Integrand size = 18, antiderivative size = 161 \[ \int x^5 \left (a+b x^3+c x^6\right )^p \, dx=\frac {\left (a+b x^3+c x^6\right )^{1+p}}{6 c (1+p)}+\frac {2^p b \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x^3+c x^6\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{2 \sqrt {b^2-4 a c}}\right )}{3 c \sqrt {b^2-4 a c} (1+p)} \]
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Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1371, 654, 638} \[ \int x^5 \left (a+b x^3+c x^6\right )^p \, dx=\frac {b 2^p \left (a+b x^3+c x^6\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-p-1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {2 c x^3+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{3 c (p+1) \sqrt {b^2-4 a c}}+\frac {\left (a+b x^3+c x^6\right )^{p+1}}{6 c (p+1)} \]
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Rule 638
Rule 654
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int x \left (a+b x+c x^2\right )^p \, dx,x,x^3\right ) \\ & = \frac {\left (a+b x^3+c x^6\right )^{1+p}}{6 c (1+p)}-\frac {b \text {Subst}\left (\int \left (a+b x+c x^2\right )^p \, dx,x,x^3\right )}{6 c} \\ & = \frac {\left (a+b x^3+c x^6\right )^{1+p}}{6 c (1+p)}+\frac {2^p b \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 c \sqrt {b^2-4 a c} (1+p)} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.01 \[ \int x^5 \left (a+b x^3+c x^6\right )^p \, dx=\frac {1}{6} x^6 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (2,-p,-p,3,-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right ) \]
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\[\int x^{5} \left (c \,x^{6}+b \,x^{3}+a \right )^{p}d x\]
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\[ \int x^5 \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{5} \,d x } \]
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Timed out. \[ \int x^5 \left (a+b x^3+c x^6\right )^p \, dx=\text {Timed out} \]
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\[ \int x^5 \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{5} \,d x } \]
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\[ \int x^5 \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{5} \,d x } \]
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Timed out. \[ \int x^5 \left (a+b x^3+c x^6\right )^p \, dx=\int x^5\,{\left (c\,x^6+b\,x^3+a\right )}^p \,d x \]
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