Integrand size = 11, antiderivative size = 98 \[ \int x^5 (a+b x)^{10} \, dx=-\frac {a^5 (a+b x)^{11}}{11 b^6}+\frac {5 a^4 (a+b x)^{12}}{12 b^6}-\frac {10 a^3 (a+b x)^{13}}{13 b^6}+\frac {5 a^2 (a+b x)^{14}}{7 b^6}-\frac {a (a+b x)^{15}}{3 b^6}+\frac {(a+b x)^{16}}{16 b^6} \]
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Time = 0.03 (sec) , antiderivative size = 98, 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} \[ \int x^5 (a+b x)^{10} \, dx=-\frac {a^5 (a+b x)^{11}}{11 b^6}+\frac {5 a^4 (a+b x)^{12}}{12 b^6}-\frac {10 a^3 (a+b x)^{13}}{13 b^6}+\frac {5 a^2 (a+b x)^{14}}{7 b^6}+\frac {(a+b x)^{16}}{16 b^6}-\frac {a (a+b x)^{15}}{3 b^6} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^5 (a+b x)^{10}}{b^5}+\frac {5 a^4 (a+b x)^{11}}{b^5}-\frac {10 a^3 (a+b x)^{12}}{b^5}+\frac {10 a^2 (a+b x)^{13}}{b^5}-\frac {5 a (a+b x)^{14}}{b^5}+\frac {(a+b x)^{15}}{b^5}\right ) \, dx \\ & = -\frac {a^5 (a+b x)^{11}}{11 b^6}+\frac {5 a^4 (a+b x)^{12}}{12 b^6}-\frac {10 a^3 (a+b x)^{13}}{13 b^6}+\frac {5 a^2 (a+b x)^{14}}{7 b^6}-\frac {a (a+b x)^{15}}{3 b^6}+\frac {(a+b x)^{16}}{16 b^6} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.35 \[ \int x^5 (a+b x)^{10} \, dx=\frac {a^{10} x^6}{6}+\frac {10}{7} a^9 b x^7+\frac {45}{8} a^8 b^2 x^8+\frac {40}{3} a^7 b^3 x^9+21 a^6 b^4 x^{10}+\frac {252}{11} a^5 b^5 x^{11}+\frac {35}{2} a^4 b^6 x^{12}+\frac {120}{13} a^3 b^7 x^{13}+\frac {45}{14} a^2 b^8 x^{14}+\frac {2}{3} a b^9 x^{15}+\frac {b^{10} x^{16}}{16} \]
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Time = 0.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.15
method | result | size |
gosper | \(\frac {1}{6} a^{10} x^{6}+\frac {10}{7} a^{9} b \,x^{7}+\frac {45}{8} a^{8} b^{2} x^{8}+\frac {40}{3} a^{7} b^{3} x^{9}+21 a^{6} b^{4} x^{10}+\frac {252}{11} a^{5} b^{5} x^{11}+\frac {35}{2} a^{4} b^{6} x^{12}+\frac {120}{13} a^{3} b^{7} x^{13}+\frac {45}{14} a^{2} b^{8} x^{14}+\frac {2}{3} a \,b^{9} x^{15}+\frac {1}{16} b^{10} x^{16}\) | \(113\) |
default | \(\frac {1}{6} a^{10} x^{6}+\frac {10}{7} a^{9} b \,x^{7}+\frac {45}{8} a^{8} b^{2} x^{8}+\frac {40}{3} a^{7} b^{3} x^{9}+21 a^{6} b^{4} x^{10}+\frac {252}{11} a^{5} b^{5} x^{11}+\frac {35}{2} a^{4} b^{6} x^{12}+\frac {120}{13} a^{3} b^{7} x^{13}+\frac {45}{14} a^{2} b^{8} x^{14}+\frac {2}{3} a \,b^{9} x^{15}+\frac {1}{16} b^{10} x^{16}\) | \(113\) |
norman | \(\frac {1}{6} a^{10} x^{6}+\frac {10}{7} a^{9} b \,x^{7}+\frac {45}{8} a^{8} b^{2} x^{8}+\frac {40}{3} a^{7} b^{3} x^{9}+21 a^{6} b^{4} x^{10}+\frac {252}{11} a^{5} b^{5} x^{11}+\frac {35}{2} a^{4} b^{6} x^{12}+\frac {120}{13} a^{3} b^{7} x^{13}+\frac {45}{14} a^{2} b^{8} x^{14}+\frac {2}{3} a \,b^{9} x^{15}+\frac {1}{16} b^{10} x^{16}\) | \(113\) |
risch | \(\frac {1}{6} a^{10} x^{6}+\frac {10}{7} a^{9} b \,x^{7}+\frac {45}{8} a^{8} b^{2} x^{8}+\frac {40}{3} a^{7} b^{3} x^{9}+21 a^{6} b^{4} x^{10}+\frac {252}{11} a^{5} b^{5} x^{11}+\frac {35}{2} a^{4} b^{6} x^{12}+\frac {120}{13} a^{3} b^{7} x^{13}+\frac {45}{14} a^{2} b^{8} x^{14}+\frac {2}{3} a \,b^{9} x^{15}+\frac {1}{16} b^{10} x^{16}\) | \(113\) |
