Integrand size = 15, antiderivative size = 49 \[ \int x^2 \left (b x+c x^2\right )^p \, dx=\frac {x^3 \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-p,3+p,4+p,-\frac {c x}{b}\right )}{3+p} \]
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Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {688, 68, 66} \[ \int x^2 \left (b x+c x^2\right )^p \, dx=\frac {x^3 \left (\frac {c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-p,p+3,p+4,-\frac {c x}{b}\right )}{p+3} \]
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Rule 66
Rule 68
Rule 688
Rubi steps \begin{align*} \text {integral}& = \left (x^{-p} (b+c x)^{-p} \left (b x+c x^2\right )^p\right ) \int x^{2+p} (b+c x)^p \, dx \\ & = \left (x^{-p} \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p\right ) \int x^{2+p} \left (1+\frac {c x}{b}\right )^p \, dx \\ & = \frac {x^3 \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,3+p;4+p;-\frac {c x}{b}\right )}{3+p} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.96 \[ \int x^2 \left (b x+c x^2\right )^p \, dx=\frac {x^3 (x (b+c x))^p \left (1+\frac {c x}{b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,3+p,4+p,-\frac {c x}{b}\right )}{3+p} \]
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\[\int x^{2} \left (c \,x^{2}+b x \right )^{p}d x\]
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\[ \int x^2 \left (b x+c x^2\right )^p \, dx=\int { {\left (c x^{2} + b x\right )}^{p} x^{2} \,d x } \]
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\[ \int x^2 \left (b x+c x^2\right )^p \, dx=\int x^{2} \left (x \left (b + c x\right )\right )^{p}\, dx \]
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\[ \int x^2 \left (b x+c x^2\right )^p \, dx=\int { {\left (c x^{2} + b x\right )}^{p} x^{2} \,d x } \]
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\[ \int x^2 \left (b x+c x^2\right )^p \, dx=\int { {\left (c x^{2} + b x\right )}^{p} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \left (b x+c x^2\right )^p \, dx=\int x^2\,{\left (c\,x^2+b\,x\right )}^p \,d x \]
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