Integrand size = 19, antiderivative size = 73 \[ \int (d x)^m \left (b x+c x^2\right )^{5/2} \, dx=\frac {2 b^2 \left (-\frac {c x}{b}\right )^{-\frac {1}{2}-m} (d x)^m (b+c x) \left (b x+c x^2\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-\frac {5}{2}-m,\frac {9}{2},1+\frac {c x}{b}\right )}{7 c^3 x^2} \]
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Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {688, 69, 67} \[ \int (d x)^m \left (b x+c x^2\right )^{5/2} \, dx=\frac {2 b^2 (b+c x) \left (b x+c x^2\right )^{5/2} (d x)^m \left (-\frac {c x}{b}\right )^{-m-\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-m-\frac {5}{2},\frac {9}{2},\frac {c x}{b}+1\right )}{7 c^3 x^2} \]
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Rule 67
Rule 69
Rule 688
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-\frac {5}{2}-m} (d x)^m \left (b x+c x^2\right )^{5/2}\right ) \int x^{\frac {5}{2}+m} (b+c x)^{5/2} \, dx}{(b+c x)^{5/2}} \\ & = \frac {\left (b^2 \left (-\frac {c x}{b}\right )^{-\frac {1}{2}-m} (d x)^m \left (b x+c x^2\right )^{5/2}\right ) \int \left (-\frac {c x}{b}\right )^{\frac {5}{2}+m} (b+c x)^{5/2} \, dx}{c^2 x^2 (b+c x)^{5/2}} \\ & = \frac {2 b^2 \left (-\frac {c x}{b}\right )^{-\frac {1}{2}-m} (d x)^m (b+c x) \left (b x+c x^2\right )^{5/2} \, _2F_1\left (\frac {7}{2},-\frac {5}{2}-m;\frac {9}{2};1+\frac {c x}{b}\right )}{7 c^3 x^2} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96 \[ \int (d x)^m \left (b x+c x^2\right )^{5/2} \, dx=\frac {2 b^2 \left (-\frac {c x}{b}\right )^{-\frac {1}{2}-m} (d x)^m (b+c x)^3 \sqrt {x (b+c x)} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-\frac {5}{2}-m,\frac {9}{2},1+\frac {c x}{b}\right )}{7 c^3} \]
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\[\int \left (d x \right )^{m} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}d x\]
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\[ \int (d x)^m \left (b x+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {5}{2}} \left (d x\right )^{m} \,d x } \]
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\[ \int (d x)^m \left (b x+c x^2\right )^{5/2} \, dx=\int \left (d x\right )^{m} \left (x \left (b + c x\right )\right )^{\frac {5}{2}}\, dx \]
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\[ \int (d x)^m \left (b x+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {5}{2}} \left (d x\right )^{m} \,d x } \]
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\[ \int (d x)^m \left (b x+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {5}{2}} \left (d x\right )^{m} \,d x } \]
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Timed out. \[ \int (d x)^m \left (b x+c x^2\right )^{5/2} \, dx=\int {\left (c\,x^2+b\,x\right )}^{5/2}\,{\left (d\,x\right )}^m \,d x \]
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