Optimal. Leaf size=134 \[ \frac{45 a^4 x^{2/3}}{2 b^7}+\frac{9 a^2 x^{4/3}}{2 b^5}-\frac{3 a^8}{2 b^9 \left (a+b \sqrt [3]{x}\right )^2}+\frac{24 a^7}{b^9 \left (a+b \sqrt [3]{x}\right )}-\frac{63 a^5 \sqrt [3]{x}}{b^8}-\frac{10 a^3 x}{b^6}+\frac{84 a^6 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{9 a x^{5/3}}{5 b^4}+\frac{x^2}{2 b^3} \]
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Rubi [A] time = 0.100282, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{45 a^4 x^{2/3}}{2 b^7}+\frac{9 a^2 x^{4/3}}{2 b^5}-\frac{3 a^8}{2 b^9 \left (a+b \sqrt [3]{x}\right )^2}+\frac{24 a^7}{b^9 \left (a+b \sqrt [3]{x}\right )}-\frac{63 a^5 \sqrt [3]{x}}{b^8}-\frac{10 a^3 x}{b^6}+\frac{84 a^6 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{9 a x^{5/3}}{5 b^4}+\frac{x^2}{2 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b \sqrt [3]{x}\right )^3} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^8}{(a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{21 a^5}{b^8}+\frac{15 a^4 x}{b^7}-\frac{10 a^3 x^2}{b^6}+\frac{6 a^2 x^3}{b^5}-\frac{3 a x^4}{b^4}+\frac{x^5}{b^3}+\frac{a^8}{b^8 (a+b x)^3}-\frac{8 a^7}{b^8 (a+b x)^2}+\frac{28 a^6}{b^8 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 a^8}{2 b^9 \left (a+b \sqrt [3]{x}\right )^2}+\frac{24 a^7}{b^9 \left (a+b \sqrt [3]{x}\right )}-\frac{63 a^5 \sqrt [3]{x}}{b^8}+\frac{45 a^4 x^{2/3}}{2 b^7}-\frac{10 a^3 x}{b^6}+\frac{9 a^2 x^{4/3}}{2 b^5}-\frac{9 a x^{5/3}}{5 b^4}+\frac{x^2}{2 b^3}+\frac{84 a^6 \log \left (a+b \sqrt [3]{x}\right )}{b^9}\\ \end{align*}
Mathematica [A] time = 0.102238, size = 120, normalized size = 0.9 \[ \frac{225 a^4 b^2 x^{2/3}+45 a^2 b^4 x^{4/3}-100 a^3 b^3 x-\frac{15 a^8}{\left (a+b \sqrt [3]{x}\right )^2}+\frac{240 a^7}{a+b \sqrt [3]{x}}-630 a^5 b \sqrt [3]{x}+840 a^6 \log \left (a+b \sqrt [3]{x}\right )-18 a b^5 x^{5/3}+5 b^6 x^2}{10 b^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 111, normalized size = 0.8 \begin{align*} -{\frac{3\,{a}^{8}}{2\,{b}^{9}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}+24\,{\frac{{a}^{7}}{{b}^{9} \left ( a+b\sqrt [3]{x} \right ) }}-63\,{\frac{{a}^{5}\sqrt [3]{x}}{{b}^{8}}}+{\frac{45\,{a}^{4}}{2\,{b}^{7}}{x}^{{\frac{2}{3}}}}-10\,{\frac{{a}^{3}x}{{b}^{6}}}+{\frac{9\,{a}^{2}}{2\,{b}^{5}}{x}^{{\frac{4}{3}}}}-{\frac{9\,a}{5\,{b}^{4}}{x}^{{\frac{5}{3}}}}+{\frac{{x}^{2}}{2\,{b}^{3}}}+84\,{\frac{{a}^{6}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{9}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976367, size = 197, normalized size = 1.47 \begin{align*} \frac{84 \, a^{6} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{9}} + \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{6}}{2 \, b^{9}} - \frac{24 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a}{5 \, b^{9}} + \frac{21 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{2}}{b^{9}} - \frac{56 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{3}}{b^{9}} + \frac{105 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{4}}{b^{9}} - \frac{168 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{5}}{b^{9}} + \frac{24 \, a^{7}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{9}} - \frac{3 \, a^{8}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51047, size = 440, normalized size = 3.28 \begin{align*} \frac{5 \, b^{12} x^{4} - 90 \, a^{3} b^{9} x^{3} - 195 \, a^{6} b^{6} x^{2} + 170 \, a^{9} b^{3} x + 225 \, a^{12} + 840 \,{\left (a^{6} b^{6} x^{2} + 2 \, a^{9} b^{3} x + a^{12}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) - 3 \,{\left (6 \, a b^{11} x^{3} - 63 \, a^{4} b^{8} x^{2} - 224 \, a^{7} b^{5} x - 140 \, a^{10} b^{2}\right )} x^{\frac{2}{3}} + 15 \,{\left (3 \, a^{2} b^{10} x^{3} - 36 \, a^{5} b^{7} x^{2} - 98 \, a^{8} b^{4} x - 56 \, a^{11} b\right )} x^{\frac{1}{3}}}{10 \,{\left (b^{15} x^{2} + 2 \, a^{3} b^{12} x + a^{6} b^{9}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.04475, size = 493, normalized size = 3.68 \begin{align*} \begin{cases} \frac{840 a^{8} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{1260 a^{8}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{1680 a^{7} b \sqrt [3]{x} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{1680 a^{7} b \sqrt [3]{x}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{840 a^{6} b^{2} x^{\frac{2}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} - \frac{280 a^{5} b^{3} x}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{70 a^{4} b^{4} x^{\frac{4}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} - \frac{28 a^{3} b^{5} x^{\frac{5}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{14 a^{2} b^{6} x^{2}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} - \frac{8 a b^{7} x^{\frac{7}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{5 b^{8} x^{\frac{8}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21395, size = 151, normalized size = 1.13 \begin{align*} \frac{84 \, a^{6} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{9}} + \frac{3 \,{\left (16 \, a^{7} b x^{\frac{1}{3}} + 15 \, a^{8}\right )}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{9}} + \frac{5 \, b^{15} x^{2} - 18 \, a b^{14} x^{\frac{5}{3}} + 45 \, a^{2} b^{13} x^{\frac{4}{3}} - 100 \, a^{3} b^{12} x + 225 \, a^{4} b^{11} x^{\frac{2}{3}} - 630 \, a^{5} b^{10} x^{\frac{1}{3}}}{10 \, b^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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