Integrand size = 19, antiderivative size = 65 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^3} \, dx=-\frac {b d \left (a+\frac {b}{c+d x^2}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,\frac {c \left (a+\frac {b}{c+d x^2}\right )}{b+a c}\right )}{2 (b+a c)^2 (1+p)} \] Output:
-1/2*b*d*(a+b/(d*x^2+c))^(p+1)*hypergeom([2, p+1],[2+p],c*(a+b/(d*x^2+c))/ (a*c+b))/(a*c+b)^2/(p+1)
\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^3} \, dx=\int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^3} \, dx \] Input:
Integrate[(a + b/(c + d*x^2))^p/x^3,x]
Output:
Integrate[(a + b/(c + d*x^2))^p/x^3, x]
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.
\(\displaystyle \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^3} \, dx\) |
\(\Big \downarrow \) 2057 |
\(\displaystyle \int \frac {\left (\frac {a c+a d x^2+b}{c+d x^2}\right )^p}{x^3}dx\) |
\(\Big \downarrow \) 2053 |
\(\displaystyle \frac {1}{2} \int \frac {\left (\frac {a d x^2+b+a c}{d x^2+c}\right )^p}{x^4}dx^2\) |
Input:
Int[(a + b/(c + d*x^2))^p/x^3,x]
Output:
$Aborted
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]]
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]
\[\int \frac {\left (a +\frac {b}{d \,x^{2}+c}\right )^{p}}{x^{3}}d x\]
Input:
int((a+b/(d*x^2+c))^p/x^3,x)
Output:
int((a+b/(d*x^2+c))^p/x^3,x)
\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^3} \, dx=\int { \frac {{\left (a + \frac {b}{d x^{2} + c}\right )}^{p}}{x^{3}} \,d x } \] Input:
integrate((a+b/(d*x^2+c))^p/x^3,x, algorithm="fricas")
Output:
integral(((a*d*x^2 + a*c + b)/(d*x^2 + c))^p/x^3, x)
Timed out. \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^3} \, dx=\text {Timed out} \] Input:
integrate((a+b/(d*x**2+c))**p/x**3,x)
Output:
Timed out
\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^3} \, dx=\int { \frac {{\left (a + \frac {b}{d x^{2} + c}\right )}^{p}}{x^{3}} \,d x } \] Input:
integrate((a+b/(d*x^2+c))^p/x^3,x, algorithm="maxima")
Output:
integrate((a + b/(d*x^2 + c))^p/x^3, x)
\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^3} \, dx=\int { \frac {{\left (a + \frac {b}{d x^{2} + c}\right )}^{p}}{x^{3}} \,d x } \] Input:
integrate((a+b/(d*x^2+c))^p/x^3,x, algorithm="giac")
Output:
integrate((a + b/(d*x^2 + c))^p/x^3, x)
Timed out. \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^3} \, dx=\int \frac {{\left (a+\frac {b}{d\,x^2+c}\right )}^p}{x^3} \,d x \] Input:
int((a + b/(c + d*x^2))^p/x^3,x)
Output:
int((a + b/(c + d*x^2))^p/x^3, x)
\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^p}{x^3} \, dx=\int \frac {\left (a d \,x^{2}+a c +b \right )^{p}}{\left (d \,x^{2}+c \right )^{p} x^{3}}d x \] Input:
int((a+b/(d*x^2+c))^p/x^3,x)
Output:
int((a*c + a*d*x**2 + b)**p/((c + d*x**2)**p*x**3),x)