Optimal. Leaf size=49 \[ -\frac{3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac{3 a (a+b x)^{2/3}}{b^3}+\frac{3 (a+b x)^{5/3}}{5 b^3} \]
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Rubi [A] time = 0.0133071, antiderivative size = 49, 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} \[ -\frac{3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac{3 a (a+b x)^{2/3}}{b^3}+\frac{3 (a+b x)^{5/3}}{5 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{(a+b x)^{4/3}} \, dx &=\int \left (\frac{a^2}{b^2 (a+b x)^{4/3}}-\frac{2 a}{b^2 \sqrt [3]{a+b x}}+\frac{(a+b x)^{2/3}}{b^2}\right ) \, dx\\ &=-\frac{3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac{3 a (a+b x)^{2/3}}{b^3}+\frac{3 (a+b x)^{5/3}}{5 b^3}\\ \end{align*}
Mathematica [A] time = 0.0900145, size = 34, normalized size = 0.69 \[ \frac{3 \left (-9 a^2-3 a b x+b^2 x^2\right )}{5 b^3 \sqrt [3]{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 32, normalized size = 0.7 \begin{align*} -{\frac{-3\,{b}^{2}{x}^{2}+9\,abx+27\,{a}^{2}}{5\,{b}^{3}}{\frac{1}{\sqrt [3]{bx+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06022, size = 55, normalized size = 1.12 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{5}{3}}}{5 \, b^{3}} - \frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}} a}{b^{3}} - \frac{3 \, a^{2}}{{\left (b x + a\right )}^{\frac{1}{3}} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52898, size = 88, normalized size = 1.8 \begin{align*} \frac{3 \,{\left (b^{2} x^{2} - 3 \, a b x - 9 \, a^{2}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{5 \,{\left (b^{4} x + a b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.52779, size = 534, normalized size = 10.9 \begin{align*} - \frac{27 a^{\frac{29}{3}} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{27 a^{\frac{29}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac{63 a^{\frac{26}{3}} b x \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{81 a^{\frac{26}{3}} b x}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac{42 a^{\frac{23}{3}} b^{2} x^{2} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{81 a^{\frac{23}{3}} b^{2} x^{2}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac{3 a^{\frac{20}{3}} b^{3} x^{3} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{27 a^{\frac{20}{3}} b^{3} x^{3}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{3 a^{\frac{17}{3}} b^{4} x^{4} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 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 = 1.17738, size = 62, normalized size = 1.27 \begin{align*} -\frac{3 \, a^{2}}{{\left (b x + a\right )}^{\frac{1}{3}} b^{3}} + \frac{3 \,{\left ({\left (b x + a\right )}^{\frac{5}{3}} b^{12} - 5 \,{\left (b x + a\right )}^{\frac{2}{3}} a b^{12}\right )}}{5 \, b^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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