Integrand size = 29, antiderivative size = 252 \[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=A x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {1}{3} C x^3 \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {B \left (a+c x^2\right )^{1+p} \left (d+f x^2\right )^q \left (\frac {c \left (d+f x^2\right )}{c d-a f}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,-\frac {f \left (a+c x^2\right )}{c d-a f}\right )}{2 c (1+p)} \]
A*x*(c*x^2+a)^p*(f*x^2+d)^q*AppellF1(1/2,-p,-q,3/2,-c*x^2/a,-f*x^2/d)/((1+ c*x^2/a)^p)/((1+f*x^2/d)^q)+1/3*C*x^3*(c*x^2+a)^p*(f*x^2+d)^q*AppellF1(3/2 ,-p,-q,5/2,-c*x^2/a,-f*x^2/d)/((1+c*x^2/a)^p)/((1+f*x^2/d)^q)+1/2*B*(c*x^2 +a)^(p+1)*(f*x^2+d)^q*hypergeom([-q, p+1],[2+p],-f*(c*x^2+a)/(-a*f+c*d))/c /(p+1)/((c*(f*x^2+d)/(-a*f+c*d))^q)
Time = 0.40 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.20 \[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=\frac {1}{6} x \left (a+c x^2\right )^p \left (d+f x^2\right )^q \left (3 B x \left (1+\frac {c x^2}{a}\right )^{-p} \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (1,-p,-q,2,-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {18 a A d \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )}{3 a d \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+2 x^2 \left (c d p \operatorname {AppellF1}\left (\frac {3}{2},1-p,-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+a f q \operatorname {AppellF1}\left (\frac {3}{2},-p,1-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )\right )}+2 C x^2 \left (1+\frac {c x^2}{a}\right )^{-p} \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )\right ) \]
(x*(a + c*x^2)^p*(d + f*x^2)^q*((3*B*x*AppellF1[1, -p, -q, 2, -((c*x^2)/a) , -((f*x^2)/d)])/((1 + (c*x^2)/a)^p*(1 + (f*x^2)/d)^q) + (18*a*A*d*AppellF 1[1/2, -p, -q, 3/2, -((c*x^2)/a), -((f*x^2)/d)])/(3*a*d*AppellF1[1/2, -p, -q, 3/2, -((c*x^2)/a), -((f*x^2)/d)] + 2*x^2*(c*d*p*AppellF1[3/2, 1 - p, - q, 5/2, -((c*x^2)/a), -((f*x^2)/d)] + a*f*q*AppellF1[3/2, -p, 1 - q, 5/2, -((c*x^2)/a), -((f*x^2)/d)])) + (2*C*x^2*AppellF1[3/2, -p, -q, 5/2, -((c*x ^2)/a), -((f*x^2)/d)])/((1 + (c*x^2)/a)^p*(1 + (f*x^2)/d)^q)))/6
Time = 0.62 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {7293, 2009}
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.
\(\displaystyle \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (A \left (a+c x^2\right )^p \left (d+f x^2\right )^q+B x \left (a+c x^2\right )^p \left (d+f x^2\right )^q+C x^2 \left (a+c x^2\right )^p \left (d+f x^2\right )^q\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle A x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \left (d+f x^2\right )^q \left (\frac {f x^2}{d}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {1}{3} C x^3 \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \left (d+f x^2\right )^q \left (\frac {f x^2}{d}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {B \left (a+c x^2\right )^{p+1} \left (d+f x^2\right )^q \left (\frac {c \left (d+f x^2\right )}{c d-a f}\right )^{-q} \operatorname {Hypergeometric2F1}\left (p+1,-q,p+2,-\frac {f \left (c x^2+a\right )}{c d-a f}\right )}{2 c (p+1)}\) |
(A*x*(a + c*x^2)^p*(d + f*x^2)^q*AppellF1[1/2, -p, -q, 3/2, -((c*x^2)/a), -((f*x^2)/d)])/((1 + (c*x^2)/a)^p*(1 + (f*x^2)/d)^q) + (C*x^3*(a + c*x^2)^ p*(d + f*x^2)^q*AppellF1[3/2, -p, -q, 5/2, -((c*x^2)/a), -((f*x^2)/d)])/(3 *(1 + (c*x^2)/a)^p*(1 + (f*x^2)/d)^q) + (B*(a + c*x^2)^(1 + p)*(d + f*x^2) ^q*Hypergeometric2F1[1 + p, -q, 2 + p, -((f*(a + c*x^2))/(c*d - a*f))])/(2 *c*(1 + p)*((c*(d + f*x^2))/(c*d - a*f))^q)
3.4.100.3.1 Defintions of rubi rules used
\[\int \left (c \,x^{2}+a \right )^{p} \left (C \,x^{2}+B x +A \right ) \left (f \,x^{2}+d \right )^{q}d x\]
\[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (c x^{2} + a\right )}^{p} {\left (f x^{2} + d\right )}^{q} \,d x } \]
Timed out. \[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=\text {Timed out} \]
\[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (c x^{2} + a\right )}^{p} {\left (f x^{2} + d\right )}^{q} \,d x } \]
\[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (c x^{2} + a\right )}^{p} {\left (f x^{2} + d\right )}^{q} \,d x } \]
Timed out. \[ \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx=\int {\left (c\,x^2+a\right )}^p\,{\left (f\,x^2+d\right )}^q\,\left (C\,x^2+B\,x+A\right ) \,d x \]