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

Optimal. Leaf size=52 \[ \frac{1}{2} x^2 \log ^2\left (c \left (b x^n\right )^p\right )-\frac{1}{2} n p x^2 \log \left (c \left (b x^n\right )^p\right )+\frac{1}{4} n^2 p^2 x^2 \]

[Out]

(n^2*p^2*x^2)/4 - (n*p*x^2*Log[c*(b*x^n)^p])/2 + (x^2*Log[c*(b*x^n)^p]^2)/2

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Rubi [A]  time = 0.0446348, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2305, 2304, 2445} \[ \frac{1}{2} x^2 \log ^2\left (c \left (b x^n\right )^p\right )-\frac{1}{2} n p x^2 \log \left (c \left (b x^n\right )^p\right )+\frac{1}{4} n^2 p^2 x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(n^2*p^2*x^2)/4 - (n*p*x^2*Log[c*(b*x^n)^p])/2 + (x^2*Log[c*(b*x^n)^p]^2)/2

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 \log ^2\left (c \left (b x^n\right )^p\right ) \, dx &=\operatorname{Subst}\left (\int x \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}{2} x^2 \log ^2\left (c \left (b x^n\right )^p\right )-\operatorname{Subst}\left ((n p) \int x \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}{4} n^2 p^2 x^2-\frac{1}{2} n p x^2 \log \left (c \left (b x^n\right )^p\right )+\frac{1}{2} x^2 \log ^2\left (c \left (b x^n\right )^p\right )\\ \end{align*}

Mathematica [A]  time = 0.0058748, size = 43, normalized size = 0.83 \[ \frac{1}{4} x^2 \left (2 \log ^2\left (c \left (b x^n\right )^p\right )-2 n p \log \left (c \left (b x^n\right )^p\right )+n^2 p^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(n^2*p^2 - 2*n*p*Log[c*(b*x^n)^p] + 2*Log[c*(b*x^n)^p]^2))/4

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \begin{align*} \int x \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*ln(c*(b*x^n)^p)^2,x)

[Out]

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

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Maxima [A]  time = 1.18958, size = 62, normalized size = 1.19 \begin{align*} \frac{1}{4} \, n^{2} p^{2} x^{2} - \frac{1}{2} \, n p x^{2} \log \left (\left (b x^{n}\right )^{p} c\right ) + \frac{1}{2} \, x^{2} \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*log(c*(b*x^n)^p)^2,x, algorithm="maxima")

[Out]

1/4*n^2*p^2*x^2 - 1/2*n*p*x^2*log((b*x^n)^p*c) + 1/2*x^2*log((b*x^n)^p*c)^2

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Fricas [B]  time = 0.94347, size = 292, normalized size = 5.62 \begin{align*} \frac{1}{2} \, n^{2} p^{2} x^{2} \log \left (x\right )^{2} + \frac{1}{4} \, n^{2} p^{2} x^{2} - \frac{1}{2} \, n p^{2} x^{2} \log \left (b\right ) + \frac{1}{2} \, p^{2} x^{2} \log \left (b\right )^{2} + \frac{1}{2} \, x^{2} \log \left (c\right )^{2} - \frac{1}{2} \,{\left (n p x^{2} - 2 \, p x^{2} \log \left (b\right )\right )} \log \left (c\right ) - \frac{1}{2} \,{\left (n^{2} p^{2} x^{2} - 2 \, n p^{2} x^{2} \log \left (b\right ) - 2 \, n p x^{2} \log \left (c\right )\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*n^2*p^2*x^2*log(x)^2 + 1/4*n^2*p^2*x^2 - 1/2*n*p^2*x^2*log(b) + 1/2*p^2*x^2*log(b)^2 + 1/2*x^2*log(c)^2 -
1/2*(n*p*x^2 - 2*p*x^2*log(b))*log(c) - 1/2*(n^2*p^2*x^2 - 2*n*p^2*x^2*log(b) - 2*n*p*x^2*log(c))*log(x)

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Sympy [B]  time = 2.81901, size = 133, normalized size = 2.56 \begin{align*} \frac{n^{2} p^{2} x^{2} \log{\left (x \right )}^{2}}{2} - \frac{n^{2} p^{2} x^{2} \log{\left (x \right )}}{2} + \frac{n^{2} p^{2} x^{2}}{4} + n p^{2} x^{2} \log{\left (b \right )} \log{\left (x \right )} - \frac{n p^{2} x^{2} \log{\left (b \right )}}{2} + n p x^{2} \log{\left (c \right )} \log{\left (x \right )} - \frac{n p x^{2} \log{\left (c \right )}}{2} + \frac{p^{2} x^{2} \log{\left (b \right )}^{2}}{2} + p x^{2} \log{\left (b \right )} \log{\left (c \right )} + \frac{x^{2} \log{\left (c \right )}^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

n**2*p**2*x**2*log(x)**2/2 - n**2*p**2*x**2*log(x)/2 + n**2*p**2*x**2/4 + n*p**2*x**2*log(b)*log(x) - n*p**2*x
**2*log(b)/2 + n*p*x**2*log(c)*log(x) - n*p*x**2*log(c)/2 + p**2*x**2*log(b)**2/2 + p*x**2*log(b)*log(c) + x**
2*log(c)**2/2

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Giac [B]  time = 1.26043, size = 151, normalized size = 2.9 \begin{align*} \frac{1}{2} \, n^{2} p^{2} x^{2} \log \left (x\right )^{2} - \frac{1}{2} \, n^{2} p^{2} x^{2} \log \left (x\right ) + n p^{2} x^{2} \log \left (b\right ) \log \left (x\right ) + \frac{1}{4} \, n^{2} p^{2} x^{2} - \frac{1}{2} \, n p^{2} x^{2} \log \left (b\right ) + \frac{1}{2} \, p^{2} x^{2} \log \left (b\right )^{2} + n p x^{2} \log \left (c\right ) \log \left (x\right ) - \frac{1}{2} \, n p x^{2} \log \left (c\right ) + p x^{2} \log \left (b\right ) \log \left (c\right ) + \frac{1}{2} \, x^{2} \log \left (c\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*n^2*p^2*x^2*log(x)^2 - 1/2*n^2*p^2*x^2*log(x) + n*p^2*x^2*log(b)*log(x) + 1/4*n^2*p^2*x^2 - 1/2*n*p^2*x^2*
log(b) + 1/2*p^2*x^2*log(b)^2 + n*p*x^2*log(c)*log(x) - 1/2*n*p*x^2*log(c) + p*x^2*log(b)*log(c) + 1/2*x^2*log
(c)^2