3.11 \(\int F^x (a+b F^x)^n \, dx\)

Optimal. Leaf size=24 \[ \frac{\left (a+b F^x\right )^{n+1}}{b (n+1) \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.0229501, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2246, 32} \[ \frac{\left (a+b F^x\right )^{n+1}}{b (n+1) \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^x*(a + b*F^x)^n,x]

[Out]

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

Rule 2246

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),
x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,
c, d, e, n, p}, x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int F^x \left (a+b F^x\right )^n \, dx &=\frac{\operatorname{Subst}\left (\int (a+b x)^n \, dx,x,F^x\right )}{\log (F)}\\ &=\frac{\left (a+b F^x\right )^{1+n}}{b (1+n) \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0240604, size = 24, normalized size = 1. \[ \frac{\left (a+b F^x\right )^{n+1}}{b n \log (F)+b \log (F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^x*(a + b*F^x)^n,x]

[Out]

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

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Maple [A]  time = 0.003, size = 25, normalized size = 1. \begin{align*}{\frac{ \left ( a+b{F}^{x} \right ) ^{1+n}}{b \left ( 1+n \right ) \ln \left ( F \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^x*(a+b*F^x)^n,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^x*(a+b*F^x)^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55113, size = 62, normalized size = 2.58 \begin{align*} \frac{{\left (F^{x} b + a\right )}{\left (F^{x} b + a\right )}^{n}}{{\left (b n + b\right )} \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^x*(a+b*F^x)^n,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.425, size = 82, normalized size = 3.42 \begin{align*} \begin{cases} \frac{x}{a} & \text{for}\: F = 1 \wedge b = 0 \wedge n = -1 \\x \left (a + b\right )^{n} & \text{for}\: F = 1 \\\frac{F^{x} a^{n}}{\log{\left (F \right )}} & \text{for}\: b = 0 \\\frac{\log{\left (F^{x} + \frac{a}{b} \right )}}{b \log{\left (F \right )}} & \text{for}\: n = -1 \\\frac{F^{x} b \left (F^{x} b + a\right )^{n}}{b n \log{\left (F \right )} + b \log{\left (F \right )}} + \frac{a \left (F^{x} b + a\right )^{n}}{b n \log{\left (F \right )} + b \log{\left (F \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**x*(a+b*F**x)**n,x)

[Out]

Piecewise((x/a, Eq(F, 1) & Eq(b, 0) & Eq(n, -1)), (x*(a + b)**n, Eq(F, 1)), (F**x*a**n/log(F), Eq(b, 0)), (log
(F**x + a/b)/(b*log(F)), Eq(n, -1)), (F**x*b*(F**x*b + a)**n/(b*n*log(F) + b*log(F)) + a*(F**x*b + a)**n/(b*n*
log(F) + b*log(F)), True))

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Giac [A]  time = 1.26729, size = 32, normalized size = 1.33 \begin{align*} \frac{{\left (F^{x} b + a\right )}^{n + 1}}{b{\left (n + 1\right )} \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^x*(a+b*F^x)^n,x, algorithm="giac")

[Out]

(F^x*b + a)^(n + 1)/(b*(n + 1)*log(F))