Optimal. Leaf size=34 \[ -\frac {3 a (a+b x)^{2/3}}{2 b^2}+\frac {3 (a+b x)^{5/3}}{5 b^2} \]
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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 = {45}
\begin {gather*} \frac {3 (a+b x)^{5/3}}{5 b^2}-\frac {3 a (a+b x)^{2/3}}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x}{\sqrt [3]{a+b x}} \, dx &=\int \left (-\frac {a}{b \sqrt [3]{a+b x}}+\frac {(a+b x)^{2/3}}{b}\right ) \, dx\\ &=-\frac {3 a (a+b x)^{2/3}}{2 b^2}+\frac {3 (a+b x)^{5/3}}{5 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 0.71 \begin {gather*} \frac {3 (a+b x)^{2/3} (-3 a+2 b x)}{10 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 26, normalized size = 0.76
method | result | size |
gosper | \(-\frac {3 \left (b x +a \right )^{\frac {2}{3}} \left (-2 b x +3 a \right )}{10 b^{2}}\) | \(21\) |
trager | \(-\frac {3 \left (b x +a \right )^{\frac {2}{3}} \left (-2 b x +3 a \right )}{10 b^{2}}\) | \(21\) |
risch | \(-\frac {3 \left (b x +a \right )^{\frac {2}{3}} \left (-2 b x +3 a \right )}{10 b^{2}}\) | \(21\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {5}{3}}}{5}-\frac {3 a \left (b x +a \right )^{\frac {2}{3}}}{2}}{b^{2}}\) | \(26\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {5}{3}}}{5}-\frac {3 a \left (b x +a \right )^{\frac {2}{3}}}{2}}{b^{2}}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 26, normalized size = 0.76 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {5}{3}}}{5 \, b^{2}} - \frac {3 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.75, size = 20, normalized size = 0.59 \begin {gather*} \frac {3 \, {\left (2 \, b x - 3 \, a\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{10 \, 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. 162 vs.
\(2 (31) = 62\).
time = 0.57, size = 162, normalized size = 4.76 \begin {gather*} - \frac {9 a^{\frac {11}{3}} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac {9 a^{\frac {11}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} - \frac {3 a^{\frac {8}{3}} b x \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac {9 a^{\frac {8}{3}} b x}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac {6 a^{\frac {5}{3}} b^{2} x^{2} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.12, size = 25, normalized size = 0.74 \begin {gather*} \frac {3 \, {\left (2 \, {\left (b x + a\right )}^{\frac {5}{3}} - 5 \, {\left (b x + a\right )}^{\frac {2}{3}} a\right )}}{10 \, b^{2}} \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 {15\,a\,{\left (a+b\,x\right )}^{2/3}-6\,{\left (a+b\,x\right )}^{5/3}}{10\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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