Integrand size = 19, antiderivative size = 61 \[ \int (d x)^{5/2} \left (b x+c x^2\right )^p \, dx=\frac {2 x (d x)^{5/2} \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-p,\frac {7}{2}+p,\frac {9}{2}+p,-\frac {c x}{b}\right )}{7+2 p} \]
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Time = 0.02 (sec) , antiderivative size = 61, 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, 68, 66} \[ \int (d x)^{5/2} \left (b x+c x^2\right )^p \, dx=\frac {2 x (d x)^{5/2} \left (\frac {c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-p,p+\frac {7}{2},p+\frac {9}{2},-\frac {c x}{b}\right )}{2 p+7} \]
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Rule 66
Rule 68
Rule 688
Rubi steps \begin{align*} \text {integral}& = \left (x^{-\frac {5}{2}-p} (d x)^{5/2} (b+c x)^{-p} \left (b x+c x^2\right )^p\right ) \int x^{\frac {5}{2}+p} (b+c x)^p \, dx \\ & = \left (x^{-\frac {5}{2}-p} (d x)^{5/2} \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p\right ) \int x^{\frac {5}{2}+p} \left (1+\frac {c x}{b}\right )^p \, dx \\ & = \frac {2 x (d x)^{5/2} \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,\frac {7}{2}+p;\frac {9}{2}+p;-\frac {c x}{b}\right )}{7+2 p} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.95 \[ \int (d x)^{5/2} \left (b x+c x^2\right )^p \, dx=\frac {x (d x)^{5/2} (x (b+c x))^p \left (1+\frac {c x}{b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {7}{2}+p,\frac {9}{2}+p,-\frac {c x}{b}\right )}{\frac {7}{2}+p} \]
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\[\int \left (d x \right )^{\frac {5}{2}} \left (c \,x^{2}+b x \right )^{p}d x\]
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\[ \int (d x)^{5/2} \left (b x+c x^2\right )^p \, dx=\int { \left (d x\right )^{\frac {5}{2}} {\left (c x^{2} + b x\right )}^{p} \,d x } \]
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\[ \int (d x)^{5/2} \left (b x+c x^2\right )^p \, dx=\int \left (d x\right )^{\frac {5}{2}} \left (x \left (b + c x\right )\right )^{p}\, dx \]
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\[ \int (d x)^{5/2} \left (b x+c x^2\right )^p \, dx=\int { \left (d x\right )^{\frac {5}{2}} {\left (c x^{2} + b x\right )}^{p} \,d x } \]
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\[ \int (d x)^{5/2} \left (b x+c x^2\right )^p \, dx=\int { \left (d x\right )^{\frac {5}{2}} {\left (c x^{2} + b x\right )}^{p} \,d x } \]
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Timed out. \[ \int (d x)^{5/2} \left (b x+c x^2\right )^p \, dx=\int {\left (c\,x^2+b\,x\right )}^p\,{\left (d\,x\right )}^{5/2} \,d x \]
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