\(\int (d x)^m (a+b \log (c x^n))^p \, dx\) [167]

Optimal result

Integrand size = 18, antiderivative size = 106 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {e^{-\frac {a (1+m)}{b n}} (d x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \Gamma \left (1+p,-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{d (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2347, 2212} \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {(d x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{d (m+1)} \]

[In]

Int[(d*x)^m*(a + b*Log[c*x^n])^p,x]

[Out]

((d*x)^(1 + m)*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(d*E^((a*(1 + m))/(b*
n))*(1 + m)*(c*x^n)^((1 + m)/n)*(-(((1 + m)*(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 ((d x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int e^{\frac {(1+m) x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{d n} \\ & = \frac {e^{-\frac {a (1+m)}{b n}} (d x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \Gamma \left (1+p,-\frac {(1+m) \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 {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{d (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.01 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {e^{-\frac {(1+m) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} x^{-m} (d x)^m \Gamma \left (1+p,-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m} \]

[In]

Integrate[(d*x)^m*(a + b*Log[c*x^n])^p,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((d*x)^m*(b*log(c*x^n) + a)^p, x)

Sympy [F]

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

[In]

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

[Out]

Integral((d*x)**m*(a + b*log(c*x**n))**p, x)

Maxima [F(-2)]

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

[In]

integrate((d*x)^m*(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]

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

[In]

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

[Out]

integrate((d*x)^m*(b*log(c*x^n) + a)^p, x)

Mupad [F(-1)]

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

[In]

int((d*x)^m*(a + b*log(c*x^n))^p,x)

[Out]

int((d*x)^m*(a + b*log(c*x^n))^p, x)