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

Optimal. Leaf size=52 \[ \frac{1}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )-\frac{2}{9} n p x^3 \log \left (c \left (b x^n\right )^p\right )+\frac{2}{27} n^2 p^2 x^3 \]

[Out]

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

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Rubi [A]  time = 0.0708018, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2305, 2304, 2445} \[ \frac{1}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )-\frac{2}{9} n p x^3 \log \left (c \left (b x^n\right )^p\right )+\frac{2}{27} n^2 p^2 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2305

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]

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 ^2\left (c \left (b x^n\right )^p\right ) \, dx &=\operatorname{Subst}\left (\int x^2 \log ^2\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}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )-\operatorname{Subst}\left (\frac{1}{3} (2 n p) \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{2}{27} n^2 p^2 x^3-\frac{2}{9} n p x^3 \log \left (c \left (b x^n\right )^p\right )+\frac{1}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )\\ \end{align*}

Mathematica [A]  time = 0.0027718, size = 52, normalized size = 1. \[ \frac{1}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )-\frac{2}{9} n p x^3 \log \left (c \left (b x^n\right )^p\right )+\frac{2}{27} n^2 p^2 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( \ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.16307, size = 62, normalized size = 1.19 \begin{align*} \frac{2}{27} \, n^{2} p^{2} x^{3} - \frac{2}{9} \, n p x^{3} \log \left (\left (b x^{n}\right )^{p} c\right ) + \frac{1}{3} \, x^{3} \log \left (\left (b x^{n}\right )^{p} c\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 0.734404, size = 293, normalized size = 5.63 \begin{align*} \frac{1}{3} \, n^{2} p^{2} x^{3} \log \left (x\right )^{2} + \frac{2}{27} \, n^{2} p^{2} x^{3} - \frac{2}{9} \, n p^{2} x^{3} \log \left (b\right ) + \frac{1}{3} \, p^{2} x^{3} \log \left (b\right )^{2} + \frac{1}{3} \, x^{3} \log \left (c\right )^{2} - \frac{2}{9} \,{\left (n p x^{3} - 3 \, p x^{3} \log \left (b\right )\right )} \log \left (c\right ) - \frac{2}{9} \,{\left (n^{2} p^{2} x^{3} - 3 \, n p^{2} x^{3} \log \left (b\right ) - 3 \, n p x^{3} \log \left (c\right )\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 8.06876, size = 150, normalized size = 2.88 \begin{align*} \frac{n^{2} p^{2} x^{3} \log{\left (x \right )}^{2}}{3} - \frac{2 n^{2} p^{2} x^{3} \log{\left (x \right )}}{9} + \frac{2 n^{2} p^{2} x^{3}}{27} + \frac{2 n p^{2} x^{3} \log{\left (b \right )} \log{\left (x \right )}}{3} - \frac{2 n p^{2} x^{3} \log{\left (b \right )}}{9} + \frac{2 n p x^{3} \log{\left (c \right )} \log{\left (x \right )}}{3} - \frac{2 n p x^{3} \log{\left (c \right )}}{9} + \frac{p^{2} x^{3} \log{\left (b \right )}^{2}}{3} + \frac{2 p x^{3} \log{\left (b \right )} \log{\left (c \right )}}{3} + \frac{x^{3} \log{\left (c \right )}^{2}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.30326, size = 155, normalized size = 2.98 \begin{align*} \frac{1}{3} \, n^{2} p^{2} x^{3} \log \left (x\right )^{2} - \frac{2}{9} \, n^{2} p^{2} x^{3} \log \left (x\right ) + \frac{2}{3} \, n p^{2} x^{3} \log \left (b\right ) \log \left (x\right ) + \frac{2}{27} \, n^{2} p^{2} x^{3} - \frac{2}{9} \, n p^{2} x^{3} \log \left (b\right ) + \frac{1}{3} \, p^{2} x^{3} \log \left (b\right )^{2} + \frac{2}{3} \, n p x^{3} \log \left (c\right ) \log \left (x\right ) - \frac{2}{9} \, n p x^{3} \log \left (c\right ) + \frac{2}{3} \, p x^{3} \log \left (b\right ) \log \left (c\right ) + \frac{1}{3} \, x^{3} \log \left (c\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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