\(\int \frac {(a+b x^2)^p (c+d x^2)^q (e+f x^2)^2}{x^4} \, dx\) [84]

Optimal result

Integrand size = 31, antiderivative size = 406 \[ \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2}{x^4} \, dx=-\frac {e^2 \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{3 a c x^3}-\frac {e (6 a c f-b c e (1-2 p)-a d e (1-2 q)) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{3 a^2 c^2 x}-\frac {\left (b^2 c^2 e^2 \left (1-4 p^2\right )-a b c e (6 c f (1+2 p)-d e (1-2 p-2 q-8 p q))-a^2 \left (3 c^2 f^2+6 c d e f (1+2 q)-d^2 e^2 \left (1-4 q^2\right )\right )\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{3 a^2 c^2}+\frac {b d e (6 a c f-b c e (1-2 p)-a d e (1-2 q)) (3+2 p+2 q) x^3 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{9 a^2 c^2} \] Output:

-1/3*e^2*(b*x^2+a)^(p+1)*(d*x^2+c)^(1+q)/a/c/x^3-1/3*e*(6*a*c*f-b*c*e*(1-2 
*p)-a*d*e*(1-2*q))*(b*x^2+a)^(p+1)*(d*x^2+c)^(1+q)/a^2/c^2/x-1/3*(b^2*c^2* 
e^2*(-4*p^2+1)-a*b*c*e*(6*c*f*(1+2*p)-d*e*(-8*p*q-2*p-2*q+1))-a^2*(3*c^2*f 
^2+6*c*d*e*f*(1+2*q)-d^2*e^2*(-4*q^2+1)))*x*(b*x^2+a)^p*(d*x^2+c)^q*Appell 
F1(1/2,-p,-q,3/2,-b*x^2/a,-d*x^2/c)/a^2/c^2/((1+b*x^2/a)^p)/((1+d*x^2/c)^q 
)+1/9*b*d*e*(6*a*c*f-b*c*e*(1-2*p)-a*d*e*(1-2*q))*(3+2*p+2*q)*x^3*(b*x^2+a 
)^p*(d*x^2+c)^q*AppellF1(3/2,-p,-q,5/2,-b*x^2/a,-d*x^2/c)/a^2/c^2/((1+b*x^ 
2/a)^p)/((1+d*x^2/c)^q)
 

Mathematica [A] (warning: unable to verify)

Time = 0.46 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2}{x^4} \, dx=\frac {\left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (-e^2 \left (1+\frac {b x^2}{a}\right )^{-p} \left (1+\frac {d x^2}{c}\right )^{-q} \operatorname {AppellF1}\left (-\frac {3}{2},-p,-q,-\frac {1}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-6 e f x^2 \left (1+\frac {b x^2}{a}\right )^{-p} \left (1+\frac {d x^2}{c}\right )^{-q} \operatorname {AppellF1}\left (-\frac {1}{2},-p,-q,\frac {1}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\frac {9 a c f^2 x^4 \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{3 a c \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+2 x^2 \left (b c p \operatorname {AppellF1}\left (\frac {3}{2},1-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+a d q \operatorname {AppellF1}\left (\frac {3}{2},-p,1-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )}\right )}{3 x^3} \] Input:

Integrate[((a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^2)/x^4,x]
 

Output:

((a + b*x^2)^p*(c + d*x^2)^q*(-((e^2*AppellF1[-3/2, -p, -q, -1/2, -((b*x^2 
)/a), -((d*x^2)/c)])/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)) - (6*e*f*x^2*A 
ppellF1[-1/2, -p, -q, 1/2, -((b*x^2)/a), -((d*x^2)/c)])/((1 + (b*x^2)/a)^p 
*(1 + (d*x^2)/c)^q) + (9*a*c*f^2*x^4*AppellF1[1/2, -p, -q, 3/2, -((b*x^2)/ 
a), -((d*x^2)/c)])/(3*a*c*AppellF1[1/2, -p, -q, 3/2, -((b*x^2)/a), -((d*x^ 
2)/c)] + 2*x^2*(b*c*p*AppellF1[3/2, 1 - p, -q, 5/2, -((b*x^2)/a), -((d*x^2 
)/c)] + a*d*q*AppellF1[3/2, -p, 1 - q, 5/2, -((b*x^2)/a), -((d*x^2)/c)]))) 
)/(3*x^3)
 

Rubi [A] (warning: unable to verify)

Time = 2.24 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.56, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {448, 445, 25, 406, 334, 334, 333, 395, 395, 394, 445, 406, 334, 334, 333, 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.

