\(\int \frac {\log ^{-1+q}(c x^n)}{x (a x^m+b \log ^q(c x^n))^2} \, dx\) [7]

Optimal result

Integrand size = 32, antiderivative size = 32 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=-\frac {1}{b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )}-\frac {a m \text {Int}\left (\frac {x^{-1+m}}{\left (a x^m+b \log ^q\left (c x^n\right )\right )^2},x\right )}{b n q} \]

[Out]

-a*m*CannotIntegrate(x^(-1+m)/(a*x^m+b*ln(c*x^n)^q)^2,x)/b/n/q-1/b/n/q/(a*x^m+b*ln(c*x^n)^q)

Rubi [N/A]

Not integrable

Time = 0.19 (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 = {} \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=\int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx \]

[In]

Int[Log[c*x^n]^(-1 + q)/(x*(a*x^m + b*Log[c*x^n]^q)^2),x]

[Out]

-(1/(b*n*q*(a*x^m + b*Log[c*x^n]^q))) - (a*m*Defer[Int][x^(-1 + m)/(a*x^m + b*Log[c*x^n]^q)^2, x])/(b*n*q)

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )}-\frac {(a m) \int \frac {x^{-1+m}}{\left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx}{b n q} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=\int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx \]

[In]

Integrate[Log[c*x^n]^(-1 + q)/(x*(a*x^m + b*Log[c*x^n]^q)^2),x]

[Out]

Integrate[Log[c*x^n]^(-1 + q)/(x*(a*x^m + b*Log[c*x^n]^q)^2), x]

Maple [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00

\[\int \frac {\ln \left (c \,x^{n}\right )^{-1+q}}{x \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{2}}d x\]

[In]

int(ln(c*x^n)^(-1+q)/x/(a*x^m+b*ln(c*x^n)^q)^2,x)

[Out]

int(ln(c*x^n)^(-1+q)/x/(a*x^m+b*ln(c*x^n)^q)^2,x)

Fricas [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.78 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=\int { \frac {\log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{2} x} \,d x } \]

[In]

integrate(log(c*x^n)^(-1+q)/x/(a*x^m+b*log(c*x^n)^q)^2,x, algorithm="fricas")

[Out]

integral(log(c*x^n)^(q - 1)/(2*a*b*x*x^m*log(c*x^n)^q + a^2*x*x^(2*m) + b^2*x*log(c*x^n)^(2*q)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(ln(c*x**n)**(-1+q)/x/(a*x**m+b*ln(c*x**n)**q)**2,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 240, normalized size of antiderivative = 7.50 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=\int { \frac {\log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{2} x} \,d x } \]

[In]

integrate(log(c*x^n)^(-1+q)/x/(a*x^m+b*log(c*x^n)^q)^2,x, algorithm="maxima")

[Out]

1/(a*b*m*x^m*log(x^n) - (n*q - m*log(c))*a*b*x^m + (b^2*m*log(x^n) - (n*q - m*log(c))*b^2)*(log(c) + log(x^n))
^q) + integrate(-(m*n*(q - 1) - m^2*log(c) - m^2*log(x^n))/(a*b*m^2*x*x^m*log(x^n)^2 - 2*(m*n*q - m^2*log(c))*
a*b*x*x^m*log(x^n) + (n^2*q^2 - 2*m*n*q*log(c) + m^2*log(c)^2)*a*b*x*x^m + (b^2*m^2*x*log(x^n)^2 - 2*(m*n*q -
m^2*log(c))*b^2*x*log(x^n) + (n^2*q^2 - 2*m*n*q*log(c) + m^2*log(c)^2)*b^2*x)*(log(c) + log(x^n))^q), x)

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=\int { \frac {\log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{2} x} \,d x } \]

[In]

integrate(log(c*x^n)^(-1+q)/x/(a*x^m+b*log(c*x^n)^q)^2,x, algorithm="giac")

[Out]

integrate(log(c*x^n)^(q - 1)/((a*x^m + b*log(c*x^n)^q)^2*x), x)

Mupad [N/A]

Not integrable

Time = 1.47 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=\int \frac {{\ln \left (c\,x^n\right )}^{q-1}}{x\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^2} \,d x \]

[In]

int(log(c*x^n)^(q - 1)/(x*(a*x^m + b*log(c*x^n)^q)^2),x)

[Out]

int(log(c*x^n)^(q - 1)/(x*(a*x^m + b*log(c*x^n)^q)^2), x)