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

Optimal. Leaf size=41 \[ \frac{\left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1}}{b d e n (p+1) \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.0646444, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2246, 32} \[ \frac{\left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1}}{b d e n (p+1) \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [F]  time = 0.289949, size = 0, normalized size = 0. \[ \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 42, normalized size = 1. \begin{align*}{\frac{ \left ( a+b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n} \right ) ^{1+p}}{bden \left ( 1+p \right ) \ln \left ( F \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a+b*(F^(e*(d*x+c)))^n)^(1+p)/b/d/e/n/(1+p)/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^(e*(d*x+c)))^n*(a+b*(F^(e*(d*x+c)))^n)^p,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56138, size = 122, normalized size = 2.98 \begin{align*} \frac{{\left (F^{d e n x + c e n} b + a\right )}{\left (F^{d e n x + c e n} b + a\right )}^{p}}{{\left (b d e n p + b d e n\right )} \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^(d*e*n*x + c*e*n)*b + a)*(F^(d*e*n*x + c*e*n)*b + a)^p/((b*d*e*n*p + b*d*e*n)*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**(e*(d*x+c)))**n*(a+b*(F**(e*(d*x+c)))**n)**p,x)

[Out]

Timed out

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Giac [A]  time = 1.35188, size = 58, normalized size = 1.41 \begin{align*} \frac{{\left (F^{d n x e + c n e} b + a\right )}^{p + 1} e^{\left (-1\right )}}{b d n{\left (p + 1\right )} \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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