Integrand size = 20, antiderivative size = 91 \[ \int \frac {(e x)^m (A+B x)}{a+c x^2} \, dx=\frac {A (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {c x^2}{a}\right )}{a e (1+m)}+\frac {B (e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\frac {c x^2}{a}\right )}{a e^2 (2+m)} \]
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Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {822, 371} \[ \int \frac {(e x)^m (A+B x)}{a+c x^2} \, dx=\frac {A (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {c x^2}{a}\right )}{a e (m+1)}+\frac {B (e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},-\frac {c x^2}{a}\right )}{a e^2 (m+2)} \]
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Rule 371
Rule 822
Rubi steps \begin{align*} \text {integral}& = A \int \frac {(e x)^m}{a+c x^2} \, dx+\frac {B \int \frac {(e x)^{1+m}}{a+c x^2} \, dx}{e} \\ & = \frac {A (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {c x^2}{a}\right )}{a e (1+m)}+\frac {B (e x)^{2+m} \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};-\frac {c x^2}{a}\right )}{a e^2 (2+m)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.90 \[ \int \frac {(e x)^m (A+B x)}{a+c x^2} \, dx=\frac {x (e x)^m \left (B (1+m) x \operatorname {Hypergeometric2F1}\left (1,1+\frac {m}{2},2+\frac {m}{2},-\frac {c x^2}{a}\right )+A (2+m) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {c x^2}{a}\right )\right )}{a (1+m) (2+m)} \]
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\[\int \frac {\left (e x \right )^{m} \left (B x +A \right )}{c \,x^{2}+a}d x\]
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\[ \int \frac {(e x)^m (A+B x)}{a+c x^2} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{m}}{c x^{2} + a} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.70 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.08 \[ \int \frac {(e x)^m (A+B x)}{a+c x^2} \, dx=\frac {A e^{m} m x^{m + 1} \Phi \left (\frac {c x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {A e^{m} x^{m + 1} \Phi \left (\frac {c x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {B e^{m} m x^{m + 2} \Phi \left (\frac {c x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{4 a \Gamma \left (\frac {m}{2} + 2\right )} + \frac {B e^{m} x^{m + 2} \Phi \left (\frac {c x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{2 a \Gamma \left (\frac {m}{2} + 2\right )} \]
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\[ \int \frac {(e x)^m (A+B x)}{a+c x^2} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{m}}{c x^{2} + a} \,d x } \]
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\[ \int \frac {(e x)^m (A+B x)}{a+c x^2} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{m}}{c x^{2} + a} \,d x } \]
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Timed out. \[ \int \frac {(e x)^m (A+B x)}{a+c x^2} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x\right )}{c\,x^2+a} \,d x \]
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