\(\int \frac {(c+d x)^m}{a+b (F^{g (e+f x)})^n} \, dx\) [73]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {(c+d x)^m}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\text {Int}\left (\frac {(c+d x)^m}{a+b \left (F^{e g+f g x}\right )^n},x\right ) \]

[Out]

Unintegrable((d*x+c)^m/(a+b*(F^(f*g*x+e*g))^n),x)

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {(c+d x)^m}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\int \frac {(c+d x)^m}{a+b \left (F^{g (e+f x)}\right )^n} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x)^m}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\int \frac {(c+d x)^m}{a+b \left (F^{g (e+f x)}\right )^n} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {(c+d x)^m}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {(c+d x)^m}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\int \frac {\left (c + d x\right )^{m}}{a + b \left (F^{e g + f g x}\right )^{n}}\, dx \]

[In]

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

[Out]

Integral((c + d*x)**m/(a + b*(F**(e*g + f*g*x))**n), x)

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {(c+d x)^m}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x)^m}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x)^m}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\int \frac {{\left (c+d\,x\right )}^m}{a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n} \,d x \]

[In]

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

[Out]

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