\(\int (g+h x)^m (a+b \log (c (d (e+f x)^p)^q))^n \, dx\) [513]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\text {Int}\left ((g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 28, 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 = {} \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\int { {\left (h x + g\right )}^{m} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F(-2)]

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

[In]

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

[In]

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

Mupad [N/A]

Not integrable

Time = 1.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\int {\left (g+h\,x\right )}^m\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^n \,d x \]

[In]

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

[Out]

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