Integrand size = 11, antiderivative size = 32 \[ \int \frac {x}{(a+b x)^{2/3}} \, dx=-\frac {3 a \sqrt [3]{a+b x}}{b^2}+\frac {3 (a+b x)^{4/3}}{4 b^2} \]
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Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x}{(a+b x)^{2/3}} \, dx=\frac {3 (a+b x)^{4/3}}{4 b^2}-\frac {3 a \sqrt [3]{a+b x}}{b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a}{b (a+b x)^{2/3}}+\frac {\sqrt [3]{a+b x}}{b}\right ) \, dx \\ & = -\frac {3 a \sqrt [3]{a+b x}}{b^2}+\frac {3 (a+b x)^{4/3}}{4 b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {x}{(a+b x)^{2/3}} \, dx=\frac {3 (-3 a+b x) \sqrt [3]{a+b x}}{4 b^2} \]
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Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66
method | result | size |
gosper | \(-\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +3 a \right )}{4 b^{2}}\) | \(21\) |
trager | \(-\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +3 a \right )}{4 b^{2}}\) | \(21\) |
risch | \(-\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +3 a \right )}{4 b^{2}}\) | \(21\) |
pseudoelliptic | \(-\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +3 a \right )}{4 b^{2}}\) | \(21\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {4}{3}}}{4}-3 a \left (b x +a \right )^{\frac {1}{3}}}{b^{2}}\) | \(26\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {4}{3}}}{4}-3 a \left (b x +a \right )^{\frac {1}{3}}}{b^{2}}\) | \(26\) |
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none
Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.59 \[ \int \frac {x}{(a+b x)^{2/3}} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (b x - 3 \, a\right )}}{4 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (29) = 58\).
Time = 1.10 (sec) , antiderivative size = 162, normalized size of antiderivative = 5.06 \[ \int \frac {x}{(a+b x)^{2/3}} \, dx=- \frac {9 a^{\frac {10}{3}} \sqrt [3]{1 + \frac {b x}{a}}}{4 a^{2} b^{2} + 4 a b^{3} x} + \frac {9 a^{\frac {10}{3}}}{4 a^{2} b^{2} + 4 a b^{3} x} - \frac {6 a^{\frac {7}{3}} b x \sqrt [3]{1 + \frac {b x}{a}}}{4 a^{2} b^{2} + 4 a b^{3} x} + \frac {9 a^{\frac {7}{3}} b x}{4 a^{2} b^{2} + 4 a b^{3} x} + \frac {3 a^{\frac {4}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac {b x}{a}}}{4 a^{2} b^{2} + 4 a b^{3} x} \]
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none
Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {x}{(a+b x)^{2/3}} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {4}{3}}}{4 \, b^{2}} - \frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}} a}{b^{2}} \]
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none
Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {x}{(a+b x)^{2/3}} \, dx=\frac {3 \, {\left ({\left (b x + a\right )}^{\frac {4}{3}} - 4 \, {\left (b x + a\right )}^{\frac {1}{3}} a\right )}}{4 \, b^{2}} \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {x}{(a+b x)^{2/3}} \, dx=-\frac {12\,a\,{\left (a+b\,x\right )}^{1/3}-3\,{\left (a+b\,x\right )}^{4/3}}{4\,b^2} \]
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