\(\int x^4 (a+b x^2)^p (c+d x^2)^q (e+f x^2)^2 \, dx\) [80]

Optimal result

Integrand size = 31, antiderivative size = 1256 \[ \int x^4 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2 \, dx =\text {Too large to display} \] Output:

-e*(b*c*f*(5+2*p)+a*d*f*(5+2*q)-b*d*e*(7+2*p+2*q))*x*(b*x^2+a)^(p+1)*(d*x^ 
2+c)^(1+q)/b^2/d^2/(5+2*p+2*q)/(7+2*p+2*q)+f*(a*d*(5+2*q)*(b*c*f*(7+2*p)+a 
*d*f*(7+2*q)-b*d*e*(9+2*p+2*q))+b*(4*a*c*d*f*p*(1+q)+b*c*(5+2*p)*(c*f*(7+2 
*p)-d*e*(9+2*p+2*q))))*x*(b*x^2+a)^(p+1)*(d*x^2+c)^(1+q)/b^3/d^3/(5+2*p+2* 
q)/(7+2*p+2*q)/(9+2*p+2*q)+e*f*x^3*(b*x^2+a)^(p+1)*(d*x^2+c)^(1+q)/b/d/(7+ 
2*p+2*q)-f*(b*c*f*(7+2*p)+a*d*f*(7+2*q)-b*d*e*(9+2*p+2*q))*x^3*(b*x^2+a)^( 
p+1)*(d*x^2+c)^(1+q)/b^2/d^2/(7+2*p+2*q)/(9+2*p+2*q)+f^2*x^5*(b*x^2+a)^(p+ 
1)*(d*x^2+c)^(1+q)/b/d/(9+2*p+2*q)+a*c*e*(b*c*f*(5+2*p)+a*d*f*(5+2*q)-b*d* 
e*(7+2*p+2*q))*x*(b*x^2+a)^p*(d*x^2+c)^q*AppellF1(1/2,-p,-q,3/2,-b*x^2/a,- 
d*x^2/c)/b^2/d^2/(5+2*p+2*q)/(7+2*p+2*q)/((1+b*x^2/a)^p)/((1+d*x^2/c)^q)-a 
*c*f*(a^2*d^2*f*(4*q^2+24*q+35)+b^2*c*(5+2*p)*(c*f*(7+2*p)-d*e*(9+2*p+2*q) 
)-a*b*d*(d*e*(5+2*q)*(9+2*p+2*q)-c*f*(8*p*q+14*p+14*q+35)))*x*(b*x^2+a)^p* 
(d*x^2+c)^q*AppellF1(1/2,-p,-q,3/2,-b*x^2/a,-d*x^2/c)/b^3/d^3/(5+2*p+2*q)/ 
(7+2*p+2*q)/(9+2*p+2*q)/((1+b*x^2/a)^p)/((1+d*x^2/c)^q)+1/3*e*(a^2*d^2*f*( 
4*q^2+16*q+15)+b^2*c*(3+2*p)*(c*f*(5+2*p)-d*e*(7+2*p+2*q))-a*b*d*(d*e*(3+2 
*q)*(7+2*p+2*q)-c*f*(8*p*q+10*p+10*q+15)))*x^3*(b*x^2+a)^p*(d*x^2+c)^q*App 
ellF1(3/2,-p,-q,5/2,-b*x^2/a,-d*x^2/c)/b^2/d^2/(5+2*p+2*q)/(7+2*p+2*q)/((1 
+b*x^2/a)^p)/((1+d*x^2/c)^q)-1/3*f*(a^3*d^3*f*(8*q^3+60*q^2+142*q+105)+b^3 
*c^2*(4*p^2+16*p+15)*(c*f*(7+2*p)-d*e*(9+2*p+2*q))-a^2*b*d^2*(5+2*q)*(d*e* 
(3+2*q)*(9+2*p+2*q)-c*f*(21+14*q+2*p*(7+6*q)))+a*b^2*c*d*(c*f*(5+2*p)*(...
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.13 \[ \int x^4 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2 \, dx=\frac {1}{315} x^5 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} \left (63 e^2 \operatorname {AppellF1}\left (\frac {5}{2},-p,-q,\frac {7}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+5 f x^2 \left (18 e \operatorname {AppellF1}\left (\frac {7}{2},-p,-q,\frac {9}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+7 f x^2 \operatorname {AppellF1}\left (\frac {9}{2},-p,-q,\frac {11}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right ) \] Input:

