\(\int \frac {(d x^m+e \log ^{-1+q}(c x^n)) (a x^m+b \log ^q(c x^n))^p}{x} \, dx\) [30]

Optimal result

Integrand size = 40, antiderivative size = 40 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^{1+p}}{b n (1+p) q}+\left (d-\frac {a e m}{b n q}\right ) \text {Int}\left (x^{-1+m} \left (a x^m+b \log ^q\left (c x^n\right )\right )^p,x\right ) \] Output:

e*(a*x^m+b*ln(c*x^n)^q)^(p+1)/b/n/(p+1)/q+(d-a*e*m/b/n/q)*Defer(Int)(x^(-1 
+m)*(a*x^m+b*ln(c*x^n)^q)^p,x)
 

Mathematica [N/A]

Not integrable

Time = 2.52 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx \] Input:

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

Output:

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

Rubi [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {3025, 7299}

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.

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

\(\Big \downarrow \) 3025

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 3025

Int[((Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_)^(m_.))^(p_.)*(Log[(c_.) 
*(x_)^(n_.)]^(r_.)*(e_.) + (d_.)*(x_)^(m_.)))/(x_), x_Symbol] :> Simp[e*((a 
*x^m + b*Log[c*x^n]^q)^(p + 1)/(b*n*q*(p + 1))), x] - Simp[(a*e*m - b*d*n*q 
)/(b*n*q)   Int[x^(m - 1)*(a*x^m + b*Log[c*x^n]^q)^p, x], x] /; FreeQ[{a, b 
, c, d, e, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1] && NeQ[a*e*m - 
 b*d*n*q, 0]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00

\[\int \frac {\left (d \,x^{m}+e \ln \left (c \,x^{n}\right )^{-1+q}\right ) \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{p}}{x}d x\]

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\int { \frac {{\left (d x^{m} + e \log \left (c x^{n}\right )^{q - 1}\right )} {\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{p}}{x} \,d x } \] Input:

integrate((d*x^m+e*log(c*x^n)^(-1+q))*(a*x^m+b*log(c*x^n)^q)^p/x,x, algori 
thm="fricas")
 

Output:

integral((d*x^m + e*log(c*x^n)^(q - 1))*(a*x^m + b*log(c*x^n)^q)^p/x, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\text {Timed out} \] Input:

integrate((d*x**m+e*ln(c*x**n)**(-1+q))*(a*x**m+b*ln(c*x**n)**q)**p/x,x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((d*x^m+e*log(c*x^n)^(-1+q))*(a*x^m+b*log(c*x^n)^q)^p/x,x, algori 
thm="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. \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((d*x^m+e*log(c*x^n)^(-1+q))*(a*x^m+b*log(c*x^n)^q)^p/x,x, algori 
thm="giac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:Unable to divide, perhaps due to rounding error%%%{1,[0 
,0,2,5,2,0,5,0,2,1,2,2,1]%%%}+%%%{-2,[0,0,2,4,2,1,5,0,1,1,2,2,1]%%%}+%%%{5 
,[0,0,2,4,2,
 

Mupad [N/A]

Not integrable

Time = 25.88 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\int \frac {{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^p\,\left (d\,x^m+e\,{\ln \left (c\,x^n\right )}^{q-1}\right )}{x} \,d x \] Input:

int(((a*x^m + b*log(c*x^n)^q)^p*(d*x^m + e*log(c*x^n)^(q - 1)))/x,x)
 

Output:

int(((a*x^m + b*log(c*x^n)^q)^p*(d*x^m + e*log(c*x^n)^(q - 1)))/x, x)
 

Reduce [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 655, normalized size of antiderivative = 16.38 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx =\text {Too large to display} \] Input:

int((d*x^m+e*log(c*x^n)^(-1+q))*(a*x^m+b*log(c*x^n)^q)^p/x,x)
 

Output:

((log(x**n*c)**q*b + x**m*a)**p*log(x**n*c)**q*b*e*m - x**m*(log(x**n*c)** 
q*b + x**m*a)**p*a*e*m*p + x**m*(log(x**n*c)**q*b + x**m*a)**p*b*d*n*p*q + 
 x**m*(log(x**n*c)**q*b + x**m*a)**p*b*d*n*q + int((x**(2*m)*(log(x**n*c)* 
*q*b + x**m*a)**p)/(log(x**n*c)**q*b*x + x**m*a*x),x)*a**2*e*m**2*p**2 + i 
nt((x**(2*m)*(log(x**n*c)**q*b + x**m*a)**p)/(log(x**n*c)**q*b*x + x**m*a* 
x),x)*a**2*e*m**2*p - int((x**(2*m)*(log(x**n*c)**q*b + x**m*a)**p)/(log(x 
**n*c)**q*b*x + x**m*a*x),x)*a*b*d*m*n*p**2*q - int((x**(2*m)*(log(x**n*c) 
**q*b + x**m*a)**p)/(log(x**n*c)**q*b*x + x**m*a*x),x)*a*b*d*m*n*p*q + int 
((x**m*(log(x**n*c)**q*b + x**m*a)**p*log(x**n*c)**q)/(log(x**n*c)**q*log( 
x**n*c)*b*x + x**m*log(x**n*c)*a*x),x)*a*b*e*m*n*p**2*q + int((x**m*(log(x 
**n*c)**q*b + x**m*a)**p*log(x**n*c)**q)/(log(x**n*c)**q*log(x**n*c)*b*x + 
 x**m*log(x**n*c)*a*x),x)*a*b*e*m*n*p*q - int((x**m*(log(x**n*c)**q*b + x* 
*m*a)**p*log(x**n*c)**q)/(log(x**n*c)**q*log(x**n*c)*b*x + x**m*log(x**n*c 
)*a*x),x)*b**2*d*n**2*p**2*q**2 - int((x**m*(log(x**n*c)**q*b + x**m*a)**p 
*log(x**n*c)**q)/(log(x**n*c)**q*log(x**n*c)*b*x + x**m*log(x**n*c)*a*x),x 
)*b**2*d*n**2*p*q**2)/(b*m*n*q*(p + 1))