\(\int x (a+b \log (c x^n))^p \, dx\) [169]

Optimal result

Integrand size = 14, antiderivative size = 89 \[ \int x \left (a+b \log \left (c x^n\right )\right )^p \, dx=2^{-1-p} e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \]

[Out]

2^(-1-p)*x^2*GAMMA(p+1,-2*(a+b*ln(c*x^n))/b/n)*(a+b*ln(c*x^n))^p/exp(2*a/b/n)/((c*x^n)^(2/n))/(((-a-b*ln(c*x^n
))/b/n)^p)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2347, 2212} \[ \int x \left (a+b \log \left (c x^n\right )\right )^p \, dx=2^{-p-1} x^2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]

[In]

Int[x*(a + b*Log[c*x^n])^p,x]

[Out]

(2^(-1 - p)*x^2*Gamma[1 + p, (-2*(a + b*Log[c*x^n]))/(b*n)]*(a + b*Log[c*x^n])^p)/(E^((2*a)/(b*n))*(c*x^n)^(2/
n)*(-((a + b*Log[c*x^n])/(b*n)))^p)

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int e^{\frac {2 x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = 2^{-1-p} e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int x \left (a+b \log \left (c x^n\right )\right )^p \, dx=2^{-1-p} e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \]

[In]

Integrate[x*(a + b*Log[c*x^n])^p,x]

[Out]

(2^(-1 - p)*x^2*Gamma[1 + p, (-2*(a + b*Log[c*x^n]))/(b*n)]*(a + b*Log[c*x^n])^p)/(E^((2*a)/(b*n))*(c*x^n)^(2/
n)*(-((a + b*Log[c*x^n])/(b*n)))^p)

Maple [F]

\[\int x {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int x \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{p} x \,d x } \]

[In]

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

[Out]

integral((b*log(c*x^n) + a)^p*x, x)

Sympy [F]

\[ \int x \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int x \left (a + b \log {\left (c x^{n} \right )}\right )^{p}\, dx \]

[In]

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

[Out]

Integral(x*(a + b*log(c*x**n))**p, x)

Maxima [F(-2)]

Exception generated. \[ \int x \left (a+b \log \left (c x^n\right )\right )^p \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

Giac [F]

\[ \int x \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{p} x \,d x } \]

[In]

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

[Out]

integrate((b*log(c*x^n) + a)^p*x, x)

Mupad [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int x\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \]

[In]

int(x*(a + b*log(c*x^n))^p,x)

[Out]

int(x*(a + b*log(c*x^n))^p, x)