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

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {1}{(c+d x) \left (a+b \left (F^{e g+f g x}\right )^n\right )^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx \]

Verification is Not applicable to the result.

[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*} \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 (c+d x)} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.01, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx \]

Verification is Not applicable to the result.

[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]

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} d x + a^{2} c + {\left (b^{2} d x + b^{2} c\right )} {\left (F^{f g x + e g}\right )}^{2 \, n} + 2 \, {\left (a b d x + a b c\right )} {\left (F^{f g x + e g}\right )}^{n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2} {\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right )^{2} \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{a^{2} d f g n x \log \relax (F) + a^{2} c f g n \log \relax (F) + {\left ({\left (F^{e g}\right )}^{n} a b d f g n x \log \relax (F) + {\left (F^{e g}\right )}^{n} a b c f g n \log \relax (F)\right )} {\left (F^{f g x}\right )}^{n}} + \int \frac {d f g n x \log \relax (F) + c f g n \log \relax (F) + d}{a^{2} d^{2} f g n x^{2} \log \relax (F) + 2 \, a^{2} c d f g n x \log \relax (F) + a^{2} c^{2} f g n \log \relax (F) + {\left ({\left (F^{e g}\right )}^{n} a b d^{2} f g n x^{2} \log \relax (F) + 2 \, {\left (F^{e g}\right )}^{n} a b c d f g n x \log \relax (F) + {\left (F^{e g}\right )}^{n} a b c^{2} f g n \log \relax (F)\right )} {\left (F^{f g x}\right )}^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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*x))^n) + 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*x))^n), x)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^2\,\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{a^{2} c f g n \log {\relax (F )} + a^{2} d f g n x \log {\relax (F )} + \left (a b c f g n \log {\relax (F )} + a b d f g n x \log {\relax (F )}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}} + \frac {\int \frac {d}{a c^{2} + 2 a c d x + a d^{2} x^{2} + b c^{2} e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}} + 2 b c d x e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}} + b d^{2} x^{2} e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}}}\, dx + \int \frac {c f g n \log {\relax (F )}}{a c^{2} + 2 a c d x + a d^{2} x^{2} + b c^{2} e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}} + 2 b c d x e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}} + b d^{2} x^{2} e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}}}\, dx + \int \frac {d f g n x \log {\relax (F )}}{a c^{2} + 2 a c d x + a d^{2} x^{2} + b c^{2} e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}} + 2 b c d x e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}} + b d^{2} x^{2} e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}}}\, dx}{a f g n \log {\relax (F )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________