Integrand size = 11, antiderivative size = 147 \[ \int x^8 (a+b x)^{10} \, dx=\frac {a^8 (a+b x)^{11}}{11 b^9}-\frac {2 a^7 (a+b x)^{12}}{3 b^9}+\frac {28 a^6 (a+b x)^{13}}{13 b^9}-\frac {4 a^5 (a+b x)^{14}}{b^9}+\frac {14 a^4 (a+b x)^{15}}{3 b^9}-\frac {7 a^3 (a+b x)^{16}}{2 b^9}+\frac {28 a^2 (a+b x)^{17}}{17 b^9}-\frac {4 a (a+b x)^{18}}{9 b^9}+\frac {(a+b x)^{19}}{19 b^9} \]
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Time = 0.05 (sec) , antiderivative size = 147, 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} \[ \int x^8 (a+b x)^{10} \, dx=\frac {a^8 (a+b x)^{11}}{11 b^9}-\frac {2 a^7 (a+b x)^{12}}{3 b^9}+\frac {28 a^6 (a+b x)^{13}}{13 b^9}-\frac {4 a^5 (a+b x)^{14}}{b^9}+\frac {14 a^4 (a+b x)^{15}}{3 b^9}-\frac {7 a^3 (a+b x)^{16}}{2 b^9}+\frac {28 a^2 (a+b x)^{17}}{17 b^9}+\frac {(a+b x)^{19}}{19 b^9}-\frac {4 a (a+b x)^{18}}{9 b^9} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^8 (a+b x)^{10}}{b^8}-\frac {8 a^7 (a+b x)^{11}}{b^8}+\frac {28 a^6 (a+b x)^{12}}{b^8}-\frac {56 a^5 (a+b x)^{13}}{b^8}+\frac {70 a^4 (a+b x)^{14}}{b^8}-\frac {56 a^3 (a+b x)^{15}}{b^8}+\frac {28 a^2 (a+b x)^{16}}{b^8}-\frac {8 a (a+b x)^{17}}{b^8}+\frac {(a+b x)^{18}}{b^8}\right ) \, dx \\ & = \frac {a^8 (a+b x)^{11}}{11 b^9}-\frac {2 a^7 (a+b x)^{12}}{3 b^9}+\frac {28 a^6 (a+b x)^{13}}{13 b^9}-\frac {4 a^5 (a+b x)^{14}}{b^9}+\frac {14 a^4 (a+b x)^{15}}{3 b^9}-\frac {7 a^3 (a+b x)^{16}}{2 b^9}+\frac {28 a^2 (a+b x)^{17}}{17 b^9}-\frac {4 a (a+b x)^{18}}{9 b^9}+\frac {(a+b x)^{19}}{19 b^9} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.85 \[ \int x^8 (a+b x)^{10} \, dx=\frac {a^{10} x^9}{9}+a^9 b x^{10}+\frac {45}{11} a^8 b^2 x^{11}+10 a^7 b^3 x^{12}+\frac {210}{13} a^6 b^4 x^{13}+18 a^5 b^5 x^{14}+14 a^4 b^6 x^{15}+\frac {15}{2} a^3 b^7 x^{16}+\frac {45}{17} a^2 b^8 x^{17}+\frac {5}{9} a b^9 x^{18}+\frac {b^{10} x^{19}}{19} \]
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Time = 0.18 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.76
method | result | size |
gosper | \(\frac {1}{9} a^{10} x^{9}+a^{9} b \,x^{10}+\frac {45}{11} a^{8} b^{2} x^{11}+10 a^{7} b^{3} x^{12}+\frac {210}{13} a^{6} b^{4} x^{13}+18 a^{5} b^{5} x^{14}+14 a^{4} b^{6} x^{15}+\frac {15}{2} a^{3} b^{7} x^{16}+\frac {45}{17} a^{2} b^{8} x^{17}+\frac {5}{9} a \,b^{9} x^{18}+\frac {1}{19} b^{10} x^{19}\) | \(112\) |
default | \(\frac {1}{9} a^{10} x^{9}+a^{9} b \,x^{10}+\frac {45}{11} a^{8} b^{2} x^{11}+10 a^{7} b^{3} x^{12}+\frac {210}{13} a^{6} b^{4} x^{13}+18 a^{5} b^{5} x^{14}+14 a^{4} b^{6} x^{15}+\frac {15}{2} a^{3} b^{7} x^{16}+\frac {45}{17} a^{2} b^{8} x^{17}+\frac {5}{9} a \,b^{9} x^{18}+\frac {1}{19} b^{10} x^{19}\) | \(112\) |
norman | \(\frac {1}{9} a^{10} x^{9}+a^{9} b \,x^{10}+\frac {45}{11} a^{8} b^{2} x^{11}+10 a^{7} b^{3} x^{12}+\frac {210}{13} a^{6} b^{4} x^{13}+18 a^{5} b^{5} x^{14}+14 a^{4} b^{6} x^{15}+\frac {15}{2} a^{3} b^{7} x^{16}+\frac {45}{17} a^{2} b^{8} x^{17}+\frac {5}{9} a \,b^{9} x^{18}+\frac {1}{19} b^{10} x^{19}\) | \(112\) |
