Integrand size = 17, antiderivative size = 125 \[ \int \frac {\left (a+c x^2\right )^p}{d+e x} \, dx=\frac {x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d}-\frac {e \left (a+c x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {e^2 \left (a+c x^2\right )}{c d^2+a e^2}\right )}{2 \left (c d^2+a e^2\right ) (1+p)} \]
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Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {771, 441, 440, 455, 70} \[ \int \frac {\left (a+c x^2\right )^p}{d+e x} \, dx=\frac {x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d}-\frac {e \left (a+c x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {e^2 \left (c x^2+a\right )}{c d^2+a e^2}\right )}{2 (p+1) \left (a e^2+c d^2\right )} \]
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Rule 70
Rule 440
Rule 441
Rule 455
Rule 771
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d \left (a+c x^2\right )^p}{d^2-e^2 x^2}+\frac {e x \left (a+c x^2\right )^p}{-d^2+e^2 x^2}\right ) \, dx \\ & = d \int \frac {\left (a+c x^2\right )^p}{d^2-e^2 x^2} \, dx+e \int \frac {x \left (a+c x^2\right )^p}{-d^2+e^2 x^2} \, dx \\ & = \frac {1}{2} e \text {Subst}\left (\int \frac {(a+c x)^p}{-d^2+e^2 x} \, dx,x,x^2\right )+\left (d \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {c x^2}{a}\right )^p}{d^2-e^2 x^2} \, dx \\ & = \frac {x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d}-\frac {e \left (a+c x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;\frac {e^2 \left (a+c x^2\right )}{c d^2+a e^2}\right )}{2 \left (c d^2+a e^2\right ) (1+p)} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+c x^2\right )^p}{d+e x} \, dx=\frac {\left (\frac {e \left (-\sqrt {-\frac {a}{c}}+x\right )}{d+e x}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{c}}+x\right )}{d+e x}\right )^{-p} \left (a+c x^2\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {d-\sqrt {-\frac {a}{c}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{c}} e}{d+e x}\right )}{2 e p} \]
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\[\int \frac {\left (c \,x^{2}+a \right )^{p}}{e x +d}d x\]
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\[ \int \frac {\left (a+c x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p}}{e x + d} \,d x } \]
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\[ \int \frac {\left (a+c x^2\right )^p}{d+e x} \, dx=\int \frac {\left (a + c x^{2}\right )^{p}}{d + e x}\, dx \]
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\[ \int \frac {\left (a+c x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p}}{e x + d} \,d x } \]
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\[ \int \frac {\left (a+c x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p}}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {\left (a+c x^2\right )^p}{d+e x} \, dx=\int \frac {{\left (c\,x^2+a\right )}^p}{d+e\,x} \,d x \]
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