Integrand size = 19, antiderivative size = 57 \[ \int \frac {\left (a+b x^2\right )^p}{\left (c+d x^2\right )^2} \, dx=\frac {x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,2,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c^2} \]
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Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {441, 440} \[ \int \frac {\left (a+b x^2\right )^p}{\left (c+d x^2\right )^2} \, dx=\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,2,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c^2} \]
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Rule 440
Rule 441
Rubi steps \begin{align*} \text {integral}& = \left (\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {b x^2}{a}\right )^p}{\left (c+d x^2\right )^2} \, dx \\ & = \frac {x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} F_1\left (\frac {1}{2};-p,2;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(162\) vs. \(2(57)=114\).
Time = 0.22 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.84 \[ \int \frac {\left (a+b x^2\right )^p}{\left (c+d x^2\right )^2} \, dx=-\frac {3 a c x \left (a+b x^2\right )^p \operatorname {AppellF1}\left (\frac {1}{2},-p,2,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{\left (c+d x^2\right )^2 \left (-3 a c \operatorname {AppellF1}\left (\frac {1}{2},-p,2,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-2 x^2 \left (b c p \operatorname {AppellF1}\left (\frac {3}{2},1-p,2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-2 a d \operatorname {AppellF1}\left (\frac {3}{2},-p,3,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )} \]
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\[\int \frac {\left (b \,x^{2}+a \right )^{p}}{\left (d \,x^{2}+c \right )^{2}}d x\]
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\[ \int \frac {\left (a+b x^2\right )^p}{\left (c+d x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p}}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^p}{\left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x^2\right )^p}{\left (c+d x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p}}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a+b x^2\right )^p}{\left (c+d x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p}}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^p}{\left (c+d x^2\right )^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^p}{{\left (d\,x^2+c\right )}^2} \,d x \]
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