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

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{1}{(c+d x) \left (a+b \left (F^{e g+f g x}\right )^n\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.126782, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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 = 0.988705, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx \]

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]

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Maple [A]  time = 0.469, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{2} \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{a^{2} d f g n x \log \left (F\right ) + a^{2} c f g n \log \left (F\right ) +{\left ({\left (F^{e g}\right )}^{n} a b d f g n x \log \left (F\right ) +{\left (F^{e g}\right )}^{n} a b c f g n \log \left (F\right )\right )}{\left (F^{f g x}\right )}^{n}} + \int \frac{d f g n x \log \left (F\right ) + c f g n \log \left (F\right ) + d}{a^{2} d^{2} f g n x^{2} \log \left (F\right ) + 2 \, a^{2} c d f g n x \log \left (F\right ) + a^{2} c^{2} f g n \log \left (F\right ) +{\left ({\left (F^{e g}\right )}^{n} a b d^{2} f g n x^{2} \log \left (F\right ) + 2 \,{\left (F^{e g}\right )}^{n} a b c d f g n x \log \left (F\right ) +{\left (F^{e g}\right )}^{n} a b c^{2} f g n \log \left (F\right )\right )}{\left (F^{f g x}\right )}^{n}}\,{d x} \end{align*}

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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)