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

Output:

(x^5*(a + b*x^2)^p*(c + d*x^2)^q*(63*e^2*AppellF1[5/2, -p, -q, 7/2, -((b*x 
^2)/a), -((d*x^2)/c)] + 5*f*x^2*(18*e*AppellF1[7/2, -p, -q, 9/2, -((b*x^2) 
/a), -((d*x^2)/c)] + 7*f*x^2*AppellF1[9/2, -p, -q, 11/2, -((b*x^2)/a), -(( 
d*x^2)/c)])))/(315*(1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)
 

Rubi [A] (warning: unable to verify)

Time = 4.67 (sec) , antiderivative size = 1171, normalized size of antiderivative = 0.93, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {448, 444, 444, 406, 334, 334, 333, 395, 395, 394, 444, 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.

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

\(\Big \downarrow \) 448

\(\displaystyle \frac {f \int x^6 \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (f x^2+e\right )dx}{e^2}+e \int x^4 \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (f x^2+e\right )dx\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {f \left (\frac {f x^5 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\int x^4 \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left ((b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9)) x^2+5 a c f\right )dx}{b d (2 p+2 q+9)}\right )}{e^2}+e \left (\frac {f x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left ((b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x^2+3 a c f\right )dx}{b d (2 p+2 q+7)}\right )\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {f \left (\frac {f x^5 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+7)+b c f (2 p+7)-b d e (2 p+2 q+9))}{b d (2 p+2 q+7)}-\frac {\int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left ((a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x^2+3 a c (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))\right )dx}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}+e \left (\frac {f x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))}{b d (2 p+2 q+5)}-\frac {\int \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left ((a d (2 q+3) (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) x^2+a c (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7))\right )dx}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {f \left (\frac {f x^5 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+7)+b c f (2 p+7)-b d e (2 p+2 q+9))}{b d (2 p+2 q+7)}-\frac {\int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left ((a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x^2+3 a c (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))\right )dx}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}+e \left (\frac {f x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))}{b d (2 p+2 q+5)}-\frac {a c (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7)) \int \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx+(a d (2 q+3) (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )\)

\(\Big \downarrow \) 334

\(\displaystyle \frac {f \left (\frac {f x^5 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+7)+b c f (2 p+7)-b d e (2 p+2 q+9))}{b d (2 p+2 q+7)}-\frac {\int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left ((a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x^2+3 a c (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))\right )dx}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}+e \left (\frac {f x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))}{b d (2 p+2 q+5)}-\frac {a c \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7)) \int \left (\frac {b x^2}{a}+1\right )^p \left (d x^2+c\right )^qdx+(a d (2 q+3) (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )\)

\(\Big \downarrow \) 334

\(\displaystyle \frac {f \left (\frac {f x^5 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+7)+b c f (2 p+7)-b d e (2 p+2 q+9))}{b d (2 p+2 q+7)}-\frac {\int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left ((a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x^2+3 a c (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))\right )dx}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}+e \left (\frac {f x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))}{b d (2 p+2 q+5)}-\frac {a c \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7)) \int \left (\frac {b x^2}{a}+1\right )^p \left (\frac {d x^2}{c}+1\right )^qdx+(a d (2 q+3) (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )\)

\(\Big \downarrow \) 333

\(\displaystyle e \left (\frac {f x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))}{b d (2 p+2 q+5)}-\frac {(a d (2 q+3) (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx+a c x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )+\frac {f \left (\frac {f x^5 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+7)+b c f (2 p+7)-b d e (2 p+2 q+9))}{b d (2 p+2 q+7)}-\frac {\int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left ((a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x^2+3 a c (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))\right )dx}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}\)

\(\Big \downarrow \) 395

\(\displaystyle e \left (\frac {f x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))}{b d (2 p+2 q+5)}-\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} (a d (2 q+3) (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) \int x^2 \left (\frac {b x^2}{a}+1\right )^p \left (d x^2+c\right )^qdx+a c x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )+\frac {f \left (\frac {f x^5 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+7)+b c f (2 p+7)-b d e (2 p+2 q+9))}{b d (2 p+2 q+7)}-\frac {\int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left ((a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x^2+3 a c (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))\right )dx}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}\)

