Integrand size = 13, antiderivative size = 110 \[ \int x^5 (a+b x)^{9/2} \, dx=-\frac {2 a^5 (a+b x)^{11/2}}{11 b^6}+\frac {10 a^4 (a+b x)^{13/2}}{13 b^6}-\frac {4 a^3 (a+b x)^{15/2}}{3 b^6}+\frac {20 a^2 (a+b x)^{17/2}}{17 b^6}-\frac {10 a (a+b x)^{19/2}}{19 b^6}+\frac {2 (a+b x)^{21/2}}{21 b^6} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 110, 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} \[ \int x^5 (a+b x)^{9/2} \, dx=-\frac {2 a^5 (a+b x)^{11/2}}{11 b^6}+\frac {10 a^4 (a+b x)^{13/2}}{13 b^6}-\frac {4 a^3 (a+b x)^{15/2}}{3 b^6}+\frac {20 a^2 (a+b x)^{17/2}}{17 b^6}+\frac {2 (a+b x)^{21/2}}{21 b^6}-\frac {10 a (a+b x)^{19/2}}{19 b^6} \]
[In]
[Out]
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^5 (a+b x)^{9/2}}{b^5}+\frac {5 a^4 (a+b x)^{11/2}}{b^5}-\frac {10 a^3 (a+b x)^{13/2}}{b^5}+\frac {10 a^2 (a+b x)^{15/2}}{b^5}-\frac {5 a (a+b x)^{17/2}}{b^5}+\frac {(a+b x)^{19/2}}{b^5}\right ) \, dx \\ & = -\frac {2 a^5 (a+b x)^{11/2}}{11 b^6}+\frac {10 a^4 (a+b x)^{13/2}}{13 b^6}-\frac {4 a^3 (a+b x)^{15/2}}{3 b^6}+\frac {20 a^2 (a+b x)^{17/2}}{17 b^6}-\frac {10 a (a+b x)^{19/2}}{19 b^6}+\frac {2 (a+b x)^{21/2}}{21 b^6} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int x^5 (a+b x)^{9/2} \, dx=\frac {2 (a+b x)^{11/2} \left (-256 a^5+1408 a^4 b x-4576 a^3 b^2 x^2+11440 a^2 b^3 x^3-24310 a b^4 x^4+46189 b^5 x^5\right )}{969969 b^6} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.59
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (-46189 b^{5} x^{5}+24310 a \,b^{4} x^{4}-11440 a^{2} b^{3} x^{3}+4576 a^{3} b^{2} x^{2}-1408 a^{4} b x +256 a^{5}\right )}{969969 b^{6}}\) | \(65\) |
pseudoelliptic | \(-\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (-46189 b^{5} x^{5}+24310 a \,b^{4} x^{4}-11440 a^{2} b^{3} x^{3}+4576 a^{3} b^{2} x^{2}-1408 a^{4} b x +256 a^{5}\right )}{969969 b^{6}}\) | \(65\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {21}{2}}}{21}-\frac {10 a \left (b x +a \right )^{\frac {19}{2}}}{19}+\frac {20 a^{2} \left (b x +a \right )^{\frac {17}{2}}}{17}-\frac {4 a^{3} \left (b x +a \right )^{\frac {15}{2}}}{3}+\frac {10 a^{4} \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {2 a^{5} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{6}}\) | \(74\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {21}{2}}}{21}-\frac {10 a \left (b x +a \right )^{\frac {19}{2}}}{19}+\frac {20 a^{2} \left (b x +a \right )^{\frac {17}{2}}}{17}-\frac {4 a^{3} \left (b x +a \right )^{\frac {15}{2}}}{3}+\frac {10 a^{4} \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {2 a^{5} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{6}}\) | \(74\) |
trager | \(-\frac {2 \left (-46189 b^{10} x^{10}-206635 a \,b^{9} x^{9}-351780 a^{2} b^{8} x^{8}-271414 a^{3} b^{7} x^{7}-80773 a^{4} b^{6} x^{6}-63 a^{5} b^{5} x^{5}+70 a^{6} b^{4} x^{4}-80 a^{7} b^{3} x^{3}+96 a^{8} b^{2} x^{2}-128 a^{9} b x +256 a^{10}\right ) \sqrt {b x +a}}{969969 b^{6}}\) | \(120\) |
risch | \(-\frac {2 \left (-46189 b^{10} x^{10}-206635 a \,b^{9} x^{9}-351780 a^{2} b^{8} x^{8}-271414 a^{3} b^{7} x^{7}-80773 a^{4} b^{6} x^{6}-63 a^{5} b^{5} x^{5}+70 a^{6} b^{4} x^{4}-80 a^{7} b^{3} x^{3}+96 a^{8} b^{2} x^{2}-128 a^{9} b x +256 a^{10}\right ) \sqrt {b x +a}}{969969 b^{6}}\) | \(120\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.08 \[ \int x^5 (a+b x)^{9/2} \, dx=\frac {2 \, {\left (46189 \, b^{10} x^{10} + 206635 \, a b^{9} x^{9} + 351780 \, a^{2} b^{8} x^{8} + 271414 \, a^{3} b^{7} x^{7} + 80773 \, a^{4} b^{6} x^{6} + 63 \, a^{5} b^{5} x^{5} - 70 \, a^{6} b^{4} x^{4} + 80 \, a^{7} b^{3} x^{3} - 96 \, a^{8} b^{2} x^{2} + 128 \, a^{9} b x - 256 \, a^{10}\right )} \sqrt {b x + a}}{969969 \, b^{6}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (105) = 210\).
