\(\int (f+g x)^m (a+b \log (c (d+e x)^n))^n \, dx\) [169]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

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

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F(-2)]

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

[In]

integrate((g*x+f)^m*(a+b*log(c*(e*x+d)^n))^n,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 [F(-2)]

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

[In]

integrate((g*x+f)^m*(a+b*log(c*(e*x+d)^n))^n,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,[0,0,7,4,0,5,0,3,5,0,0,0]%%%}+%%%{5,[0,0,6,4,0,4,1,3,5,0,0,0]%%%}+%%%{2,[0
,0,6,3,1,5,0,3

Mupad [N/A]

Not integrable

Time = 1.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int (f+g x)^m \left (a+b \log \left (c (d+e x)^n\right )\right )^n \, dx=\int {\left (f+g\,x\right )}^m\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^n \,d x \]

[In]

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

[Out]

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