\(\int \frac {(a+\frac {b}{x^2})^p (c+\frac {d}{x^2})^q}{x^3} \, dx\) [202]

Optimal result

Integrand size = 22, antiderivative size = 70 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^3} \, dx=-\frac {\left (a+\frac {b}{x^2}\right )^{1+p} \left (c+\frac {d}{x^2}\right )^{1+q} \operatorname {Hypergeometric2F1}\left (1,2+p+q,2+p,-\frac {d \left (a+\frac {b}{x^2}\right )}{b c-a d}\right )}{2 (b c-a d) (1+p)} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.07 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.57 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^3} \, dx=-\frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \left (1+\frac {a x^2}{b}\right )^{-p} \left (d+c x^2\right ) \left (1+\frac {c x^2}{d}\right )^p \operatorname {Hypergeometric2F1}\left (-p,-1-p-q,-p-q,\frac {(b c-a d) x^2}{b \left (d+c x^2\right )}\right )}{2 d (1+p+q) x^2} \] Input:

Integrate[((a + b/x^2)^p*(c + d/x^2)^q)/x^3,x]
 

Output:

-1/2*((a + b/x^2)^p*(c + d/x^2)^q*(d + c*x^2)*(1 + (c*x^2)/d)^p*Hypergeome 
tric2F1[-p, -1 - p - q, -p - q, ((b*c - a*d)*x^2)/(b*(d + c*x^2))])/(d*(1 
+ p + q)*x^2*(1 + (a*x^2)/b)^p)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.21, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {946, 80, 79}

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.

\(\displaystyle \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^3} \, dx\)

\(\Big \downarrow \) 946

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle -\frac {\left (a+\frac {b}{x^2}\right )^{p+1} \left (c+\frac {d}{x^2}\right )^q \left (\frac {b \left (c+\frac {d}{x^2}\right )}{b c-a d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (p+1,-q,p+2,-\frac {d \left (a+\frac {b}{x^2}\right )}{b c-a d}\right )}{2 b (p+1)}\)

Input:

Int[((a + b/x^2)^p*(c + d/x^2)^q)/x^3,x]
 

Output:

-1/2*((a + b/x^2)^(1 + p)*(c + d/x^2)^q*Hypergeometric2F1[1 + p, -q, 2 + p 
, -((d*(a + b/x^2))/(b*c - a*d))])/(b*(1 + p)*((b*(c + d/x^2))/(b*c - a*d) 
)^q)
 

Defintions of rubi rules used

rule 79

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

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 946

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^n], 
x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m - n 
+ 1, 0]
 

Maple [F]

\[\int \frac {\left (a +\frac {b}{x^{2}}\right )^{p} \left (c +\frac {d}{x^{2}}\right )^{q}}{x^{3}}d x\]

Input:

int((a+b/x^2)^p*(c+d/x^2)^q/x^3,x)
 

Output:

int((a+b/x^2)^p*(c+d/x^2)^q/x^3,x)
 

Fricas [F]

\[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^3} \, dx=\int { \frac {{\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q}}{x^{3}} \,d x } \] Input:

integrate((a+b/x^2)^p*(c+d/x^2)^q/x^3,x, algorithm="fricas")
 

Output:

integral(((a*x^2 + b)/x^2)^p*((c*x^2 + d)/x^2)^q/x^3, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^3} \, dx=\text {Timed out} \] Input:

integrate((a+b/x**2)**p*(c+d/x**2)**q/x**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^3} \, dx=\int { \frac {{\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q}}{x^{3}} \,d x } \] Input:

integrate((a+b/x^2)^p*(c+d/x^2)^q/x^3,x, algorithm="maxima")
 

Output:

integrate((a + b/x^2)^p*(c + d/x^2)^q/x^3, x)
 

Giac [F]

\[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^3} \, dx=\int { \frac {{\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q}}{x^{3}} \,d x } \] Input:

integrate((a+b/x^2)^p*(c+d/x^2)^q/x^3,x, algorithm="giac")
 

