Integrand size = 23, antiderivative size = 90 \[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=-\frac {d \left (d^2-e^2 x^2\right )^p}{2 e^2 p}-\frac {e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},1-p,\frac {5}{2},\frac {e^2 x^2}{d^2}\right )}{3 d^2} \]
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Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {799, 778, 267, 372, 371} \[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=-\frac {e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},1-p,\frac {5}{2},\frac {e^2 x^2}{d^2}\right )}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^p}{2 e^2 p} \]
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Rule 267
Rule 371
Rule 372
Rule 778
Rule 799
Rubi steps \begin{align*} \text {integral}& = \frac {\int x \left (d^2 e-d e^2 x\right ) \left (d^2-e^2 x^2\right )^{-1+p} \, dx}{d e} \\ & = d \int x \left (d^2-e^2 x^2\right )^{-1+p} \, dx-e \int x^2 \left (d^2-e^2 x^2\right )^{-1+p} \, dx \\ & = -\frac {d \left (d^2-e^2 x^2\right )^p}{2 e^2 p}-\frac {\left (e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int x^2 \left (1-\frac {e^2 x^2}{d^2}\right )^{-1+p} \, dx}{d^2} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^p}{2 e^2 p}-\frac {e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {3}{2},1-p;\frac {5}{2};\frac {e^2 x^2}{d^2}\right )}{3 d^2} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.63 \[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\frac {2^{-1+p} \left (1+\frac {e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (2 e (1+p) x \left (\frac {1}{2}+\frac {e x}{2 d}\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )+(d-e x) \left (1-\frac {e^2 x^2}{d^2}\right )^p \operatorname {Hypergeometric2F1}\left (1-p,1+p,2+p,\frac {d-e x}{2 d}\right )\right )}{e^2 (1+p)} \]
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\[\int \frac {x \left (-e^{2} x^{2}+d^{2}\right )^{p}}{e x +d}d x\]
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\[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x}{e x + d} \,d x } \]
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Result contains complex when optimal does not.
Time = 4.64 (sec) , antiderivative size = 440, normalized size of antiderivative = 4.89 \[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\begin {cases} \frac {0^{p} d^{2 p + 1} \log {\left (\frac {d^{2}}{e^{2} x^{2}} \right )}}{2 e^{2}} - \frac {0^{p} d^{2 p + 1} \log {\left (\frac {d^{2}}{e^{2} x^{2}} - 1 \right )}}{2 e^{2}} - \frac {0^{p} d^{2 p + 1} \operatorname {acoth}{\left (\frac {d}{e x} \right )}}{e^{2}} + \frac {0^{p} d^{2 p + 1} x}{d e} - \frac {e^{2 p - 1} p x^{2 p + 1} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {d^{2 p + 1} x^{2} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{2} \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {0^{p} d^{2 p + 1} \log {\left (\frac {d^{2}}{e^{2} x^{2}} \right )}}{2 e^{2}} - \frac {0^{p} d^{2 p + 1} \log {\left (- \frac {d^{2}}{e^{2} x^{2}} + 1 \right )}}{2 e^{2}} - \frac {0^{p} d^{2 p + 1} \operatorname {atanh}{\left (\frac {d}{e x} \right )}}{e^{2}} + \frac {0^{p} d^{2 p + 1} x}{d e} - \frac {e^{2 p - 1} p x^{2 p + 1} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {d^{2 p + 1} x^{2} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{2} \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases} \]
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\[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x}{e x + d} \,d x } \]
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\[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\int \frac {x\,{\left (d^2-e^2\,x^2\right )}^p}{d+e\,x} \,d x \]
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