\(\Big \downarrow \) 395

\(\displaystyle e \left (\frac {f x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))}{b d (2 p+2 q+5)}-\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} (a d (2 q+3) (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) \int x^2 \left (\frac {b x^2}{a}+1\right )^p \left (\frac {d x^2}{c}+1\right )^qdx+a c x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )+\frac {f \left (\frac {f x^5 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+7)+b c f (2 p+7)-b d e (2 p+2 q+9))}{b d (2 p+2 q+7)}-\frac {\int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left ((a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x^2+3 a c (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))\right )dx}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}\)

\(\Big \downarrow \) 394

\(\displaystyle \frac {f \left (\frac {f x^5 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+7)+b c f (2 p+7)-b d e (2 p+2 q+9))}{b d (2 p+2 q+7)}-\frac {\int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left ((a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x^2+3 a c (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))\right )dx}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}+e \left (\frac {f x^3 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))}{b d (2 p+2 q+5)}-\frac {a c x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))+\frac {1}{3} x^3 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) (a d (2 q+3) (a d f (2 q+5)+b c f (2 p+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7))))}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )\)

\(\Big \downarrow \) 444

\(\displaystyle e \left (\frac {f x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^p \left (d x^2+c\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 (\frac {b x^2}{a}+1\right )^{-p}+\frac {1}{3} (a d (2 q+3) (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) x^3 \left (b x^2+a\right )^p \left (d x^2+c\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 ) \left (\frac {b x^2}{a}+1\right )^{-p}}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )+\frac {f \left (\frac {f x^5 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {(b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9)) x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {\int \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (\left (b c \left (c \left (4 p^2+16 p+15\right ) (c f (2 p+7)-d e (2 p+2 q+9)) b^2+4 a d p (q+1) (2 c f (2 p+5)-d e (2 p+2 q+9)) b+4 a^2 d^2 f p \left (2 q^2+9 q+7\right )\right )+a d (2 q+3) (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9))))\right ) x^2+a c (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9))))\right )dx}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}\)

\(\Big \downarrow \) 406

\(\displaystyle e \left (\frac {f x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^p \left (d x^2+c\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 (\frac {b x^2}{a}+1\right )^{-p}+\frac {1}{3} (a d (2 q+3) (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) x^3 \left (b x^2+a\right )^p \left (d x^2+c\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 ) \left (\frac {b x^2}{a}+1\right )^{-p}}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )+\frac {f \left (\frac {f x^5 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {(b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9)) x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) \int \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx+\left (b c \left (c \left (4 p^2+16 p+15\right ) (c f (2 p+7)-d e (2 p+2 q+9)) b^2+4 a d p (q+1) (2 c f (2 p+5)-d e (2 p+2 q+9)) b+4 a^2 d^2 f p \left (2 q^2+9 q+7\right )\right )+a d (2 q+3) (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9))))\right ) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}\)

\(\Big \downarrow \) 334

\(\displaystyle e \left (\frac {f x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^p \left (d x^2+c\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 (\frac {b x^2}{a}+1\right )^{-p}+\frac {1}{3} (a d (2 q+3) (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) x^3 \left (b x^2+a\right )^p \left (d x^2+c\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 ) \left (\frac {b x^2}{a}+1\right )^{-p}}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )+\frac {f \left (\frac {f x^5 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {(b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9)) x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) \left (b x^2+a\right )^p \int \left (\frac {b x^2}{a}+1\right )^p \left (d x^2+c\right )^qdx \left (\frac {b x^2}{a}+1\right )^{-p}+\left (b c \left (c \left (4 p^2+16 p+15\right ) (c f (2 p+7)-d e (2 p+2 q+9)) b^2+4 a d p (q+1) (2 c f (2 p+5)-d e (2 p+2 q+9)) b+4 a^2 d^2 f p \left (2 q^2+9 q+7\right )\right )+a d (2 q+3) (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9))))\right ) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}\)

