Optimal. Leaf size=468 \[ \frac{3 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^9 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (2 p+9)}-\frac{12 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^8 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (p+4)}+\frac{84 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^7 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (2 p+7)}-\frac{84 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^6 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (p+3)}+\frac{210 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^5 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (2 p+5)}-\frac{84 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^4 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (p+2)}+\frac{84 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^3 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (2 p+3)}-\frac{12 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^2 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (p+1)}+\frac{3 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (2 p+1)} \]
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Rubi [A] time = 0.224245, antiderivative size = 468, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {1356, 266, 43} \[ \frac{3 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^9 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (2 p+9)}-\frac{12 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^8 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (p+4)}+\frac{84 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^7 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (2 p+7)}-\frac{84 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^6 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (p+3)}+\frac{210 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^5 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (2 p+5)}-\frac{84 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^4 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (p+2)}+\frac{84 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^3 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (2 p+3)}-\frac{12 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right )^2 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (p+1)}+\frac{3 a^9 \left (\frac{b \sqrt [3]{x}}{a}+1\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 1356
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p x^2 \, dx &=\left (\left (1+\frac{b \sqrt [3]{x}}{a}\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p\right ) \int \left (1+\frac{b \sqrt [3]{x}}{a}\right )^{2 p} x^2 \, dx\\ &=\left (3 \left (1+\frac{b \sqrt [3]{x}}{a}\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p\right ) \operatorname{Subst}\left (\int x^8 \left (1+\frac{b x}{a}\right )^{2 p} \, dx,x,\sqrt [3]{x}\right )\\ &=\left (3 \left (1+\frac{b \sqrt [3]{x}}{a}\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p\right ) \operatorname{Subst}\left (\int \left (\frac{a^8 \left (1+\frac{b x}{a}\right )^{2 p}}{b^8}-\frac{8 a^8 \left (1+\frac{b x}{a}\right )^{1+2 p}}{b^8}+\frac{28 a^8 \left (1+\frac{b x}{a}\right )^{2+2 p}}{b^8}-\frac{56 a^8 \left (1+\frac{b x}{a}\right )^{3+2 p}}{b^8}+\frac{70 a^8 \left (1+\frac{b x}{a}\right )^{4+2 p}}{b^8}-\frac{56 a^8 \left (1+\frac{b x}{a}\right )^{5+2 p}}{b^8}+\frac{28 a^8 \left (1+\frac{b x}{a}\right )^{6+2 p}}{b^8}-\frac{8 a^8 \left (1+\frac{b x}{a}\right )^{7+2 p}}{b^8}+\frac{a^8 \left (1+\frac{b x}{a}\right )^{8+2 p}}{b^8}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^9 \left (1+\frac{b \sqrt [3]{x}}{a}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (1+2 p)}-\frac{12 a^9 \left (1+\frac{b \sqrt [3]{x}}{a}\right )^2 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (1+p)}+\frac{84 a^9 \left (1+\frac{b \sqrt [3]{x}}{a}\right )^3 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (3+2 p)}-\frac{84 a^9 \left (1+\frac{b \sqrt [3]{x}}{a}\right )^4 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (2+p)}+\frac{210 a^9 \left (1+\frac{b \sqrt [3]{x}}{a}\right )^5 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (5+2 p)}-\frac{84 a^9 \left (1+\frac{b \sqrt [3]{x}}{a}\right )^6 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (3+p)}+\frac{84 a^9 \left (1+\frac{b \sqrt [3]{x}}{a}\right )^7 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (7+2 p)}-\frac{12 a^9 \left (1+\frac{b \sqrt [3]{x}}{a}\right )^8 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (4+p)}+\frac{3 a^9 \left (1+\frac{b \sqrt [3]{x}}{a}\right )^9 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^9 (9+2 p)}\\ \end{align*}