\(\displaystyle \int \frac {\left (e+f x^2\right )^2 \left (a+b x^2\right )^p \left (c+d x^2\right )^q}{x^4} \, dx\)

\(\Big \downarrow \) 448

\(\displaystyle \frac {f \int \frac {\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (f x^2+e\right )}{x^2}dx}{e^2}+e \int \frac {\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (f x^2+e\right )}{x^4}dx\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {f \left (-\frac {\int -\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (b d e (2 p+2 q+3) x^2+b c e (2 p+1)+a (c f+d (2 q e+e))\right )dx}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}+e \left (-\frac {\int -\frac {\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (b d e (2 p+2 q+1) x^2+3 a c f-b c e (1-2 p)-a d (e-2 e q)\right )}{x^2}dx}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {f \left (\frac {\int \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (b d e (2 p+2 q+3) x^2+a c f+b c e (2 p+1)+a d (2 q e+e)\right )dx}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}+e \left (\frac {\int \frac {\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (b d e (2 p+2 q+1) x^2+3 a c f-b c e (1-2 p)-a d e (1-2 q)\right )}{x^2}dx}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {f \left (\frac {(a c f+a d (2 e q+e)+b c e (2 p+1)) \int \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx+b d e (2 p+2 q+3) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}+e \left (\frac {\int \frac {\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (b d e (2 p+2 q+1) x^2+3 a c f-b c e (1-2 p)-a d e (1-2 q)\right )}{x^2}dx}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )\)

\(\Big \downarrow \) 334

\(\displaystyle \frac {f \left (\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} (a c f+a d (2 e q+e)+b c e (2 p+1)) \int \left (\frac {b x^2}{a}+1\right )^p \left (d x^2+c\right )^qdx+b d e (2 p+2 q+3) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}+e \left (\frac {\int \frac {\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (b d e (2 p+2 q+1) x^2+3 a c f-b c e (1-2 p)-a d e (1-2 q)\right )}{x^2}dx}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )\)

\(\Big \downarrow \) 334

\(\displaystyle \frac {f \left (\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} (a c f+a d (2 e q+e)+b c e (2 p+1)) \int \left (\frac {b x^2}{a}+1\right )^p \left (\frac {d x^2}{c}+1\right )^qdx+b d e (2 p+2 q+3) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}+e \left (\frac {\int \frac {\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (b d e (2 p+2 q+1) x^2+3 a c f-b c e (1-2 p)-a d e (1-2 q)\right )}{x^2}dx}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )\)

\(\Big \downarrow \) 333

\(\displaystyle \frac {f \left (\frac {b d e (2 p+2 q+3) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx+x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a c f+a d (2 e q+e)+b c e (2 p+1))}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}+e \left (\frac {\int \frac {\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (b d e (2 p+2 q+1) x^2+3 a c f-b c e (1-2 p)-a d e (1-2 q)\right )}{x^2}dx}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )\)

\(\Big \downarrow \) 395

\(\displaystyle \frac {f \left (\frac {b d e (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int x^2 \left (\frac {b x^2}{a}+1\right )^p \left (d x^2+c\right )^qdx+x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a c f+a d (2 e q+e)+b c e (2 p+1))}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}+e \left (\frac {\int \frac {\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (b d e (2 p+2 q+1) x^2+3 a c f-b c e (1-2 p)-a d e (1-2 q)\right )}{x^2}dx}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )\)

\(\Big \downarrow \) 395

\(\displaystyle \frac {f \left (\frac {b d e (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \int x^2 \left (\frac {b x^2}{a}+1\right )^p \left (\frac {d x^2}{c}+1\right )^qdx+x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a c f+a d (2 e q+e)+b c e (2 p+1))}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}+e \left (\frac {\int \frac {\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (b d e (2 p+2 q+1) x^2+3 a c f-b c e (1-2 p)-a d e (1-2 q)\right )}{x^2}dx}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )\)

\(\Big \downarrow \) 394

\(\displaystyle e \left (\frac {\int \frac {\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (b d e (2 p+2 q+1) x^2+3 a c f-b c e (1-2 p)-a d e (1-2 q)\right )}{x^2}dx}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )+\frac {f \left (\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a c f+a d (2 e q+e)+b c e (2 p+1))+\frac {1}{3} b d e x^3 (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}\)

