Integrand size = 18, antiderivative size = 130 \[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=-\frac {2^{1+p} \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x^3+c x^6\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{2 \sqrt {b^2-4 a c}}\right )}{3 \sqrt {b^2-4 a c} (1+p)} \]
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Time = 0.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1366, 638} \[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=-\frac {2^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x^3+c x^6\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {2 c x^3+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{3 (p+1) \sqrt {b^2-4 a c}} \]
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Rule 638
Rule 1366
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \left (a+b x+c x^2\right )^p \, dx,x,x^3\right ) \\ & = -\frac {2^{1+p} \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x^3+c x^6\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{2 \sqrt {b^2-4 a c}}\right )}{3 \sqrt {b^2-4 a c} (1+p)} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.06 \[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=\frac {2^{-1+p} \left (b-\sqrt {b^2-4 a c}+2 c x^3\right ) \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {-b+\sqrt {b^2-4 a c}-2 c x^3}{2 \sqrt {b^2-4 a c}}\right )}{3 c (1+p)} \]
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\[\int x^{2} \left (c \,x^{6}+b \,x^{3}+a \right )^{p}d x\]
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\[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=\text {Timed out} \]
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\[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2} \,d x } \]
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\[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=\int x^2\,{\left (c\,x^6+b\,x^3+a\right )}^p \,d x \]
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