Integrand size = 13, antiderivative size = 91 \[ \int x^4 (a+b x)^{9/2} \, dx=\frac {2 a^4 (a+b x)^{11/2}}{11 b^5}-\frac {8 a^3 (a+b x)^{13/2}}{13 b^5}+\frac {4 a^2 (a+b x)^{15/2}}{5 b^5}-\frac {8 a (a+b x)^{17/2}}{17 b^5}+\frac {2 (a+b x)^{19/2}}{19 b^5} \]
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Time = 0.02 (sec) , antiderivative size = 91, 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} \[ \int x^4 (a+b x)^{9/2} \, dx=\frac {2 a^4 (a+b x)^{11/2}}{11 b^5}-\frac {8 a^3 (a+b x)^{13/2}}{13 b^5}+\frac {4 a^2 (a+b x)^{15/2}}{5 b^5}+\frac {2 (a+b x)^{19/2}}{19 b^5}-\frac {8 a (a+b x)^{17/2}}{17 b^5} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^4 (a+b x)^{9/2}}{b^4}-\frac {4 a^3 (a+b x)^{11/2}}{b^4}+\frac {6 a^2 (a+b x)^{13/2}}{b^4}-\frac {4 a (a+b x)^{15/2}}{b^4}+\frac {(a+b x)^{17/2}}{b^4}\right ) \, dx \\ & = \frac {2 a^4 (a+b x)^{11/2}}{11 b^5}-\frac {8 a^3 (a+b x)^{13/2}}{13 b^5}+\frac {4 a^2 (a+b x)^{15/2}}{5 b^5}-\frac {8 a (a+b x)^{17/2}}{17 b^5}+\frac {2 (a+b x)^{19/2}}{19 b^5} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.63 \[ \int x^4 (a+b x)^{9/2} \, dx=\frac {2 (a+b x)^{11/2} \left (128 a^4-704 a^3 b x+2288 a^2 b^2 x^2-5720 a b^3 x^3+12155 b^4 x^4\right )}{230945 b^5} \]
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Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.59
method | result | size |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (12155 b^{4} x^{4}-5720 a \,b^{3} x^{3}+2288 a^{2} b^{2} x^{2}-704 a^{3} b x +128 a^{4}\right )}{230945 b^{5}}\) | \(54\) |
pseudoelliptic | \(\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (12155 b^{4} x^{4}-5720 a \,b^{3} x^{3}+2288 a^{2} b^{2} x^{2}-704 a^{3} b x +128 a^{4}\right )}{230945 b^{5}}\) | \(54\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {19}{2}}}{19}-\frac {8 a \left (b x +a \right )^{\frac {17}{2}}}{17}+\frac {4 a^{2} \left (b x +a \right )^{\frac {15}{2}}}{5}-\frac {8 a^{3} \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 a^{4} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{5}}\) | \(62\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {19}{2}}}{19}-\frac {8 a \left (b x +a \right )^{\frac {17}{2}}}{17}+\frac {4 a^{2} \left (b x +a \right )^{\frac {15}{2}}}{5}-\frac {8 a^{3} \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 a^{4} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{5}}\) | \(62\) |
trager | \(\frac {2 \left (12155 b^{9} x^{9}+55055 a \,x^{8} b^{8}+95238 a^{2} x^{7} b^{7}+75086 x^{6} a^{3} b^{6}+23063 a^{4} x^{5} b^{5}+35 a^{5} b^{4} x^{4}-40 a^{6} b^{3} x^{3}+48 a^{7} b^{2} x^{2}-64 a^{8} b x +128 a^{9}\right ) \sqrt {b x +a}}{230945 b^{5}}\) | \(109\) |
risch | \(\frac {2 \left (12155 b^{9} x^{9}+55055 a \,x^{8} b^{8}+95238 a^{2} x^{7} b^{7}+75086 x^{6} a^{3} b^{6}+23063 a^{4} x^{5} b^{5}+35 a^{5} b^{4} x^{4}-40 a^{6} b^{3} x^{3}+48 a^{7} b^{2} x^{2}-64 a^{8} b x +128 a^{9}\right ) \sqrt {b x +a}}{230945 b^{5}}\) | \(109\) |
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none
Time = 0.22 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.19 \[ \int x^4 (a+b x)^{9/2} \, dx=\frac {2 \, {\left (12155 \, b^{9} x^{9} + 55055 \, a b^{8} x^{8} + 95238 \, a^{2} b^{7} x^{7} + 75086 \, a^{3} b^{6} x^{6} + 23063 \, a^{4} b^{5} x^{5} + 35 \, a^{5} b^{4} x^{4} - 40 \, a^{6} b^{3} x^{3} + 48 \, a^{7} b^{2} x^{2} - 64 \, a^{8} b x + 128 \, a^{9}\right )} \sqrt {b x + a}}{230945 \, b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (87) = 174\).