Time = 1.80 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.14 \[ \int x^5 (a+b x)^{9/2} \, dx=\begin {cases} - \frac {512 a^{10} \sqrt {a + b x}}{969969 b^{6}} + \frac {256 a^{9} x \sqrt {a + b x}}{969969 b^{5}} - \frac {64 a^{8} x^{2} \sqrt {a + b x}}{323323 b^{4}} + \frac {160 a^{7} x^{3} \sqrt {a + b x}}{969969 b^{3}} - \frac {20 a^{6} x^{4} \sqrt {a + b x}}{138567 b^{2}} + \frac {6 a^{5} x^{5} \sqrt {a + b x}}{46189 b} + \frac {2098 a^{4} x^{6} \sqrt {a + b x}}{12597} + \frac {3796 a^{3} b x^{7} \sqrt {a + b x}}{6783} + \frac {1640 a^{2} b^{2} x^{8} \sqrt {a + b x}}{2261} + \frac {170 a b^{3} x^{9} \sqrt {a + b x}}{399} + \frac {2 b^{4} x^{10} \sqrt {a + b x}}{21} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{6}}{6} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.78 \[ \int x^5 (a+b x)^{9/2} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {21}{2}}}{21 \, b^{6}} - \frac {10 \, {\left (b x + a\right )}^{\frac {19}{2}} a}{19 \, b^{6}} + \frac {20 \, {\left (b x + a\right )}^{\frac {17}{2}} a^{2}}{17 \, b^{6}} - \frac {4 \, {\left (b x + a\right )}^{\frac {15}{2}} a^{3}}{3 \, b^{6}} + \frac {10 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{4}}{13 \, b^{6}} - \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{5}}{11 \, b^{6}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 637 vs. \(2 (86) = 172\).
Time = 0.29 (sec) , antiderivative size = 637, normalized size of antiderivative = 5.79 \[ \int x^5 (a+b x)^{9/2} \, dx=\frac {2 \, {\left (\frac {4199 \, {\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^{5}}{b^{5}} + \frac {4845 \, {\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^{4}}{b^{5}} + \frac {4522 \, {\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^{3}}{b^{5}} + \frac {266 \, {\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^{2}}{b^{5}} + \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 )} a}{b^{5}} + \frac {3 \, {\left (46189 \, {\left (b x + a\right )}^{\frac {21}{2}} - 510510 \, {\left (b x + a\right )}^{\frac {19}{2}} a + 2567565 \, {\left (b x + a\right )}^{\frac {17}{2}} a^{2} - 7759752 \, {\left (b x + a\right )}^{\frac {15}{2}} a^{3} + 15668730 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{4} - 22221108 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{5} + 22632610 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{6} - 16628040 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{7} + 8729721 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{8} - 3233230 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{9} + 969969 \, \sqrt {b x + a} a^{10}\right )}}{b^{5}}\right )}}{2909907 \, b} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.78 \[ \int x^5 (a+b x)^{9/2} \, dx=\frac {2\,{\left (a+b\,x\right )}^{21/2}}{21\,b^6}-\frac {2\,a^5\,{\left (a+b\,x\right )}^{11/2}}{11\,b^6}+\frac {10\,a^4\,{\left (a+b\,x\right )}^{13/2}}{13\,b^6}-\frac {4\,a^3\,{\left (a+b\,x\right )}^{15/2}}{3\,b^6}+\frac {20\,a^2\,{\left (a+b\,x\right )}^{17/2}}{17\,b^6}-\frac {10\,a\,{\left (a+b\,x\right )}^{19/2}}{19\,b^6} \]
[In]
[Out]