3.223 \(\int x^2 \log (c (b x^n)^p) \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{3} x^3 \log \left (c \left (b x^n\right )^p\right )-\frac {1}{9} n p x^3 \]

[Out]

-1/9*n*p*x^3+1/3*x^3*ln(c*(b*x^n)^p)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2304, 2445} \[ \frac {1}{3} x^3 \log \left (c \left (b x^n\right )^p\right )-\frac {1}{9} n p x^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*Log[c*(b*x^n)^p],x]

[Out]

-(n*p*x^3)/9 + (x^3*Log[c*(b*x^n)^p])/3

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
]

Rule 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin {align*} \int x^2 \log \left (c \left (b x^n\right )^p\right ) \, dx &=\operatorname {Subst}\left (\int x^2 \log \left (b^p c x^{n p}\right ) \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=-\frac {1}{9} n p x^3+\frac {1}{3} x^3 \log \left (c \left (b x^n\right )^p\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 27, normalized size = 1.00 \[ \frac {1}{3} x^3 \log \left (c \left (b x^n\right )^p\right )-\frac {1}{9} n p x^3 \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Log[c*(b*x^n)^p],x]

[Out]

-1/9*(n*p*x^3) + (x^3*Log[c*(b*x^n)^p])/3

________________________________________________________________________________________

fricas [A]  time = 0.73, size = 32, normalized size = 1.19 \[ \frac {1}{3} \, n p x^{3} \log \relax (x) - \frac {1}{9} \, n p x^{3} + \frac {1}{3} \, p x^{3} \log \relax (b) + \frac {1}{3} \, x^{3} \log \relax (c) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*n*p*x^3*log(x) - 1/9*n*p*x^3 + 1/3*p*x^3*log(b) + 1/3*x^3*log(c)

________________________________________________________________________________________

giac [A]  time = 0.35, size = 32, normalized size = 1.19 \[ \frac {1}{3} \, n p x^{3} \log \relax (x) - \frac {1}{9} \, n p x^{3} + \frac {1}{3} \, p x^{3} \log \relax (b) + \frac {1}{3} \, x^{3} \log \relax (c) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*n*p*x^3*log(x) - 1/9*n*p*x^3 + 1/3*p*x^3*log(b) + 1/3*x^3*log(c)

________________________________________________________________________________________

maple [F]  time = 0.56, size = 0, normalized size = 0.00 \[ \int x^{2} \ln \left (c \left (b \,x^{n}\right )^{p}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*ln(c*(b*x^n)^p),x)

[Out]

int(x^2*ln(c*(b*x^n)^p),x)

________________________________________________________________________________________

maxima [A]  time = 1.10, size = 23, normalized size = 0.85 \[ -\frac {1}{9} \, n p x^{3} + \frac {1}{3} \, x^{3} \log \left (\left (b x^{n}\right )^{p} c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*n*p*x^3 + 1/3*x^3*log((b*x^n)^p*c)

________________________________________________________________________________________

mupad [B]  time = 3.84, size = 23, normalized size = 0.85 \[ \frac {x^3\,\ln \left (c\,{\left (b\,x^n\right )}^p\right )}{3}-\frac {n\,p\,x^3}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*log(c*(b*x^n)^p),x)

[Out]

(x^3*log(c*(b*x^n)^p))/3 - (n*p*x^3)/9

________________________________________________________________________________________

sympy [A]  time = 2.21, size = 37, normalized size = 1.37 \[ \frac {n p x^{3} \log {\relax (x )}}{3} - \frac {n p x^{3}}{9} + \frac {p x^{3} \log {\relax (b )}}{3} + \frac {x^{3} \log {\relax (c )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*ln(c*(b*x**n)**p),x)

[Out]

n*p*x**3*log(x)/3 - n*p*x**3/9 + p*x**3*log(b)/3 + x**3*log(c)/3

________________________________________________________________________________________