parallelrisch | \(\frac {1}{6} a^{10} x^{6}+\frac {10}{7} a^{9} b \,x^{7}+\frac {45}{8} a^{8} b^{2} x^{8}+\frac {40}{3} a^{7} b^{3} x^{9}+21 a^{6} b^{4} x^{10}+\frac {252}{11} a^{5} b^{5} x^{11}+\frac {35}{2} a^{4} b^{6} x^{12}+\frac {120}{13} a^{3} b^{7} x^{13}+\frac {45}{14} a^{2} b^{8} x^{14}+\frac {2}{3} a \,b^{9} x^{15}+\frac {1}{16} b^{10} x^{16}\) | \(113\) |
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Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int x^5 (a+b x)^{10} \, dx=\frac {1}{16} \, b^{10} x^{16} + \frac {2}{3} \, a b^{9} x^{15} + \frac {45}{14} \, a^{2} b^{8} x^{14} + \frac {120}{13} \, a^{3} b^{7} x^{13} + \frac {35}{2} \, a^{4} b^{6} x^{12} + \frac {252}{11} \, a^{5} b^{5} x^{11} + 21 \, a^{6} b^{4} x^{10} + \frac {40}{3} \, a^{7} b^{3} x^{9} + \frac {45}{8} \, a^{8} b^{2} x^{8} + \frac {10}{7} \, a^{9} b x^{7} + \frac {1}{6} \, a^{10} x^{6} \]
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Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.36 \[ \int x^5 (a+b x)^{10} \, dx=\frac {a^{10} x^{6}}{6} + \frac {10 a^{9} b x^{7}}{7} + \frac {45 a^{8} b^{2} x^{8}}{8} + \frac {40 a^{7} b^{3} x^{9}}{3} + 21 a^{6} b^{4} x^{10} + \frac {252 a^{5} b^{5} x^{11}}{11} + \frac {35 a^{4} b^{6} x^{12}}{2} + \frac {120 a^{3} b^{7} x^{13}}{13} + \frac {45 a^{2} b^{8} x^{14}}{14} + \frac {2 a b^{9} x^{15}}{3} + \frac {b^{10} x^{16}}{16} \]
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Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int x^5 (a+b x)^{10} \, dx=\frac {1}{16} \, b^{10} x^{16} + \frac {2}{3} \, a b^{9} x^{15} + \frac {45}{14} \, a^{2} b^{8} x^{14} + \frac {120}{13} \, a^{3} b^{7} x^{13} + \frac {35}{2} \, a^{4} b^{6} x^{12} + \frac {252}{11} \, a^{5} b^{5} x^{11} + 21 \, a^{6} b^{4} x^{10} + \frac {40}{3} \, a^{7} b^{3} x^{9} + \frac {45}{8} \, a^{8} b^{2} x^{8} + \frac {10}{7} \, a^{9} b x^{7} + \frac {1}{6} \, a^{10} x^{6} \]
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Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int x^5 (a+b x)^{10} \, dx=\frac {1}{16} \, b^{10} x^{16} + \frac {2}{3} \, a b^{9} x^{15} + \frac {45}{14} \, a^{2} b^{8} x^{14} + \frac {120}{13} \, a^{3} b^{7} x^{13} + \frac {35}{2} \, a^{4} b^{6} x^{12} + \frac {252}{11} \, a^{5} b^{5} x^{11} + 21 \, a^{6} b^{4} x^{10} + \frac {40}{3} \, a^{7} b^{3} x^{9} + \frac {45}{8} \, a^{8} b^{2} x^{8} + \frac {10}{7} \, a^{9} b x^{7} + \frac {1}{6} \, a^{10} x^{6} \]
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Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int x^5 (a+b x)^{10} \, dx=\frac {a^{10}\,x^6}{6}+\frac {10\,a^9\,b\,x^7}{7}+\frac {45\,a^8\,b^2\,x^8}{8}+\frac {40\,a^7\,b^3\,x^9}{3}+21\,a^6\,b^4\,x^{10}+\frac {252\,a^5\,b^5\,x^{11}}{11}+\frac {35\,a^4\,b^6\,x^{12}}{2}+\frac {120\,a^3\,b^7\,x^{13}}{13}+\frac {45\,a^2\,b^8\,x^{14}}{14}+\frac {2\,a\,b^9\,x^{15}}{3}+\frac {b^{10}\,x^{16}}{16} \]
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