Integrand size = 24, antiderivative size = 167 \[ \int (A+B x) \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx=A x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {B \left (a+c x^2\right )^{1+p} \left (d+f x^2\right )^q \left (\frac {c \left (d+f x^2\right )}{c d-a f}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,-\frac {f \left (a+c x^2\right )}{c d-a f}\right )}{2 c (1+p)} \]
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Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1024, 441, 440, 455, 72, 71} \[ \int (A+B x) \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx=A x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \left (d+f x^2\right )^q \left (\frac {f x^2}{d}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {B \left (a+c x^2\right )^{p+1} \left (d+f x^2\right )^q \left (\frac {c \left (d+f x^2\right )}{c d-a f}\right )^{-q} \operatorname {Hypergeometric2F1}\left (p+1,-q,p+2,-\frac {f \left (c x^2+a\right )}{c d-a f}\right )}{2 c (p+1)} \]
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Rule 71
Rule 72
Rule 440
Rule 441
Rule 455
Rule 1024
Rubi steps \begin{align*} \text {integral}& = A \int \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx+B \int x \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx \\ & = \frac {1}{2} B \text {Subst}\left (\int (a+c x)^p (d+f x)^q \, dx,x,x^2\right )+\left (A \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \left (1+\frac {c x^2}{a}\right )^p \left (d+f x^2\right )^q \, dx \\ & = \frac {1}{2} \left (B \left (d+f x^2\right )^q \left (\frac {c \left (d+f x^2\right )}{c d-a f}\right )^{-q}\right ) \text {Subst}\left (\int (a+c x)^p \left (\frac {c d}{c d-a f}+\frac {c f x}{c d-a f}\right )^q \, dx,x,x^2\right )+\left (A \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}\right ) \int \left (1+\frac {c x^2}{a}\right )^p \left (1+\frac {f x^2}{d}\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} F_1\left (\frac {1}{2};-p,-q;\frac {3}{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} \, _2F_1\left (1+p,-q;2+p;-\frac {f \left (a+c x^2\right )}{c d-a f}\right )}{2 c (1+p)} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.41 \[ \int (A+B x) \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx=\frac {1}{2} x \left (a+c x^2\right )^p \left (d+f x^2\right )^q \left (B x \left (1+\frac {c x^2}{a}\right )^{-p} \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (1,-p,-q,2,-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {6 a A d \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )}{3 a d \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+2 x^2 \left (c d p \operatorname {AppellF1}\left (\frac {3}{2},1-p,-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+a f q \operatorname {AppellF1}\left (\frac {3}{2},-p,1-q,\frac {5}{2},-\frac {c x^2}{a},-\frac {f x^2}{d}\right )\right )}\right ) \]
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\[\int \left (B x +A \right ) \left (c \,x^{2}+a \right )^{p} \left (f \,x^{2}+d \right )^{q}d x\]
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\[ \int (A+B x) \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx=\int { {\left (B x + A\right )} {\left (c x^{2} + a\right )}^{p} {\left (f x^{2} + d\right )}^{q} \,d x } \]
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Timed out. \[ \int (A+B x) \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx=\text {Timed out} \]
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\[ \int (A+B x) \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx=\int { {\left (B x + A\right )} {\left (c x^{2} + a\right )}^{p} {\left (f x^{2} + d\right )}^{q} \,d x } \]
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\[ \int (A+B x) \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx=\int { {\left (B x + A\right )} {\left (c x^{2} + a\right )}^{p} {\left (f x^{2} + d\right )}^{q} \,d x } \]
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Timed out. \[ \int (A+B x) \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx=\int {\left (c\,x^2+a\right )}^p\,{\left (f\,x^2+d\right )}^q\,\left (A+B\,x\right ) \,d x \]
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