Integrand size = 13, antiderivative size = 48 \[ \int \frac {x^m}{(a+b x)^{5/2}} \, dx=-\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-m,-\frac {1}{2},1+\frac {b x}{a}\right )}{3 b (a+b x)^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {69, 67} \[ \int \frac {x^m}{(a+b x)^{5/2}} \, dx=-\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-m,-\frac {1}{2},\frac {b x}{a}+1\right )}{3 b (a+b x)^{3/2}} \]
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Rule 67
Rule 69
Rubi steps \begin{align*} \text {integral}& = \left (x^m \left (-\frac {b x}{a}\right )^{-m}\right ) \int \frac {\left (-\frac {b x}{a}\right )^m}{(a+b x)^{5/2}} \, dx \\ & = -\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (-\frac {3}{2},-m;-\frac {1}{2};1+\frac {b x}{a}\right )}{3 b (a+b x)^{3/2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{(a+b x)^{5/2}} \, dx=-\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-m,-\frac {1}{2},1+\frac {b x}{a}\right )}{3 b (a+b x)^{3/2}} \]
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\[\int \frac {x^{m}}{\left (b x +a \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {x^m}{(a+b x)^{5/2}} \, dx=\int { \frac {x^{m}}{{\left (b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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Result contains complex when optimal does not.
Time = 2.67 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.75 \[ \int \frac {x^m}{(a+b x)^{5/2}} \, dx=\frac {x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{a^{\frac {5}{2}} \Gamma \left (m + 2\right )} \]
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\[ \int \frac {x^m}{(a+b x)^{5/2}} \, dx=\int { \frac {x^{m}}{{\left (b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x^m}{(a+b x)^{5/2}} \, dx=\int { \frac {x^{m}}{{\left (b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^m}{(a+b x)^{5/2}} \, dx=\int \frac {x^m}{{\left (a+b\,x\right )}^{5/2}} \,d x \]
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