\(\int \frac {(a+b x^2)^p (c+d x^2)^q}{x^2} \, dx\) [1644]

Optimal result

Integrand size = 22, antiderivative size = 82 \[ \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )^q}{x^2} \, dx=-\frac {\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} \operatorname {AppellF1}\left (-\frac {1}{2},-p,-q,\frac {1}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{x} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )^q}{x^2} \, dx=-\frac {\left (a+b x^2\right )^p \left (\frac {a+b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (\frac {c+d x^2}{c}\right )^{-q} \operatorname {AppellF1}\left (-\frac {1}{2},-p,-q,\frac {1}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{x} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {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.

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

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 394

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

Input:

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

Output:

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

((c + d*x**2)**q*(a + b*x**2)**p*a*d + (c + d*x**2)**q*(a + b*x**2)**p*b*c 
 - 4*int(((c + d*x**2)**q*(a + b*x**2)**p*x**2)/(2*a**2*c*d*q - a**2*c*d + 
 2*a**2*d**2*q*x**2 - a**2*d**2*x**2 + 2*a*b*c**2*p - a*b*c**2 + 2*a*b*c*d 
*p*x**2 + 2*a*b*c*d*q*x**2 - 2*a*b*c*d*x**2 + 2*a*b*d**2*q*x**4 - a*b*d**2 
*x**4 + 2*b**2*c**2*p*x**2 - b**2*c**2*x**2 + 2*b**2*c*d*p*x**4 - b**2*c*d 
*x**4),x)*a**2*b*d**3*p*q*x + 2*int(((c + d*x**2)**q*(a + b*x**2)**p*x**2) 
/(2*a**2*c*d*q - a**2*c*d + 2*a**2*d**2*q*x**2 - a**2*d**2*x**2 + 2*a*b*c* 
*2*p - a*b*c**2 + 2*a*b*c*d*p*x**2 + 2*a*b*c*d*q*x**2 - 2*a*b*c*d*x**2 + 2 
*a*b*d**2*q*x**4 - a*b*d**2*x**4 + 2*b**2*c**2*p*x**2 - b**2*c**2*x**2 + 2 
*b**2*c*d*p*x**4 - b**2*c*d*x**4),x)*a**2*b*d**3*p*x - 4*int(((c + d*x**2) 
**q*(a + b*x**2)**p*x**2)/(2*a**2*c*d*q - a**2*c*d + 2*a**2*d**2*q*x**2 - 
a**2*d**2*x**2 + 2*a*b*c**2*p - a*b*c**2 + 2*a*b*c*d*p*x**2 + 2*a*b*c*d*q* 
x**2 - 2*a*b*c*d*x**2 + 2*a*b*d**2*q*x**4 - a*b*d**2*x**4 + 2*b**2*c**2*p* 
x**2 - b**2*c**2*x**2 + 2*b**2*c*d*p*x**4 - b**2*c*d*x**4),x)*a*b**2*c*d** 
2*p**2*x + 2*int(((c + d*x**2)**q*(a + b*x**2)**p*x**2)/(2*a**2*c*d*q - a* 
*2*c*d + 2*a**2*d**2*q*x**2 - a**2*d**2*x**2 + 2*a*b*c**2*p - a*b*c**2 + 2 
*a*b*c*d*p*x**2 + 2*a*b*c*d*q*x**2 - 2*a*b*c*d*x**2 + 2*a*b*d**2*q*x**4 - 
a*b*d**2*x**4 + 2*b**2*c**2*p*x**2 - b**2*c**2*x**2 + 2*b**2*c*d*p*x**4 - 
b**2*c*d*x**4),x)*a*b**2*c*d**2*p*x - 4*int(((c + d*x**2)**q*(a + b*x**2)* 
*p*x**2)/(2*a**2*c*d*q - a**2*c*d + 2*a**2*d**2*q*x**2 - a**2*d**2*x**2...