\(\int \frac {(g+h x)^m}{(a+b \log (c (d (e+f x)^p)^q))^2} \, dx\) [508]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\text {Int}\left (\frac {(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2},x\right ) \]

[Out]

Unintegrable((h*x+g)^m/(a+b*ln(c*(d*(f*x+e)^p)^q))^2,x)

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \frac {(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int \frac {(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx \]

[In]

Int[(g + h*x)^m/(a + b*Log[c*(d*(e + f*x)^p)^q])^2,x]

[Out]

Defer[Int][(g + h*x)^m/(a + b*Log[c*(d*(e + f*x)^p)^q])^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int \frac {(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx \]

[In]

Integrate[(g + h*x)^m/(a + b*Log[c*(d*(e + f*x)^p)^q])^2,x]

[Out]

Integrate[(g + h*x)^m/(a + b*Log[c*(d*(e + f*x)^p)^q])^2, x]

Maple [N/A]

Not integrable

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

\[\int \frac {\left (h x +g \right )^{m}}{{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{2}}d x\]

[In]

int((h*x+g)^m/(a+b*ln(c*(d*(f*x+e)^p)^q))^2,x)

[Out]

int((h*x+g)^m/(a+b*ln(c*(d*(f*x+e)^p)^q))^2,x)

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int { \frac {{\left (h x + g\right )}^{m}}{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((h*x+g)^m/(a+b*log(c*(d*(f*x+e)^p)^q))^2,x, algorithm="fricas")

[Out]

integral((h*x + g)^m/(b^2*log(((f*x + e)^p*d)^q*c)^2 + 2*a*b*log(((f*x + e)^p*d)^q*c) + a^2), x)

Sympy [F(-2)]

Exception generated. \[ \int \frac {(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate((h*x+g)**m/(a+b*ln(c*(d*(f*x+e)**p)**q))**2,x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [N/A]

Not integrable

Time = 1.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 6.29 \[ \int \frac {(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int { \frac {{\left (h x + g\right )}^{m}}{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((h*x+g)^m/(a+b*log(c*(d*(f*x+e)^p)^q))^2,x, algorithm="maxima")

[Out]

-(f*x + e)*(h*x + g)^m/(b^2*f*p*q*log(((f*x + e)^p)^q) + a*b*f*p*q + (f*p*q^2*log(d) + f*p*q*log(c))*b^2) + in
tegrate((f*h*(m + 1)*x + e*h*m + f*g)*(h*x + g)^m/(a*b*f*g*p*q + (f*g*p*q^2*log(d) + f*g*p*q*log(c))*b^2 + (a*
b*f*h*p*q + (f*h*p*q^2*log(d) + f*h*p*q*log(c))*b^2)*x + (b^2*f*h*p*q*x + b^2*f*g*p*q)*log(((f*x + e)^p)^q)),
x)

Giac [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int { \frac {{\left (h x + g\right )}^{m}}{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((h*x+g)^m/(a+b*log(c*(d*(f*x+e)^p)^q))^2,x, algorithm="giac")

[Out]

integrate((h*x + g)^m/(b*log(((f*x + e)^p*d)^q*c) + a)^2, x)

Mupad [N/A]

Not integrable

Time = 1.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int \frac {{\left (g+h\,x\right )}^m}{{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2} \,d x \]

[In]

int((g + h*x)^m/(a + b*log(c*(d*(e + f*x)^p)^q))^2,x)

[Out]

int((g + h*x)^m/(a + b*log(c*(d*(e + f*x)^p)^q))^2, x)