Integrand size = 11, antiderivative size = 96 \[ \int x^5 (a+b x)^7 \, dx=-\frac {a^5 (a+b x)^8}{8 b^6}+\frac {5 a^4 (a+b x)^9}{9 b^6}-\frac {a^3 (a+b x)^{10}}{b^6}+\frac {10 a^2 (a+b x)^{11}}{11 b^6}-\frac {5 a (a+b x)^{12}}{12 b^6}+\frac {(a+b x)^{13}}{13 b^6} \]
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Time = 0.03 (sec) , antiderivative size = 96, 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} \[ \int x^5 (a+b x)^7 \, dx=-\frac {a^5 (a+b x)^8}{8 b^6}+\frac {5 a^4 (a+b x)^9}{9 b^6}-\frac {a^3 (a+b x)^{10}}{b^6}+\frac {10 a^2 (a+b x)^{11}}{11 b^6}+\frac {(a+b x)^{13}}{13 b^6}-\frac {5 a (a+b x)^{12}}{12 b^6} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^5 (a+b x)^7}{b^5}+\frac {5 a^4 (a+b x)^8}{b^5}-\frac {10 a^3 (a+b x)^9}{b^5}+\frac {10 a^2 (a+b x)^{10}}{b^5}-\frac {5 a (a+b x)^{11}}{b^5}+\frac {(a+b x)^{12}}{b^5}\right ) \, dx \\ & = -\frac {a^5 (a+b x)^8}{8 b^6}+\frac {5 a^4 (a+b x)^9}{9 b^6}-\frac {a^3 (a+b x)^{10}}{b^6}+\frac {10 a^2 (a+b x)^{11}}{11 b^6}-\frac {5 a (a+b x)^{12}}{12 b^6}+\frac {(a+b x)^{13}}{13 b^6} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96 \[ \int x^5 (a+b x)^7 \, dx=\frac {a^7 x^6}{6}+a^6 b x^7+\frac {21}{8} a^5 b^2 x^8+\frac {35}{9} a^4 b^3 x^9+\frac {7}{2} a^3 b^4 x^{10}+\frac {21}{11} a^2 b^5 x^{11}+\frac {7}{12} a b^6 x^{12}+\frac {b^7 x^{13}}{13} \]
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Time = 0.17 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(\frac {1}{13} b^{7} x^{13}+\frac {7}{12} a \,b^{6} x^{12}+\frac {21}{11} a^{2} b^{5} x^{11}+\frac {7}{2} a^{3} b^{4} x^{10}+\frac {35}{9} a^{4} b^{3} x^{9}+\frac {21}{8} a^{5} b^{2} x^{8}+a^{6} b \,x^{7}+\frac {1}{6} a^{7} x^{6}\) | \(79\) |
default | \(\frac {1}{13} b^{7} x^{13}+\frac {7}{12} a \,b^{6} x^{12}+\frac {21}{11} a^{2} b^{5} x^{11}+\frac {7}{2} a^{3} b^{4} x^{10}+\frac {35}{9} a^{4} b^{3} x^{9}+\frac {21}{8} a^{5} b^{2} x^{8}+a^{6} b \,x^{7}+\frac {1}{6} a^{7} x^{6}\) | \(79\) |
norman | \(\frac {1}{13} b^{7} x^{13}+\frac {7}{12} a \,b^{6} x^{12}+\frac {21}{11} a^{2} b^{5} x^{11}+\frac {7}{2} a^{3} b^{4} x^{10}+\frac {35}{9} a^{4} b^{3} x^{9}+\frac {21}{8} a^{5} b^{2} x^{8}+a^{6} b \,x^{7}+\frac {1}{6} a^{7} x^{6}\) | \(79\) |
risch | \(\frac {1}{13} b^{7} x^{13}+\frac {7}{12} a \,b^{6} x^{12}+\frac {21}{11} a^{2} b^{5} x^{11}+\frac {7}{2} a^{3} b^{4} x^{10}+\frac {35}{9} a^{4} b^{3} x^{9}+\frac {21}{8} a^{5} b^{2} x^{8}+a^{6} b \,x^{7}+\frac {1}{6} a^{7} x^{6}\) | \(79\) |
parallelrisch | \(\frac {1}{13} b^{7} x^{13}+\frac {7}{12} a \,b^{6} x^{12}+\frac {21}{11} a^{2} b^{5} x^{11}+\frac {7}{2} a^{3} b^{4} x^{10}+\frac {35}{9} a^{4} b^{3} x^{9}+\frac {21}{8} a^{5} b^{2} x^{8}+a^{6} b \,x^{7}+\frac {1}{6} a^{7} x^{6}\) | \(79\) |
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Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.81 \[ \int x^5 (a+b x)^7 \, dx=\frac {1}{13} \, b^{7} x^{13} + \frac {7}{12} \, a b^{6} x^{12} + \frac {21}{11} \, a^{2} b^{5} x^{11} + \frac {7}{2} \, a^{3} b^{4} x^{10} + \frac {35}{9} \, a^{4} b^{3} x^{9} + \frac {21}{8} \, a^{5} b^{2} x^{8} + a^{6} b x^{7} + \frac {1}{6} \, a^{7} x^{6} \]
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Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int x^5 (a+b x)^7 \, dx=\frac {a^{7} x^{6}}{6} + a^{6} b x^{7} + \frac {21 a^{5} b^{2} x^{8}}{8} + \frac {35 a^{4} b^{3} x^{9}}{9} + \frac {7 a^{3} b^{4} x^{10}}{2} + \frac {21 a^{2} b^{5} x^{11}}{11} + \frac {7 a b^{6} x^{12}}{12} + \frac {b^{7} x^{13}}{13} \]
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Time = 0.23 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.81 \[ \int x^5 (a+b x)^7 \, dx=\frac {1}{13} \, b^{7} x^{13} + \frac {7}{12} \, a b^{6} x^{12} + \frac {21}{11} \, a^{2} b^{5} x^{11} + \frac {7}{2} \, a^{3} b^{4} x^{10} + \frac {35}{9} \, a^{4} b^{3} x^{9} + \frac {21}{8} \, a^{5} b^{2} x^{8} + a^{6} b x^{7} + \frac {1}{6} \, a^{7} x^{6} \]
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Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.81 \[ \int x^5 (a+b x)^7 \, dx=\frac {1}{13} \, b^{7} x^{13} + \frac {7}{12} \, a b^{6} x^{12} + \frac {21}{11} \, a^{2} b^{5} x^{11} + \frac {7}{2} \, a^{3} b^{4} x^{10} + \frac {35}{9} \, a^{4} b^{3} x^{9} + \frac {21}{8} \, a^{5} b^{2} x^{8} + a^{6} b x^{7} + \frac {1}{6} \, a^{7} x^{6} \]
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Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.81 \[ \int x^5 (a+b x)^7 \, dx=\frac {a^7\,x^6}{6}+a^6\,b\,x^7+\frac {21\,a^5\,b^2\,x^8}{8}+\frac {35\,a^4\,b^3\,x^9}{9}+\frac {7\,a^3\,b^4\,x^{10}}{2}+\frac {21\,a^2\,b^5\,x^{11}}{11}+\frac {7\,a\,b^6\,x^{12}}{12}+\frac {b^7\,x^{13}}{13} \]
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