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

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

[Out]

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

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Rubi [A]  time = 0.184984, 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. \[ \text{Int}\left (\frac{1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^2},x\right ) \]

Verification is Not applicable to the result.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 1.5098, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.219, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{2} \left ( dx+c \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{a^{2} d^{2} f g n x^{2} \log \left (F\right ) + 2 \, a^{2} c d f g n x \log \left (F\right ) + a^{2} c^{2} f g n \log \left (F\right ) +{\left ({\left (F^{e g}\right )}^{n} a b d^{2} f g n x^{2} \log \left (F\right ) + 2 \,{\left (F^{e g}\right )}^{n} a b c d f g n x \log \left (F\right ) +{\left (F^{e g}\right )}^{n} a b c^{2} f g n \log \left (F\right )\right )}{\left (F^{f g x}\right )}^{n}} + \int \frac{d f g n x \log \left (F\right ) + c f g n \log \left (F\right ) + 2 \, d}{a^{2} d^{3} f g n x^{3} \log \left (F\right ) + 3 \, a^{2} c d^{2} f g n x^{2} \log \left (F\right ) + 3 \, a^{2} c^{2} d f g n x \log \left (F\right ) + a^{2} c^{3} f g n \log \left (F\right ) +{\left ({\left (F^{e g}\right )}^{n} a b d^{3} f g n x^{3} \log \left (F\right ) + 3 \,{\left (F^{e g}\right )}^{n} a b c d^{2} f g n x^{2} \log \left (F\right ) + 3 \,{\left (F^{e g}\right )}^{n} a b c^{2} d f g n x \log \left (F\right ) +{\left (F^{e g}\right )}^{n} a b c^{3} f g n \log \left (F\right )\right )}{\left (F^{f g x}\right )}^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(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) + integrate((d*f*g*n*x*log(F) +
c*f*g*n*log(F) + 2*d)/(a^2*d^3*f*g*n*x^3*log(F) + 3*a^2*c*d^2*f*g*n*x^2*log(F) +
3*a^2*c^2*d*f*g*n*x*log(F) + a^2*c^3*f*g*n*log(F) + ((F^(e*g))^n*a*b*d^3*f*g*n*
x^3*log(F) + 3*(F^(e*g))^n*a*b*c*d^2*f*g*n*x^2*log(F) + 3*(F^(e*g))^n*a*b*c^2*d*
f*g*n*x*log(F) + (F^(e*g))^n*a*b*c^3*f*g*n*log(F))*(F^(f*g*x))^n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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