Integrand size = 17, antiderivative size = 93 \[ \int \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\frac {d x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}-\frac {(a d-b c (3+2 p)) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )}{b (3+2 p)} \]
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Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {396, 252, 251} \[ \int \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c-\frac {a d}{2 b p+3 b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )+\frac {d x \left (a+b x^2\right )^{p+1}}{b (2 p+3)} \]
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Rule 251
Rule 252
Rule 396
Rubi steps \begin{align*} \text {integral}& = \frac {d x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}-\left (-c+\frac {a d}{3 b+2 b p}\right ) \int \left (a+b x^2\right )^p \, dx \\ & = \frac {d x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}-\left (\left (-c+\frac {a d}{3 b+2 b p}\right ) \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \left (1+\frac {b x^2}{a}\right )^p \, dx \\ & = \frac {d x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}+\left (c-\frac {a d}{3 b+2 b p}\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97 \[ \int \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\frac {x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (d \left (a+b x^2\right ) \left (1+\frac {b x^2}{a}\right )^p+(-a d+b c (3+2 p)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )\right )}{b (3+2 p)} \]
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\[\int \left (b \,x^{2}+a \right )^{p} \left (d \,x^{2}+c \right )d x\]
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\[ \int \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\int { {\left (d x^{2} + c\right )} {\left (b x^{2} + a\right )}^{p} \,d x } \]
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Result contains complex when optimal does not.
Time = 4.76 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.57 \[ \int \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=a^{p} c x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} + \frac {a^{p} d x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3} \]
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\[ \int \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\int { {\left (d x^{2} + c\right )} {\left (b x^{2} + a\right )}^{p} \,d x } \]
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\[ \int \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\int { {\left (d x^{2} + c\right )} {\left (b x^{2} + a\right )}^{p} \,d x } \]
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Timed out. \[ \int \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\int {\left (b\,x^2+a\right )}^p\,\left (d\,x^2+c\right ) \,d x \]
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