risch | \(\frac {1}{9} a^{10} x^{9}+a^{9} b \,x^{10}+\frac {45}{11} a^{8} b^{2} x^{11}+10 a^{7} b^{3} x^{12}+\frac {210}{13} a^{6} b^{4} x^{13}+18 a^{5} b^{5} x^{14}+14 a^{4} b^{6} x^{15}+\frac {15}{2} a^{3} b^{7} x^{16}+\frac {45}{17} a^{2} b^{8} x^{17}+\frac {5}{9} a \,b^{9} x^{18}+\frac {1}{19} b^{10} x^{19}\) | \(112\) |
parallelrisch | \(\frac {1}{9} a^{10} x^{9}+a^{9} b \,x^{10}+\frac {45}{11} a^{8} b^{2} x^{11}+10 a^{7} b^{3} x^{12}+\frac {210}{13} a^{6} b^{4} x^{13}+18 a^{5} b^{5} x^{14}+14 a^{4} b^{6} x^{15}+\frac {15}{2} a^{3} b^{7} x^{16}+\frac {45}{17} a^{2} b^{8} x^{17}+\frac {5}{9} a \,b^{9} x^{18}+\frac {1}{19} b^{10} x^{19}\) | \(112\) |
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Time = 0.22 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.76 \[ \int x^8 (a+b x)^{10} \, dx=\frac {1}{19} \, b^{10} x^{19} + \frac {5}{9} \, a b^{9} x^{18} + \frac {45}{17} \, a^{2} b^{8} x^{17} + \frac {15}{2} \, a^{3} b^{7} x^{16} + 14 \, a^{4} b^{6} x^{15} + 18 \, a^{5} b^{5} x^{14} + \frac {210}{13} \, a^{6} b^{4} x^{13} + 10 \, a^{7} b^{3} x^{12} + \frac {45}{11} \, a^{8} b^{2} x^{11} + a^{9} b x^{10} + \frac {1}{9} \, a^{10} x^{9} \]
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Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.86 \[ \int x^8 (a+b x)^{10} \, dx=\frac {a^{10} x^{9}}{9} + a^{9} b x^{10} + \frac {45 a^{8} b^{2} x^{11}}{11} + 10 a^{7} b^{3} x^{12} + \frac {210 a^{6} b^{4} x^{13}}{13} + 18 a^{5} b^{5} x^{14} + 14 a^{4} b^{6} x^{15} + \frac {15 a^{3} b^{7} x^{16}}{2} + \frac {45 a^{2} b^{8} x^{17}}{17} + \frac {5 a b^{9} x^{18}}{9} + \frac {b^{10} x^{19}}{19} \]
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Time = 0.21 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.76 \[ \int x^8 (a+b x)^{10} \, dx=\frac {1}{19} \, b^{10} x^{19} + \frac {5}{9} \, a b^{9} x^{18} + \frac {45}{17} \, a^{2} b^{8} x^{17} + \frac {15}{2} \, a^{3} b^{7} x^{16} + 14 \, a^{4} b^{6} x^{15} + 18 \, a^{5} b^{5} x^{14} + \frac {210}{13} \, a^{6} b^{4} x^{13} + 10 \, a^{7} b^{3} x^{12} + \frac {45}{11} \, a^{8} b^{2} x^{11} + a^{9} b x^{10} + \frac {1}{9} \, a^{10} x^{9} \]
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Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.76 \[ \int x^8 (a+b x)^{10} \, dx=\frac {1}{19} \, b^{10} x^{19} + \frac {5}{9} \, a b^{9} x^{18} + \frac {45}{17} \, a^{2} b^{8} x^{17} + \frac {15}{2} \, a^{3} b^{7} x^{16} + 14 \, a^{4} b^{6} x^{15} + 18 \, a^{5} b^{5} x^{14} + \frac {210}{13} \, a^{6} b^{4} x^{13} + 10 \, a^{7} b^{3} x^{12} + \frac {45}{11} \, a^{8} b^{2} x^{11} + a^{9} b x^{10} + \frac {1}{9} \, a^{10} x^{9} \]
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Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.76 \[ \int x^8 (a+b x)^{10} \, dx=\frac {a^{10}\,x^9}{9}+a^9\,b\,x^{10}+\frac {45\,a^8\,b^2\,x^{11}}{11}+10\,a^7\,b^3\,x^{12}+\frac {210\,a^6\,b^4\,x^{13}}{13}+18\,a^5\,b^5\,x^{14}+14\,a^4\,b^6\,x^{15}+\frac {15\,a^3\,b^7\,x^{16}}{2}+\frac {45\,a^2\,b^8\,x^{17}}{17}+\frac {5\,a\,b^9\,x^{18}}{9}+\frac {b^{10}\,x^{19}}{19} \]
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