\(\int \frac {(g+h x)^m}{a+b \log (c (d (e+f x)^p)^q)} \, dx\) [507]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

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

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(g+h x)^m}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int { \frac {{\left (h x + g\right )}^{m}}{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 13.57 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {(g+h x)^m}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int \frac {\left (g + h x\right )^{m}}{a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.93 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(g+h x)^m}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int { \frac {{\left (h x + g\right )}^{m}}{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

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

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(g+h x)^m}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int \frac {{\left (g+h\,x\right )}^m}{a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )} \,d x \]

[In]

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

[Out]

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