Output:

integrate((a + b/x^2)^p*(c + d/x^2)^q/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^3} \, dx=\int \frac {{\left (a+\frac {b}{x^2}\right )}^p\,{\left (c+\frac {d}{x^2}\right )}^q}{x^3} \,d x \] Input:

int(((a + b/x^2)^p*(c + d/x^2)^q)/x^3,x)
 

Output:

int(((a + b/x^2)^p*(c + d/x^2)^q)/x^3, x)
 

Reduce [F]

\[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^3} \, dx=\text {too large to display} \] Input:

int((a+b/x^2)^p*(c+d/x^2)^q/x^3,x)
 

Output:

( - (c*x**2 + d)**q*(a*x**2 + b)**p*a*c*p*x**2 - (c*x**2 + d)**q*(a*x**2 + 
 b)**p*a*c*q*x**2 - (c*x**2 + d)**q*(a*x**2 + b)**p*a*d*q - (c*x**2 + d)** 
q*(a*x**2 + b)**p*b*c*p + 2*x**(2*p + 2*q)*int(((c*x**2 + d)**q*(a*x**2 + 
b)**p)/(x**(2*p + 2*q)*a**2*c*d*p*q*x**5 + x**(2*p + 2*q)*a**2*c*d*q**2*x* 
*5 + x**(2*p + 2*q)*a**2*c*d*q*x**5 + x**(2*p + 2*q)*a**2*d**2*p*q*x**3 + 
x**(2*p + 2*q)*a**2*d**2*q**2*x**3 + x**(2*p + 2*q)*a**2*d**2*q*x**3 + x** 
(2*p + 2*q)*a*b*c**2*p**2*x**5 + x**(2*p + 2*q)*a*b*c**2*p*q*x**5 + x**(2* 
p + 2*q)*a*b*c**2*p*x**5 + x**(2*p + 2*q)*a*b*c*d*p**2*x**3 + 2*x**(2*p + 
2*q)*a*b*c*d*p*q*x**3 + x**(2*p + 2*q)*a*b*c*d*p*x**3 + x**(2*p + 2*q)*a*b 
*c*d*q**2*x**3 + x**(2*p + 2*q)*a*b*c*d*q*x**3 + x**(2*p + 2*q)*a*b*d**2*p 
*q*x + x**(2*p + 2*q)*a*b*d**2*q**2*x + x**(2*p + 2*q)*a*b*d**2*q*x + x**( 
2*p + 2*q)*b**2*c**2*p**2*x**3 + x**(2*p + 2*q)*b**2*c**2*p*q*x**3 + x**(2 
*p + 2*q)*b**2*c**2*p*x**3 + x**(2*p + 2*q)*b**2*c*d*p**2*x + x**(2*p + 2* 
q)*b**2*c*d*p*q*x + x**(2*p + 2*q)*b**2*c*d*p*x),x)*a**3*d**3*p**2*q**2*x* 
*2 + 2*x**(2*p + 2*q)*int(((c*x**2 + d)**q*(a*x**2 + b)**p)/(x**(2*p + 2*q 
)*a**2*c*d*p*q*x**5 + x**(2*p + 2*q)*a**2*c*d*q**2*x**5 + x**(2*p + 2*q)*a 
**2*c*d*q*x**5 + x**(2*p + 2*q)*a**2*d**2*p*q*x**3 + x**(2*p + 2*q)*a**2*d 
**2*q**2*x**3 + x**(2*p + 2*q)*a**2*d**2*q*x**3 + x**(2*p + 2*q)*a*b*c**2* 
p**2*x**5 + x**(2*p + 2*q)*a*b*c**2*p*q*x**5 + x**(2*p + 2*q)*a*b*c**2*p*x 
**5 + x**(2*p + 2*q)*a*b*c*d*p**2*x**3 + 2*x**(2*p + 2*q)*a*b*c*d*p*q*x...