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

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 = 0.0734713, 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, Rules used = {2247, 2246, 32} \[ \frac{F^{-c} \left (a+b F^{c+d x}\right )^{n+1}}{b d (n+1) \log (F)} \]

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])

Rule 2247

Int[((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.)*((G_)^((h_.)*((f_.) + (g_.)*(x_))))^(m_.),
x_Symbol] :> Dist[(G^(h*(f + g*x)))^m/(F^(e*(c + d*x)))^n, Int[(F^(e*(c + d*x)))^n*(a + b*(F^(e*(c + d*x)))^n)
^p, x], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[d*e*n*Log[F], g*h*m*Log[G]]

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

Mathematica [A]  time = 0.0428595, 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 = 0.025, size = 81, normalized size = 2.3 \begin{align*}{\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 ) }}}{b{F}^{c}d\ln \left ( F \right ) \left ( 1+n \right ) }} \end{align*}

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/(F^c)/ln(F)/b/d/(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. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59747, size = 100, normalized size = 2.78 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(d*x)*(a+b*F^(d*x+c))^n,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. \begin{align*} \text{Timed out} \end{align*}

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 [A]  time = 1.26914, size = 82, normalized size = 2.28 \begin{align*} \frac{{\left (F^{d x} F^{c} b + a\right )}^{n} F^{d x} F^{c} b +{\left (F^{d x} F^{c} b + a\right )}^{n} a}{{\left (b n + b\right )} F^{c} d \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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