Optimal. Leaf size=59 \[ \frac{3 a^2 \left (a+b x^2\right )^{5/3}}{10 b^3}+\frac{3 \left (a+b x^2\right )^{11/3}}{22 b^3}-\frac{3 a \left (a+b x^2\right )^{8/3}}{8 b^3} \]
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Rubi [A] time = 0.0359452, antiderivative size = 59, 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} \[ \frac{3 a^2 \left (a+b x^2\right )^{5/3}}{10 b^3}+\frac{3 \left (a+b x^2\right )^{11/3}}{22 b^3}-\frac{3 a \left (a+b x^2\right )^{8/3}}{8 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a+b x^2\right )^{2/3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (a+b x)^{2/3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^{2/3}}{b^2}-\frac{2 a (a+b x)^{5/3}}{b^2}+\frac{(a+b x)^{8/3}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac{3 a^2 \left (a+b x^2\right )^{5/3}}{10 b^3}-\frac{3 a \left (a+b x^2\right )^{8/3}}{8 b^3}+\frac{3 \left (a+b x^2\right )^{11/3}}{22 b^3}\\ \end{align*}
Mathematica [A] time = 0.0165565, size = 39, normalized size = 0.66 \[ \frac{3 \left (a+b x^2\right )^{5/3} \left (9 a^2-15 a b x^2+20 b^2 x^4\right )}{440 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 36, normalized size = 0.6 \begin{align*}{\frac{60\,{b}^{2}{x}^{4}-45\,ab{x}^{2}+27\,{a}^{2}}{440\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.96401, size = 63, normalized size = 1.07 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{11}{3}}}{22 \, b^{3}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{8}{3}} a}{8 \, b^{3}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} a^{2}}{10 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72563, size = 105, normalized size = 1.78 \begin{align*} \frac{3 \,{\left (20 \, b^{3} x^{6} + 5 \, a b^{2} x^{4} - 6 \, a^{2} b x^{2} + 9 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{440 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.8706, size = 700, normalized size = 11.86 \begin{align*} \frac{27 a^{\frac{35}{3}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} - \frac{27 a^{\frac{35}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac{63 a^{\frac{32}{3}} b x^{2} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} - \frac{81 a^{\frac{32}{3}} b x^{2}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac{42 a^{\frac{29}{3}} b^{2} x^{4} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} - \frac{81 a^{\frac{29}{3}} b^{2} x^{4}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac{78 a^{\frac{26}{3}} b^{3} x^{6} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} - \frac{27 a^{\frac{26}{3}} b^{3} x^{6}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac{207 a^{\frac{23}{3}} b^{4} x^{8} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac{195 a^{\frac{20}{3}} b^{5} x^{10} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} + \frac{60 a^{\frac{17}{3}} b^{6} x^{12} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{440 a^{8} b^{3} + 1320 a^{7} b^{4} x^{2} + 1320 a^{6} b^{5} x^{4} + 440 a^{5} b^{6} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.1507, size = 58, normalized size = 0.98 \begin{align*} \frac{3 \,{\left (20 \,{\left (b x^{2} + a\right )}^{\frac{11}{3}} - 55 \,{\left (b x^{2} + a\right )}^{\frac{8}{3}} a + 44 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} a^{2}\right )}}{440 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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