Integrand size = 18, antiderivative size = 58 \[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x} \, dx=-\frac {F^a \left (c+d x^n\right )^{1+b \log (F)} \operatorname {Hypergeometric2F1}\left (1,1+b \log (F),2+b \log (F),\frac {c+d x^n}{c}\right )}{c n (1+b \log (F))} \] Output:
-F^a*(c+d*x^n)^(1+b*ln(F))*hypergeom([1, 1+b*ln(F)],[2+b*ln(F)],(c+d*x^n)/ c)/c/n/(1+b*ln(F))
Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.86 \[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x} \, dx=-\frac {F^{a+b \log \left (c+d x^n\right )} \left (-1+\operatorname {Hypergeometric2F1}\left (1,b \log (F),1+b \log (F),1+\frac {d x^n}{c}\right )\right )}{b n \log (F)} \] Input:
Integrate[F^(a + b*Log[c + d*x^n])/x,x]
Output:
-((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]))
Time = 0.34 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2704, 27, 798, 75}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x} \, dx\) |
\(\Big \downarrow \) 2704 |
\(\displaystyle \int \frac {F^a \left (c+d x^n\right )^{b \log (F)}}{x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle F^a \int \frac {\left (d x^n+c\right )^{b \log (F)}}{x}dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {F^a \int x^{-n} \left (d x^n+c\right )^{b \log (F)}dx^n}{n}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle -\frac {F^a \left (c+d x^n\right )^{b \log (F)+1} \operatorname {Hypergeometric2F1}\left (1,b \log (F)+1,b \log (F)+2,\frac {d x^n}{c}+1\right )}{c n (b \log (F)+1)}\) |
Input:
Int[F^(a + b*Log[c + d*x^n])/x,x]
Output:
-((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])))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
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]
\[\int \frac {F^{a +b \ln \left (c +d \,x^{n}\right )}}{x}d x\]
Input:
int(F^(a+b*ln(c+d*x^n))/x,x)
Output:
int(F^(a+b*ln(c+d*x^n))/x,x)
\[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x} \, dx=\int { \frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x} \,d x } \] Input:
integrate(F^(a+b*log(c+d*x^n))/x,x, algorithm="fricas")
Output:
integral(F^(b*log(d*x^n + c) + a)/x, x)
\[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x} \, dx=\int \frac {F^{a + b \log {\left (c + d x^{n} \right )}}}{x}\, dx \] Input:
integrate(F**(a+b*ln(c+d*x**n))/x,x)
Output:
Integral(F**(a + b*log(c + d*x**n))/x, x)
\[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x} \, dx=\int { \frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x} \,d x } \] Input:
integrate(F^(a+b*log(c+d*x^n))/x,x, algorithm="maxima")
Output:
integrate(F^(b*log(d*x^n + c) + a)/x, x)
\[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x} \, dx=\int { \frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x} \,d x } \] Input:
integrate(F^(a+b*log(c+d*x^n))/x,x, algorithm="giac")
Output:
integrate(F^(b*log(d*x^n + c) + a)/x, x)
Timed out. \[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x} \, dx=\int \frac {F^{a+b\,\ln \left (c+d\,x^n\right )}}{x} \,d x \] Input:
int(F^(a + b*log(c + d*x^n))/x,x)
Output:
int(F^(a + b*log(c + d*x^n))/x, x)
\[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x} \, dx=\frac {f^{a} \left (f^{\mathrm {log}\left (x^{n} d +c \right ) b}+\left (\int \frac {f^{\mathrm {log}\left (x^{n} d +c \right ) b}}{x^{n} d x +c x}d x \right ) \mathrm {log}\left (f \right ) b c n \right )}{\mathrm {log}\left (f \right ) b n} \] Input:
int(F^(a+b*log(c+d*x^n))/x,x)
Output:
(f**a*(f**(log(x**n*d + c)*b) + int(f**(log(x**n*d + c)*b)/(x**n*d*x + c*x ),x)*log(f)*b*c*n))/(log(f)*b*n)