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

Optimal. Leaf size=27 \[ \frac{1}{2} x^2 \log \left (c \left (b x^n\right )^p\right )-\frac{1}{4} n p x^2 \]

[Out]

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

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Rubi [A]  time = 0.016509, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2304, 2445} \[ \frac{1}{2} x^2 \log \left (c \left (b x^n\right )^p\right )-\frac{1}{4} n p x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0009574, size = 27, normalized size = 1. \[ \frac{1}{2} x^2 \log \left (c \left (b x^n\right )^p\right )-\frac{1}{4} n p x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int x\ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.16532, size = 31, normalized size = 1.15 \begin{align*} -\frac{1}{4} \, n p x^{2} + \frac{1}{2} \, x^{2} \log \left (\left (b x^{n}\right )^{p} c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.824537, size = 95, normalized size = 3.52 \begin{align*} \frac{1}{2} \, n p x^{2} \log \left (x\right ) - \frac{1}{4} \, n p x^{2} + \frac{1}{2} \, p x^{2} \log \left (b\right ) + \frac{1}{2} \, x^{2} \log \left (c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.05761, size = 37, normalized size = 1.37 \begin{align*} \frac{n p x^{2} \log{\left (x \right )}}{2} - \frac{n p x^{2}}{4} + \frac{p x^{2} \log{\left (b \right )}}{2} + \frac{x^{2} \log{\left (c \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.30976, size = 43, normalized size = 1.59 \begin{align*} \frac{1}{2} \, n p x^{2} \log \left (x\right ) - \frac{1}{4} \, n p x^{2} + \frac{1}{2} \, p x^{2} \log \left (b\right ) + \frac{1}{2} \, x^{2} \log \left (c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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