Integrand size = 22, antiderivative size = 145 \[ \int \frac {(e x)^m (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {A (e x)^{1+m} \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1+m}{2},\frac {3+m}{2},-\frac {c x^2}{a}\right )}{a e (1+m) \sqrt {a+c x^2}}+\frac {B (e x)^{2+m} \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},-\frac {c x^2}{a}\right )}{a e^2 (2+m) \sqrt {a+c x^2}} \]
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Time = 0.05 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {822, 372, 371} \[ \int \frac {(e x)^m (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {A \sqrt {\frac {c x^2}{a}+1} (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {m+1}{2},\frac {m+3}{2},-\frac {c x^2}{a}\right )}{a e (m+1) \sqrt {a+c x^2}}+\frac {B \sqrt {\frac {c x^2}{a}+1} (e x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {m+2}{2},\frac {m+4}{2},-\frac {c x^2}{a}\right )}{a e^2 (m+2) \sqrt {a+c x^2}} \]
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Rule 371
Rule 372
Rule 822
Rubi steps \begin{align*} \text {integral}& = A \int \frac {(e x)^m}{\left (a+c x^2\right )^{3/2}} \, dx+\frac {B \int \frac {(e x)^{1+m}}{\left (a+c x^2\right )^{3/2}} \, dx}{e} \\ & = \frac {\left (A \sqrt {1+\frac {c x^2}{a}}\right ) \int \frac {(e x)^m}{\left (1+\frac {c x^2}{a}\right )^{3/2}} \, dx}{a \sqrt {a+c x^2}}+\frac {\left (B \sqrt {1+\frac {c x^2}{a}}\right ) \int \frac {(e x)^{1+m}}{\left (1+\frac {c x^2}{a}\right )^{3/2}} \, dx}{a e \sqrt {a+c x^2}} \\ & = \frac {A (e x)^{1+m} \sqrt {1+\frac {c x^2}{a}} \, _2F_1\left (\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};-\frac {c x^2}{a}\right )}{a e (1+m) \sqrt {a+c x^2}}+\frac {B (e x)^{2+m} \sqrt {1+\frac {c x^2}{a}} \, _2F_1\left (\frac {3}{2},\frac {2+m}{2};\frac {4+m}{2};-\frac {c x^2}{a}\right )}{a e^2 (2+m) \sqrt {a+c x^2}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.77 \[ \int \frac {(e x)^m (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {x (e x)^m \sqrt {1+\frac {c x^2}{a}} \left (B (1+m) x \operatorname {Hypergeometric2F1}\left (\frac {3}{2},1+\frac {m}{2},2+\frac {m}{2},-\frac {c x^2}{a}\right )+A (2+m) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1+m}{2},\frac {3+m}{2},-\frac {c x^2}{a}\right )\right )}{a (1+m) (2+m) \sqrt {a+c x^2}} \]
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\[\int \frac {\left (e x \right )^{m} \left (B x +A \right )}{\left (c \,x^{2}+a \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {(e x)^m (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{m}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Result contains complex when optimal does not.
Time = 5.13 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.76 \[ \int \frac {(e x)^m (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {A e^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {B e^{m} x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {m}{2} + 2\right )} \]
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\[ \int \frac {(e x)^m (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{m}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(e x)^m (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{m}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^m (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x\right )}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
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