Optimal. Leaf size=34 \[ -\frac {3 a (a+b x)^{5/3}}{5 b^2}+\frac {3 (a+b x)^{8/3}}{8 b^2} \]
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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 = {45}
\begin {gather*} \frac {3 (a+b x)^{8/3}}{8 b^2}-\frac {3 a (a+b x)^{5/3}}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
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 = 0.02, size = 35, normalized size = 1.03 \begin {gather*} \frac {3 (a+b x)^{2/3} \left (-3 a^2+2 a b x+5 b^2 x^2\right )}{40 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 26, normalized size = 0.76
method | result | size |
gosper | \(-\frac {3 \left (b x +a \right )^{\frac {5}{3}} \left (-5 b x +3 a \right )}{40 b^{2}}\) | \(21\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {8}{3}}}{8}-\frac {3 a \left (b x +a \right )^{\frac {5}{3}}}{5}}{b^{2}}\) | \(26\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {8}{3}}}{8}-\frac {3 a \left (b x +a \right )^{\frac {5}{3}}}{5}}{b^{2}}\) | \(26\) |
trager | \(-\frac {3 \left (-5 x^{2} b^{2}-2 a b x +3 a^{2}\right ) \left (b x +a \right )^{\frac {2}{3}}}{40 b^{2}}\) | \(32\) |
risch | \(-\frac {3 \left (-5 x^{2} b^{2}-2 a b x +3 a^{2}\right ) \left (b x +a \right )^{\frac {2}{3}}}{40 b^{2}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 26, normalized size = 0.76 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 31, normalized size = 0.91 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 202 vs.
\(2 (31) = 62\).
time = 0.56, size = 202, normalized size = 5.94 \begin {gather*} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (26) = 52\).
time = 0.75, size = 68, normalized size = 2.00 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} -\frac {24\,a\,{\left (a+b\,x\right )}^{5/3}-15\,{\left (a+b\,x\right )}^{8/3}}{40\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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