3.66.1 \(\int \frac {12-54 x^2+12 x^2 \log (3 x)}{x} \, dx\)

Optimal. Leaf size=23 \[ 3 \left (3+(1-x)^2+2 x+x^2\right ) (-5+\log (3 x)) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {14, 2304} \begin {gather*} -30 x^2+6 x^2 \log (3 x)+12 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12 - 54*x^2 + 12*x^2*Log[3*x])/x,x]

[Out]

-30*x^2 + 12*Log[x] + 6*x^2*Log[3*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 (-2+9 x^2\right )}{x}+12 x \log (3 x)\right ) \, dx\\ &=-\left (6 \int \frac {-2+9 x^2}{x} \, dx\right )+12 \int x \log (3 x) \, dx\\ &=-3 x^2+6 x^2 \log (3 x)-6 \int \left (-\frac {2}{x}+9 x\right ) \, dx\\ &=-30 x^2+12 \log (x)+6 x^2 \log (3 x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 19, normalized size = 0.83 \begin {gather*} -30 x^2+12 \log (x)+6 x^2 \log (3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12 - 54*x^2 + 12*x^2*Log[3*x])/x,x]

[Out]

-30*x^2 + 12*Log[x] + 6*x^2*Log[3*x]

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 17, normalized size = 0.74 \begin {gather*} -30 \, x^{2} + 6 \, {\left (x^{2} + 2\right )} \log \left (3 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^2*log(3*x)-54*x^2+12)/x,x, algorithm="fricas")

[Out]

-30*x^2 + 6*(x^2 + 2)*log(3*x)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 19, normalized size = 0.83 \begin {gather*} 6 \, x^{2} \log \left (3 \, x\right ) - 30 \, x^{2} + 12 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^2*log(3*x)-54*x^2+12)/x,x, algorithm="giac")

[Out]

6*x^2*log(3*x) - 30*x^2 + 12*log(x)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 20, normalized size = 0.87

method	result	size
risch	\(6 x^{2} \ln \left (3 x \right )-30 x^{2}+12 \ln \relax (x )\)	\(20\)
derivativedivides	\(6 x^{2} \ln \left (3 x \right )-30 x^{2}+12 \ln \left (3 x \right )\)	\(22\)
default	\(6 x^{2} \ln \left (3 x \right )-30 x^{2}+12 \ln \left (3 x \right )\)	\(22\)
norman	\(6 x^{2} \ln \left (3 x \right )-30 x^{2}+12 \ln \left (3 x \right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^2*ln(3*x)-54*x^2+12)/x,x,method=_RETURNVERBOSE)

[Out]

6*x^2*ln(3*x)-30*x^2+12*ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 19, normalized size = 0.83 \begin {gather*} 6 \, x^{2} \log \left (3 \, x\right ) - 30 \, x^{2} + 12 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^2*log(3*x)-54*x^2+12)/x,x, algorithm="maxima")

[Out]

6*x^2*log(3*x) - 30*x^2 + 12*log(x)

________________________________________________________________________________________

mupad [B]  time = 4.13, size = 24, normalized size = 1.04 \begin {gather*} 12\,\ln \relax (x)+6\,x^2\,\ln \relax (x)+6\,x^2\,\ln \relax (3)-30\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^2*log(3*x) - 54*x^2 + 12)/x,x)

[Out]

12*log(x) + 6*x^2*log(x) + 6*x^2*log(3) - 30*x^2

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 19, normalized size = 0.83 \begin {gather*} 6 x^{2} \log {\left (3 x \right )} - 30 x^{2} + 12 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x**2*ln(3*x)-54*x**2+12)/x,x)

[Out]

6*x**2*log(3*x) - 30*x**2 + 12*log(x)

________________________________________________________________________________________