Integrand size = 13, antiderivative size = 38 \[ \int \frac {x}{\sqrt [3]{-a+b x}} \, dx=\frac {3 a (-a+b x)^{2/3}}{2 b^2}+\frac {3 (-a+b x)^{5/3}}{5 b^2} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int \frac {x}{\sqrt [3]{-a+b x}} \, dx=\frac {3 (b x-a)^{5/3}}{5 b^2}+\frac {3 a (b x-a)^{2/3}}{2 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68 \[ \int \frac {x}{\sqrt [3]{-a+b x}} \, dx=\frac {3 (-a+b x)^{2/3} (3 a+2 b x)}{10 b^2} \]
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Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61
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 | \(\frac {3 \left (2 b x +3 a \right ) \left (b x -a \right )^{\frac {2}{3}}}{10 b^{2}}\) | \(23\) |
pseudoelliptic | \(\frac {3 \left (2 b x +3 a \right ) \left (b x -a \right )^{\frac {2}{3}}}{10 b^{2}}\) | \(23\) |
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 | \(\frac {\frac {3 \left (b x -a \right )^{\frac {5}{3}}}{5}+\frac {3 a \left (b x -a \right )^{\frac {2}{3}}}{2}}{b^{2}}\) | \(30\) |
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Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.58 \[ \int \frac {x}{\sqrt [3]{-a+b x}} \, dx=\frac {3 \, {\left (2 \, b x + 3 \, a\right )} {\left (b x - a\right )}^{\frac {2}{3}}}{10 \, b^{2}} \]
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Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 486, normalized size of antiderivative = 12.79 \[ \int \frac {x}{\sqrt [3]{-a+b x}} \, dx=\begin {cases} - \frac {9 a^{\frac {11}{3}} \left (-1 + \frac {b x}{a}\right )^{\frac {2}{3}} e^{\frac {i \pi }{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} - \frac {9 a^{\frac {11}{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} + \frac {3 a^{\frac {8}{3}} b x \left (-1 + \frac {b x}{a}\right )^{\frac {2}{3}} e^{\frac {i \pi }{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} + \frac {9 a^{\frac {8}{3}} b x}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} + \frac {6 a^{\frac {5}{3}} b^{2} x^{2} \left (-1 + \frac {b x}{a}\right )^{\frac {2}{3}} e^{\frac {i \pi }{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {9 a^{\frac {11}{3}} \left (1 - \frac {b x}{a}\right )^{\frac {2}{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} - \frac {9 a^{\frac {11}{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} - \frac {3 a^{\frac {8}{3}} b x \left (1 - \frac {b x}{a}\right )^{\frac {2}{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} + \frac {9 a^{\frac {8}{3}} b x}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} - \frac {6 a^{\frac {5}{3}} b^{2} x^{2} \left (1 - \frac {b x}{a}\right )^{\frac {2}{3}}}{- 10 a^{2} b^{2} e^{\frac {i \pi }{3}} + 10 a b^{3} x e^{\frac {i \pi }{3}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int \frac {x}{\sqrt [3]{-a+b x}} \, dx=\frac {3 \, {\left (b x - a\right )}^{\frac {5}{3}}}{5 \, b^{2}} + \frac {3 \, {\left (b x - a\right )}^{\frac {2}{3}} a}{2 \, b^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \frac {x}{\sqrt [3]{-a+b x}} \, dx=\frac {3 \, {\left (2 \, {\left (b x - a\right )}^{\frac {5}{3}} + 5 \, {\left (b x - a\right )}^{\frac {2}{3}} a\right )}}{10 \, b^{2}} \]
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Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \frac {x}{\sqrt [3]{-a+b x}} \, dx=\frac {15\,a\,{\left (b\,x-a\right )}^{2/3}+6\,{\left (b\,x-a\right )}^{5/3}}{10\,b^2} \]
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