Integrand size = 13, antiderivative size = 53 \[ \int x^2 (a+b x)^{3/2} \, dx=\frac {2 a^2 (a+b x)^{5/2}}{5 b^3}-\frac {4 a (a+b x)^{7/2}}{7 b^3}+\frac {2 (a+b x)^{9/2}}{9 b^3} \]
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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} \[ \int x^2 (a+b x)^{3/2} \, dx=\frac {2 a^2 (a+b x)^{5/2}}{5 b^3}+\frac {2 (a+b x)^{9/2}}{9 b^3}-\frac {4 a (a+b x)^{7/2}}{7 b^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 (a+b x)^{3/2}}{b^2}-\frac {2 a (a+b x)^{5/2}}{b^2}+\frac {(a+b x)^{7/2}}{b^2}\right ) \, dx \\ & = \frac {2 a^2 (a+b x)^{5/2}}{5 b^3}-\frac {4 a (a+b x)^{7/2}}{7 b^3}+\frac {2 (a+b x)^{9/2}}{9 b^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.66 \[ \int x^2 (a+b x)^{3/2} \, dx=\frac {2 (a+b x)^{5/2} \left (8 a^2-20 a b x+35 b^2 x^2\right )}{315 b^3} \]
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Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (35 b^{2} x^{2}-20 a b x +8 a^{2}\right )}{315 b^{3}}\) | \(32\) |
pseudoelliptic | \(\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (35 b^{2} x^{2}-20 a b x +8 a^{2}\right )}{315 b^{3}}\) | \(32\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {4 a \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{3}}\) | \(38\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {4 a \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{3}}\) | \(38\) |
trager | \(\frac {2 \left (35 b^{4} x^{4}+50 a \,b^{3} x^{3}+3 a^{2} b^{2} x^{2}-4 a^{3} b x +8 a^{4}\right ) \sqrt {b x +a}}{315 b^{3}}\) | \(54\) |
risch | \(\frac {2 \left (35 b^{4} x^{4}+50 a \,b^{3} x^{3}+3 a^{2} b^{2} x^{2}-4 a^{3} b x +8 a^{4}\right ) \sqrt {b x +a}}{315 b^{3}}\) | \(54\) |
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Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int x^2 (a+b x)^{3/2} \, dx=\frac {2 \, {\left (35 \, b^{4} x^{4} + 50 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} - 4 \, a^{3} b x + 8 \, a^{4}\right )} \sqrt {b x + a}}{315 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 733 vs. \(2 (49) = 98\).
Time = 1.22 (sec) , antiderivative size = 733, normalized size of antiderivative = 13.83 \[ \int x^2 (a+b x)^{3/2} \, dx=\frac {16 a^{\frac {25}{2}} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {25}{2}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {40 a^{\frac {23}{2}} b x \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {23}{2}} b x}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {30 a^{\frac {21}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {21}{2}} b^{2} x^{2}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {110 a^{\frac {19}{2}} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {19}{2}} b^{3} x^{3}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {380 a^{\frac {17}{2}} b^{4} x^{4} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {516 a^{\frac {15}{2}} b^{5} x^{5} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {310 a^{\frac {13}{2}} b^{6} x^{6} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {70 a^{\frac {11}{2}} b^{7} x^{7} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} \]
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Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int x^2 (a+b x)^{3/2} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {9}{2}}}{9 \, b^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {7}{2}} a}{7 \, b^{3}} + \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2}}{5 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (41) = 82\).
Time = 0.31 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.94 \[ \int x^2 (a+b x)^{3/2} \, dx=\frac {2 \, {\left (\frac {21 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} a^{2}}{b^{2}} + \frac {18 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a}{b^{2}} + \frac {35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}}{b^{2}}\right )}}{315 \, b} \]
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Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int x^2 (a+b x)^{3/2} \, dx=\frac {70\,{\left (a+b\,x\right )}^{9/2}-180\,a\,{\left (a+b\,x\right )}^{7/2}+126\,a^2\,{\left (a+b\,x\right )}^{5/2}}{315\,b^3} \]
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