\(\int \frac {\log (x^{-n} (a+b x^n))}{c+d x} \, dx\) [398]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{c+d x} \, dx=\text {Int}\left (\frac {\log \left (b+a x^{-n}\right )}{c+d x},x\right ) \]

[Out]

Unintegrable(ln(b+a/(x^n))/(d*x+c),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{c+d x} \, dx=\int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{c+d x} \, dx \]

[In]

Int[Log[(a + b*x^n)/x^n]/(c + d*x),x]

[Out]

Defer[Int][Log[b + a/x^n]/(c + d*x), x]

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

Mathematica [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{c+d x} \, dx=\int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{c+d x} \, dx \]

[In]

Integrate[Log[(a + b*x^n)/x^n]/(c + d*x),x]

[Out]

Integrate[Log[(a + b*x^n)/x^n]/(c + d*x), x]

Maple [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

int(ln((a+b*x^n)/(x^n))/(d*x+c),x)

[Out]

int(ln((a+b*x^n)/(x^n))/(d*x+c),x)

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{c+d x} \, dx=\int { \frac {\log \left (\frac {b x^{n} + a}{x^{n}}\right )}{d x + c} \,d x } \]

[In]

integrate(log((a+b*x^n)/(x^n))/(d*x+c),x, algorithm="fricas")

[Out]

integral(log((b*x^n + a)/x^n)/(d*x + c), x)

Sympy [N/A]

Not integrable

Time = 23.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{c+d x} \, dx=\int \frac {\log {\left (a x^{- n} + b \right )}}{c + d x}\, dx \]

[In]

integrate(ln((a+b*x**n)/(x**n))/(d*x+c),x)

[Out]

Integral(log(a/x**n + b)/(c + d*x), x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{c+d x} \, dx=\int { \frac {\log \left (\frac {b x^{n} + a}{x^{n}}\right )}{d x + c} \,d x } \]

[In]

integrate(log((a+b*x^n)/(x^n))/(d*x+c),x, algorithm="maxima")

[Out]

integrate(log((b*x^n + a)/x^n)/(d*x + c), x)

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{c+d x} \, dx=\int { \frac {\log \left (\frac {b x^{n} + a}{x^{n}}\right )}{d x + c} \,d x } \]

[In]

integrate(log((a+b*x^n)/(x^n))/(d*x+c),x, algorithm="giac")

[Out]

integrate(log((b*x^n + a)/x^n)/(d*x + c), x)

Mupad [N/A]

Not integrable

Time = 1.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{c+d x} \, dx=\int \frac {\ln \left (\frac {a+b\,x^n}{x^n}\right )}{c+d\,x} \,d x \]

[In]

int(log((a + b*x^n)/x^n)/(c + d*x),x)

[Out]

int(log((a + b*x^n)/x^n)/(c + d*x), x)