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

Optimal. Leaf size=31 \[ \frac {\left (a+b F^{c+d x}\right )^{n+1}}{b d (n+1) \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2246, 32} \[ \frac {\left (a+b F^{c+d x}\right )^{n+1}}{b d (n+1) \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 30, normalized size = 0.97 \[ \frac {\left (a+b F^{c+d x}\right )^{n+1}}{b d n \log (F)+b d \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 39, normalized size = 1.26 \[ \frac {{\left (F^{d x + c} b + a\right )} {\left (F^{d x + c} b + a\right )}^{n}}{{\left (b d n + b d\right )} \log \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.44, size = 31, normalized size = 1.00 \[ \frac {{\left (F^{d x + c} b + a\right )}^{n + 1}}{b d {\left (n + 1\right )} \log \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 32, normalized size = 1.03 \[ \frac {\left (b \,F^{d x +c}+a \right )^{n +1}}{\left (n +1\right ) b d \ln \relax (F )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.44, size = 31, normalized size = 1.00 \[ \frac {{\left (F^{d x + c} b + a\right )}^{n + 1}}{b d {\left (n + 1\right )} \log \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.43, size = 52, normalized size = 1.68 \[ {\left (a+F^{c+d\,x}\,b\right )}^n\,\left (\frac {F^{c+d\,x}}{d\,\ln \relax (F)\,\left (n+1\right )}+\frac {a}{b\,d\,\ln \relax (F)\,\left (n+1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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