Integrand size = 51, antiderivative size = 14 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=\frac {(a+b x)^{16}}{16 b} \]
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Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {2084, 32} \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=\frac {(a+b x)^{16}}{16 b} \]
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Rule 32
Rule 2084
Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^{15} \, dx \\ & = \frac {(a+b x)^{16}}{16 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=\frac {(a+b x)^{16}}{16 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(163\) vs. \(2(12)=24\).
Time = 0.05 (sec) , antiderivative size = 164, normalized size of antiderivative = 11.71
method | result | size |
default | \(a^{15} x +\frac {15}{2} a^{14} b \,x^{2}+35 a^{13} b^{2} x^{3}+\frac {455}{4} a^{12} b^{3} x^{4}+273 a^{11} b^{4} x^{5}+\frac {1001}{2} a^{10} b^{5} x^{6}+715 a^{9} b^{6} x^{7}+\frac {6435}{8} a^{8} b^{7} x^{8}+715 a^{7} b^{8} x^{9}+\frac {1001}{2} a^{6} b^{9} x^{10}+273 a^{5} b^{10} x^{11}+\frac {455}{4} a^{4} b^{11} x^{12}+35 a^{3} b^{12} x^{13}+\frac {15}{2} a^{2} b^{13} x^{14}+a \,b^{14} x^{15}+\frac {1}{16} b^{15} x^{16}\) | \(164\) |
norman | \(a^{15} x +\frac {15}{2} a^{14} b \,x^{2}+35 a^{13} b^{2} x^{3}+\frac {455}{4} a^{12} b^{3} x^{4}+273 a^{11} b^{4} x^{5}+\frac {1001}{2} a^{10} b^{5} x^{6}+715 a^{9} b^{6} x^{7}+\frac {6435}{8} a^{8} b^{7} x^{8}+715 a^{7} b^{8} x^{9}+\frac {1001}{2} a^{6} b^{9} x^{10}+273 a^{5} b^{10} x^{11}+\frac {455}{4} a^{4} b^{11} x^{12}+35 a^{3} b^{12} x^{13}+\frac {15}{2} a^{2} b^{13} x^{14}+a \,b^{14} x^{15}+\frac {1}{16} b^{15} x^{16}\) | \(164\) |
risch | \(a^{15} x +\frac {15}{2} a^{14} b \,x^{2}+35 a^{13} b^{2} x^{3}+\frac {455}{4} a^{12} b^{3} x^{4}+273 a^{11} b^{4} x^{5}+\frac {1001}{2} a^{10} b^{5} x^{6}+715 a^{9} b^{6} x^{7}+\frac {6435}{8} a^{8} b^{7} x^{8}+715 a^{7} b^{8} x^{9}+\frac {1001}{2} a^{6} b^{9} x^{10}+273 a^{5} b^{10} x^{11}+\frac {455}{4} a^{4} b^{11} x^{12}+35 a^{3} b^{12} x^{13}+\frac {15}{2} a^{2} b^{13} x^{14}+a \,b^{14} x^{15}+\frac {1}{16} b^{15} x^{16}\) | \(164\) |
parallelrisch | \(a^{15} x +\frac {15}{2} a^{14} b \,x^{2}+35 a^{13} b^{2} x^{3}+\frac {455}{4} a^{12} b^{3} x^{4}+273 a^{11} b^{4} x^{5}+\frac {1001}{2} a^{10} b^{5} x^{6}+715 a^{9} b^{6} x^{7}+\frac {6435}{8} a^{8} b^{7} x^{8}+715 a^{7} b^{8} x^{9}+\frac {1001}{2} a^{6} b^{9} x^{10}+273 a^{5} b^{10} x^{11}+\frac {455}{4} a^{4} b^{11} x^{12}+35 a^{3} b^{12} x^{13}+\frac {15}{2} a^{2} b^{13} x^{14}+a \,b^{14} x^{15}+\frac {1}{16} b^{15} x^{16}\) | \(164\) |
gosper | \(\frac {x \left (b^{15} x^{15}+16 a \,b^{14} x^{14}+120 a^{2} b^{13} x^{13}+560 a^{3} b^{12} x^{12}+1820 a^{4} b^{11} x^{11}+4368 a^{5} b^{10} x^{10}+8008 a^{6} b^{9} x^{9}+11440 a^{7} b^{8} x^{8}+12870 a^{8} b^{7} x^{7}+11440 a^{9} b^{6} x^{6}+8008 a^{10} b^{5} x^{5}+4368 a^{11} b^{4} x^{4}+1820 a^{12} b^{3} x^{3}+560 a^{13} b^{2} x^{2}+120 a^{14} b x +16 a^{15}\right )}{16}\) | \(165\) |
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Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (12) = 24\).
