3.1.62 \(\int \frac {1}{(a+b (F^{g (e+f x)})^n)^3 (c+d x)} \, dx\) [62]

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {1}{\left (a+b \left (F^{e g+f g x}\right )^n\right )^3 (c+d x)},x\right ) \]

[Out]

Unintegrable(1/(a+b*(F^(f*g*x+e*g))^n)^3/(d*x+c),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 )^3 (c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )^{3} \left (d x +c \right )}\, dx\]

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),x)

[Out]

int(1/(a+b*(F^(g*(f*x+e)))^n)^3/(d*x+c),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)^3/(d*x+c),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^(g*n*e)*b*d*f*g*n*x*log(F) + 2*F^(g*n*e)*b*c*f*g*n*log(F
) + F^(g*n*e)*b*d)*F^(f*g*n*x) + a*d)/(a^4*d^2*f^2*g^2*n^2*x^2*log(F)^2 + 2*a^4*c*d*f^2*g^2*n^2*x*log(F)^2 + a
^4*c^2*f^2*g^2*n^2*log(F)^2 + (F^(2*g*n*e)*a^2*b^2*d^2*f^2*g^2*n^2*x^2*log(F)^2 + 2*F^(2*g*n*e)*a^2*b^2*c*d*f^
2*g^2*n^2*x*log(F)^2 + F^(2*g*n*e)*a^2*b^2*c^2*f^2*g^2*n^2*log(F)^2)*F^(2*f*g*n*x) + 2*(F^(g*n*e)*a^3*b*d^2*f^
2*g^2*n^2*x^2*log(F)^2 + 2*F^(g*n*e)*a^3*b*c*d*f^2*g^2*n^2*x*log(F)^2 + F^(g*n*e)*a^3*b*c^2*f^2*g^2*n^2*log(F)
^2)*F^(f*g*n*x)) + integrate(1/2*(2*d^2*f^2*g^2*n^2*x^2*log(F)^2 + 2*c^2*f^2*g^2*n^2*log(F)^2 + 3*c*d*f*g*n*lo
g(F) + 2*d^2 + (4*c*d*f^2*g^2*n^2*log(F)^2 + 3*d^2*f*g*n*log(F))*x)/(a^3*d^3*f^2*g^2*n^2*x^3*log(F)^2 + 3*a^3*
c*d^2*f^2*g^2*n^2*x^2*log(F)^2 + 3*a^3*c^2*d*f^2*g^2*n^2*x*log(F)^2 + a^3*c^3*f^2*g^2*n^2*log(F)^2 + (F^(g*n*e
)*a^2*b*d^3*f^2*g^2*n^2*x^3*log(F)^2 + 3*F^(g*n*e)*a^2*b*c*d^2*f^2*g^2*n^2*x^2*log(F)^2 + 3*F^(g*n*e)*a^2*b*c^
2*d*f^2*g^2*n^2*x*log(F)^2 + F^(g*n*e)*a^2*b*c^3*f^2*g^2*n^2*log(F)^2)*F^(f*g*n*x)), 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)^3/(d*x+c),x, algorithm="fricas")

[Out]

integral(1/(a^3*d*x + a^3*c + (b^3*d*x + b^3*c)*(F^(f*g*x + g*e))^(3*n) + 3*(a*b^2*d*x + a*b^2*c)*(F^(f*g*x +
g*e))^(2*n) + 3*(a^2*b*d*x + a^2*b*c)*(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*} \frac {3 a c f g n \log {\left (F \right )} + 3 a d f g n x \log {\left (F \right )} + a d + \left (2 b c f g n \log {\left (F \right )} + 2 b d f g n x \log {\left (F \right )} + b d\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{2 a^{4} c^{2} f^{2} g^{2} n^{2} \log {\left (F \right )}^{2} + 4 a^{4} c d f^{2} g^{2} n^{2} x \log {\left (F \right )}^{2} + 2 a^{4} d^{2} f^{2} g^{2} n^{2} x^{2} \log {\left (F \right )}^{2} + \left (2 a^{2} b^{2} c^{2} f^{2} g^{2} n^{2} \log {\left (F \right )}^{2} + 4 a^{2} b^{2} c d f^{2} g^{2} n^{2} x \log {\left (F \right )}^{2} + 2 a^{2} b^{2} d^{2} f^{2} g^{2} n^{2} x^{2} \log {\left (F \right )}^{2}\right ) \left (F^{g \left (e + f x\right )}\right )^{2 n} + \left (4 a^{3} b c^{2} f^{2} g^{2} n^{2} \log {\left (F \right )}^{2} + 8 a^{3} b c d f^{2} g^{2} n^{2} x \log {\left (F \right )}^{2} + 4 a^{3} b d^{2} f^{2} g^{2} n^{2} x^{2} \log {\left (F \right )}^{2}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}} + \frac {\int \frac {2 d^{2}}{a c^{3} + 3 a c^{2} d x + 3 a c d^{2} x^{2} + a d^{3} x^{3} + b c^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c^{2} d x e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c d^{2} x^{2} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + b d^{3} x^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\, dx + \int \frac {2 c^{2} f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}}{a c^{3} + 3 a c^{2} d x + 3 a c d^{2} x^{2} + a d^{3} x^{3} + b c^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c^{2} d x e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c d^{2} x^{2} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + b d^{3} x^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\, dx + \int \frac {3 c d f g n \log {\left (F \right )}}{a c^{3} + 3 a c^{2} d x + 3 a c d^{2} x^{2} + a d^{3} x^{3} + b c^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c^{2} d x e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c d^{2} x^{2} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + b d^{3} x^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\, dx + \int \frac {3 d^{2} f g n x \log {\left (F \right )}}{a c^{3} + 3 a c^{2} d x + 3 a c d^{2} x^{2} + a d^{3} x^{3} + b c^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c^{2} d x e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c d^{2} x^{2} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + b d^{3} x^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\, dx + \int \frac {2 d^{2} f^{2} g^{2} n^{2} x^{2} \log {\left (F \right )}^{2}}{a c^{3} + 3 a c^{2} d x + 3 a c d^{2} x^{2} + a d^{3} x^{3} + b c^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c^{2} d x e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c d^{2} x^{2} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + b d^{3} x^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\, dx + \int \frac {4 c d f^{2} g^{2} n^{2} x \log {\left (F \right )}^{2}}{a c^{3} + 3 a c^{2} d x + 3 a c d^{2} x^{2} + a d^{3} x^{3} + b c^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c^{2} d x e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c d^{2} x^{2} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + b d^{3} x^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\, dx}{2 a^{2} f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} \end {gather*}

