Optimal. Leaf size=38 \[ \frac {3 a (-a+b x)^{2/3}}{2 b^2}+\frac {3 (-a+b x)^{5/3}}{5 b^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 = {45}
\begin {gather*} \frac {3 (b x-a)^{5/3}}{5 b^2}+\frac {3 a (b x-a)^{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 = 0.01, size = 26, normalized size = 0.68 \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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 3.96, size = 308, normalized size = 8.11 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {3 a^{\frac {2}{3}} \left (3 a^2 \left (-1^{\frac {2}{3}}+\left (\frac {-a+b x}{a}\right )^{\frac {2}{3}}\right )+a b x \left (3-1^{\frac {2}{3}}-\left (\frac {-a+b x}{a}\right )^{\frac {2}{3}}\right )-2 b^2 x^2 \left (\frac {-a+b x}{a}\right )^{\frac {2}{3}}\right )}{10 b^2 \left (a-b x\right )},\text {Abs}\left [\frac {b x}{a}\right ]>1\right \}\right \},\frac {-9 a^{\frac {11}{3}} \left (1-\frac {b x}{a}\right )^{\frac {2}{3}}}{10 E^{\frac {I \text {Pi}}{3}} a^2 b^2-10 E^{\frac {I \text {Pi}}{3}} a b^3 x}+\frac {9 a^{\frac {11}{3}}}{10 E^{\frac {I \text {Pi}}{3}} a^2 b^2-10 E^{\frac {I \text {Pi}}{3}} a b^3 x}-\frac {9 a^{\frac {8}{3}} b x}{10 E^{\frac {I \text {Pi}}{3}} a^2 b^2-10 E^{\frac {I \text {Pi}}{3}} a b^3 x}+\frac {3 a^{\frac {8}{3}} b x \left (1-\frac {b x}{a}\right )^{\frac {2}{3}}}{10 E^{\frac {I \text {Pi}}{3}} a^2 b^2-10 E^{\frac {I \text {Pi}}{3}} 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 E^{\frac {I \text {Pi}}{3}} a^2 b^2-10 E^{\frac {I \text {Pi}}{3}} a b^3 x}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 30, normalized size = 0.79
method | result | size |
gosper | \(\frac {3 \left (2 b x +3 a \right ) \left (b x -a \right )^{\frac {2}{3}}}{10 b^{2}}\) | \(23\) |
trager | \(\frac {3 \left (2 b x +3 a \right ) \left (b x -a \right )^{\frac {2}{3}}}{10 b^{2}}\) | \(23\) |
risch | \(-\frac {3 \left (-b x +a \right ) \left (2 b x +3 a \right )}{10 b^{2} \left (b x -a \right )^{\frac {1}{3}}}\) | \(29\) |
derivativedivides | \(\frac {\frac {3 \left (b x -a \right )^{\frac {5}{3}}}{5}+\frac {3 a \left (b x -a \right )^{\frac {2}{3}}}{2}}{b^{2}}\) | \(30\) |
default | \(\frac {\frac {3 \left (b x -a \right )^{\frac {5}{3}}}{5}+\frac {3 a \left (b x -a \right )^{\frac {2}{3}}}{2}}{b^{2}}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 30, normalized size = 0.79 \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 = 0.30, size = 22, normalized size = 0.58 \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 [C] Result contains complex when optimal does not.
time = 0.62, size = 486, normalized size = 12.79 \begin {gather*} \begin {cases} - \frac {9 a^{\frac {11}{3}} \left (-1 + \frac {b x}{a}\right )^{\frac {2}{3}} e^{\frac {i \pi }{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} - \frac {9 a^{\frac {11}{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} + \frac {3 a^{\frac {8}{3}} b x \left (-1 + \frac {b x}{a}\right )^{\frac {2}{3}} e^{\frac {i \pi }{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} + \frac {9 a^{\frac {8}{3}} b x}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} + \frac {6 a^{\frac {5}{3}} b^{2} x^{2} \left (-1 + \frac {b x}{a}\right )^{\frac {2}{3}} e^{\frac {i \pi }{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {9 a^{\frac {11}{3}} \left (1 - \frac {b x}{a}\right )^{\frac {2}{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} - \frac {9 a^{\frac {11}{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} - \frac {3 a^{\frac {8}{3}} b x \left (1 - \frac {b x}{a}\right )^{\frac {2}{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} + \frac {9 a^{\frac {8}{3}} b x}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} - \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} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 44, normalized size = 1.16 \begin {gather*} \frac {3 \left (\frac {1}{5} \left (\left (-a+b x\right )^{\frac {1}{3}}\right )^{2} \left (-a+b x\right )+\frac {1}{2} \left (\left (-a+b x\right )^{\frac {1}{3}}\right )^{2} a\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 29, normalized size = 0.76 \begin {gather*} \frac {15\,a\,{\left (b\,x-a\right )}^{2/3}+6\,{\left (b\,x-a\right )}^{5/3}}{10\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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