\(\int \frac {(f+g x)^m}{a+b \log (c (d+e x)^n)} \, dx\) [163]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {(f+g x)^m}{a+b \log \left (c (d+e x)^n\right )} \, dx=\text {Int}\left (\frac {(f+g x)^m}{a+b \log \left (c (d+e x)^n\right )},x\right ) \]

[Out]

Unintegrable((g*x+f)^m/(a+b*ln(c*(e*x+d)^n)),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {(f+g x)^m}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {(f+g x)^m}{a+b \log \left (c (d+e x)^n\right )} \, dx \]

[In]

Int[(f + g*x)^m/(a + b*Log[c*(d + e*x)^n]),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {(f+g x)^m}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {(f+g x)^m}{a+b \log \left (c (d+e x)^n\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

\[\int \frac {\left (g x +f \right )^{m}}{a +b \ln \left (c \left (e x +d \right )^{n}\right )}d x\]

[In]

int((g*x+f)^m/(a+b*ln(c*(e*x+d)^n)),x)

[Out]

int((g*x+f)^m/(a+b*ln(c*(e*x+d)^n)),x)

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {(f+g x)^m}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \frac {{\left (g x + f\right )}^{m}}{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

integral((g*x + f)^m/(b*log((e*x + d)^n*c) + a), x)

Sympy [N/A]

Not integrable

Time = 31.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {(f+g x)^m}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {\left (f + g x\right )^{m}}{a + b \log {\left (c \left (d + e x\right )^{n} \right )}}\, dx \]

[In]

integrate((g*x+f)**m/(a+b*ln(c*(e*x+d)**n)),x)

[Out]

Integral((f + g*x)**m/(a + b*log(c*(d + e*x)**n)), x)

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {(f+g x)^m}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \frac {{\left (g x + f\right )}^{m}}{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

integrate((g*x + f)^m/(b*log((e*x + d)^n*c) + a), x)

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {(f+g x)^m}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \frac {{\left (g x + f\right )}^{m}}{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

integrate((g*x + f)^m/(b*log((e*x + d)^n*c) + a), x)

Mupad [N/A]

Not integrable

Time = 1.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {(f+g x)^m}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {{\left (f+g\,x\right )}^m}{a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )} \,d x \]

[In]

int((f + g*x)^m/(a + b*log(c*(d + e*x)^n)),x)

[Out]

int((f + g*x)^m/(a + b*log(c*(d + e*x)^n)), x)