\(\Big \downarrow \) 334

\(\displaystyle e \left (\frac {f x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^p \left (d x^2+c\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 (\frac {b x^2}{a}+1\right )^{-p}+\frac {1}{3} (a d (2 q+3) (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) x^3 \left (b x^2+a\right )^p \left (d x^2+c\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 ) \left (\frac {b x^2}{a}+1\right )^{-p}}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )+\frac {f \left (\frac {f x^5 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {(b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9)) x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) \left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \int \left (\frac {b x^2}{a}+1\right )^p \left (\frac {d x^2}{c}+1\right )^qdx \left (\frac {b x^2}{a}+1\right )^{-p}+\left (b c \left (c \left (4 p^2+16 p+15\right ) (c f (2 p+7)-d e (2 p+2 q+9)) b^2+4 a d p (q+1) (2 c f (2 p+5)-d e (2 p+2 q+9)) b+4 a^2 d^2 f p \left (2 q^2+9 q+7\right )\right )+a d (2 q+3) (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9))))\right ) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}\)

\(\Big \downarrow \) 333

\(\displaystyle e \left (\frac {f x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^p \left (d x^2+c\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 (\frac {b x^2}{a}+1\right )^{-p}+\frac {1}{3} (a d (2 q+3) (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) x^3 \left (b x^2+a\right )^p \left (d x^2+c\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 ) \left (\frac {b x^2}{a}+1\right )^{-p}}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )+\frac {f \left (\frac {f x^5 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {(b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9)) x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x \left (b x^2+a\right )^p \left (d x^2+c\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 (\frac {b x^2}{a}+1\right )^{-p}+\left (b c \left (c \left (4 p^2+16 p+15\right ) (c f (2 p+7)-d e (2 p+2 q+9)) b^2+4 a d p (q+1) (2 c f (2 p+5)-d e (2 p+2 q+9)) b+4 a^2 d^2 f p \left (2 q^2+9 q+7\right )\right )+a d (2 q+3) (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9))))\right ) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}\)

\(\Big \downarrow \) 395

\(\displaystyle e \left (\frac {f x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^p \left (d x^2+c\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 (\frac {b x^2}{a}+1\right )^{-p}+\frac {1}{3} (a d (2 q+3) (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) x^3 \left (b x^2+a\right )^p \left (d x^2+c\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 ) \left (\frac {b x^2}{a}+1\right )^{-p}}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )+\frac {f \left (\frac {f x^5 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {(b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9)) x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x \left (b x^2+a\right )^p \left (d x^2+c\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 (\frac {b x^2}{a}+1\right )^{-p}+\left (b c \left (c \left (4 p^2+16 p+15\right ) (c f (2 p+7)-d e (2 p+2 q+9)) b^2+4 a d p (q+1) (2 c f (2 p+5)-d e (2 p+2 q+9)) b+4 a^2 d^2 f p \left (2 q^2+9 q+7\right )\right )+a d (2 q+3) (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9))))\right ) \left (b x^2+a\right )^p \int x^2 \left (\frac {b x^2}{a}+1\right )^p \left (d x^2+c\right )^qdx \left (\frac {b x^2}{a}+1\right )^{-p}}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}\)

\(\Big \downarrow \) 395

\(\displaystyle e \left (\frac {f x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^p \left (d x^2+c\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 (\frac {b x^2}{a}+1\right )^{-p}+\frac {1}{3} (a d (2 q+3) (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) x^3 \left (b x^2+a\right )^p \left (d x^2+c\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 ) \left (\frac {b x^2}{a}+1\right )^{-p}}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )+\frac {f \left (\frac {f x^5 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {(b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9)) x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x \left (b x^2+a\right )^p \left (d x^2+c\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 (\frac {b x^2}{a}+1\right )^{-p}+\left (b c \left (c \left (4 p^2+16 p+15\right ) (c f (2 p+7)-d e (2 p+2 q+9)) b^2+4 a d p (q+1) (2 c f (2 p+5)-d e (2 p+2 q+9)) b+4 a^2 d^2 f p \left (2 q^2+9 q+7\right )\right )+a d (2 q+3) (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9))))\right ) \left (b x^2+a\right )^p \left (d x^2+c\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 \left (\frac {b x^2}{a}+1\right )^{-p}}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}\)

