\(\int (g x)^m (a+b \log (c x^n))^p (d+e \log (f x^r)) \, dx\) [184]

Optimal result

Integrand size = 28, antiderivative size = 355 \[ \int (g x)^m \left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right ) \, dx=-\frac {e e^{-\frac {a (1+m)}{b n}} r (g x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \Gamma \left (2+p,-\frac {a (1+m)}{b n}-\frac {(1+m) \log \left (c x^n\right )}{n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{g (1+m)^2}-\frac {e e^{-\frac {a (1+m)}{b n}} r (g x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \Gamma \left (1+p,-\frac {a (1+m)}{b n}-\frac {(1+m) \log \left (c x^n\right )}{n}\right ) \left (a+b \log \left (c x^n\right )\right )^{1+p} \left (-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{b g (1+m) n}+\frac {e^{-\frac {a (1+m)}{b n}} (g x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \Gamma \left (1+p,-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{g (1+m)} \] Output:

-e*r*(g*x)^(1+m)*GAMMA(2+p,-a*(1+m)/b/n-(1+m)*ln(c*x^n)/n)*(a+b*ln(c*x^n)) 
^p/exp(a*(1+m)/b/n)/g/(1+m)^2/((c*x^n)^((1+m)/n))/((-(1+m)*(a+b*ln(c*x^n)) 
/b/n)^p)-e*r*(g*x)^(1+m)*GAMMA(p+1,-a*(1+m)/b/n-(1+m)*ln(c*x^n)/n)*(a+b*ln 
(c*x^n))^(p+1)/b/exp(a*(1+m)/b/n)/g/(1+m)/n/((c*x^n)^((1+m)/n))/((-(1+m)*( 
a+b*ln(c*x^n))/b/n)^p)+(g*x)^(1+m)*GAMMA(p+1,-(1+m)*(a+b*ln(c*x^n))/b/n)*( 
a+b*ln(c*x^n))^p*(d+e*ln(f*x^r))/exp(a*(1+m)/b/n)/g/(1+m)/((c*x^n)^((1+m)/ 
n))/((-(1+m)*(a+b*ln(c*x^n))/b/n)^p)
 

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.50 \[ \int (g x)^m \left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right ) \, dx=-\frac {e^{-\frac {(1+m) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} x^{-m} (g x)^m \left (a+b \log \left (c x^n\right )\right )^{-1+p} \left (-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{1-p} \left (-b e n r \Gamma \left (2+p,-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+(1+m) \Gamma \left (1+p,-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (b d n-a e r-b e r \log \left (c x^n\right )+b e n \log \left (f x^r\right )\right )\right )}{(1+m)^3} \] Input:

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

Output:

-(((g*x)^m*(a + b*Log[c*x^n])^(-1 + p)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b* 
n)))^(1 - p)*(-(b*e*n*r*Gamma[2 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n)) 
]) + (1 + m)*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))]*(b*d*n - 
a*e*r - b*e*r*Log[c*x^n] + b*e*n*Log[f*x^r])))/(E^(((1 + m)*(a - b*n*Log[x 
] + b*Log[c*x^n]))/(b*n))*(1 + m)^3*x^m))
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.83, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2813, 27, 31, 27, 2033, 3039, 7281, 7111}

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.

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

\(\Big \downarrow \) 2813

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 31

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2033

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

\(\Big \downarrow \) 3039

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

\(\Big \downarrow \) 7281

\(\displaystyle \frac {e r (g x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \int \Gamma \left (p+1,-\frac {\log \left (c x^n\right ) (m+1)}{n}-\frac {a (m+1)}{b n}\right )d\left (-\frac {\log \left (c x^n\right ) (m+1)}{n}-\frac {a (m+1)}{b n}\right )}{g (m+1)^2}+\frac {(g x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{g (m+1)}\)

\(\Big \downarrow \) 7111

\(\displaystyle \frac {(g x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{g (m+1)}+\frac {e r (g x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \left (\left (-\frac {a (m+1)}{b n}-\frac {(m+1) \log \left (c x^n\right )}{n}\right ) \Gamma \left (p+1,-\frac {\log \left (c x^n\right ) (m+1)}{n}-\frac {a (m+1)}{b n}\right )-\Gamma \left (p+2,-\frac {\log \left (c x^n\right ) (m+1)}{n}-\frac {a (m+1)}{b n}\right )\right )}{g (m+1)^2}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 31

