\(\int \frac {\log ^{-1+q}(c x^n)}{x (a x^m+b \log ^q(c x^n))} \, dx\) [6]

Optimal result

Integrand size = 32, antiderivative size = 32 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\frac {\log \left (a x^m+b \log ^q\left (c x^n\right )\right )}{b n q}-\frac {a m \text {Int}\left (\frac {x^{-1+m}}{a x^m+b \log ^q\left (c x^n\right )},x\right )}{b n q} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \frac {\log \left (a x^m+b \log ^q\left (c x^n\right )\right )}{b n q}-\frac {(a m) \int \frac {x^{-1+m}}{a x^m+b \log ^q\left (c x^n\right )} \, dx}{b n q} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\int { \frac {\log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} x} \,d x } \]

[In]

integrate(log(c*x^n)^(-1+q)/x/(a*x^m+b*log(c*x^n)^q),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.34 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\int { \frac {\log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} x} \,d x } \]

[In]

integrate(log(c*x^n)^(-1+q)/x/(a*x^m+b*log(c*x^n)^q),x, algorithm="maxima")

[Out]

-a*integrate(x^m/(a*b*x*x^m*log(c) + a*b*x*x^m*log(x^n) + (b^2*x*log(c) + b^2*x*log(x^n))*(log(c) + log(x^n))^
q), x) + log(log(c) + log(x^n))/(b*n)

Giac [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\int { \frac {\log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} x} \,d x } \]

[In]

integrate(log(c*x^n)^(-1+q)/x/(a*x^m+b*log(c*x^n)^q),x, algorithm="giac")

[Out]

integrate(log(c*x^n)^(q - 1)/((a*x^m + b*log(c*x^n)^q)*x), x)

Mupad [N/A]

Not integrable

Time = 1.43 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\int \frac {{\ln \left (c\,x^n\right )}^{q-1}}{x\,\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )} \,d x \]

[In]

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

[Out]

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