3.2.79 \(\int x^2 (a+b \log (c x^n))^p (d+e \log (f x^r)) \, dx\) [179]

3.2.79.1 Optimal result

Integrand size = 26, antiderivative size = 298 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right ) \, dx=-3^{-2-p} e e^{-\frac {3 a}{b n}} r x^3 \left (c x^n\right )^{-3/n} \Gamma \left (2+p,-\frac {3 a}{b n}-\frac {3 \log \left (c x^n\right )}{n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}-\frac {3^{-1-p} e e^{-\frac {3 a}{b n}} r x^3 \left (c x^n\right )^{-3/n} \Gamma \left (1+p,-\frac {3 a}{b n}-\frac {3 \log \left (c x^n\right )}{n}\right ) \left (a+b \log \left (c x^n\right )\right )^{1+p} \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{b n}+3^{-1-p} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right ) \]

output

-3^(-2-p)*e*r*x^3*GAMMA(2+p,-3*a/b/n-3*ln(c*x^n)/n)*(a+b*ln(c*x^n))^p/exp( 
3*a/b/n)/((c*x^n)^(3/n))/(((-a-b*ln(c*x^n))/b/n)^p)-3^(-1-p)*e*r*x^3*GAMMA 
(p+1,-3*a/b/n-3*ln(c*x^n)/n)*(a+b*ln(c*x^n))^(p+1)/b/exp(3*a/b/n)/n/((c*x^ 
n)^(3/n))/(((-a-b*ln(c*x^n))/b/n)^p)+3^(-1-p)*x^3*GAMMA(p+1,-3*(a+b*ln(c*x 
^n))/b/n)*(a+b*ln(c*x^n))^p*(d+e*ln(f*x^r))/exp(3*a/b/n)/((c*x^n)^(3/n))/( 
((-a-b*ln(c*x^n))/b/n)^p)
 

3.2.79.2 Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.52 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right ) \, dx=-3^{-2-p} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \left (a+b \log \left (c x^n\right )\right )^{-1+p} \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{1-p} \left (-b e n r \Gamma \left (2+p,-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+3 \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (b d n-a e r-b e r \log \left (c x^n\right )+b e n \log \left (f x^r\right )\right )\right ) \]

input

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

output

-((3^(-2 - p)*x^3*(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, (-3*(a + b*Log[c*x^n]))/(b*n)]) + 3*Gamm 
a[1 + p, (-3*(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^((3*a)/(b*n))*(c*x^n)^(3/n)))
 

3.2.79.3 Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.82, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2813, 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.

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

\(\Big \downarrow \) 2813

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 31

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

\(\Big \downarrow \) 2033

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

\(\Big \downarrow \) 3039

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

\(\Big \downarrow \) 7281

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

\(\Big \downarrow \) 7111

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

input

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

output

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

3.2.79.3.1 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]]
 

3.2.79.4 Maple [F]

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

input

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

output

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

3.2.79.5 Fricas [F]

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

input

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

output

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

3.2.79.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.2.79.7 Maxima [F(-2)]

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

input

integrate(x^2*(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
 

3.2.79.8 Giac [F]

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

input

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

output

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

3.2.79.9 Mupad [F(-1)]

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

input

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

output

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