\(\int \frac {(a+\frac {b}{x^2})^p (c+\frac {d}{x^2})^q}{x^5} \, dx\) [203]

Optimal result

Integrand size = 22, antiderivative size = 134 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^5} \, dx=-\frac {\left (a+\frac {b}{x^2}\right )^{1+p} \left (c+\frac {d}{x^2}\right )^{1+q}}{2 b d (2+p+q)}+\frac {(b c (1+p)+a d (1+q)) \left (a+\frac {b}{x^2}\right )^{1+p} \left (c+\frac {d}{x^2}\right )^{1+q} \operatorname {Hypergeometric2F1}\left (1,2+p+q,2+p,-\frac {d \left (a+\frac {b}{x^2}\right )}{b c-a d}\right )}{2 b d (b c-a d) (1+p) (2+p+q)} \] Output:

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

Mathematica [A] (verified)

Time = 5.21 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^5} \, dx=\frac {\left (a+\frac {b}{x^2}\right )^{1+p} \left (c+\frac {d}{x^2}\right )^q \left (-b \left (c+\frac {d}{x^2}\right )+\frac {(b c (1+p)+a d (1+q)) \left (\frac {b \left (d+c x^2\right )}{(b c-a d) x^2}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,-\frac {d \left (b+a x^2\right )}{(b c-a d) x^2}\right )}{1+p}\right )}{2 b^2 d (2+p+q)} \] Input:

Integrate[((a + b/x^2)^p*(c + d/x^2)^q)/x^5,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {948, 90, 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.

\(\displaystyle \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^5} \, dx\)

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

Int[((a + b/x^2)^p*(c + d/x^2)^q)/x^5,x]
 

Output:

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

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 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [F]

\[\int \frac {\left (a +\frac {b}{x^{2}}\right )^{p} \left (c +\frac {d}{x^{2}}\right )^{q}}{x^{5}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^5} \, dx=\int { \frac {{\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q}}{x^{5}} \,d x } \] Input:

integrate((a+b/x^2)^p*(c+d/x^2)^q/x^5,x, algorithm="fricas")
 

Output:

integral(((a*x^2 + b)/x^2)^p*((c*x^2 + d)/x^2)^q/x^5, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^5} \, dx=\text {Timed out} \] Input:

integrate((a+b/x**2)**p*(c+d/x**2)**q/x**5,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^5} \, dx=\int { \frac {{\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q}}{x^{5}} \,d x } \] Input:

integrate((a+b/x^2)^p*(c+d/x^2)^q/x^5,x, algorithm="maxima")
 

Output:

integrate((a + b/x^2)^p*(c + d/x^2)^q/x^5, x)
 

Giac [F]

\[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^5} \, dx=\int { \frac {{\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q}}{x^{5}} \,d x } \] Input:

integrate((a+b/x^2)^p*(c+d/x^2)^q/x^5,x, algorithm="giac")
 

Output:

integrate((a + b/x^2)^p*(c + d/x^2)^q/x^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^5} \, dx=\int \frac {{\left (a+\frac {b}{x^2}\right )}^p\,{\left (c+\frac {d}{x^2}\right )}^q}{x^5} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{x^5} \, dx=\text {too large to display} \] Input:

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

Output:

((c*x**2 + d)**q*(a*x**2 + b)**p*a**2*c*d*p*x**4 - (c*x**2 + d)**q*(a*x**2 
 + b)**p*a**2*d**2*p*q*x**2 + (c*x**2 + d)**q*(a*x**2 + b)**p*a*b*c**2*q*x 
**4 - (c*x**2 + d)**q*(a*x**2 + b)**p*a*b*c*d*p**2*x**2 - (c*x**2 + d)**q* 
(a*x**2 + b)**p*a*b*c*d*q**2*x**2 - (c*x**2 + d)**q*(a*x**2 + b)**p*a*b*d* 
*2*p*q - (c*x**2 + d)**q*(a*x**2 + b)**p*a*b*d**2*q**2 - (c*x**2 + d)**q*( 
a*x**2 + b)**p*a*b*d**2*q - (c*x**2 + d)**q*(a*x**2 + b)**p*b**2*c**2*p*q* 
x**2 - (c*x**2 + d)**q*(a*x**2 + b)**p*b**2*c*d*p**2 - (c*x**2 + d)**q*(a* 
x**2 + b)**p*b**2*c*d*p*q - (c*x**2 + d)**q*(a*x**2 + b)**p*b**2*c*d*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**2*q*x**5 + 2*x**(2*p + 2*q)*a**2*c*d*p*q**2*x**5 + 3*x**(2*p + 2*q 
)*a**2*c*d*p*q*x**5 + x**(2*p + 2*q)*a**2*c*d*q**3*x**5 + 3*x**(2*p + 2*q) 
*a**2*c*d*q**2*x**5 + 2*x**(2*p + 2*q)*a**2*c*d*q*x**5 + x**(2*p + 2*q)*a* 
*2*d**2*p**2*q*x**3 + 2*x**(2*p + 2*q)*a**2*d**2*p*q**2*x**3 + 3*x**(2*p + 
 2*q)*a**2*d**2*p*q*x**3 + x**(2*p + 2*q)*a**2*d**2*q**3*x**3 + 3*x**(2*p 
+ 2*q)*a**2*d**2*q**2*x**3 + 2*x**(2*p + 2*q)*a**2*d**2*q*x**3 + x**(2*p + 
 2*q)*a*b*c**2*p**3*x**5 + 2*x**(2*p + 2*q)*a*b*c**2*p**2*q*x**5 + 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**2*x**5 + 3*x** 
(2*p + 2*q)*a*b*c**2*p*q*x**5 + 2*x**(2*p + 2*q)*a*b*c**2*p*x**5 + x**(2*p 
 + 2*q)*a*b*c*d*p**3*x**3 + 3*x**(2*p + 2*q)*a*b*c*d*p**2*q*x**3 + 3*x**(2 
*p + 2*q)*a*b*c*d*p**2*x**3 + 3*x**(2*p + 2*q)*a*b*c*d*p*q**2*x**3 + 6*...