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

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

[Out]

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

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Rubi [A]  time = 0.137254, 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 = {} \[ \int \frac{1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.116607, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.094, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) \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)/(d*x+c),x)

[Out]

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

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Maxima [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 )}{\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)/(d*x+c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a d x +{\left (b d x + b c\right )}{\left (F^{f g x + e g}\right )}^{n} + a c}, 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)/(d*x+c),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}{\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)/(d*x+c),x, algorithm="giac")

[Out]

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