\(\int F^{a+b \log (c+d x^n)} \, dx\) [581]

Optimal result

Integrand size = 14, antiderivative size = 56 \[ \int F^{a+b \log \left (c+d x^n\right )} \, dx=F^a x \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)} \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-b \log (F),1+\frac {1}{n},-\frac {d x^n}{c}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2306, 12, 252, 251} \[ \int F^{a+b \log \left (c+d x^n\right )} \, dx=x F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac {d x^n}{c}+1\right )^{-b \log (F)} \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-b \log (F),1+\frac {1}{n},-\frac {d x^n}{c}\right ) \]

[In]

Int[F^(a + b*Log[c + d*x^n]),x]

[Out]

(F^a*x*(c + d*x^n)^(b*Log[F])*Hypergeometric2F1[n^(-1), -(b*Log[F]), 1 + n^(-1), -((d*x^n)/c)])/(1 + (d*x^n)/c
)^(b*Log[F])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 2306

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rubi steps \begin{align*} \text {integral}& = \int F^a \left (c+d x^n\right )^{b \log (F)} \, dx \\ & = F^a \int \left (c+d x^n\right )^{b \log (F)} \, dx \\ & = \left (F^a \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)}\right ) \int \left (1+\frac {d x^n}{c}\right )^{b \log (F)} \, dx \\ & = F^a x \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)} \, _2F_1\left (\frac {1}{n},-b \log (F);1+\frac {1}{n};-\frac {d x^n}{c}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.48 \[ \int F^{a+b \log \left (c+d x^n\right )} \, dx=-\frac {F^{a+b \log \left (c+d x^n\right )} x \left (-\frac {d x^n}{c}\right )^{-1/n} \left (c+d x^n\right ) \operatorname {Hypergeometric2F1}\left (\frac {-1+n}{n},1+b \log (F),2+b \log (F),1+\frac {d x^n}{c}\right )}{c n (1+b \log (F))} \]

[In]

Integrate[F^(a + b*Log[c + d*x^n]),x]

[Out]

-((F^(a + b*Log[c + d*x^n])*x*(c + d*x^n)*Hypergeometric2F1[(-1 + n)/n, 1 + b*Log[F], 2 + b*Log[F], 1 + (d*x^n
)/c])/(c*n*(-((d*x^n)/c))^n^(-1)*(1 + b*Log[F])))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(F^(b*log(d*x^n + c) + a), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.04 \[ \int F^{a+b \log \left (c+d x^n\right )} \, dx=\frac {F^a\,x\,{\left (c+d\,x^n\right )}^{b\,\ln \left (F\right )}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{n},-b\,\ln \left (F\right );\ \frac {1}{n}+1;\ -\frac {d\,x^n}{c}\right )}{{\left (\frac {d\,x^n}{c}+1\right )}^{b\,\ln \left (F\right )}} \]

[In]

int(F^(a + b*log(c + d*x^n)),x)

[Out]

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