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),x)

[Out]

(3*a*c*f*g*n*log(F) + 3*a*d*f*g*n*x*log(F) + a*d + (2*b*c*f*g*n*log(F) + 2*b*d*f*g*n*x*log(F) + b*d)*(F**(g*(e
+ f*x)))**n)/(2*a**4*c**2*f**2*g**2*n**2*log(F)**2 + 4*a**4*c*d*f**2*g**2*n**2*x*log(F)**2 + 2*a**4*d**2*f**2
*g**2*n**2*x**2*log(F)**2 + (2*a**2*b**2*c**2*f**2*g**2*n**2*log(F)**2 + 4*a**2*b**2*c*d*f**2*g**2*n**2*x*log(
F)**2 + 2*a**2*b**2*d**2*f**2*g**2*n**2*x**2*log(F)**2)*(F**(g*(e + f*x)))**(2*n) + (4*a**3*b*c**2*f**2*g**2*n
**2*log(F)**2 + 8*a**3*b*c*d*f**2*g**2*n**2*x*log(F)**2 + 4*a**3*b*d**2*f**2*g**2*n**2*x**2*log(F)**2)*(F**(g*
(e + f*x)))**n) + (Integral(2*d**2/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b*c**3*exp(e*g*n*l
og(F))*exp(f*g*n*x*log(F)) + 3*b*c**2*d*x*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + 3*b*c*d**2*x**2*exp(e*g*n*lo
g(F))*exp(f*g*n*x*log(F)) + b*d**3*x**3*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(2*c**2*f**2*g**2
*n**2*log(F)**2/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b*c**3*exp(e*g*n*log(F))*exp(f*g*n*x*
log(F)) + 3*b*c**2*d*x*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + 3*b*c*d**2*x**2*exp(e*g*n*log(F))*exp(f*g*n*x*l
og(F)) + b*d**3*x**3*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(3*c*d*f*g*n*log(F)/(a*c**3 + 3*a*c*
*2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b*c**3*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + 3*b*c**2*d*x*exp(e*g*n
*log(F))*exp(f*g*n*x*log(F)) + 3*b*c*d**2*x**2*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + b*d**3*x**3*exp(e*g*n*l
og(F))*exp(f*g*n*x*log(F))), x) + Integral(3*d**2*f*g*n*x*log(F)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*
d**3*x**3 + b*c**3*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + 3*b*c**2*d*x*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))
+ 3*b*c*d**2*x**2*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + b*d**3*x**3*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))),
x) + Integral(2*d**2*f**2*g**2*n**2*x**2*log(F)**2/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b*
c**3*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + 3*b*c**2*d*x*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + 3*b*c*d**2*x
**2*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + b*d**3*x**3*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(
4*c*d*f**2*g**2*n**2*x*log(F)**2/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b*c**3*exp(e*g*n*log
(F))*exp(f*g*n*x*log(F)) + 3*b*c**2*d*x*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + 3*b*c*d**2*x**2*exp(e*g*n*log(
F))*exp(f*g*n*x*log(F)) + b*d**3*x**3*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x))/(2*a**2*f**2*g**2*n**2*log(F
)**2)

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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)^3/(d*x+c),x, algorithm="giac")

[Out]

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

[Out]

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

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