Optimal. Leaf size=51 \[ \frac{3 a^2 \sqrt [3]{a+b x}}{b^3}+\frac{3 (a+b x)^{7/3}}{7 b^3}-\frac{3 a (a+b x)^{4/3}}{2 b^3} \]
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Rubi [A] time = 0.0119599, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{3 a^2 \sqrt [3]{a+b x}}{b^3}+\frac{3 (a+b x)^{7/3}}{7 b^3}-\frac{3 a (a+b x)^{4/3}}{2 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{(a+b x)^{2/3}} \, dx &=\int \left (\frac{a^2}{b^2 (a+b x)^{2/3}}-\frac{2 a \sqrt [3]{a+b x}}{b^2}+\frac{(a+b x)^{4/3}}{b^2}\right ) \, dx\\ &=\frac{3 a^2 \sqrt [3]{a+b x}}{b^3}-\frac{3 a (a+b x)^{4/3}}{2 b^3}+\frac{3 (a+b x)^{7/3}}{7 b^3}\\ \end{align*}
Mathematica [A] time = 0.0884474, size = 35, normalized size = 0.69 \[ \frac{3 \sqrt [3]{a+b x} \left (9 a^2-3 a b x+2 b^2 x^2\right )}{14 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 32, normalized size = 0.6 \begin{align*}{\frac{6\,{b}^{2}{x}^{2}-9\,abx+27\,{a}^{2}}{14\,{b}^{3}}\sqrt [3]{bx+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06806, size = 55, normalized size = 1.08 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{7}{3}}}{7 \, b^{3}} - \frac{3 \,{\left (b x + a\right )}^{\frac{4}{3}} a}{2 \, b^{3}} + \frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}} a^{2}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57083, size = 76, normalized size = 1.49 \begin{align*} \frac{3 \,{\left (2 \, b^{2} x^{2} - 3 \, a b x + 9 \, a^{2}\right )}{\left (b x + a\right )}^{\frac{1}{3}}}{14 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.45954, size = 600, normalized size = 11.76 \begin{align*} \frac{27 a^{\frac{31}{3}} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{31}{3}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{72 a^{\frac{28}{3}} b x \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{28}{3}} b x}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{60 a^{\frac{25}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{25}{3}} b^{2} x^{2}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{18 a^{\frac{22}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{22}{3}} b^{3} x^{3}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{9 a^{\frac{19}{3}} b^{4} x^{4} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{6 a^{\frac{16}{3}} b^{5} x^{5} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19898, size = 50, normalized size = 0.98 \begin{align*} \frac{3 \,{\left (2 \,{\left (b x + a\right )}^{\frac{7}{3}} - 7 \,{\left (b x + a\right )}^{\frac{4}{3}} a + 14 \,{\left (b x + a\right )}^{\frac{1}{3}} a^{2}\right )}}{14 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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