left (x^3\right )}{2}+\frac {\ln \left (x^3\right )}{x}+\frac {3\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x)/2 - log(x^3) + 3*x^3 + 3)/x^2,x)

[Out]

log(x^3)/2 + log(x^3)/x + (3*x^2)/2

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sympy [A]  time = 0.11, size = 19, normalized size = 0.61 \begin {gather*} \frac {3 x^{2}}{2} + \frac {3 \log {\relax (x )}}{2} + \frac {\log {\left (x^{3} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x**2/2 + 3*log(x)/2 + log(x**3)/x

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