Integrand size = 13, antiderivative size = 48 \[ \int x^m (a+b x)^{3/2} \, dx=\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} (a+b x)^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-m,\frac {7}{2},1+\frac {b x}{a}\right )}{5 b} \]
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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} \[ \int x^m (a+b x)^{3/2} \, dx=\frac {2 x^m (a+b x)^{5/2} \left (-\frac {b x}{a}\right )^{-m} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-m,\frac {7}{2},\frac {b x}{a}+1\right )}{5 b} \]
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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 \left (-\frac {b x}{a}\right )^m (a+b x)^{3/2} \, dx \\ & = \frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} (a+b x)^{5/2} \, _2F_1\left (\frac {5}{2},-m;\frac {7}{2};1+\frac {b x}{a}\right )}{5 b} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int x^m (a+b x)^{3/2} \, dx=\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} (a+b x)^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-m,\frac {7}{2},1+\frac {b x}{a}\right )}{5 b} \]
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\[\int x^{m} \left (b x +a \right )^{\frac {3}{2}}d x\]
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\[ \int x^m (a+b x)^{3/2} \, dx=\int { {\left (b x + a\right )}^{\frac {3}{2}} x^{m} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.99 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.77 \[ \int x^m (a+b x)^{3/2} \, dx=\frac {a^{\frac {3}{2}} x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\Gamma \left (m + 2\right )} \]
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\[ \int x^m (a+b x)^{3/2} \, dx=\int { {\left (b x + a\right )}^{\frac {3}{2}} x^{m} \,d x } \]
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\[ \int x^m (a+b x)^{3/2} \, dx=\int { {\left (b x + a\right )}^{\frac {3}{2}} x^{m} \,d x } \]
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Timed out. \[ \int x^m (a+b x)^{3/2} \, dx=\int x^m\,{\left (a+b\,x\right )}^{3/2} \,d x \]
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