\(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^n}{g+h x} \, dx\) [517]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n}{g+h x} \, dx=\int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n}{g+h x} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n}{g+h x} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{n}}{h x + g} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 6.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n}{g+h x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{n}}{g + h x}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^n/(h*x+g),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 = 0.55 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n}{g+h x} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{n}}{h x + g} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n}{g+h x} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^n}{g+h\,x} \,d x \]

[In]

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

[Out]

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