Time = 0.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 11.64 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=\frac {1}{16} \, b^{15} x^{16} + a b^{14} x^{15} + \frac {15}{2} \, a^{2} b^{13} x^{14} + 35 \, a^{3} b^{12} x^{13} + \frac {455}{4} \, a^{4} b^{11} x^{12} + 273 \, a^{5} b^{10} x^{11} + \frac {1001}{2} \, a^{6} b^{9} x^{10} + 715 \, a^{7} b^{8} x^{9} + \frac {6435}{8} \, a^{8} b^{7} x^{8} + 715 \, a^{9} b^{6} x^{7} + \frac {1001}{2} \, a^{10} b^{5} x^{6} + 273 \, a^{11} b^{4} x^{5} + \frac {455}{4} \, a^{12} b^{3} x^{4} + 35 \, a^{13} b^{2} x^{3} + \frac {15}{2} \, a^{14} b x^{2} + a^{15} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (8) = 16\).
Time = 0.04 (sec) , antiderivative size = 185, normalized size of antiderivative = 13.21 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=a^{15} x + \frac {15 a^{14} b x^{2}}{2} + 35 a^{13} b^{2} x^{3} + \frac {455 a^{12} b^{3} x^{4}}{4} + 273 a^{11} b^{4} x^{5} + \frac {1001 a^{10} b^{5} x^{6}}{2} + 715 a^{9} b^{6} x^{7} + \frac {6435 a^{8} b^{7} x^{8}}{8} + 715 a^{7} b^{8} x^{9} + \frac {1001 a^{6} b^{9} x^{10}}{2} + 273 a^{5} b^{10} x^{11} + \frac {455 a^{4} b^{11} x^{12}}{4} + 35 a^{3} b^{12} x^{13} + \frac {15 a^{2} b^{13} x^{14}}{2} + a b^{14} x^{15} + \frac {b^{15} x^{16}}{16} \]
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Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (12) = 24\).
Time = 0.21 (sec) , antiderivative size = 592, normalized size of antiderivative = 42.29 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=\frac {1}{16} \, b^{15} x^{16} + a b^{14} x^{15} + \frac {75}{14} \, a^{2} b^{13} x^{14} + \frac {125}{13} \, a^{3} b^{12} x^{13} + 100 \, a^{6} b^{9} x^{10} + \frac {1000}{7} \, a^{9} b^{6} x^{7} + \frac {125}{4} \, a^{12} b^{3} x^{4} + a^{15} x + \frac {1}{2} \, {\left (b^{5} x^{6} + 6 \, a b^{4} x^{5} + 15 \, a^{2} b^{3} x^{4} + 20 \, a^{3} b^{2} x^{3} + 15 \, a^{4} b x^{2}\right )} a^{10} + \frac {25}{56} \, {\left (21 \, b^{5} x^{8} + 120 \, a b^{4} x^{7} + 280 \, a^{2} b^{3} x^{6} + 336 \, a^{3} b^{2} x^{5}\right )} a^{8} b^{2} + \frac {5}{3} \, {\left (18 \, b^{5} x^{10} + 100 \, a b^{4} x^{9} + 225 \, a^{2} b^{3} x^{8}\right )} a^{6} b^{4} + \frac {25}{11} \, {\left (11 \, b^{5} x^{12} + 60 \, a b^{4} x^{11}\right )} a^{4} b^{6} + \frac {1}{462} \, {\left (126 \, b^{10} x^{11} + 1386 \, a b^{9} x^{10} + 