3.3.23 \(\int x^2 \log (c (b x^n)^p) \, dx\) [223]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {2341, 2495} \begin {gather*} \frac {1}{3} x^3 \log \left (c \left (b x^n\right )^p\right )-\frac {1}{9} n p x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[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

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 2495

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 &=\text {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*}

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

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

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Maple [F]
time = 0.02, 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)

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Maxima [A]
time = 0.28, size = 23, normalized size = 0.85 \begin {gather*} -\frac {1}{9} \, n p x^{3} + \frac {1}{3} \, x^{3} \log \left (\left (b x^{n}\right )^{p} c\right ) \end {gather*}

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)

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Fricas [A]
time = 0.34, size = 32, normalized size = 1.19 \begin {gather*} \frac {1}{3} \, n p x^{3} \log \left (x\right ) - \frac {1}{9} \, n p x^{3} + \frac {1}{3} \, p x^{3} \log \left (b\right ) + \frac {1}{3} \, x^{3} \log \left (c\right ) \end {gather*}

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)

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Sympy [A]
time = 0.41, size = 22, normalized size = 0.81 \begin {gather*} - \frac {n p x^{3}}{9} + \frac {x^{3} \log {\left (c \left (b x^{n}\right )^{p} \right )}}{3} \end {gather*}

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**3/9 + x**3*log(c*(b*x**n)**p)/3

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Giac [A]
time = 3.53, size = 32, normalized size = 1.19 \begin {gather*} \frac {1}{3} \, n p x^{3} \log \left (x\right ) - \frac {1}{9} \, n p x^{3} + \frac {1}{3} \, p x^{3} \log \left (b\right ) + \frac {1}{3} \, x^{3} \log \left (c\right ) \end {gather*}

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)

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

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

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