\(\int \log (f x^m) (a+b \log (c (d+e x)^n))^p \, dx\) [377]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx=\text {Int}\left (\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^p,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 23, 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 = {} \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx=\int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx=\int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{p} \log \left (f x^{m}\right ) \,d x } \]

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F(-2)]

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

[In]

integrate(log(f*x^m)*(a+b*log(c*(e*x+d)^n))^p,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 = 0.45 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{p} \log \left (f x^{m}\right ) \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx=\int \ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^p \,d x \]

[In]

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

[Out]

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