\(\int x^2 \log ^2(c x) \, dx\) [9]

Optimal result

Integrand size = 10, antiderivative size = 32 \[ \int x^2 \log ^2(c x) \, dx=\frac {2 x^3}{27}-\frac {2}{9} x^3 \log (c x)+\frac {1}{3} x^3 \log ^2(c x) \]

[Out]

2/27*x^3-2/9*x^3*ln(c*x)+1/3*x^3*ln(c*x)^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2342, 2341} \[ \int x^2 \log ^2(c x) \, dx=\frac {1}{3} x^3 \log ^2(c x)-\frac {2}{9} x^3 \log (c x)+\frac {2 x^3}{27} \]

[In]

Int[x^2*Log[c*x]^2,x]

[Out]

(2*x^3)/27 - (2*x^3*Log[c*x])/9 + (x^3*Log[c*x]^2)/3

Rule 2341

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
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \log ^2(c x)-\frac {2}{3} \int x^2 \log (c x) \, dx \\ & = \frac {2 x^3}{27}-\frac {2}{9} x^3 \log (c x)+\frac {1}{3} x^3 \log ^2(c x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int x^2 \log ^2(c x) \, dx=\frac {2 x^3}{27}-\frac {2}{9} x^3 \log (c x)+\frac {1}{3} x^3 \log ^2(c x) \]

[In]

Integrate[x^2*Log[c*x]^2,x]

[Out]

(2*x^3)/27 - (2*x^3*Log[c*x])/9 + (x^3*Log[c*x]^2)/3

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84

method	result	size
norman	\(\frac {2 x^{3}}{27}-\frac {2 x^{3} \ln \left (x c \right )}{9}+\frac {x^{3} \ln \left (x c \right )^{2}}{3}\)	\(27\)
risch	\(\frac {2 x^{3}}{27}-\frac {2 x^{3} \ln \left (x c \right )}{9}+\frac {x^{3} \ln \left (x c \right )^{2}}{3}\)	\(27\)
parallelrisch	\(\frac {2 x^{3}}{27}-\frac {2 x^{3} \ln \left (x c \right )}{9}+\frac {x^{3} \ln \left (x c \right )^{2}}{3}\)	\(27\)
parts	\(\frac {x^{3} \ln \left (x c \right )^{2}}{3}-\frac {2 \left (\frac {x^{3} c^{3} \ln \left (x c \right )}{3}-\frac {x^{3} c^{3}}{9}\right )}{3 c^{3}}\)	\(39\)
derivativedivides	\(\frac {\frac {x^{3} c^{3} \ln \left (x c \right )^{2}}{3}-\frac {2 x^{3} c^{3} \ln \left (x c \right )}{9}+\frac {2 x^{3} c^{3}}{27}}{c^{3}}\)	\(40\)
default	\(\frac {\frac {x^{3} c^{3} \ln \left (x c \right )^{2}}{3}-\frac {2 x^{3} c^{3} \ln \left (x c \right )}{9}+\frac {2 x^{3} c^{3}}{27}}{c^{3}}\)	\(40\)

[In]

int(x^2*ln(x*c)^2,x,method=_RETURNVERBOSE)

[Out]

2/27*x^3-2/9*x^3*ln(x*c)+1/3*x^3*ln(x*c)^2

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int x^2 \log ^2(c x) \, dx=\frac {1}{3} \, x^{3} \log \left (c x\right )^{2} - \frac {2}{9} \, x^{3} \log \left (c x\right ) + \frac {2}{27} \, x^{3} \]

[In]

integrate(x^2*log(c*x)^2,x, algorithm="fricas")

[Out]

1/3*x^3*log(c*x)^2 - 2/9*x^3*log(c*x) + 2/27*x^3

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int x^2 \log ^2(c x) \, dx=\frac {x^{3} \log {\left (c x \right )}^{2}}{3} - \frac {2 x^{3} \log {\left (c x \right )}}{9} + \frac {2 x^{3}}{27} \]

[In]

integrate(x**2*ln(c*x)**2,x)

[Out]

x**3*log(c*x)**2/3 - 2*x**3*log(c*x)/9 + 2*x**3/27

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int x^2 \log ^2(c x) \, dx=\frac {1}{27} \, {\left (9 \, \log \left (c x\right )^{2} - 6 \, \log \left (c x\right ) + 2\right )} x^{3} \]

[In]

integrate(x^2*log(c*x)^2,x, algorithm="maxima")

[Out]

1/27*(9*log(c*x)^2 - 6*log(c*x) + 2)*x^3

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int x^2 \log ^2(c x) \, dx=\frac {1}{3} \, x^{3} \log \left (c x\right )^{2} - \frac {2}{9} \, x^{3} \log \left (c x\right ) + \frac {2}{27} \, x^{3} \]

[In]

integrate(x^2*log(c*x)^2,x, algorithm="giac")

[Out]

1/3*x^3*log(c*x)^2 - 2/9*x^3*log(c*x) + 2/27*x^3

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int x^2 \log ^2(c x) \, dx=\frac {x^3\,\left (9\,{\ln \left (c\,x\right )}^2-6\,\ln \left (c\,x\right )+2\right )}{27} \]

[In]

int(x^2*log(c*x)^2,x)

[Out]

(x^3*(9*log(c*x)^2 - 6*log(c*x) + 2))/27