\(\Big \downarrow \) 445

\(\displaystyle e \left (\frac {-\frac {\int \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (-d (3 c f-d e (1-2 q)) (2 q+1) a^2-b c (3 c f (2 p+1)-d e (-8 q p-2 p-2 q+1)) a-b d (3 a c f-b c e (1-2 p)-a d e (1-2 q)) (2 p+2 q+3) x^2+b^2 c^2 e \left (1-4 p^2\right )\right )dx}{a c}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c x}}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )+\frac {f \left (\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a c f+a d (2 e q+e)+b c e (2 p+1))+\frac {1}{3} b d e x^3 (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}\)

\(\Big \downarrow \) 406

\(\displaystyle e \left (\frac {-\frac {\left (a^2 (-d) (2 q+1) (3 c f-d e (1-2 q))-a b c (3 c f (2 p+1)-d e (-8 p q-2 p-2 q+1))+b^2 c^2 e \left (1-4 p^2\right )\right ) \int \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx-b d (2 p+2 q+3) (3 a c f-a d e (1-2 q)-b c e (1-2 p)) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{a c}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c x}}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )+\frac {f \left (\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a c f+a d (2 e q+e)+b c e (2 p+1))+\frac {1}{3} b d e x^3 (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}\)

\(\Big \downarrow \) 334

\(\displaystyle e \left (\frac {-\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a^2 (-d) (2 q+1) (3 c f-d e (1-2 q))-a b c (3 c f (2 p+1)-d e (-8 p q-2 p-2 q+1))+b^2 c^2 e \left (1-4 p^2\right )\right ) \int \left (\frac {b x^2}{a}+1\right )^p \left (d x^2+c\right )^qdx-b d (2 p+2 q+3) (3 a c f-a d e (1-2 q)-b c e (1-2 p)) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{a c}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c x}}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )+\frac {f \left (\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a c f+a d (2 e q+e)+b c e (2 p+1))+\frac {1}{3} b d e x^3 (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}\)

\(\Big \downarrow \) 334

\(\displaystyle e \left (\frac {-\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \left (a^2 (-d) (2 q+1) (3 c f-d e (1-2 q))-a b c (3 c f (2 p+1)-d e (-8 p q-2 p-2 q+1))+b^2 c^2 e \left (1-4 p^2\right )\right ) \int \left (\frac {b x^2}{a}+1\right )^p \left (\frac {d x^2}{c}+1\right )^qdx-b d (2 p+2 q+3) (3 a c f-a d e (1-2 q)-b c e (1-2 p)) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{a c}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c x}}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )+\frac {f \left (\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a c f+a d (2 e q+e)+b c e (2 p+1))+\frac {1}{3} b d e x^3 (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}\)

\(\Big \downarrow \) 333

\(\displaystyle e \left (\frac {-\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) \left (a^2 (-d) (2 q+1) (3 c f-d e (1-2 q))-a b c (3 c f (2 p+1)-d e (-8 p q-2 p-2 q+1))+b^2 c^2 e \left (1-4 p^2\right )\right )-b d (2 p+2 q+3) (3 a c f-a d e (1-2 q)-b c e (1-2 p)) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{a c}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c x}}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )+\frac {f \left (\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a c f+a d (2 e q+e)+b c e (2 p+1))+\frac {1}{3} b d e x^3 (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}\)

\(\Big \downarrow \) 395

\(\displaystyle e \left (\frac {-\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) \left (a^2 (-d) (2 q+1) (3 c f-d e (1-2 q))-a b c (3 c f (2 p+1)-d e (-8 p q-2 p-2 q+1))+b^2 c^2 e \left (1-4 p^2\right )\right )-b d (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} (3 a c f-a d e (1-2 q)-b c e (1-2 p)) \int x^2 \left (\frac {b x^2}{a}+1\right )^p \left (d x^2+c\right )^qdx}{a c}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c x}}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )+\frac {f \left (\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a c f+a d (2 e q+e)+b c e (2 p+1))+\frac {1}{3} b d e x^3 (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}\)

