3.1.51 \(\int \frac {1}{(a+b (F^{g (e+f x)})^n) (c+d x)^2} \, dx\) [51]

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

[Out]

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

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Rubi [A]
time = 0.09, 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 = {} \begin {gather*} \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.37, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right ) \left (d x +c \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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