\(\Big \downarrow \) 394

\(\displaystyle e \left (\frac {f x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7)) x \left (b x^2+a\right )^p \left (d x^2+c\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 (\frac {b x^2}{a}+1\right )^{-p}+\frac {1}{3} (a d (2 q+3) (b c f (2 p+5)+a d f (2 q+5)-b d e (2 p+2 q+7))+b (4 a c d f p (q+1)+b c (2 p+3) (c f (2 p+5)-d e (2 p+2 q+7)))) x^3 \left (b x^2+a\right )^p \left (d x^2+c\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 ) \left (\frac {b x^2}{a}+1\right )^{-p}}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}\right )+\frac {f \left (\frac {f x^5 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+9)}-\frac {\frac {(b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9)) x^3 \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+7)}-\frac {\frac {(a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^{q+1}}{b d (2 p+2 q+5)}-\frac {a c (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9)))) x \left (b x^2+a\right )^p \left (d x^2+c\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 (\frac {b x^2}{a}+1\right )^{-p}+\frac {1}{3} \left (b c \left (c \left (4 p^2+16 p+15\right ) (c f (2 p+7)-d e (2 p+2 q+9)) b^2+4 a d p (q+1) (2 c f (2 p+5)-d e (2 p+2 q+9)) b+4 a^2 d^2 f p \left (2 q^2+9 q+7\right )\right )+a d (2 q+3) (a d (2 q+5) (b c f (2 p+7)+a d f (2 q+7)-b d e (2 p+2 q+9))+b (4 a c d f p (q+1)+b c (2 p+5) (c f (2 p+7)-d e (2 p+2 q+9))))\right ) x^3 \left (b x^2+a\right )^p \left (d x^2+c\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 ) \left (\frac {b x^2}{a}+1\right )^{-p}}{b d (2 p+2 q+5)}}{b d (2 p+2 q+7)}}{b d (2 p+2 q+9)}\right )}{e^2}\)

Input:

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

Output:

e*((f*x^3*(a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q))/(b*d*(7 + 2*p + 2*q)) - 
 (((b*c*f*(5 + 2*p) + a*d*f*(5 + 2*q) - b*d*e*(7 + 2*p + 2*q))*x*(a + b*x^ 
2)^(1 + p)*(c + d*x^2)^(1 + q))/(b*d*(5 + 2*p + 2*q)) - ((a*c*(b*c*f*(5 + 
2*p) + a*d*f*(5 + 2*q) - b*d*e*(7 + 2*p + 2*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) + ((a*d*(3 + 2*q)*(b*c*f*(5 + 2*p) + a*d*f*(5 + 2* 
q) - b*d*e*(7 + 2*p + 2*q)) + b*(4*a*c*d*f*p*(1 + q) + b*c*(3 + 2*p)*(c*f* 
(5 + 2*p) - d*e*(7 + 2*p + 2*q))))*x^3*(a + b*x^2)^p*(c + d*x^2)^q*AppellF 
1[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))/(b*d*(5 + 2*p + 2*q)))/(b*d*(7 + 2*p + 2*q))) + (f*((f*x^5 
*(a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q))/(b*d*(9 + 2*p + 2*q)) - (((b*c*f 
*(7 + 2*p) + a*d*f*(7 + 2*q) - b*d*e*(9 + 2*p + 2*q))*x^3*(a + b*x^2)^(1 + 
 p)*(c + d*x^2)^(1 + q))/(b*d*(7 + 2*p + 2*q)) - (((a*d*(5 + 2*q)*(b*c*f*( 
7 + 2*p) + a*d*f*(7 + 2*q) - b*d*e*(9 + 2*p + 2*q)) + b*(4*a*c*d*f*p*(1 + 
q) + b*c*(5 + 2*p)*(c*f*(7 + 2*p) - d*e*(9 + 2*p + 2*q))))*x*(a + b*x^2)^( 
1 + p)*(c + d*x^2)^(1 + q))/(b*d*(5 + 2*p + 2*q)) - ((a*c*(a*d*(5 + 2*q)*( 
b*c*f*(7 + 2*p) + a*d*f*(7 + 2*q) - b*d*e*(9 + 2*p + 2*q)) + b*(4*a*c*d*f* 
p*(1 + q) + b*c*(5 + 2*p)*(c*f*(7 + 2*p) - d*e*(9 + 2*p + 2*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*c*(4*a^2*d^2*f*p*(7 + 9...
 

Defintions of rubi rules used

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 444

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((f^2*x^8 + 2*e*f*x^6 + e^2*x^4)*(b*x^2 + a)^p*(d*x^2 + c)^q, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(x^4*(b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)^2,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(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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