3.2.83 \(\int \frac {(a+b \log (c x^n))^p (d+e \log (f x^r))}{x^2} \, dx\) [183]

3.2.83.1 Optimal result

Integrand size = 26, antiderivative size = 260 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=-\frac {e e^{\frac {a}{b n}} r \left (c x^n\right )^{\frac {1}{n}} \Gamma \left (2+p,\frac {a}{b n}+\frac {\log \left (c x^n\right )}{n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x}+\frac {e e^{\frac {a}{b n}} r \left (c x^n\right )^{\frac {1}{n}} \Gamma \left (1+p,\frac {a}{b n}+\frac {\log \left (c x^n\right )}{n}\right ) \left (a+b \log \left (c x^n\right )\right )^{1+p} \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{b n x}-\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \Gamma \left (1+p,\frac {a+b \log \left (c x^n\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{x} \]

output

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

3.2.83.2 Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.54 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=-\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \left (a+b \log \left (c x^n\right )\right )^{-1+p} \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{1-p} \left (b e n r \Gamma \left (2+p,\frac {a+b \log \left (c x^n\right )}{b n}\right )+\Gamma \left (1+p,\frac {a+b \log \left (c x^n\right )}{b n}\right ) \left (b d n-a e r-b e r \log \left (c x^n\right )+b e n \log \left (f x^r\right )\right )\right )}{x} \]

input

Integrate[((a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/x^2,x]
 

output

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

3.2.83.3 Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.83, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, 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.

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

\(\Big \downarrow \) 2813

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 31

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

\(\Big \downarrow \) 2033

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

\(\Big \downarrow \) 3039

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

\(\Big \downarrow \) 7281

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

\(\Big \downarrow \) 7111

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

input

Int[((a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/x^2,x]
 

output

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

3.2.83.3.1 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]]
 

3.2.83.4 Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p} \left (d +e \ln \left (f \,x^{r}\right )\right )}{x^{2}}d x\]

input

int((a+b*ln(c*x^n))^p*(d+e*ln(f*x^r))/x^2,x)
 

output

int((a+b*ln(c*x^n))^p*(d+e*ln(f*x^r))/x^2,x)
 

3.2.83.5 Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=\int { \frac {{\left (e \log \left (f x^{r}\right ) + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{x^{2}} \,d x } \]

input

integrate((a+b*log(c*x^n))^p*(d+e*log(f*x^r))/x^2,x, algorithm="fricas")
 

output

integral((e*log(f*x^r) + d)*(b*log(c*x^n) + a)^p/x^2, x)
 

3.2.83.6 Sympy [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{p} \left (d + e \log {\left (f x^{r} \right )}\right )}{x^{2}}\, dx \]

input

integrate((a+b*ln(c*x**n))**p*(d+e*ln(f*x**r))/x**2,x)
 

output

Integral((a + b*log(c*x**n))**p*(d + e*log(f*x**r))/x**2, x)
 

3.2.83.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=\text {Exception raised: RuntimeError} \]

input

integrate((a+b*log(c*x^n))^p*(d+e*log(f*x^r))/x^2,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
 

3.2.83.8 Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=\int { \frac {{\left (e \log \left (f x^{r}\right ) + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{x^{2}} \,d x } \]

input

integrate((a+b*log(c*x^n))^p*(d+e*log(f*x^r))/x^2,x, algorithm="giac")
 

output

integrate((e*log(f*x^r) + d)*(b*log(c*x^n) + a)^p/x^2, x)
 

3.2.83.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=\int \frac {\left (d+e\,\ln \left (f\,x^r\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p}{x^2} \,d x \]

input

int(((d + e*log(f*x^r))*(a + b*log(c*x^n))^p)/x^2,x)
 

output

int(((d + e*log(f*x^r))*(a + b*log(c*x^n))^p)/x^2, x)