3.3.31 \(\int x \log ^2(c (b x^n)^p) \, dx\) [231]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, 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 = {2342, 2341, 2495} \begin {gather*} \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 \end {gather*}

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 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]

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 \log ^2\left (c \left (b x^n\right )^p\right ) \, dx &=\text {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 )-\text {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*}

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Mathematica [A]
time = 0.01, size = 43, normalized size = 0.83 \begin {gather*} \frac {1}{4} x^2 \left (n^2 p^2-2 n p \log \left (c \left (b x^n\right )^p\right )+2 \log ^2\left (c \left (b x^n\right )^p\right )\right ) \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int x \ln \left (c \left (b \,x^{n}\right )^{p}\right )^{2}\, dx\]

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 = 0.27, size = 46, normalized size = 0.88 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (46) = 92\).
time = 0.34, size = 113, normalized size = 2.17 \begin {gather*} \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 {gather*}

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 [A]
time = 0.42, size = 46, normalized size = 0.88 \begin {gather*} \frac {n^{2} p^{2} x^{2}}{4} - \frac {n p x^{2} \log {\left (c \left (b x^{n}\right )^{p} \right )}}{2} + \frac {x^{2} \log {\left (c \left (b x^{n}\right )^{p} \right )}^{2}}{2} \end {gather*}

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**2/4 - 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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (46) = 92\).
time = 7.38, size = 112, normalized size = 2.15 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 3.85, size = 46, normalized size = 0.88 \begin {gather*} \frac {n^2\,p^2\,x^2}{4}-\frac {n\,p\,x^2\,\ln \left (c\,{\left (b\,x^n\right )}^p\right )}{2}+\frac {x^2\,{\ln \left (c\,{\left (b\,x^n\right )}^p\right )}^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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