\(\int \frac {(a+b \log (c (d+e x)^n))^n}{f+g x} \, dx\) [174]

Optimal result

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

[Out]

Unintegrable((a+b*ln(c*(e*x+d)^n))^n/(g*x+f),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 = {} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^n}{f+g x} \, dx=\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^n}{f+g x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^n}{f+g x} \, dx=\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^n}{f+g x} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.87 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^n}{f+g x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{n}}{f + g x}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^n}{f+g x} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

Giac [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^n}{f+g x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{n}}{g x + f} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^n}{f+g x} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^n}{f+g\,x} \,d x \]

[In]

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

[Out]

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