Optimal. Leaf size=53 \[ \frac{3 a^2 (a+b x)^{4/3}}{4 b^3}+\frac{3 (a+b x)^{10/3}}{10 b^3}-\frac{6 a (a+b x)^{7/3}}{7 b^3} \]
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Rubi [A] time = 0.0123199, antiderivative size = 53, 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} \[ \frac{3 a^2 (a+b x)^{4/3}}{4 b^3}+\frac{3 (a+b x)^{10/3}}{10 b^3}-\frac{6 a (a+b x)^{7/3}}{7 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x^2 \sqrt [3]{a+b x} \, dx &=\int \left (\frac{a^2 \sqrt [3]{a+b x}}{b^2}-\frac{2 a (a+b x)^{4/3}}{b^2}+\frac{(a+b x)^{7/3}}{b^2}\right ) \, dx\\ &=\frac{3 a^2 (a+b x)^{4/3}}{4 b^3}-\frac{6 a (a+b x)^{7/3}}{7 b^3}+\frac{3 (a+b x)^{10/3}}{10 b^3}\\ \end{align*}
Mathematica [A] time = 0.0277879, size = 35, normalized size = 0.66 \[ \frac{3 (a+b x)^{4/3} \left (9 a^2-12 a b x+14 b^2 x^2\right )}{140 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.6 \begin{align*}{\frac{42\,{b}^{2}{x}^{2}-36\,abx+27\,{a}^{2}}{140\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{4}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01767, size = 55, normalized size = 1.04 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{10}{3}}}{10 \, b^{3}} - \frac{6 \,{\left (b x + a\right )}^{\frac{7}{3}} a}{7 \, b^{3}} + \frac{3 \,{\left (b x + a\right )}^{\frac{4}{3}} a^{2}}{4 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78826, size = 100, normalized size = 1.89 \begin{align*} \frac{3 \,{\left (14 \, b^{3} x^{3} + 2 \, a b^{2} x^{2} - 3 \, a^{2} b x + 9 \, a^{3}\right )}{\left (b x + a\right )}^{\frac{1}{3}}}{140 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.47257, size = 666, normalized size = 12.57 \begin{align*} \frac{27 a^{\frac{34}{3}} \sqrt [3]{1 + \frac{b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{34}{3}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac{72 a^{\frac{31}{3}} b x \sqrt [3]{1 + \frac{b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{31}{3}} b x}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac{60 a^{\frac{28}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac{b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{28}{3}} b^{2} x^{2}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac{60 a^{\frac{25}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac{b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{25}{3}} b^{3} x^{3}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac{135 a^{\frac{22}{3}} b^{4} x^{4} \sqrt [3]{1 + \frac{b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac{132 a^{\frac{19}{3}} b^{5} x^{5} \sqrt [3]{1 + \frac{b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac{42 a^{\frac{16}{3}} b^{6} x^{6} \sqrt [3]{1 + \frac{b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 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 = 1.12951, size = 50, normalized size = 0.94 \begin{align*} \frac{3 \,{\left (14 \,{\left (b x + a\right )}^{\frac{10}{3}} - 40 \,{\left (b x + a\right )}^{\frac{7}{3}} a + 35 \,{\left (b x + a\right )}^{\frac{4}{3}} a^{2}\right )}}{140 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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