\(\int F^{a+b (c+d x)^n} \, dx\) [363]

Optimal result

Integrand size = 13, antiderivative size = 50 \[ \int F^{a+b (c+d x)^n} \, dx=-\frac {F^a (c+d x) \Gamma \left (\frac {1}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{-1/n}}{d n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2239} \[ \int F^{a+b (c+d x)^n} \, dx=-\frac {F^a (c+d x) \left (-b \log (F) (c+d x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b (c+d x)^n \log (F)\right )}{d n} \]

[In]

Int[F^(a + b*(c + d*x)^n),x]

[Out]

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

Rule 2239

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a)*(c + d*x)*(Gamma[1/n, (-b)*(c + d
*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log[F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps \begin{align*} \text {integral}& = -\frac {F^a (c+d x) \Gamma \left (\frac {1}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{-1/n}}{d n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int F^{a+b (c+d x)^n} \, dx=-\frac {F^a (c+d x) \Gamma \left (\frac {1}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{-1/n}}{d n} \]

[In]

Integrate[F^(a + b*(c + d*x)^n),x]

[Out]

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

Maple [F]

\[\int F^{a +b \left (d x +c \right )^{n}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int F^{a+b (c+d x)^n} \, dx=\int { F^{{\left (d x + c\right )}^{n} b + a} \,d x } \]

[In]

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

[Out]

integral(F^((d*x + c)^n*b + a), x)

Sympy [F]

\[ \int F^{a+b (c+d x)^n} \, dx=\int F^{a + b \left (c + d x\right )^{n}}\, dx \]

[In]

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

[Out]

Integral(F**(a + b*(c + d*x)**n), x)

Maxima [F]

\[ \int F^{a+b (c+d x)^n} \, dx=\int { F^{{\left (d x + c\right )}^{n} b + a} \,d x } \]

[In]

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

[Out]

integrate(F^((d*x + c)^n*b + a), x)

Giac [F]

\[ \int F^{a+b (c+d x)^n} \, dx=\int { F^{{\left (d x + c\right )}^{n} b + a} \,d x } \]

[In]

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

[Out]

integrate(F^((d*x + c)^n*b + a), x)

Mupad [F(-1)]

Timed out. \[ \int F^{a+b (c+d x)^n} \, dx=\int F^{a+b\,{\left (c+d\,x\right )}^n} \,d x \]

[In]

int(F^(a + b*(c + d*x)^n),x)

[Out]

int(F^(a + b*(c + d*x)^n), x)