Integrand size = 11, antiderivative size = 34 \[ \int x (a+b x)^{9/2} \, dx=-\frac {2 a (a+b x)^{11/2}}{11 b^2}+\frac {2 (a+b x)^{13/2}}{13 b^2} \]
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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} \[ \int x (a+b x)^{9/2} \, dx=\frac {2 (a+b x)^{13/2}}{13 b^2}-\frac {2 a (a+b x)^{11/2}}{11 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a (a+b x)^{9/2}}{b}+\frac {(a+b x)^{11/2}}{b}\right ) \, dx \\ & = -\frac {2 a (a+b x)^{11/2}}{11 b^2}+\frac {2 (a+b x)^{13/2}}{13 b^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int x (a+b x)^{9/2} \, dx=\frac {2 (a+b x)^{11/2} (-2 a+11 b x)}{143 b^2} \]
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Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.62
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (-11 b x +2 a \right )}{143 b^{2}}\) | \(21\) |
pseudoelliptic | \(-\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (-11 b x +2 a \right )}{143 b^{2}}\) | \(21\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {2 a \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{2}}\) | \(26\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {2 a \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{2}}\) | \(26\) |
trager | \(-\frac {2 \left (-11 b^{6} x^{6}-53 a \,x^{5} b^{5}-100 a^{2} x^{4} b^{4}-90 a^{3} x^{3} b^{3}-35 a^{4} x^{2} b^{2}-a^{5} x b +2 a^{6}\right ) \sqrt {b x +a}}{143 b^{2}}\) | \(76\) |
risch | \(-\frac {2 \left (-11 b^{6} x^{6}-53 a \,x^{5} b^{5}-100 a^{2} x^{4} b^{4}-90 a^{3} x^{3} b^{3}-35 a^{4} x^{2} b^{2}-a^{5} x b +2 a^{6}\right ) \sqrt {b x +a}}{143 b^{2}}\) | \(76\) |
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (26) = 52\).
Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.18 \[ \int x (a+b x)^{9/2} \, dx=\frac {2 \, {\left (11 \, b^{6} x^{6} + 53 \, a b^{5} x^{5} + 100 \, a^{2} b^{4} x^{4} + 90 \, a^{3} b^{3} x^{3} + 35 \, a^{4} b^{2} x^{2} + a^{5} b x - 2 \, a^{6}\right )} \sqrt {b x + a}}{143 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (31) = 62\).
Time = 0.98 (sec) , antiderivative size = 146, normalized size of antiderivative = 4.29 \[ \int x (a+b x)^{9/2} \, dx=\begin {cases} - \frac {4 a^{6} \sqrt {a + b x}}{143 b^{2}} + \frac {2 a^{5} x \sqrt {a + b x}}{143 b} + \frac {70 a^{4} x^{2} \sqrt {a + b x}}{143} + \frac {180 a^{3} b x^{3} \sqrt {a + b x}}{143} + \frac {200 a^{2} b^{2} x^{4} \sqrt {a + b x}}{143} + \frac {106 a b^{3} x^{5} \sqrt {a + b x}}{143} + \frac {2 b^{4} x^{6} \sqrt {a + b x}}{13} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{2}}{2} & \text {otherwise} \end {cases} \]
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none
Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int x (a+b x)^{9/2} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {13}{2}}}{13 \, b^{2}} - \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}} a}{11 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (26) = 52\).
Time = 0.32 (sec) , antiderivative size = 347, normalized size of antiderivative = 10.21 \[ \int x (a+b x)^{9/2} \, dx=\frac {2 \, {\left (\frac {3003 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} a^{5}}{b} + \frac {3003 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} a^{4}}{b} + \frac {2574 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a^{3}}{b} + \frac {286 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} a^{2}}{b} + \frac {65 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} a}{b} + \frac {3 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )}}{b}\right )}}{9009 \, b} \]
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Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74 \[ \int x (a+b x)^{9/2} \, dx=-\frac {26\,a\,{\left (a+b\,x\right )}^{11/2}-22\,{\left (a+b\,x\right )}^{13/2}}{143\,b^2} \]
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