Optimal. Leaf size=34 \[ \frac {3 (a+b x)^{8/3}}{8 b^2}-\frac {3 a (a+b x)^{5/3}}{5 b^2} \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \[ \frac {3 (a+b x)^{8/3}}{8 b^2}-\frac {3 a (a+b x)^{5/3}}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int x (a+b x)^{2/3} \, dx &=\int \left (-\frac {a (a+b x)^{2/3}}{b}+\frac {(a+b x)^{5/3}}{b}\right ) \, dx\\ &=-\frac {3 a (a+b x)^{5/3}}{5 b^2}+\frac {3 (a+b x)^{8/3}}{8 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.71 \[ \frac {3 (a+b x)^{5/3} (5 b x-3 a)}{40 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 31, normalized size = 0.91 \[ \frac {3 \, {\left (5 \, b^{2} x^{2} + 2 \, a b x - 3 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{40 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.89, size = 68, normalized size = 2.00 \[ \frac {3 \, {\left (\frac {4 \, {\left (2 \, {\left (b x + a\right )}^{\frac {5}{3}} - 5 \, {\left (b x + a\right )}^{\frac {2}{3}} a\right )} a}{b} + \frac {5 \, {\left (b x + a\right )}^{\frac {8}{3}} - 16 \, {\left (b x + a\right )}^{\frac {5}{3}} a + 20 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{2}}{b}\right )}}{40 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.62 \[ -\frac {3 \left (b x +a \right )^{\frac {5}{3}} \left (-5 b x +3 a \right )}{40 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 26, normalized size = 0.76 \[ \frac {3 \, {\left (b x + a\right )}^{\frac {8}{3}}}{8 \, b^{2}} - \frac {3 \, {\left (b x + a\right )}^{\frac {5}{3}} a}{5 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 25, normalized size = 0.74 \[ -\frac {24\,a\,{\left (a+b\,x\right )}^{5/3}-15\,{\left (a+b\,x\right )}^{8/3}}{40\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.28, size = 202, normalized size = 5.94 \[ - \frac {9 a^{\frac {14}{3}} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac {9 a^{\frac {14}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} - \frac {3 a^{\frac {11}{3}} b x \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac {9 a^{\frac {11}{3}} b x}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac {21 a^{\frac {8}{3}} b^{2} x^{2} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac {15 a^{\frac {5}{3}} b^{3} x^{3} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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