\(\int \frac {(a+\frac {b}{c+d x^2})^p}{x} \, dx\) [215]

Optimal result

Integrand size = 19, antiderivative size = 118 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x} \, dx=-\frac {c \left (a+\frac {b}{c+d x^2}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {c \left (a+\frac {b}{c+d x^2}\right )}{b+a c}\right )}{2 (b+a c) (1+p)}+\frac {\left (a+\frac {b}{c+d x^2}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b}{a \left (c+d x^2\right )}\right )}{2 a (1+p)} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [F]

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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2053

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2053

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

rule 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

Maple [F]

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

Input:

int((a+b/(d*x^2+c))^p/x,x)
 

Output:

int((a+b/(d*x^2+c))^p/x,x)
 

Fricas [F]

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

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

Output:

integral(((a*d*x^2 + a*c + b)/(d*x^2 + c))^p/x, x)
 

Sympy [F]

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

integrate((a+b/(d*x**2+c))**p/x,x)
 

Output:

Integral(((a*c + a*d*x**2 + b)/(c + d*x**2))**p/x, x)
 

Maxima [F]

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

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

Output:

integrate((a + b/(d*x^2 + c))^p/x, x)
 

Giac [F]

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

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

Output:

integrate((a + b/(d*x^2 + c))^p/x, x)
 

Mupad [F(-1)]

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

int((a + b/(c + d*x^2))^p/x,x)
 

Output:

int((a + b/(c + d*x^2))^p/x, x)
 

Reduce [F]

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

int((a+b/(d*x^2+c))^p/x,x)
 

Output:

int((a*c + a*d*x**2 + b)**p/((c + d*x**2)**p*x),x)