Integrand size = 11, antiderivative size = 132 \[ \int x^7 (a+b x)^{10} \, dx=-\frac {a^7 (a+b x)^{11}}{11 b^8}+\frac {7 a^6 (a+b x)^{12}}{12 b^8}-\frac {21 a^5 (a+b x)^{13}}{13 b^8}+\frac {5 a^4 (a+b x)^{14}}{2 b^8}-\frac {7 a^3 (a+b x)^{15}}{3 b^8}+\frac {21 a^2 (a+b x)^{16}}{16 b^8}-\frac {7 a (a+b x)^{17}}{17 b^8}+\frac {(a+b x)^{18}}{18 b^8} \]
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Time = 0.04 (sec) , antiderivative size = 132, 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} \[ \int x^7 (a+b x)^{10} \, dx=-\frac {a^7 (a+b x)^{11}}{11 b^8}+\frac {7 a^6 (a+b x)^{12}}{12 b^8}-\frac {21 a^5 (a+b x)^{13}}{13 b^8}+\frac {5 a^4 (a+b x)^{14}}{2 b^8}-\frac {7 a^3 (a+b x)^{15}}{3 b^8}+\frac {21 a^2 (a+b x)^{16}}{16 b^8}+\frac {(a+b x)^{18}}{18 b^8}-\frac {7 a (a+b x)^{17}}{17 b^8} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^7 (a+b x)^{10}}{b^7}+\frac {7 a^6 (a+b x)^{11}}{b^7}-\frac {21 a^5 (a+b x)^{12}}{b^7}+\frac {35 a^4 (a+b x)^{13}}{b^7}-\frac {35 a^3 (a+b x)^{14}}{b^7}+\frac {21 a^2 (a+b x)^{15}}{b^7}-\frac {7 a (a+b x)^{16}}{b^7}+\frac {(a+b x)^{17}}{b^7}\right ) \, dx \\ & = -\frac {a^7 (a+b x)^{11}}{11 b^8}+\frac {7 a^6 (a+b x)^{12}}{12 b^8}-\frac {21 a^5 (a+b x)^{13}}{13 b^8}+\frac {5 a^4 (a+b x)^{14}}{2 b^8}-\frac {7 a^3 (a+b x)^{15}}{3 b^8}+\frac {21 a^2 (a+b x)^{16}}{16 b^8}-\frac {7 a (a+b x)^{17}}{17 b^8}+\frac {(a+b x)^{18}}{18 b^8} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.98 \[ \int x^7 (a+b x)^{10} \, dx=\frac {a^{10} x^8}{8}+\frac {10}{9} a^9 b x^9+\frac {9}{2} a^8 b^2 x^{10}+\frac {120}{11} a^7 b^3 x^{11}+\frac {35}{2} a^6 b^4 x^{12}+\frac {252}{13} a^5 b^5 x^{13}+15 a^4 b^6 x^{14}+8 a^3 b^7 x^{15}+\frac {45}{16} a^2 b^8 x^{16}+\frac {10}{17} a b^9 x^{17}+\frac {b^{10} x^{18}}{18} \]
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Time = 0.18 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86
method | result | size |
gosper | \(\frac {1}{8} a^{10} x^{8}+\frac {10}{9} a^{9} b \,x^{9}+\frac {9}{2} a^{8} b^{2} x^{10}+\frac {120}{11} a^{7} b^{3} x^{11}+\frac {35}{2} a^{6} b^{4} x^{12}+\frac {252}{13} a^{5} b^{5} x^{13}+15 a^{4} b^{6} x^{14}+8 a^{3} b^{7} x^{15}+\frac {45}{16} a^{2} b^{8} x^{16}+\frac {10}{17} a \,b^{9} x^{17}+\frac {1}{18} b^{10} x^{18}\) | \(113\) |
default | \(\frac {1}{8} a^{10} x^{8}+\frac {10}{9} a^{9} b \,x^{9}+\frac {9}{2} a^{8} b^{2} x^{10}+\frac {120}{11} a^{7} b^{3} x^{11}+\frac {35}{2} a^{6} b^{4} x^{12}+\frac {252}{13} a^{5} b^{5} x^{13}+15 a^{4} b^{6} x^{14}+8 a^{3} b^{7} x^{15}+\frac {45}{16} a^{2} b^{8} x^{16}+\frac {10}{17} a \,b^{9} x^{17}+\frac {1}{18} b^{10} x^{18}\) | \(113\) |
norman | \(\frac {1}{8} a^{10} x^{8}+\frac {10}{9} a^{9} b \,x^{9}+\frac {9}{2} a^{8} b^{2} x^{10}+\frac {120}{11} a^{7} b^{3} x^{11}+\frac {35}{2} a^{6} b^{4} x^{12}+\frac {252}{13} a^{5} b^{5} x^{13}+15 a^{4} b^{6} x^{14}+8 a^{3} b^{7} x^{15}+\frac {45}{16} a^{2} b^{8} x^{16}+\frac {10}{17} a \,b^{9} x^{17}+\frac {1}{18} b^{10} x^{18}\) | \(113\) |
