\(\int (d x^q)^m (a+b \log (c x^n))^p \, dx\) [192]

Optimal result

Integrand size = 20, antiderivative size = 114 \[ \int \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {e^{-\frac {a+a m q}{b n}} x \left (c x^n\right )^{-\frac {1+m q}{n}} \left (d x^q\right )^m \Gamma \left (1+p,-\frac {(1+m q) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m q) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m q} \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04 \[ \int \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {e^{-\frac {(1+m q) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} x^{-m q} \left (d x^q\right )^m \Gamma \left (1+p,-\frac {(1+m q) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m q) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m q} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {34, 2747, 2612}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \, dx\)

\(\Big \downarrow \) 34

\(\displaystyle x^{-m q} \left (d x^q\right )^m \int x^{m q} \left (a+b \log \left (c x^n\right )\right )^pdx\)

\(\Big \downarrow \) 2747

\(\displaystyle \frac {x \left (d x^q\right )^m \left (c x^n\right )^{-\frac {m q+1}{n}} \int \left (c x^n\right )^{\frac {m q+1}{n}} \left (a+b \log \left (c x^n\right )\right )^pd\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 2612

\(\displaystyle \frac {x \left (d x^q\right )^m e^{-\frac {a m q+a}{b n}} \left (c x^n\right )^{-\frac {m q+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m q+1}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 34

Int[(u_.)*((a_.)*(x_)^(m_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*x^m)^F 
racPart[p]/x^(m*FracPart[p]))   Int[u*x^(m*p), x], x] /; FreeQ[{a, m, p}, x 
] &&  !IntegerQ[p]
 

rule 2612

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 2747

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[(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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {d^{m} \left (x^{m q} {\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{p} a x +\left (\int \frac {x^{m q} {\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{p} \mathrm {log}\left (x^{n} c \right )}{\mathrm {log}\left (x^{n} c \right ) a b m q +\mathrm {log}\left (x^{n} c \right ) a b +\mathrm {log}\left (x^{n} c \right ) b^{2} n p +a^{2} m q +a^{2}+a b n p}d x \right ) a \,b^{2} m n p q +\left (\int \frac {x^{m q} {\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{p} \mathrm {log}\left (x^{n} c \right )}{\mathrm {log}\left (x^{n} c \right ) a b m q +\mathrm {log}\left (x^{n} c \right ) a b +\mathrm {log}\left (x^{n} c \right ) b^{2} n p +a^{2} m q +a^{2}+a b n p}d x \right ) a \,b^{2} n p +\left (\int \frac {x^{m q} {\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{p} \mathrm {log}\left (x^{n} c \right )}{\mathrm {log}\left (x^{n} c \right ) a b m q +\mathrm {log}\left (x^{n} c \right ) a b +\mathrm {log}\left (x^{n} c \right ) b^{2} n p +a^{2} m q +a^{2}+a b n p}d x \right ) b^{3} n^{2} p^{2}\right )}{a m q +b n p +a} \] Input:

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

Output:

(d**m*(x**(m*q)*(log(x**n*c)*b + a)**p*a*x + int((x**(m*q)*(log(x**n*c)*b 
+ a)**p*log(x**n*c))/(log(x**n*c)*a*b*m*q + log(x**n*c)*a*b + log(x**n*c)* 
b**2*n*p + a**2*m*q + a**2 + a*b*n*p),x)*a*b**2*m*n*p*q + int((x**(m*q)*(l 
og(x**n*c)*b + a)**p*log(x**n*c))/(log(x**n*c)*a*b*m*q + log(x**n*c)*a*b + 
 log(x**n*c)*b**2*n*p + a**2*m*q + a**2 + a*b*n*p),x)*a*b**2*n*p + int((x* 
*(m*q)*(log(x**n*c)*b + a)**p*log(x**n*c))/(log(x**n*c)*a*b*m*q + log(x**n 
*c)*a*b + log(x**n*c)*b**2*n*p + a**2*m*q + a**2 + a*b*n*p),x)*b**3*n**2*p 
**2))/(a*m*q + a + b*n*p)