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

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

[Out]

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

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Rubi [A]  time = 0.119591, 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 = {} \[ \int \frac{(c+d x)^m}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.129046, size = 0, normalized size = 0. \[ \int \frac{(c+d x)^m}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.454, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{m}}{ \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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