Integrand size = 25, antiderivative size = 153 \[ \int (g x)^m (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\frac {d (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{g (1+m)}+\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-p,\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m)} \]
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Time = 0.05 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {822, 372, 371} \[ \int (g x)^m (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\frac {e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},-p,\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2)}+\frac {d (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-p,\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{g (m+1)} \]
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Rule 371
Rule 372
Rule 822
Rubi steps \begin{align*} \text {integral}& = d \int (g x)^m \left (d^2-e^2 x^2\right )^p \, dx+\frac {e \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \, dx}{g} \\ & = \left (d \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx+\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 (g x)^{1+m} \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx}{g} \\ & = \frac {d (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{g (1+m)}+\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {2+m}{2},-p;\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.76 \[ \int (g x)^m (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\frac {x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d (2+m) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )+e (1+m) x \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-p,\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )\right )}{(1+m) (2+m)} \]
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\[\int \left (g x \right )^{m} \left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{p}d x\]
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\[ \int (g x)^m (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m} \,d x } \]
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Result contains complex when optimal does not.
Time = 3.59 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.79 \[ \int (g x)^m (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\frac {d d^{2 p} g^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {d^{2 p} e g^{m} x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} \]
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\[ \int (g x)^m (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m} \,d x } \]
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\[ \int (g x)^m (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m} \,d x } \]
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Timed out. \[ \int (g x)^m (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\int {\left (d^2-e^2\,x^2\right )}^p\,{\left (g\,x\right )}^m\,\left (d+e\,x\right ) \,d x \]
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