Integrand size = 11, antiderivative size = 34 \[ \int x (a+b x)^{4/3} \, dx=-\frac {3 a (a+b x)^{7/3}}{7 b^2}+\frac {3 (a+b x)^{10/3}}{10 b^2} \]
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Time = 0.01 (sec) , antiderivative size = 34, 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} \[ \int x (a+b x)^{4/3} \, dx=\frac {3 (a+b x)^{10/3}}{10 b^2}-\frac {3 a (a+b x)^{7/3}}{7 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a (a+b x)^{4/3}}{b}+\frac {(a+b x)^{7/3}}{b}\right ) \, dx \\ & = -\frac {3 a (a+b x)^{7/3}}{7 b^2}+\frac {3 (a+b x)^{10/3}}{10 b^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int x (a+b x)^{4/3} \, dx=\frac {3 (a+b x)^{7/3} (-3 a+7 b x)}{70 b^2} \]
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Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.62
method | result | size |
gosper | \(-\frac {3 \left (b x +a \right )^{\frac {7}{3}} \left (-7 b x +3 a \right )}{70 b^{2}}\) | \(21\) |
pseudoelliptic | \(-\frac {3 \left (b x +a \right )^{\frac {7}{3}} \left (-7 b x +3 a \right )}{70 b^{2}}\) | \(21\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {3 a \left (b x +a \right )^{\frac {7}{3}}}{7}}{b^{2}}\) | \(26\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {3 a \left (b x +a \right )^{\frac {7}{3}}}{7}}{b^{2}}\) | \(26\) |
trager | \(-\frac {3 \left (-7 b^{3} x^{3}-11 a \,b^{2} x^{2}-a^{2} b x +3 a^{3}\right ) \left (b x +a \right )^{\frac {1}{3}}}{70 b^{2}}\) | \(43\) |
risch | \(-\frac {3 \left (-7 b^{3} x^{3}-11 a \,b^{2} x^{2}-a^{2} b x +3 a^{3}\right ) \left (b x +a \right )^{\frac {1}{3}}}{70 b^{2}}\) | \(43\) |
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none
Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21 \[ \int x (a+b x)^{4/3} \, dx=\frac {3 \, {\left (7 \, b^{3} x^{3} + 11 \, a b^{2} x^{2} + a^{2} b x - 3 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{70 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (31) = 62\).
Time = 0.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.35 \[ \int x (a+b x)^{4/3} \, dx=\begin {cases} - \frac {9 a^{3} \sqrt [3]{a + b x}}{70 b^{2}} + \frac {3 a^{2} x \sqrt [3]{a + b x}}{70 b} + \frac {33 a x^{2} \sqrt [3]{a + b x}}{70} + \frac {3 b x^{3} \sqrt [3]{a + b x}}{10} & \text {for}\: b \neq 0 \\\frac {a^{\frac {4}{3}} x^{2}}{2} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int x (a+b x)^{4/3} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {10}{3}}}{10 \, b^{2}} - \frac {3 \, {\left (b x + a\right )}^{\frac {7}{3}} a}{7 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (26) = 52\).
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.47 \[ \int x (a+b x)^{4/3} \, dx=\frac {3 \, {\left (\frac {35 \, {\left ({\left (b x + a\right )}^{\frac {4}{3}} - 4 \, {\left (b x + a\right )}^{\frac {1}{3}} a\right )} a^{2}}{b} + \frac {20 \, {\left (2 \, {\left (b x + a\right )}^{\frac {7}{3}} - 7 \, {\left (b x + a\right )}^{\frac {4}{3}} a + 14 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2}\right )} a}{b} + \frac {14 \, {\left (b x + a\right )}^{\frac {10}{3}} - 60 \, {\left (b x + a\right )}^{\frac {7}{3}} a + 105 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{2} - 140 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{3}}{b}\right )}}{140 \, b} \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74 \[ \int x (a+b x)^{4/3} \, dx=-\frac {30\,a\,{\left (a+b\,x\right )}^{7/3}-21\,{\left (a+b\,x\right )}^{10/3}}{70\,b^2} \]
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