Integrand size = 13, antiderivative size = 53 \[ \int x^2 (a+b x)^{4/3} \, dx=\frac {3 a^2 (a+b x)^{7/3}}{7 b^3}-\frac {3 a (a+b x)^{10/3}}{5 b^3}+\frac {3 (a+b x)^{13/3}}{13 b^3} \]
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Time = 0.01 (sec) , antiderivative size = 53, 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} \[ \int x^2 (a+b x)^{4/3} \, dx=\frac {3 a^2 (a+b x)^{7/3}}{7 b^3}+\frac {3 (a+b x)^{13/3}}{13 b^3}-\frac {3 a (a+b x)^{10/3}}{5 b^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 (a+b x)^{4/3}}{b^2}-\frac {2 a (a+b x)^{7/3}}{b^2}+\frac {(a+b x)^{10/3}}{b^2}\right ) \, dx \\ & = \frac {3 a^2 (a+b x)^{7/3}}{7 b^3}-\frac {3 a (a+b x)^{10/3}}{5 b^3}+\frac {3 (a+b x)^{13/3}}{13 b^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.66 \[ \int x^2 (a+b x)^{4/3} \, dx=\frac {3 (a+b x)^{7/3} \left (9 a^2-21 a b x+35 b^2 x^2\right )}{455 b^3} \]
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Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(\frac {3 \left (b x +a \right )^{\frac {7}{3}} \left (35 b^{2} x^{2}-21 a b x +9 a^{2}\right )}{455 b^{3}}\) | \(32\) |
pseudoelliptic | \(\frac {3 \left (b x +a \right )^{\frac {7}{3}} \left (35 b^{2} x^{2}-21 a b x +9 a^{2}\right )}{455 b^{3}}\) | \(32\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {13}{3}}}{13}-\frac {3 a \left (b x +a \right )^{\frac {10}{3}}}{5}+\frac {3 a^{2} \left (b x +a \right )^{\frac {7}{3}}}{7}}{b^{3}}\) | \(38\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {13}{3}}}{13}-\frac {3 a \left (b x +a \right )^{\frac {10}{3}}}{5}+\frac {3 a^{2} \left (b x +a \right )^{\frac {7}{3}}}{7}}{b^{3}}\) | \(38\) |
trager | \(\frac {3 \left (35 b^{4} x^{4}+49 a \,b^{3} x^{3}+2 a^{2} b^{2} x^{2}-3 a^{3} b x +9 a^{4}\right ) \left (b x +a \right )^{\frac {1}{3}}}{455 b^{3}}\) | \(54\) |
risch | \(\frac {3 \left (35 b^{4} x^{4}+49 a \,b^{3} x^{3}+2 a^{2} b^{2} x^{2}-3 a^{3} b x +9 a^{4}\right ) \left (b x +a \right )^{\frac {1}{3}}}{455 b^{3}}\) | \(54\) |
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Time = 0.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int x^2 (a+b x)^{4/3} \, dx=\frac {3 \, {\left (35 \, b^{4} x^{4} + 49 \, a b^{3} x^{3} + 2 \, a^{2} b^{2} x^{2} - 3 \, a^{3} b x + 9 \, a^{4}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{455 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 733 vs. \(2 (49) = 98\).
Time = 1.34 (sec) , antiderivative size = 733, normalized size of antiderivative = 13.83 \[ \int x^2 (a+b x)^{4/3} \, dx=\frac {27 a^{\frac {37}{3}} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {37}{3}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {72 a^{\frac {34}{3}} b x \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {34}{3}} b x}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {60 a^{\frac {31}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {31}{3}} b^{2} x^{2}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {165 a^{\frac {28}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {28}{3}} b^{3} x^{3}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {555 a^{\frac {25}{3}} b^{4} x^{4} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {762 a^{\frac {22}{3}} b^{5} x^{5} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {462 a^{\frac {19}{3}} b^{6} x^{6} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {105 a^{\frac {16}{3}} b^{7} x^{7} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int x^2 (a+b x)^{4/3} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {13}{3}}}{13 \, b^{3}} - \frac {3 \, {\left (b x + a\right )}^{\frac {10}{3}} a}{5 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{\frac {7}{3}} a^{2}}{7 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (41) = 82\).
Time = 0.29 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.96 \[ \int x^2 (a+b x)^{4/3} \, dx=\frac {3 \, {\left (\frac {65 \, {\left (2 \, {\left (b x + a\right )}^{\frac {7}{3}} - 7 \, {\left (b x + a\right )}^{\frac {4}{3}} a + 14 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2}\right )} a^{2}}{b^{2}} + \frac {13 \, {\left (14 \, {\left (b x + a\right )}^{\frac {10}{3}} - 60 \, {\left (b x + a\right )}^{\frac {7}{3}} a + 105 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{2} - 140 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{3}\right )} a}{b^{2}} + \frac {2 \, {\left (35 \, {\left (b x + a\right )}^{\frac {13}{3}} - 182 \, {\left (b x + a\right )}^{\frac {10}{3}} a + 390 \, {\left (b x + a\right )}^{\frac {7}{3}} a^{2} - 455 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{3} + 455 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{4}\right )}}{b^{2}}\right )}}{910 \, b} \]
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Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int x^2 (a+b x)^{4/3} \, dx=\frac {105\,{\left (a+b\,x\right )}^{13/3}-273\,a\,{\left (a+b\,x\right )}^{10/3}+195\,a^2\,{\left (a+b\,x\right )}^{7/3}}{455\,b^3} \]
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