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

3.1.30.1 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

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

3.1.30.2 Mathematica [N/A]

Not integrable

Time = 1.46 (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]
 

3.1.30.3 Rubi [N/A]

Not integrable

Time = 0.47 (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
 

3.1.30.3.1 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]
 

3.1.30.4 Maple [N/A]

Not integrable

Time = 0.53 (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)
 

3.1.30.5 Fricas [N/A]

Not integrable

Time = 0.32 (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)
 

3.1.30.6 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
 

3.1.30.7 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
 

3.1.30.8 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,
 

3.1.30.9 Mupad [N/A]

Not integrable

Time = 1.89 (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)