3.63 \(\int \frac{1}{(a+b (F^{g (e+f x)})^n)^3 (c+d x)^2} \, dx\)

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{1}{(c+d x)^2 \left (a+b \left (F^{e g+f g x}\right )^n\right )^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.120864, 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 = {} \[ \int \frac{1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][1/((a + b*(F^(e*g + f*g*x))^n)^3*(c + d*x)^2), x]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2} \, dx &=\int \frac{1}{\left (a+b \left (F^{e g+f g x}\right )^n\right )^3 (c+d x)^2} \, dx\\ \end{align*}

Mathematica [A]  time = 1.43372, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.574, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{3} \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*a*d*f*g*n*x*log(F) + 3*a*c*f*g*n*log(F) + 2*((F^(e*g))^n*b*d*f*g*n*x*log(F) + (F^(e*g))^n*b*c*f*g*n*log
(F) + (F^(e*g))^n*b*d)*(F^(f*g*x))^n + 2*a*d)/(a^4*d^3*f^2*g^2*n^2*x^3*log(F)^2 + 3*a^4*c*d^2*f^2*g^2*n^2*x^2*
log(F)^2 + 3*a^4*c^2*d*f^2*g^2*n^2*x*log(F)^2 + a^4*c^3*f^2*g^2*n^2*log(F)^2 + ((F^(e*g))^(2*n)*a^2*b^2*d^3*f^
2*g^2*n^2*x^3*log(F)^2 + 3*(F^(e*g))^(2*n)*a^2*b^2*c*d^2*f^2*g^2*n^2*x^2*log(F)^2 + 3*(F^(e*g))^(2*n)*a^2*b^2*
c^2*d*f^2*g^2*n^2*x*log(F)^2 + (F^(e*g))^(2*n)*a^2*b^2*c^3*f^2*g^2*n^2*log(F)^2)*(F^(f*g*x))^(2*n) + 2*((F^(e*
g))^n*a^3*b*d^3*f^2*g^2*n^2*x^3*log(F)^2 + 3*(F^(e*g))^n*a^3*b*c*d^2*f^2*g^2*n^2*x^2*log(F)^2 + 3*(F^(e*g))^n*
a^3*b*c^2*d*f^2*g^2*n^2*x*log(F)^2 + (F^(e*g))^n*a^3*b*c^3*f^2*g^2*n^2*log(F)^2)*(F^(f*g*x))^n) + integrate((d
^2*f^2*g^2*n^2*x^2*log(F)^2 + c^2*f^2*g^2*n^2*log(F)^2 + 3*c*d*f*g*n*log(F) + 3*d^2 + (2*c*d*f^2*g^2*n^2*log(F
)^2 + 3*d^2*f*g*n*log(F))*x)/(a^3*d^4*f^2*g^2*n^2*x^4*log(F)^2 + 4*a^3*c*d^3*f^2*g^2*n^2*x^3*log(F)^2 + 6*a^3*
c^2*d^2*f^2*g^2*n^2*x^2*log(F)^2 + 4*a^3*c^3*d*f^2*g^2*n^2*x*log(F)^2 + a^3*c^4*f^2*g^2*n^2*log(F)^2 + ((F^(e*
g))^n*a^2*b*d^4*f^2*g^2*n^2*x^4*log(F)^2 + 4*(F^(e*g))^n*a^2*b*c*d^3*f^2*g^2*n^2*x^3*log(F)^2 + 6*(F^(e*g))^n*
a^2*b*c^2*d^2*f^2*g^2*n^2*x^2*log(F)^2 + 4*(F^(e*g))^n*a^2*b*c^3*d*f^2*g^2*n^2*x*log(F)^2 + (F^(e*g))^n*a^2*b*
c^4*f^2*g^2*n^2*log(F)^2)*(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^{3} d^{2} x^{2} + 2 \, a^{3} c d x + a^{3} c^{2} +{\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )}{\left (F^{f g x + e g}\right )}^{3 \, n} + 3 \,{\left (a b^{2} d^{2} x^{2} + 2 \, a b^{2} c d x + a b^{2} c^{2}\right )}{\left (F^{f g x + e g}\right )}^{2 \, n} + 3 \,{\left (a^{2} b d^{2} x^{2} + 2 \, a^{2} b c d x + a^{2} b c^{2}\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)^3/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral(1/(a^3*d^2*x^2 + 2*a^3*c*d*x + a^3*c^2 + (b^3*d^2*x^2 + 2*b^3*c*d*x + b^3*c^2)*(F^(f*g*x + e*g))^(3*n
) + 3*(a*b^2*d^2*x^2 + 2*a*b^2*c*d*x + a*b^2*c^2)*(F^(f*g*x + e*g))^(2*n) + 3*(a^2*b*d^2*x^2 + 2*a^2*b*c*d*x +
a^2*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. \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)**3/(d*x+c)**2,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 )}^{3}{\left (d x + c\right )}^{2}}\,{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)^3/(d*x+c)^2,x, algorithm="giac")

[Out]

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