\(\int (f+g x)^m \log (c (d+e x^n)^p) \, dx\) [211]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int (f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\text {Int}\left ((f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ),x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 20, 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 = {} \[ \int (f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int (f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int (f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { {\left (g x + f\right )}^{m} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int (f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 100, normalized size of antiderivative = 5.00 \[ \int (f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { {\left (g x + f\right )}^{m} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]

[In]

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

[Out]

(g*x + f)*(g*x + f)^m*log((e*x^n + d)^p)/(g*(m + 1)) + integrate((d*g*(m + 1)*x*log(c) - (e*f*n*p + (e*g*n*p -
e*g*(m + 1)*log(c))*x)*x^n)*(g*x + f)^m/(e*g*(m + 1)*x*x^n + d*g*(m + 1)*x), x)

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:Unable to divide,
perhaps due to rounding error%%%{-1,[0,0,6,3,6,0,2,2,0,1,0]%%%}+%%%{1,[0,0,6,2,6,1,2,2,0,0,1]%%%}+%%%{1,[0,0,
6,2,6,0,2,2,0,

Mupad [N/A]

Not integrable

Time = 1.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int \ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f+g\,x\right )}^m \,d x \]

[In]

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

[Out]

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