3.35 \(\int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 \, dx\)

Optimal. Leaf size=67 \[ a^2 x+\frac{2 a b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)}+\frac{b^2 \left (F^{g (e+f x)}\right )^{2 n}}{2 f g n \log (F)} \]

[Out]

a^2*x + (2*a*b*(F^(g*(e + f*x)))^n)/(f*g*n*Log[F]) + (b^2*(F^(g*(e + f*x)))^(2*n
))/(2*f*g*n*Log[F])

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Rubi [A]  time = 0.0824926, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ a^2 x+\frac{2 a b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)}+\frac{b^2 \left (F^{g (e+f x)}\right )^{2 n}}{2 f g n \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

a^2*x + (2*a*b*(F^(g*(e + f*x)))^n)/(f*g*n*Log[F]) + (b^2*(F^(g*(e + f*x)))^(2*n
))/(2*f*g*n*Log[F])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.100352, size = 52, normalized size = 0.78 \[ a^2 x+\frac{b \left (F^{g (e+f x)}\right )^n \left (4 a+b \left (F^{g (e+f x)}\right )^n\right )}{2 f g n \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

a^2*x + (b*(F^(g*(e + f*x)))^n*(4*a + b*(F^(g*(e + f*x)))^n))/(2*f*g*n*Log[F])

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Maple [A]  time = 0.023, size = 90, normalized size = 1.3 \[{\frac{{b}^{2} \left ( \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{2}}{2\,ngf\ln \left ( F \right ) }}+2\,{\frac{ab \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n}}{ngf\ln \left ( F \right ) }}+{\frac{{a}^{2}\ln \left ( \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) }{ngf\ln \left ( F \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/g/f/ln(F)/n*b^2*((F^(g*(f*x+e)))^n)^2+2*a*b*(F^(g*(f*x+e)))^n/f/g/n/ln(F)+1/
g/f/ln(F)/n*a^2*ln((F^(g*(f*x+e)))^n)

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Maxima [A]  time = 0.837926, size = 101, normalized size = 1.51 \[ a^{2} x + \frac{2 \,{\left (F^{f g x}\right )}^{n}{\left (F^{e g}\right )}^{n} a b}{f g n \log \left (F\right )} + \frac{{\left (F^{f g x}\right )}^{2 \, n}{\left (F^{e g}\right )}^{2 \, n} b^{2}}{2 \, f g n \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*x + 2*(F^(f*g*x))^n*(F^(e*g))^n*a*b/(f*g*n*log(F)) + 1/2*(F^(f*g*x))^(2*n)*(
F^(e*g))^(2*n)*b^2/(f*g*n*log(F))

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Fricas [A]  time = 0.280398, size = 82, normalized size = 1.22 \[ \frac{2 \, a^{2} f g n x \log \left (F\right ) + 4 \, F^{f g n x + e g n} a b + F^{2 \, f g n x + 2 \, e g n} b^{2}}{2 \, f g n \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*a^2*f*g*n*x*log(F) + 4*F^(f*g*n*x + e*g*n)*a*b + F^(2*f*g*n*x + 2*e*g*n)*
b^2)/(f*g*n*log(F))

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Sympy [A]  time = 0.378717, size = 94, normalized size = 1.4 \[ a^{2} x + \begin{cases} \frac{4 a b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log{\left (F \right )} + b^{2} f g n \left (F^{g \left (e + f x\right )}\right )^{2 n} \log{\left (F \right )}}{2 f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}} & \text{for}\: 2 f^{2} g^{2} n^{2} \log{\left (F \right )}^{2} \neq 0 \\x \left (2 a b + b^{2}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(F**(g*(f*x+e)))**n)**2,x)

[Out]

a**2*x + Piecewise(((4*a*b*f*g*n*(F**(g*(e + f*x)))**n*log(F) + b**2*f*g*n*(F**(
g*(e + f*x)))**(2*n)*log(F))/(2*f**2*g**2*n**2*log(F)**2), Ne(2*f**2*g**2*n**2*l
og(F)**2, 0)), (x*(2*a*b + b**2), True))

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GIAC/XCAS [A]  time = 0.282942, size = 914, normalized size = 13.64 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*x + (2*b^2*f*g*n*cos(-pi*f*g*n*x*sign(F) + pi*f*g*n*x - pi*g*n*e*sign(F) + p
i*g*n*e)*ln(abs(F))/(4*f^2*g^2*n^2*ln(abs(F))^2 + (pi*f*g*n*sign(F) - pi*f*g*n)^
2) - (pi*f*g*n*sign(F) - pi*f*g*n)*b^2*sin(-pi*f*g*n*x*sign(F) + pi*f*g*n*x - pi
*g*n*e*sign(F) + pi*g*n*e)/(4*f^2*g^2*n^2*ln(abs(F))^2 + (pi*f*g*n*sign(F) - pi*
f*g*n)^2))*e^(2*f*g*n*x*ln(abs(F)) + 2*g*n*e*ln(abs(F))) - 1/2*I*(-I*b^2*e^(I*pi
*f*g*n*x*sign(F) - I*pi*f*g*n*x + I*pi*g*n*e*sign(F) - I*pi*g*n*e)/(I*pi*f*g*n*s
ign(F) - I*pi*f*g*n + 2*f*g*n*ln(abs(F))) + I*b^2*e^(-I*pi*f*g*n*x*sign(F) + I*p
i*f*g*n*x - I*pi*g*n*e*sign(F) + I*pi*g*n*e)/(-I*pi*f*g*n*sign(F) + I*pi*f*g*n +
2*f*g*n*ln(abs(F))))*e^(2*f*g*n*x*ln(abs(F)) + 2*g*n*e*ln(abs(F))) + 4*(2*a*b*f
*g*n*cos(-1/2*pi*f*g*n*x*sign(F) + 1/2*pi*f*g*n*x - 1/2*pi*g*n*e*sign(F) + 1/2*p
i*g*n*e)*ln(abs(F))/(4*f^2*g^2*n^2*ln(abs(F))^2 + (pi*f*g*n*sign(F) - pi*f*g*n)^
2) - (pi*f*g*n*sign(F) - pi*f*g*n)*a*b*sin(-1/2*pi*f*g*n*x*sign(F) + 1/2*pi*f*g*
n*x - 1/2*pi*g*n*e*sign(F) + 1/2*pi*g*n*e)/(4*f^2*g^2*n^2*ln(abs(F))^2 + (pi*f*g
*n*sign(F) - pi*f*g*n)^2))*e^(f*g*n*x*ln(abs(F)) + g*n*e*ln(abs(F))) - 1/2*I*(-4
*I*a*b*e^(1/2*I*pi*f*g*n*x*sign(F) - 1/2*I*pi*f*g*n*x + 1/2*I*pi*g*n*e*sign(F) -
1/2*I*pi*g*n*e)/(I*pi*f*g*n*sign(F) - I*pi*f*g*n + 2*f*g*n*ln(abs(F))) + 4*I*a*
b*e^(-1/2*I*pi*f*g*n*x*sign(F) + 1/2*I*pi*f*g*n*x - 1/2*I*pi*g*n*e*sign(F) + 1/2
*I*pi*g*n*e)/(-I*pi*f*g*n*sign(F) + I*pi*f*g*n + 2*f*g*n*ln(abs(F))))*e^(f*g*n*x
*ln(abs(F)) + g*n*e*ln(abs(F)))