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

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Rubi [A]  time = 0.06, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 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 \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*}

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Mathematica [F]  time = 0.31, size = 0, normalized size = 0.00 \[ \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx \]

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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fricas [A]  time = 0.43, size = 53, normalized size = 1.29 \[ \frac {{\left (F^{d e n x + c e n} b + a\right )} {\left (F^{d e n x + c e n} b + a\right )}^{p}}{{\left (b d e n p + b d e n\right )} \log \relax (F)} \]

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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giac [A]  time = 0.50, size = 43, normalized size = 1.05 \[ \frac {{\left (F^{d n x e + c n e} b + a\right )}^{p + 1} e^{\left (-1\right )}}{b d n {\left (p + 1\right )} \log \relax (F)} \]

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))

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maple [A]  time = 0.01, size = 42, normalized size = 1.02 \[ \frac {\left (b \left (F^{\left (d x +c \right ) e}\right )^{n}+a \right )^{p +1}}{\left (p +1\right ) b d e n \ln \relax (F )} \]

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)^(p+1)/b/d/e/n/(p+1)/ln(F)

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maxima [A]  time = 0.44, size = 41, normalized size = 1.00 \[ \frac {{\left ({\left (F^{{\left (d x + c\right )} e}\right )}^{n} b + a\right )}^{p + 1}}{b d e n {\left (p + 1\right )} \log \relax (F)} \]

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]

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

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mupad [B]  time = 3.51, size = 74, normalized size = 1.80 \[ \left (\frac {{\left (F^{c\,e+d\,e\,x}\right )}^n}{d\,e\,n\,\ln \relax (F)\,\left (p+1\right )}+\frac {a}{b\,d\,e\,n\,\ln \relax (F)\,\left (p+1\right )}\right )\,{\left (a+b\,{\left (F^{c\,e+d\,e\,x}\right )}^n\right )}^p \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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