\(\int (a+\frac {b}{x^2})^p (c+\frac {d}{x^2})^q \, dx\) [196]

Optimal result

Integrand size = 19, antiderivative size = 79 \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \, dx=\left (a+\frac {b}{x^2}\right )^p \left (1+\frac {b}{a x^2}\right )^{-p} \left (c+\frac {d}{x^2}\right )^q \left (1+\frac {d}{c x^2}\right )^{-q} x \operatorname {AppellF1}\left (-\frac {1}{2},-p,-q,\frac {1}{2},-\frac {b}{a x^2},-\frac {d}{c x^2}\right ) \] Output:

(a+b/x^2)^p*(c+d/x^2)^q*x*AppellF1(-1/2,-p,-q,1/2,-b/a/x^2,-d/c/x^2)/((1+b 
/a/x^2)^p)/((1+d/c/x^2)^q)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.32 \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \, dx=-\frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q x \left (1+\frac {a x^2}{b}\right )^{-p} \left (1+\frac {c x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2}-p-q,-p,-q,\frac {3}{2}-p-q,-\frac {a x^2}{b},-\frac {c x^2}{d}\right )}{-1+2 p+2 q} \] Input:

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

Output:

-(((a + b/x^2)^p*(c + d/x^2)^q*x*AppellF1[1/2 - p - q, -p, -q, 3/2 - p - q 
, -((a*x^2)/b), -((c*x^2)/d)])/((-1 + 2*p + 2*q)*(1 + (a*x^2)/b)^p*(1 + (c 
*x^2)/d)^q))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {899, 395, 395, 394}

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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 394

\(\displaystyle x \left (a+\frac {b}{x^2}\right )^p \left (\frac {b}{a x^2}+1\right )^{-p} \left (c+\frac {d}{x^2}\right )^q \left (\frac {d}{c x^2}+1\right )^{-q} \operatorname {AppellF1}\left (-\frac {1}{2},-p,-q,\frac {1}{2},-\frac {b}{a x^2},-\frac {d}{c x^2}\right )\)

Input:

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

Output:

((a + b/x^2)^p*(c + d/x^2)^q*x*AppellF1[-1/2, -p, -q, 1/2, -(b/(a*x^2)), - 
(d/(c*x^2))])/((1 + b/(a*x^2))^p*(1 + d/(c*x^2))^q)
 

Defintions of rubi rules used

rule 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 395

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^ 
FracPart[p])   Int[(e*x)^m*(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ 
[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 
1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 899

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol 
] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, 
 b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \, dx=\frac {\left (c \,x^{2}+d \right )^{q} \left (a \,x^{2}+b \right )^{p} x +2 x^{2 p +2 q} \left (\int \frac {\left (c \,x^{2}+d \right )^{q} \left (a \,x^{2}+b \right )^{p} x^{2}}{x^{2 p +2 q} a c \,x^{4}+x^{2 p +2 q} a d \,x^{2}+x^{2 p +2 q} b c \,x^{2}+x^{2 p +2 q} b d}d x \right ) a d q +2 x^{2 p +2 q} \left (\int \frac {\left (c \,x^{2}+d \right )^{q} \left (a \,x^{2}+b \right )^{p} x^{2}}{x^{2 p +2 q} a c \,x^{4}+x^{2 p +2 q} a d \,x^{2}+x^{2 p +2 q} b c \,x^{2}+x^{2 p +2 q} b d}d x \right ) b c p +2 x^{2 p +2 q} \left (\int \frac {\left (c \,x^{2}+d \right )^{q} \left (a \,x^{2}+b \right )^{p}}{x^{2 p +2 q} a c \,x^{4}+x^{2 p +2 q} a d \,x^{2}+x^{2 p +2 q} b c \,x^{2}+x^{2 p +2 q} b d}d x \right ) b d p +2 x^{2 p +2 q} \left (\int \frac {\left (c \,x^{2}+d \right )^{q} \left (a \,x^{2}+b \right )^{p}}{x^{2 p +2 q} a c \,x^{4}+x^{2 p +2 q} a d \,x^{2}+x^{2 p +2 q} b c \,x^{2}+x^{2 p +2 q} b d}d x \right ) b d q}{x^{2 p +2 q}} \] Input:

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

Output:

((c*x**2 + d)**q*(a*x**2 + b)**p*x + 2*x**(2*p + 2*q)*int(((c*x**2 + d)**q 
*(a*x**2 + b)**p*x**2)/(x**(2*p + 2*q)*a*c*x**4 + x**(2*p + 2*q)*a*d*x**2 
+ x**(2*p + 2*q)*b*c*x**2 + x**(2*p + 2*q)*b*d),x)*a*d*q + 2*x**(2*p + 2*q 
)*int(((c*x**2 + d)**q*(a*x**2 + b)**p*x**2)/(x**(2*p + 2*q)*a*c*x**4 + x* 
*(2*p + 2*q)*a*d*x**2 + x**(2*p + 2*q)*b*c*x**2 + x**(2*p + 2*q)*b*d),x)*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*c*x**4 + x**(2*p + 2*q)*a*d*x**2 + x**(2*p + 2*q)*b*c*x**2 + x**(2*p 
 + 2*q)*b*d),x)*b*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*c*x**4 + x**(2*p + 2*q)*a*d*x**2 + x**(2*p + 2*q)*b 
*c*x**2 + x**(2*p + 2*q)*b*d),x)*b*d*q)/x**(2*p + 2*q)