risch | \(\frac {1}{8} a^{10} x^{8}+\frac {10}{9} a^{9} b \,x^{9}+\frac {9}{2} a^{8} b^{2} x^{10}+\frac {120}{11} a^{7} b^{3} x^{11}+\frac {35}{2} a^{6} b^{4} x^{12}+\frac {252}{13} a^{5} b^{5} x^{13}+15 a^{4} b^{6} x^{14}+8 a^{3} b^{7} x^{15}+\frac {45}{16} a^{2} b^{8} x^{16}+\frac {10}{17} a \,b^{9} x^{17}+\frac {1}{18} b^{10} x^{18}\) | \(113\) |
parallelrisch | \(\frac {1}{8} a^{10} x^{8}+\frac {10}{9} a^{9} b \,x^{9}+\frac {9}{2} a^{8} b^{2} x^{10}+\frac {120}{11} a^{7} b^{3} x^{11}+\frac {35}{2} a^{6} b^{4} x^{12}+\frac {252}{13} a^{5} b^{5} x^{13}+15 a^{4} b^{6} x^{14}+8 a^{3} b^{7} x^{15}+\frac {45}{16} a^{2} b^{8} x^{16}+\frac {10}{17} a \,b^{9} x^{17}+\frac {1}{18} b^{10} x^{18}\) | \(113\) |
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Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^7 (a+b x)^{10} \, dx=\frac {1}{18} \, b^{10} x^{18} + \frac {10}{17} \, a b^{9} x^{17} + \frac {45}{16} \, a^{2} b^{8} x^{16} + 8 \, a^{3} b^{7} x^{15} + 15 \, a^{4} b^{6} x^{14} + \frac {252}{13} \, a^{5} b^{5} x^{13} + \frac {35}{2} \, a^{6} b^{4} x^{12} + \frac {120}{11} \, a^{7} b^{3} x^{11} + \frac {9}{2} \, a^{8} b^{2} x^{10} + \frac {10}{9} \, a^{9} b x^{9} + \frac {1}{8} \, a^{10} x^{8} \]
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Time = 0.03 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.99 \[ \int x^7 (a+b x)^{10} \, dx=\frac {a^{10} x^{8}}{8} + \frac {10 a^{9} b x^{9}}{9} + \frac {9 a^{8} b^{2} x^{10}}{2} + \frac {120 a^{7} b^{3} x^{11}}{11} + \frac {35 a^{6} b^{4} x^{12}}{2} + \frac {252 a^{5} b^{5} x^{13}}{13} + 15 a^{4} b^{6} x^{14} + 8 a^{3} b^{7} x^{15} + \frac {45 a^{2} b^{8} x^{16}}{16} + \frac {10 a b^{9} x^{17}}{17} + \frac {b^{10} x^{18}}{18} \]
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Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^7 (a+b x)^{10} \, dx=\frac {1}{18} \, b^{10} x^{18} + \frac {10}{17} \, a b^{9} x^{17} + \frac {45}{16} \, a^{2} b^{8} x^{16} + 8 \, a^{3} b^{7} x^{15} + 15 \, a^{4} b^{6} x^{14} + \frac {252}{13} \, a^{5} b^{5} x^{13} + \frac {35}{2} \, a^{6} b^{4} x^{12} + \frac {120}{11} \, a^{7} b^{3} x^{11} + \frac {9}{2} \, a^{8} b^{2} x^{10} + \frac {10}{9} \, a^{9} b x^{9} + \frac {1}{8} \, a^{10} x^{8} \]
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Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^7 (a+b x)^{10} \, dx=\frac {1}{18} \, b^{10} x^{18} + \frac {10}{17} \, a b^{9} x^{17} + \frac {45}{16} \, a^{2} b^{8} x^{16} + 8 \, a^{3} b^{7} x^{15} + 15 \, a^{4} b^{6} x^{14} + \frac {252}{13} \, a^{5} b^{5} x^{13} + \frac {35}{2} \, a^{6} b^{4} x^{12} + \frac {120}{11} \, a^{7} b^{3} x^{11} + \frac {9}{2} \, a^{8} b^{2} x^{10} + \frac {10}{9} \, a^{9} b x^{9} + \frac {1}{8} \, a^{10} x^{8} \]
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Time = 0.05 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^7 (a+b x)^{10} \, dx=\frac {a^{10}\,x^8}{8}+\frac {10\,a^9\,b\,x^9}{9}+\frac {9\,a^8\,b^2\,x^{10}}{2}+\frac {120\,a^7\,b^3\,x^{11}}{11}+\frac {35\,a^6\,b^4\,x^{12}}{2}+\frac {252\,a^5\,b^5\,x^{13}}{13}+15\,a^4\,b^6\,x^{14}+8\,a^3\,b^7\,x^{15}+\frac {45\,a^2\,b^8\,x^{16}}{16}+\frac {10\,a\,b^9\,x^{17}}{17}+\frac {b^{10}\,x^{18}}{18} \]
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