\(\int \frac {F^{a+b \log (c+d x^n)}}{x^2} \, dx\) [505]

Optimal result

Integrand size = 18, antiderivative size = 66 \[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=-\frac {F^a \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{n},-b \log (F),-\frac {1-n}{n},-\frac {d x^n}{c}\right )}{x} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.23 \[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=-\frac {F^{a+b \log \left (c+d x^n\right )} \left (-\frac {d x^n}{c}\right )^{\frac {1}{n}} \left (c+d x^n\right ) \operatorname {Hypergeometric2F1}\left (1+\frac {1}{n},1+b \log (F),2+b \log (F),1+\frac {d x^n}{c}\right )}{c n x (1+b \log (F))} \] Input:

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

Output:

-((F^(a + b*Log[c + d*x^n])*(-((d*x^n)/c))^n^(-1)*(c + d*x^n)*Hypergeometr 
ic2F1[1 + n^(-1), 1 + b*Log[F], 2 + b*Log[F], 1 + (d*x^n)/c])/(c*n*x*(1 + 
b*Log[F])))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 66, 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 = {2704, 27, 889, 888}

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.

\(\displaystyle \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx\)

\(\Big \downarrow \) 2704

\(\displaystyle \int \frac {F^a \left (c+d x^n\right )^{b \log (F)}}{x^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle F^a \int \frac {\left (d x^n+c\right )^{b \log (F)}}{x^2}dx\)

\(\Big \downarrow \) 889

\(\displaystyle F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac {d x^n}{c}+1\right )^{-b \log (F)} \int \frac {\left (\frac {d x^n}{c}+1\right )^{b \log (F)}}{x^2}dx\)

\(\Big \downarrow \) 888

\(\displaystyle -\frac {F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac {d x^n}{c}+1\right )^{-b \log (F)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{n},-b \log (F),-\frac {1-n}{n},-\frac {d x^n}{c}\right )}{x}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 888

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

rule 889

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I 
ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(c*x) 
^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0 
] &&  !(ILtQ[p, 0] || GtQ[a, 0])
 

rule 2704

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]
 

Maple [F]

\[\int \frac {F^{a +b \ln \left (c +d \,x^{n}\right )}}{x^{2}}d x\]

Input:

int(F^(a+b*ln(c+d*x^n))/x^2,x)
 

Output:

int(F^(a+b*ln(c+d*x^n))/x^2,x)
 

Fricas [F]

\[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=\int { \frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x^{2}} \,d x } \] Input:

integrate(F^(a+b*log(c+d*x^n))/x^2,x, algorithm="fricas")
 

Output:

integral(F^(b*log(d*x^n + c) + a)/x^2, x)
 

Sympy [F]

\[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=\int \frac {F^{a + b \log {\left (c + d x^{n} \right )}}}{x^{2}}\, dx \] Input:

integrate(F**(a+b*ln(c+d*x**n))/x**2,x)
 

Output:

Integral(F**(a + b*log(c + d*x**n))/x**2, x)
 

Maxima [F]

\[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=\int { \frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x^{2}} \,d x } \] Input:

integrate(F^(a+b*log(c+d*x^n))/x^2,x, algorithm="maxima")
 

Output:

integrate(F^(b*log(d*x^n + c) + a)/x^2, x)
 

Giac [F]

\[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=\int { \frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x^{2}} \,d x } \] Input:

integrate(F^(a+b*log(c+d*x^n))/x^2,x, algorithm="giac")
 

Output:

integrate(F^(b*log(d*x^n + c) + a)/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=\int \frac {F^{a+b\,\ln \left (c+d\,x^n\right )}}{x^2} \,d x \] Input:

int(F^(a + b*log(c + d*x^n))/x^2,x)
 

Output:

int(F^(a + b*log(c + d*x^n))/x^2, x)
 

Reduce [F]

\[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=\frac {f^{a} \left (f^{\mathrm {log}\left (x^{n} d +c \right ) b}+\left (\int \frac {f^{\mathrm {log}\left (x^{n} d +c \right ) b}}{x^{n} \mathrm {log}\left (f \right ) b d n \,x^{2}-x^{n} d \,x^{2}+\mathrm {log}\left (f \right ) b c n \,x^{2}-c \,x^{2}}d x \right ) \mathrm {log}\left (f \right )^{2} b^{2} c \,n^{2} x -\left (\int \frac {f^{\mathrm {log}\left (x^{n} d +c \right ) b}}{x^{n} \mathrm {log}\left (f \right ) b d n \,x^{2}-x^{n} d \,x^{2}+\mathrm {log}\left (f \right ) b c n \,x^{2}-c \,x^{2}}d x \right ) \mathrm {log}\left (f \right ) b c n x \right )}{x \left (\mathrm {log}\left (f \right ) b n -1\right )} \] Input:

int(F^(a+b*log(c+d*x^n))/x^2,x)
 

Output:

(f**a*(f**(log(x**n*d + c)*b) + int(f**(log(x**n*d + c)*b)/(x**n*log(f)*b* 
d*n*x**2 - x**n*d*x**2 + log(f)*b*c*n*x**2 - c*x**2),x)*log(f)**2*b**2*c*n 
**2*x - int(f**(log(x**n*d + c)*b)/(x**n*log(f)*b*d*n*x**2 - x**n*d*x**2 + 
 log(f)*b*c*n*x**2 - c*x**2),x)*log(f)*b*c*n*x))/(x*(log(f)*b*n - 1))