Mathematica [A] time = 0.224451, size = 207, normalized size = 0.44 \[ \frac{3 \left (a+b \sqrt [3]{x}\right ) \left (-\frac{4 a^7 \left (a+b \sqrt [3]{x}\right )}{p+1}+\frac{28 a^6 \left (a+b \sqrt [3]{x}\right )^2}{2 p+3}-\frac{28 a^5 \left (a+b \sqrt [3]{x}\right )^3}{p+2}+\frac{70 a^4 \left (a+b \sqrt [3]{x}\right )^4}{2 p+5}-\frac{28 a^3 \left (a+b \sqrt [3]{x}\right )^5}{p+3}+\frac{28 a^2 \left (a+b \sqrt [3]{x}\right )^6}{2 p+7}+\frac{a^8}{2 p+1}-\frac{4 a \left (a+b \sqrt [3]{x}\right )^7}{p+4}+\frac{\left (a+b \sqrt [3]{x}\right )^8}{2 p+9}\right ) \left (\left (a+b \sqrt [3]{x}\right )^2\right )^p}{b^9} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.007, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}} \right ) ^{p}{x}^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06306, size = 489, normalized size = 1.04 \begin{align*} \frac{3 \,{\left ({\left (16 \, p^{8} + 288 \, p^{7} + 2184 \, p^{6} + 9072 \, p^{5} + 22449 \, p^{4} + 33642 \, p^{3} + 29531 \, p^{2} + 13698 \, p + 2520\right )} b^{9} x^{3} +{\left (16 \, p^{8} + 224 \, p^{7} + 1288 \, p^{6} + 3920 \, p^{5} + 6769 \, p^{4} + 6566 \, p^{3} + 3267 \, p^{2} + 630 \, p\right )} a b^{8} x^{\frac{8}{3}} - 8 \,{\left (8 \, p^{7} + 84 \, p^{6} + 350 \, p^{5} + 735 \, p^{4} + 812 \, p^{3} + 441 \, p^{2} + 90 \, p\right )} a^{2} b^{7} x^{\frac{7}{3}} + 28 \,{\left (8 \, p^{6} + 60 \, p^{5} + 170 \, p^{4} + 225 \, p^{3} + 137 \, p^{2} + 30 \, p\right )} a^{3} b^{6} x^{2} - 168 \,{\left (4 \, p^{5} + 20 \, p^{4} + 35 \, p^{3} + 25 \, p^{2} + 6 \, p\right )} a^{4} b^{5} x^{\frac{5}{3}} + 420 \,{\left (4 \, p^{4} + 12 \, p^{3} + 11 \, p^{2} + 3 \, p\right )} a^{5} b^{4} x^{\frac{4}{3}} - 1680 \,{\left (2 \, p^{3} + 3 \, p^{2} + p\right )} a^{6} b^{3} x + 2520 \,{\left (2 \, p^{2} + p\right )} a^{7} b^{2} x^{\frac{2}{3}} - 5040 \, a^{8} b p x^{\frac{1}{3}} + 2520 \, a^{9}\right )}{\left (b x^{\frac{1}{3}} + a\right )}^{2 \, p}}{{\left (32 \, p^{9} + 720 \, p^{8} + 6960 \, p^{7} + 37800 \, p^{6} + 126546 \, p^{5} + 269325 \, p^{4} + 361840 \, p^{3} + 293175 \, p^{2} + 128322 \, p + 22680\right )} b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.18854, size = 1354, normalized size = 2.89 \begin{align*} \frac{3 \,{\left (2520 \, a^{9} +{\left (16 \, b^{9} p^{8} + 288 \, b^{9} p^{7} + 2184 \, b^{9} p^{6} + 9072 \, b^{9} p^{5} + 22449 \, b^{9} p^{4} + 33642 \, b^{9} p^{3} + 29531 \, b^{9} p^{2} + 13698 \, b^{9} p + 2520 \, b^{9}\right )} x^{3} + 28 \,{\left (8 \, a^{3} b^{6} p^{6} + 60 \, a^{3} b^{6} p^{5} + 170 \, a^{3} b^{6} p^{4} + 225 \, a^{3} b^{6} p^{3} + 137 \, a^{3} b^{6} p^{2} + 30 \, a^{3} b^{6} p\right )} x^{2} - 1680 \,{\left (2 \, a^{6} b^{3} p^{3} + 3 \, a^{6} b^{3} p^{2} + a^{6} b^{3} p\right )} x +{\left (5040 \, a^{7} b^{2} p^{2} + 2520 \, a^{7} b^{2} p +{\left (16 \, a b^{8} p^{8} + 224 \, a b^{8} p^{7} + 1288 \, a b^{8} p^{6} + 3920 \, a b^{8} p^{5} + 6769 \, a b^{8} p^{4} + 6566 \, a b^{8} p^{3} + 3267 \, a b^{8} p^{2} + 630 \, a b^{8} p\right )} x^{2} - 168 \,{\left (4 \, a^{4} b^{5} p^{5} + 20 \, a^{4} b^{5} p^{4} + 35 \, a^{4} b^{5} p^{3} + 25 \, a^{4} b^{5} p^{2} + 6 \, a^{4} b^{5} p\right )} x\right )} x^{\frac{2}{3}} - 4 \,{\left (1260 \, a^{8} b p + 2 \,{\left (8 \, a^{2} b^{7} p^{7} + 84 \, a^{2} b^{7} p^{6} + 350 \, a^{2} b^{7} p^{5} + 735 \, a^{2} b^{7} p^{4} + 812 \, a^{2} b^{7} p^{3} + 441 \, a^{2} b^{7} p^{2} + 90 \, a^{2} b^{7} p\right )} x^{2} - 105 \,{\left (4 \, a^{5} b^{4} p^{4} + 12 \, a^{5} b^{4} p^{3} + 11 \, a^{5} b^{4} p^{2} + 3 \, a^{5} b^{4} p\right )} x\right )} x^{\frac{1}{3}}\right )}{\left (b^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + a^{2}\right )}^{p}}{32 \, b^{9} p^{9} + 720 \, b^{9} p^{8} + 6960 \, b^{9} p^{7} + 37800 \, b^{9} p^{6} + 126546 \, b^{9} p^{5} + 269325 \, b^{9} p^{4} + 361840 \, b^{9} p^{3} + 293175 \, b^{9} p^{2} + 128322 \, b^{9} p + 22680 \, b^{9}} \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 [B] time = 1.18422, size = 2111, normalized size = 4.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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