Optimal. Leaf size=224 \[ \frac{2^p \left (2 a c-b^2 (p+2)\right ) \left (a+b x^3+c x^6\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{2 c x^3+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{3 c^2 (p+1) (2 p+3) \sqrt{b^2-4 a c}}-\frac{b (p+2) \left (a+b x^3+c x^6\right )^{p+1}}{6 c^2 (p+1) (2 p+3)}+\frac{x^3 \left (a+b x^3+c x^6\right )^{p+1}}{3 c (2 p+3)} \]
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Rubi [A] time = 0.243492, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1357, 742, 640, 624} \[ \frac{2^p \left (2 a c-b^2 (p+2)\right ) \left (a+b x^3+c x^6\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{2 c x^3+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{3 c^2 (p+1) (2 p+3) \sqrt{b^2-4 a c}}-\frac{b (p+2) \left (a+b x^3+c x^6\right )^{p+1}}{6 c^2 (p+1) (2 p+3)}+\frac{x^3 \left (a+b x^3+c x^6\right )^{p+1}}{3 c (2 p+3)} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 742
Rule 640
Rule 624
Rubi steps
\begin{align*} \int x^8 \left (a+b x^3+c x^6\right )^p \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^2 \left (a+b x+c x^2\right )^p \, dx,x,x^3\right )\\ &=\frac{x^3 \left (a+b x^3+c x^6\right )^{1+p}}{3 c (3+2 p)}+\frac{\operatorname{Subst}\left (\int (-a-b (2+p) x) \left (a+b x+c x^2\right )^p \, dx,x,x^3\right )}{3 c (3+2 p)}\\ &=-\frac{b (2+p) \left (a+b x^3+c x^6\right )^{1+p}}{6 c^2 (1+p) (3+2 p)}+\frac{x^3 \left (a+b x^3+c x^6\right )^{1+p}}{3 c (3+2 p)}-\frac{\left (2 a c-b^2 (2+p)\right ) \operatorname{Subst}\left (\int \left (a+b x+c x^2\right )^p \, dx,x,x^3\right )}{6 c^2 (3+2 p)}\\ &=-\frac{b (2+p) \left (a+b x^3+c x^6\right )^{1+p}}{6 c^2 (1+p) (3+2 p)}+\frac{x^3 \left (a+b x^3+c x^6\right )^{1+p}}{3 c (3+2 p)}+\frac{2^p \left (2 a c-b^2 (2+p)\right ) \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^2 \sqrt{b^2-4 a c} (1+p) (3+2 p)}\\ \end{align*}
Mathematica [C] time = 0.203059, size = 162, normalized size = 0.72 \[ \frac{1}{9} x^9 \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (3;-p,-p;4;-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{x}^{8} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{6} + b x^{3} + a\right )}^{p} x^{8}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c x^{6} + b x^{3} + a\right )}^{p} x^{8}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{6} + b x^{3} + a\right )}^{p} x^{8}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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