Time = 1.25 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.33 \[ \int x^4 (a+b x)^{9/2} \, dx=\begin {cases} \frac {256 a^{9} \sqrt {a + b x}}{230945 b^{5}} - \frac {128 a^{8} x \sqrt {a + b x}}{230945 b^{4}} + \frac {96 a^{7} x^{2} \sqrt {a + b x}}{230945 b^{3}} - \frac {16 a^{6} x^{3} \sqrt {a + b x}}{46189 b^{2}} + \frac {14 a^{5} x^{4} \sqrt {a + b x}}{46189 b} + \frac {46126 a^{4} x^{5} \sqrt {a + b x}}{230945} + \frac {13652 a^{3} b x^{6} \sqrt {a + b x}}{20995} + \frac {1332 a^{2} b^{2} x^{7} \sqrt {a + b x}}{1615} + \frac {154 a b^{3} x^{8} \sqrt {a + b x}}{323} + \frac {2 b^{4} x^{9} \sqrt {a + b x}}{19} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{5}}{5} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.78 \[ \int x^4 (a+b x)^{9/2} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {19}{2}}}{19 \, b^{5}} - \frac {8 \, {\left (b x + a\right )}^{\frac {17}{2}} a}{17 \, b^{5}} + \frac {4 \, {\left (b x + a\right )}^{\frac {15}{2}} a^{2}}{5 \, b^{5}} - \frac {8 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{3}}{13 \, b^{5}} + \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{4}}{11 \, b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (71) = 142\).
Time = 0.31 (sec) , antiderivative size = 565, normalized size of antiderivative = 6.21 \[ \int x^4 (a+b x)^{9/2} \, dx=\frac {2 \, {\left (\frac {46189 \, {\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^{5}}{b^{4}} + \frac {104975 \, {\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^{4}}{b^{4}} + \frac {48450 \, {\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 )} a^{3}}{b^{4}} + \frac {22610 \, {\left (429 \, {\left (b x + a\right )}^{\frac {15}{2}} - 3465 \, {\left (b x + a\right )}^{\frac {13}{2}} a + 12285 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{2} - 25025 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{3} + 32175 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{4} - 27027 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} + 15015 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6} - 6435 \, \sqrt {b x + a} a^{7}\right )} a^{2}}{b^{4}} + \frac {665 \, {\left (6435 \, {\left (b x + a\right )}^{\frac {17}{2}} - 58344 \, {\left (b x + a\right )}^{\frac {15}{2}} a + 235620 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{2} - 556920 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{3} + 850850 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{4} - 875160 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{5} + 612612 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{6} - 291720 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{7} + 109395 \, \sqrt {b x + a} a^{8}\right )} a}{b^{4}} + \frac {63 \, {\left (12155 \, {\left (b x + a\right )}^{\frac {19}{2}} - 122265 \, {\left (b x + a\right )}^{\frac {17}{2}} a + 554268 \, {\left (b x + a\right )}^{\frac {15}{2}} a^{2} - 1492260 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{3} + 2645370 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{4} - 3233230 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{5} + 2771340 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{6} - 1662804 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{7} + 692835 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{8} - 230945 \, \sqrt {b x + a} a^{9}\right )}}{b^{4}}\right )}}{14549535 \, b} \]
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Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.78 \[ \int x^4 (a+b x)^{9/2} \, dx=\frac {2\,{\left (a+b\,x\right )}^{19/2}}{19\,b^5}+\frac {2\,a^4\,{\left (a+b\,x\right )}^{11/2}}{11\,b^5}-\frac {8\,a^3\,{\left (a+b\,x\right )}^{13/2}}{13\,b^5}+\frac {4\,a^2\,{\left (a+b\,x\right )}^{15/2}}{5\,b^5}-\frac {8\,a\,{\left (a+b\,x\right )}^{17/2}}{17\,b^5} \]
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