\(\int (e x)^m \Gamma (p,d (a+b \log (c x^n))) \, dx\) [202]

Optimal result

Integrand size = 20, antiderivative size = 144 \[ \int (e x)^m \Gamma \left (p,d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{1+m} \Gamma \left (p,d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {e^{-\frac {a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \Gamma \left (p,-\frac {(1+m-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (d \left (a+b \log \left (c x^n\right )\right )\right )^p \left (-\frac {(1+m-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.94 \[ \int (e x)^m \Gamma \left (p,d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^m \left (x \Gamma \left (p,d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {(1+m) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} x^{-m} \Gamma \left (p,\frac {(-1-m+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (d \left (a+b \log \left (c x^n\right )\right )\right )^p \left (\frac {(-1-m+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}\right )}{1+m} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.17, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {7132, 7271, 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.

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

\(\Big \downarrow \) 7132

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2612

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

Input:

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

Output:

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

Defintions of rubi rules used

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]
 

rule 7132

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

rule 7271

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ 
FracPart[p]/v^(m*FracPart[p]))   Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, 
x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&  !(Eq 
Q[v, x] && EqQ[m, 1])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

e**m*int(x**m*gamma(p,log(x**n*c)*b*d + a*d),x)