\(\Big \downarrow \) 395

\(\displaystyle e \left (\frac {-\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) \left (a^2 (-d) (2 q+1) (3 c f-d e (1-2 q))-a b c (3 c f (2 p+1)-d e (-8 p q-2 p-2 q+1))+b^2 c^2 e \left (1-4 p^2\right )\right )-b d (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} (3 a c f-a d e (1-2 q)-b c e (1-2 p)) \int x^2 \left (\frac {b x^2}{a}+1\right )^p \left (\frac {d x^2}{c}+1\right )^qdx}{a c}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c x}}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )+\frac {f \left (\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a c f+a d (2 e q+e)+b c e (2 p+1))+\frac {1}{3} b d e x^3 (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}\)

\(\Big \downarrow \) 394

\(\displaystyle e \left (\frac {-\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) \left (a^2 (-d) (2 q+1) (3 c f-d e (1-2 q))-a b c (3 c f (2 p+1)-d e (-8 p q-2 p-2 q+1))+b^2 c^2 e \left (1-4 p^2\right )\right )-\frac {1}{3} b d x^3 (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c x}}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\right )+\frac {f \left (\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a c f+a d (2 e q+e)+b c e (2 p+1))+\frac {1}{3} b d e x^3 (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{a c x}\right )}{e^2}\)

Input:

Int[((a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^2)/x^4,x]
 

Output:

(f*(-((e*(a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q))/(a*c*x)) + (((a*c*f + b* 
c*e*(1 + 2*p) + a*d*(e + 2*e*q))*x*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[1/ 
2, -p, -q, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/((1 + (b*x^2)/a)^p*(1 + (d*x^ 
2)/c)^q) + (b*d*e*(3 + 2*p + 2*q)*x^3*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1 
[3/2, -p, -q, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/(3*(1 + (b*x^2)/a)^p*(1 + 
(d*x^2)/c)^q))/(a*c)))/e^2 + e*(-1/3*(e*(a + b*x^2)^(1 + p)*(c + d*x^2)^(1 
 + q))/(a*c*x^3) + (-(((3*a*c*f - b*c*e*(1 - 2*p) - a*d*e*(1 - 2*q))*(a + 
b*x^2)^(1 + p)*(c + d*x^2)^(1 + q))/(a*c*x)) - (((b^2*c^2*e*(1 - 4*p^2) - 
a^2*d*(3*c*f - d*e*(1 - 2*q))*(1 + 2*q) - a*b*c*(3*c*f*(1 + 2*p) - d*e*(1 
- 2*p - 2*q - 8*p*q)))*x*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[1/2, -p, -q, 
 3/2, -((b*x^2)/a), -((d*x^2)/c)])/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q) - 
 (b*d*(3*a*c*f - b*c*e*(1 - 2*p) - a*d*e*(1 - 2*q))*(3 + 2*p + 2*q)*x^3*(a 
 + b*x^2)^p*(c + d*x^2)^q*AppellF1[3/2, -p, -q, 5/2, -((b*x^2)/a), -((d*x^ 
2)/c)])/(3*(1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q))/(a*c))/(3*a*c))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 334

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

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 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

rule 448

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[e   Int[(g*x)^m*(a + b*x 
^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] + Simp[f/e^2   Int[(g*x)^(m 
 + 2)*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] /; FreeQ[{a, 
b, c, d, e, f, g, m, p, q}, x] && IGtQ[r, 0]
 

Maple [F]

\[\int \frac {\left (b \,x^{2}+a \right )^{p} \left (d \,x^{2}+c \right )^{q} \left (f \,x^{2}+e \right )^{2}}{x^{4}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)^2/x^4,x, algorithm="fricas")
 

Output:

integral((f^2*x^4 + 2*e*f*x^2 + e^2)*(b*x^2 + a)^p*(d*x^2 + c)^q/x^4, x)
 

Sympy [F(-1)]

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

integrate((b*x**2+a)**p*(d*x**2+c)**q*(f*x**2+e)**2/x**4,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)^2/x^4,x, algorithm="maxima")
 

Output:

integrate((f*x^2 + e)^2*(b*x^2 + a)^p*(d*x^2 + c)^q/x^4, x)
                                                                                    
                                                                                    
 

Giac [F]

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

integrate((b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)^2/x^4,x, algorithm="giac")
 

Output:

integrate((f*x^2 + e)^2*(b*x^2 + a)^p*(d*x^2 + c)^q/x^4, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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