Integrand size = 13, antiderivative size = 106 \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {b^8 x^9}{9}+\frac {4}{5} a b^7 x^{10}+\frac {28}{11} a^2 b^6 x^{11}+\frac {14}{3} a^3 b^5 x^{12}+\frac {70}{13} a^4 b^4 x^{13}+4 a^5 b^3 x^{14}+\frac {28}{15} a^6 b^2 x^{15}+\frac {1}{2} a^7 b x^{16}+\frac {a^8 x^{17}}{17} \]
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Time = 0.05 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 45} \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {a^8 x^{17}}{17}+\frac {1}{2} a^7 b x^{16}+\frac {28}{15} a^6 b^2 x^{15}+4 a^5 b^3 x^{14}+\frac {70}{13} a^4 b^4 x^{13}+\frac {14}{3} a^3 b^5 x^{12}+\frac {28}{11} a^2 b^6 x^{11}+\frac {4}{5} a b^7 x^{10}+\frac {b^8 x^9}{9} \]
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Rule 45
Rule 269
Rubi steps \begin{align*} \text {integral}& = \int x^8 (b+a x)^8 \, dx \\ & = \int \left (b^8 x^8+8 a b^7 x^9+28 a^2 b^6 x^{10}+56 a^3 b^5 x^{11}+70 a^4 b^4 x^{12}+56 a^5 b^3 x^{13}+28 a^6 b^2 x^{14}+8 a^7 b x^{15}+a^8 x^{16}\right ) \, dx \\ & = \frac {b^8 x^9}{9}+\frac {4}{5} a b^7 x^{10}+\frac {28}{11} a^2 b^6 x^{11}+\frac {14}{3} a^3 b^5 x^{12}+\frac {70}{13} a^4 b^4 x^{13}+4 a^5 b^3 x^{14}+\frac {28}{15} a^6 b^2 x^{15}+\frac {1}{2} a^7 b x^{16}+\frac {a^8 x^{17}}{17} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {b^8 x^9}{9}+\frac {4}{5} a b^7 x^{10}+\frac {28}{11} a^2 b^6 x^{11}+\frac {14}{3} a^3 b^5 x^{12}+\frac {70}{13} a^4 b^4 x^{13}+4 a^5 b^3 x^{14}+\frac {28}{15} a^6 b^2 x^{15}+\frac {1}{2} a^7 b x^{16}+\frac {a^8 x^{17}}{17} \]
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Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.86
method | result | size |
gosper | \(\frac {x^{9} \left (12870 a^{8} x^{8}+109395 x^{7} b \,a^{7}+408408 a^{6} b^{2} x^{6}+875160 a^{5} b^{3} x^{5}+1178100 a^{4} x^{4} b^{4}+1021020 a^{3} b^{5} x^{3}+556920 a^{2} b^{6} x^{2}+175032 a \,b^{7} x +24310 b^{8}\right )}{218790}\) | \(91\) |
default | \(\frac {1}{9} b^{8} x^{9}+\frac {4}{5} a \,b^{7} x^{10}+\frac {28}{11} a^{2} b^{6} x^{11}+\frac {14}{3} a^{3} b^{5} x^{12}+\frac {70}{13} x^{13} b^{4} a^{4}+4 a^{5} b^{3} x^{14}+\frac {28}{15} a^{6} b^{2} x^{15}+\frac {1}{2} a^{7} b \,x^{16}+\frac {1}{17} a^{8} x^{17}\) | \(91\) |
risch | \(\frac {1}{9} b^{8} x^{9}+\frac {4}{5} a \,b^{7} x^{10}+\frac {28}{11} a^{2} b^{6} x^{11}+\frac {14}{3} a^{3} b^{5} x^{12}+\frac {70}{13} x^{13} b^{4} a^{4}+4 a^{5} b^{3} x^{14}+\frac {28}{15} a^{6} b^{2} x^{15}+\frac {1}{2} a^{7} b \,x^{16}+\frac {1}{17} a^{8} x^{17}\) | \(91\) |
parallelrisch | \(\frac {1}{9} b^{8} x^{9}+\frac {4}{5} a \,b^{7} x^{10}+\frac {28}{11} a^{2} b^{6} x^{11}+\frac {14}{3} a^{3} b^{5} x^{12}+\frac {70}{13} x^{13} b^{4} a^{4}+4 a^{5} b^{3} x^{14}+\frac {28}{15} a^{6} b^{2} x^{15}+\frac {1}{2} a^{7} b \,x^{16}+\frac {1}{17} a^{8} x^{17}\) | \(91\) |
norman | \(\frac {\frac {1}{17} a^{8} x^{24}+\frac {1}{9} b^{8} x^{16}+\frac {4}{5} a \,b^{7} x^{17}+\frac {28}{11} a^{2} b^{6} x^{18}+\frac {14}{3} a^{3} b^{5} x^{19}+\frac {70}{13} a^{4} b^{4} x^{20}+4 a^{5} b^{3} x^{21}+\frac {28}{15} a^{6} b^{2} x^{22}+\frac {1}{2} a^{7} b \,x^{23}}{x^{7}}\) | \(95\) |
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Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {1}{17} \, a^{8} x^{17} + \frac {1}{2} \, a^{7} b x^{16} + \frac {28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {14}{3} \, a^{3} b^{5} x^{12} + \frac {28}{11} \, a^{2} b^{6} x^{11} + \frac {4}{5} \, a b^{7} x^{10} + \frac {1}{9} \, b^{8} x^{9} \]
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Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.98 \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {a^{8} x^{17}}{17} + \frac {a^{7} b x^{16}}{2} + \frac {28 a^{6} b^{2} x^{15}}{15} + 4 a^{5} b^{3} x^{14} + \frac {70 a^{4} b^{4} x^{13}}{13} + \frac {14 a^{3} b^{5} x^{12}}{3} + \frac {28 a^{2} b^{6} x^{11}}{11} + \frac {4 a b^{7} x^{10}}{5} + \frac {b^{8} x^{9}}{9} \]
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Time = 0.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {1}{17} \, a^{8} x^{17} + \frac {1}{2} \, a^{7} b x^{16} + \frac {28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {14}{3} \, a^{3} b^{5} x^{12} + \frac {28}{11} \, a^{2} b^{6} x^{11} + \frac {4}{5} \, a b^{7} x^{10} + \frac {1}{9} \, b^{8} x^{9} \]
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Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {1}{17} \, a^{8} x^{17} + \frac {1}{2} \, a^{7} b x^{16} + \frac {28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {14}{3} \, a^{3} b^{5} x^{12} + \frac {28}{11} \, a^{2} b^{6} x^{11} + \frac {4}{5} \, a b^{7} x^{10} + \frac {1}{9} \, b^{8} x^{9} \]
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Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {a^8\,x^{17}}{17}+\frac {a^7\,b\,x^{16}}{2}+\frac {28\,a^6\,b^2\,x^{15}}{15}+4\,a^5\,b^3\,x^{14}+\frac {70\,a^4\,b^4\,x^{13}}{13}+\frac {14\,a^3\,b^5\,x^{12}}{3}+\frac {28\,a^2\,b^6\,x^{11}}{11}+\frac {4\,a\,b^7\,x^{10}}{5}+\frac {b^8\,x^9}{9} \]
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