\(\int \frac {1}{(f+g x) (a+b \log (c (d+e x)^n))} \, dx\) [92]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\text {Int}\left (\frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right ) \]

[Out]

Unintegrable(1/(g*x+f)/(a+b*ln(c*(e*x+d)^n)),x)

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

[In]

Int[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])),x]

[Out]

Defer[Int][1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])), x]

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

Mathematica [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

[In]

Integrate[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])),x]

[Out]

Integrate[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])), x]

Maple [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (g x +f \right ) \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}d x\]

[In]

int(1/(g*x+f)/(a+b*ln(c*(e*x+d)^n)),x)

[Out]

int(1/(g*x+f)/(a+b*ln(c*(e*x+d)^n)),x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(g*x+f)/(a+b*log(c*(e*x+d)^n)),x, algorithm="fricas")

[Out]

integral(1/(a*g*x + a*f + (b*g*x + b*f)*log((e*x + d)^n*c)), x)

Sympy [N/A]

Not integrable

Time = 1.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \left (f + g x\right )}\, dx \]

[In]

integrate(1/(g*x+f)/(a+b*ln(c*(e*x+d)**n)),x)

[Out]

Integral(1/((a + b*log(c*(d + e*x)**n))*(f + g*x)), x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(g*x+f)/(a+b*log(c*(e*x+d)^n)),x, algorithm="maxima")

[Out]

integrate(1/((g*x + f)*(b*log((e*x + d)^n*c) + a)), x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(g*x+f)/(a+b*log(c*(e*x+d)^n)),x, algorithm="giac")

[Out]

integrate(1/((g*x + f)*(b*log((e*x + d)^n*c) + a)), x)

Mupad [N/A]

Not integrable

Time = 1.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int \frac {1}{\left (f+g\,x\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )} \,d x \]

[In]

int(1/((f + g*x)*(a + b*log(c*(d + e*x)^n))),x)

[Out]

int(1/((f + g*x)*(a + b*log(c*(d + e*x)^n))), x)