\(\int \frac {1}{(a+b (F^{g (e+f x)})^n)^2 (c+d x)} \, dx\) [56]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx=\text {Int}\left (\frac {1}{\left (a+b \left (F^{e g+f g x}\right )^n\right )^2 (c+d x)},x\right ) \]

[Out]

Unintegrable(1/(a+b*(F^(f*g*x+e*g))^n)^2/(d*x+c),x)

Rubi [N/A]

Not integrable

Time = 0.09 (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 = {} \[ \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx=\int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx \]

[In]

Int[1/((a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)),x]

[Out]

Defer[Int][1/((a + b*(F^(e*g + f*g*x))^n)^2*(c + d*x)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (a+b \left (F^{e g+f g x}\right )^n\right )^2 (c+d x)} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.79 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx=\int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx \]

[In]

Integrate[1/((a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)),x]

[Out]

Integrate[1/((a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)), x]

Maple [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

\[\int \frac {1}{{\left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}^{2} \left (d x +c \right )}d x\]

[In]

int(1/(a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c),x)

[Out]

int(1/(a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68 \[ \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx=\int { \frac {1}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2} {\left (d x + c\right )}} \,d x } \]

[In]

integrate(1/(a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c),x, algorithm="fricas")

[Out]

integral(1/(a^2*d*x + a^2*c + (b^2*d*x + b^2*c)*(F^(f*g*x + e*g))^(2*n) + 2*(a*b*d*x + a*b*c)*(F^(f*g*x + e*g)
)^n), x)

Sympy [N/A]

Not integrable

Time = 3.95 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx=\int \frac {1}{\left (a + b \left (F^{e g + f g x}\right )^{n}\right )^{2} \left (c + d x\right )}\, dx \]

[In]

integrate(1/(a+b*(F**(g*(f*x+e)))**n)**2/(d*x+c),x)

[Out]

Integral(1/((a + b*(F**(e*g + f*g*x))**n)**2*(c + d*x)), x)

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 192, normalized size of antiderivative = 7.68 \[ \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx=\int { \frac {1}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2} {\left (d x + c\right )}} \,d x } \]

[In]

integrate(1/(a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c),x, algorithm="maxima")

[Out]

1/(a^2*d*f*g*n*x*log(F) + a^2*c*f*g*n*log(F) + (F^(e*g*n)*a*b*d*f*g*n*x*log(F) + F^(e*g*n)*a*b*c*f*g*n*log(F))
*F^(f*g*n*x)) + integrate((d*f*g*n*x*log(F) + c*f*g*n*log(F) + d)/(a^2*d^2*f*g*n*x^2*log(F) + 2*a^2*c*d*f*g*n*
x*log(F) + a^2*c^2*f*g*n*log(F) + (F^(e*g*n)*a*b*d^2*f*g*n*x^2*log(F) + 2*F^(e*g*n)*a*b*c*d*f*g*n*x*log(F) + F
^(e*g*n)*a*b*c^2*f*g*n*log(F))*F^(f*g*n*x)), x)

Giac [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx=\int { \frac {1}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2} {\left (d x + c\right )}} \,d x } \]

[In]

integrate(1/(a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c),x, algorithm="giac")

[Out]

integrate(1/(((F^((f*x + e)*g))^n*b + a)^2*(d*x + c)), x)

Mupad [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx=\int \frac {1}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^2\,\left (c+d\,x\right )} \,d x \]

[In]

int(1/((a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)),x)

[Out]

int(1/((a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)), x)