3.12 \(\int F^{d x} \left (a+b F^{c+d x}\right )^n \, dx\)

Optimal. Leaf size=36 \[ \frac{F^{-c} \left (a+b F^{c+d x}\right )^{n+1}}{b d (n+1) \log (F)} \]

[Out]

(a + b*F^(c + d*x))^(1 + n)/(b*d*F^c*(1 + n)*Log[F])

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Rubi [A]  time = 0.122426, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{F^{-c} \left (a+b F^{c+d x}\right )^{n+1}}{b d (n+1) \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[F^(d*x)*(a + b*F^(c + d*x))^n,x]

[Out]

(a + b*F^(c + d*x))^(1 + n)/(b*d*F^c*(1 + n)*Log[F])

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Rubi in Sympy [A]  time = 9.56589, size = 26, normalized size = 0.72 \[ \frac{F^{- c} \left (F^{c + d x} b + a\right )^{n + 1}}{b d \left (n + 1\right ) \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(d*x)*(a+b*F**(d*x+c))**n,x)

[Out]

F**(-c)*(F**(c + d*x)*b + a)**(n + 1)/(b*d*(n + 1)*log(F))

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Mathematica [A]  time = 0.0557343, size = 35, normalized size = 0.97 \[ \frac{F^{-c} \left (a+b F^{c+d x}\right )^{n+1}}{b d n \log (F)+b d \log (F)} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(d*x)*(a + b*F^(c + d*x))^n,x]

[Out]

(a + b*F^(c + d*x))^(1 + n)/(F^c*(b*d*Log[F] + b*d*n*Log[F]))

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Maple [B]  time = 0.034, size = 81, normalized size = 2.3 \[{\frac{{{\rm e}^{d\ln \left ( F \right ) x}}{{\rm e}^{n\ln \left ( a+b{{\rm e}^{c\ln \left ( F \right ) }}{{\rm e}^{d\ln \left ( F \right ) x}} \right ) }}}{d\ln \left ( F \right ) \left ( 1+n \right ) }}+{\frac{a{{\rm e}^{n\ln \left ( a+b{{\rm e}^{c\ln \left ( F \right ) }}{{\rm e}^{d\ln \left ( F \right ) x}} \right ) }}}{{F}^{c}\ln \left ( F \right ) bd \left ( 1+n \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(d*x)*(a+b*F^(d*x+c))^n,x)

[Out]

1/ln(F)/d/(1+n)*exp(d*ln(F)*x)*exp(n*ln(a+b*exp(c*ln(F))*exp(d*ln(F)*x)))+1/b/(F
^c)/d/ln(F)/(1+n)*a*exp(n*ln(a+b*exp(c*ln(F))*exp(d*ln(F)*x)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((F^(d*x + c)*b + a)^n*F^(d*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.265468, size = 68, normalized size = 1.89 \[ \frac{{\left (F^{d x + c} b + a\right )}^{n}{\left (\frac{F^{d x + c} b}{F^{c}} + \frac{a}{F^{c}}\right )}}{{\left (b d n + b d\right )} \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((F^(d*x + c)*b + a)^n*F^(d*x),x, algorithm="fricas")

[Out]

(F^(d*x + c)*b + a)^n*(F^(d*x + c)*b/F^c + a/F^c)/((b*d*n + b*d)*log(F))

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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(F**(d*x)*(a+b*F**(d*x+c))**n,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((F^(d*x + c)*b + a)^n*F^(d*x),x, algorithm="giac")

[Out]

integrate((F^(d*x + c)*b + a)^n*F^(d*x), x)