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[(b* 
x^i)^p/(a*x)^(i*p)   Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p} 
, x] &&  !IntegerQ[p]
 

rule 2033

Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[a^(m + n) 
*((b*v)^n/(a*v)^n)   Int[v^(m + n)*Fx, x], x] /; FreeQ[{a, b, m, n}, x] && 
 !IntegerQ[m] &&  !IntegerQ[n] && IntegerQ[m + n]
 

rule 2813

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_ 
.)]*(e_.))*((g_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[(g*x)^m*(a + 
b*Log[c*x^n])^p, x]}, Simp[(d + e*Log[f*x^r])   u, x] - Simp[e*r   Int[Simp 
lifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, 
x] &&  !(EqQ[p, 1] && EqQ[a, 0] && NeQ[d, 0])
 

rule 3039

Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst 
[[3]]   Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /;  !FalseQ[lst]] /; 
NonsumQ[u]
 

rule 7111

Int[Gamma[n_, (a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(Gamma[n, a 
+ b*x]/b), x] - Simp[Gamma[n + 1, a + b*x]/b, x] /; FreeQ[{a, b, n}, x]
 

rule 7281

Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] 
 Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /;  !FalseQ[lst]]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

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(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:Unable to divide, perhaps due to rounding error%%%{1,[0 
,2,2,2,0,2,0,0]%%%}+%%%{2,[0,2,2,2,0,1,0,0]%%%}+%%%{1,[0,2,2,2,0,0,0,0]%%% 
}+%%%{1,[0,2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (g x)^m \left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right ) \, dx=\text {too large to display} \] Input:

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

Output:

(g**m*(x**m*(log(x**n*c)*b + a)**p*log(x**r*f)*a*e*m*x + x**m*(log(x**n*c) 
*b + a)**p*log(x**r*f)*a*e*x + x**m*(log(x**n*c)*b + a)**p*a*d*m*x + x**m* 
(log(x**n*c)*b + a)**p*a*d*x - x**m*(log(x**n*c)*b + a)**p*a*e*r*x + x**m* 
(log(x**n*c)*b + a)**p*b*d*n*p*x + int((x**m*(log(x**n*c)*b + a)**p*log(x* 
*n*c)*log(x**r*f))/(log(x**n*c)*a*b*m**2 + 2*log(x**n*c)*a*b*m + log(x**n* 
c)*a*b + log(x**n*c)*b**2*m*n*p + log(x**n*c)*b**2*n*p + a**2*m**2 + 2*a** 
2*m + a**2 + a*b*m*n*p + a*b*n*p),x)*a*b**2*e*m**3*n*p + 3*int((x**m*(log( 
x**n*c)*b + a)**p*log(x**n*c)*log(x**r*f))/(log(x**n*c)*a*b*m**2 + 2*log(x 
**n*c)*a*b*m + log(x**n*c)*a*b + log(x**n*c)*b**2*m*n*p + log(x**n*c)*b**2 
*n*p + a**2*m**2 + 2*a**2*m + a**2 + a*b*m*n*p + a*b*n*p),x)*a*b**2*e*m**2 
*n*p + 3*int((x**m*(log(x**n*c)*b + a)**p*log(x**n*c)*log(x**r*f))/(log(x* 
*n*c)*a*b*m**2 + 2*log(x**n*c)*a*b*m + log(x**n*c)*a*b + log(x**n*c)*b**2* 
m*n*p + log(x**n*c)*b**2*n*p + a**2*m**2 + 2*a**2*m + a**2 + a*b*m*n*p + a 
*b*n*p),x)*a*b**2*e*m*n*p + int((x**m*(log(x**n*c)*b + a)**p*log(x**n*c)*l 
og(x**r*f))/(log(x**n*c)*a*b*m**2 + 2*log(x**n*c)*a*b*m + log(x**n*c)*a*b 
+ log(x**n*c)*b**2*m*n*p + log(x**n*c)*b**2*n*p + a**2*m**2 + 2*a**2*m + a 
**2 + a*b*m*n*p + a*b*n*p),x)*a*b**2*e*n*p + int((x**m*(log(x**n*c)*b + a) 
**p*log(x**n*c)*log(x**r*f))/(log(x**n*c)*a*b*m**2 + 2*log(x**n*c)*a*b*m + 
 log(x**n*c)*a*b + log(x**n*c)*b**2*m*n*p + log(x**n*c)*b**2*n*p + a**2*m* 
*2 + 2*a**2*m + a**2 + a*b*m*n*p + a*b*n*p),x)*b**3*e*m**2*n**2*p**2 + ...