3850 \, a^{2} b^{8} x^{9} + 19800 \, a^{4} b^{6} x^{7} + 27720 \, a^{6} b^{4} x^{5} + 11550 \, a^{8} b^{2} x^{3} + 330 \, {\left (6 \, b^{5} x^{7} + 35 \, a b^{4} x^{6} + 84 \, a^{2} b^{3} x^{5} + 105 \, a^{3} b^{2} x^{4}\right )} a^{4} b + 165 \, {\left (21 \, b^{5} x^{8} + 120 \, a b^{4} x^{7} + 280 \, a^{2} b^{3} x^{6}\right )} a^{3} b^{2} + 385 \, {\left (8 \, b^{5} x^{9} + 45 \, a b^{4} x^{8}\right )} a^{2} b^{3}\right )} a^{5} + \frac {5}{308} \, {\left (77 \, b^{10} x^{12} + 840 \, a b^{9} x^{11} + 4158 \, a^{2} b^{8} x^{10} + 12320 \, a^{3} b^{7} x^{9} + 23100 \, a^{4} b^{6} x^{8} + 26400 \, a^{5} b^{5} x^{7} + 15400 \, a^{6} b^{4} x^{6}\right )} a^{4} b + \frac {5}{429} \, {\left (198 \, b^{10} x^{13} + 2145 \, a b^{9} x^{12} + 10530 \, a^{2} b^{8} x^{11} + 25740 \, a^{3} b^{7} x^{10} + 28600 \, a^{4} b^{6} x^{9}\right )} a^{3} b^{2} + \frac {5}{182} \, {\left (78 \, b^{10} x^{14} + 840 \, a b^{9} x^{13} + 2275 \, a^{2} b^{8} x^{12}\right )} a^{2} b^{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (12) = 24\).
Time = 0.27 (sec) , antiderivative size = 163, normalized size of antiderivative = 11.64 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=\frac {1}{16} \, b^{15} x^{16} + a b^{14} x^{15} + \frac {15}{2} \, a^{2} b^{13} x^{14} + 35 \, a^{3} b^{12} x^{13} + \frac {455}{4} \, a^{4} b^{11} x^{12} + 273 \, a^{5} b^{10} x^{11} + \frac {1001}{2} \, a^{6} b^{9} x^{10} + 715 \, a^{7} b^{8} x^{9} + \frac {6435}{8} \, a^{8} b^{7} x^{8} + 715 \, a^{9} b^{6} x^{7} + \frac {1001}{2} \, a^{10} b^{5} x^{6} + 273 \, a^{11} b^{4} x^{5} + \frac {455}{4} \, a^{12} b^{3} x^{4} + 35 \, a^{13} b^{2} x^{3} + \frac {15}{2} \, a^{14} b x^{2} + a^{15} x \]
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Time = 0.13 (sec) , antiderivative size = 163, normalized size of antiderivative = 11.64 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=a^{15}\,x+\frac {15\,a^{14}\,b\,x^2}{2}+35\,a^{13}\,b^2\,x^3+\frac {455\,a^{12}\,b^3\,x^4}{4}+273\,a^{11}\,b^4\,x^5+\frac {1001\,a^{10}\,b^5\,x^6}{2}+715\,a^9\,b^6\,x^7+\frac {6435\,a^8\,b^7\,x^8}{8}+715\,a^7\,b^8\,x^9+\frac {1001\,a^6\,b^9\,x^{10}}{2}+273\,a^5\,b^{10}\,x^{11}+\frac {455\,a^4\,b^{11}\,x^{12}}{4}+35\,a^3\,b^{12}\,x^{13}+\frac {15\,a^2\,b^{13}\,x^{14}}{2}+a\,b^{14}\,x^{15}+\frac {b^{15}\,x^{16}}{16} \]
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