\(\int \frac {F^{a+b \log (c+d x^n)}}{x} \, dx\) [582]

Optimal result

Integrand size = 18, antiderivative size = 57 \[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x} \, dx=-\frac {F^a \left (c+d x^n\right )^{1+b \log (F)} \operatorname {Hypergeometric2F1}\left (1,1+b \log (F),2+b \log (F),1+\frac {d x^n}{c}\right )}{c n (1+b \log (F))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2306, 12, 272, 67} \[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x} \, dx=-\frac {F^a \left (c+d x^n\right )^{b \log (F)+1} \operatorname {Hypergeometric2F1}\left (1,b \log (F)+1,b \log (F)+2,\frac {d x^n}{c}+1\right )}{c n (b \log (F)+1)} \]

[In]

Int[F^(a + b*Log[c + d*x^n])/x,x]

[Out]

-((F^a*(c + d*x^n)^(1 + b*Log[F])*Hypergeometric2F1[1, 1 + b*Log[F], 2 + b*Log[F], 1 + (d*x^n)/c])/(c*n*(1 + 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 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 \frac {F^a \left (c+d x^n\right )^{b \log (F)}}{x} \, dx \\ & = F^a \int \frac {\left (c+d x^n\right )^{b \log (F)}}{x} \, dx \\ & = \frac {F^a \text {Subst}\left (\int \frac {(c+d x)^{b \log (F)}}{x} \, dx,x,x^n\right )}{n} \\ & = -\frac {F^a \left (c+d x^n\right )^{1+b \log (F)} \, _2F_1\left (1,1+b \log (F);2+b \log (F);1+\frac {d x^n}{c}\right )}{c n (1+b \log (F))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.88 \[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x} \, dx=-\frac {F^{a+b \log \left (c+d x^n\right )} \left (-1+\operatorname {Hypergeometric2F1}\left (1,b \log (F),1+b \log (F),1+\frac {d x^n}{c}\right )\right )}{b n \log (F)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(F^(a + b*log(c + d*x^n))/x,x)

[Out]

int(F^(a + b*log(c + d*x^n))/x, x)