\(\int \frac {(a+b \log (c x^n))^p (d+e \log (f x^r))}{x^3} \, dx\) [190]

Optimal result

Integrand size = 26, antiderivative size = 295 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^3} \, dx=-\frac {2^{-2-p} e e^{\frac {2 a}{b n}} r \left (c x^n\right )^{2/n} \Gamma \left (2+p,\frac {2 a}{b n}+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x^2}+\frac {2^{-1-p} e e^{\frac {2 a}{b n}} r \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac {2 a}{b n}+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (a+b \log \left (c x^n\right )\right )^{1+p} \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{b n x^2}-\frac {2^{-1-p} e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{x^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.52 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^3} \, dx=-\frac {2^{-2-p} e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right )^{-1+p} \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{1-p} \left (b e n r \Gamma \left (2+p,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+2 \Gamma \left (1+p,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (b d n-a e r-b e r \log \left (c x^n\right )+b e n \log \left (f x^r\right )\right )\right )}{x^2} \] Input:

Integrate[((a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/x^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.82, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2813, 25, 27, 31, 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.

\(\displaystyle \int \frac {\left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p}{x^3} \, dx\)

\(\Big \downarrow \) 2813

\(\displaystyle -e r \int -\frac {2^{-p-1} e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (p+1,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x^3}dx-\frac {2^{-p-1} e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+1,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle e r \int \frac {2^{-p-1} e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (p+1,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x^3}dx-\frac {2^{-p-1} e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+1,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle e 2^{-p-1} r e^{\frac {2 a}{b n}} \int \frac {\left (c x^n\right )^{2/n} \Gamma \left (p+1,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x^3}dx-\frac {2^{-p-1} e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+1,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x^2}\)

\(\Big \downarrow \) 31

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

\(\Big \downarrow \) 2033

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

\(\Big \downarrow \) 3039

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

\(\Big \downarrow \) 7281

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

\(\Big \downarrow \) 7111

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

Input:

Int[((a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/x^3,x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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]

\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p} \left (d +e \ln \left (f \,x^{r}\right )\right )}{x^{3}}d x\]

Input:

int((a+b*ln(c*x^n))^p*(d+e*ln(f*x^r))/x^3,x)
 

Output:

int((a+b*ln(c*x^n))^p*(d+e*ln(f*x^r))/x^3,x)
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^3} \, dx=\int { \frac {{\left (e \log \left (f x^{r}\right ) + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{x^{3}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^p*(d+e*log(f*x^r))/x^3,x, algorithm="fricas")
 

Output:

integral((e*log(f*x^r) + d)*(b*log(c*x^n) + a)^p/x^3, x)
 

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{p} \left (d + e \log {\left (f x^{r} \right )}\right )}{x^{3}}\, dx \] Input:

integrate((a+b*ln(c*x**n))**p*(d+e*ln(f*x**r))/x**3,x)
 

Output:

Integral((a + b*log(c*x**n))**p*(d + e*log(f*x**r))/x**3, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^3} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*log(c*x^n))^p*(d+e*log(f*x^r))/x^3,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]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^3} \, dx=\int { \frac {{\left (e \log \left (f x^{r}\right ) + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{x^{3}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^p*(d+e*log(f*x^r))/x^3,x, algorithm="giac")
 

Output:

integrate((e*log(f*x^r) + d)*(b*log(c*x^n) + a)^p/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^3} \, dx=\int \frac {\left (d+e\,\ln \left (f\,x^r\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p}{x^3} \,d x \] Input:

int(((d + e*log(f*x^r))*(a + b*log(c*x^n))^p)/x^3,x)
 

Output:

int(((d + e*log(f*x^r))*(a + b*log(c*x^n))^p)/x^3, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^3} \, dx =\text {Too large to display} \] Input:

int((a+b*log(c*x^n))^p*(d+e*log(f*x^r))/x^3,x)
 

Output:

( - 2*(log(x**n*c)*b + a)**p*log(x**r*f)*a*e - 2*(log(x**n*c)*b + a)**p*a* 
d - (log(x**n*c)*b + a)**p*a*e*r + (log(x**n*c)*b + a)**p*b*d*n*p + 4*int( 
(log(x**n*c)*b + a)**p/(2*log(x**n*c)*a*b*x**3 - log(x**n*c)*b**2*n*p*x**3 
 + 2*a**2*x**3 - a*b*n*p*x**3),x)*a**2*b*d*n*p*x**2 + 2*int((log(x**n*c)*b 
 + a)**p/(2*log(x**n*c)*a*b*x**3 - log(x**n*c)*b**2*n*p*x**3 + 2*a**2*x**3 
 - a*b*n*p*x**3),x)*a**2*b*e*n*p*r*x**2 - 4*int((log(x**n*c)*b + a)**p/(2* 
log(x**n*c)*a*b*x**3 - log(x**n*c)*b**2*n*p*x**3 + 2*a**2*x**3 - a*b*n*p*x 
**3),x)*a*b**2*d*n**2*p**2*x**2 - int((log(x**n*c)*b + a)**p/(2*log(x**n*c 
)*a*b*x**3 - log(x**n*c)*b**2*n*p*x**3 + 2*a**2*x**3 - a*b*n*p*x**3),x)*a* 
b**2*e*n**2*p**2*r*x**2 + int((log(x**n*c)*b + a)**p/(2*log(x**n*c)*a*b*x* 
*3 - log(x**n*c)*b**2*n*p*x**3 + 2*a**2*x**3 - a*b*n*p*x**3),x)*b**3*d*n** 
3*p**3*x**2 - 4*int(((log(x**n*c)*b + a)**p*log(x**n*c)*log(x**r*f))/(2*lo 
g(x**n*c)*a*b*x**3 - log(x**n*c)*b**2*n*p*x**3 + 2*a**2*x**3 - a*b*n*p*x** 
3),x)*a*b**2*e*n*p*x**2 + 2*int(((log(x**n*c)*b + a)**p*log(x**n*c)*log(x* 
*r*f))/(2*log(x**n*c)*a*b*x**3 - log(x**n*c)*b**2*n*p*x**3 + 2*a**2*x**3 - 
 a*b*n*p*x**3),x)*b**3*e*n**2*p**2*x**2)/(2*x**2*(2*a - b*n*p))