Integrand size = 11, antiderivative size = 95 \[ \int \frac {(a+b x)^7}{x^{16}} \, dx=-\frac {a^7}{15 x^{15}}-\frac {a^6 b}{2 x^{14}}-\frac {21 a^5 b^2}{13 x^{13}}-\frac {35 a^4 b^3}{12 x^{12}}-\frac {35 a^3 b^4}{11 x^{11}}-\frac {21 a^2 b^5}{10 x^{10}}-\frac {7 a b^6}{9 x^9}-\frac {b^7}{8 x^8} \]
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Time = 0.02 (sec) , antiderivative size = 95, 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} \[ \int \frac {(a+b x)^7}{x^{16}} \, dx=-\frac {a^7}{15 x^{15}}-\frac {a^6 b}{2 x^{14}}-\frac {21 a^5 b^2}{13 x^{13}}-\frac {35 a^4 b^3}{12 x^{12}}-\frac {35 a^3 b^4}{11 x^{11}}-\frac {21 a^2 b^5}{10 x^{10}}-\frac {7 a b^6}{9 x^9}-\frac {b^7}{8 x^8} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^7}{x^{16}}+\frac {7 a^6 b}{x^{15}}+\frac {21 a^5 b^2}{x^{14}}+\frac {35 a^4 b^3}{x^{13}}+\frac {35 a^3 b^4}{x^{12}}+\frac {21 a^2 b^5}{x^{11}}+\frac {7 a b^6}{x^{10}}+\frac {b^7}{x^9}\right ) \, dx \\ & = -\frac {a^7}{15 x^{15}}-\frac {a^6 b}{2 x^{14}}-\frac {21 a^5 b^2}{13 x^{13}}-\frac {35 a^4 b^3}{12 x^{12}}-\frac {35 a^3 b^4}{11 x^{11}}-\frac {21 a^2 b^5}{10 x^{10}}-\frac {7 a b^6}{9 x^9}-\frac {b^7}{8 x^8} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^7}{x^{16}} \, dx=-\frac {a^7}{15 x^{15}}-\frac {a^6 b}{2 x^{14}}-\frac {21 a^5 b^2}{13 x^{13}}-\frac {35 a^4 b^3}{12 x^{12}}-\frac {35 a^3 b^4}{11 x^{11}}-\frac {21 a^2 b^5}{10 x^{10}}-\frac {7 a b^6}{9 x^9}-\frac {b^7}{8 x^8} \]
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Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83
method | result | size |
norman | \(\frac {-\frac {1}{15} a^{7}-\frac {1}{2} a^{6} b x -\frac {21}{13} a^{5} b^{2} x^{2}-\frac {35}{12} a^{4} b^{3} x^{3}-\frac {35}{11} a^{3} b^{4} x^{4}-\frac {21}{10} a^{2} b^{5} x^{5}-\frac {7}{9} a \,b^{6} x^{6}-\frac {1}{8} b^{7} x^{7}}{x^{15}}\) | \(79\) |
risch | \(\frac {-\frac {1}{15} a^{7}-\frac {1}{2} a^{6} b x -\frac {21}{13} a^{5} b^{2} x^{2}-\frac {35}{12} a^{4} b^{3} x^{3}-\frac {35}{11} a^{3} b^{4} x^{4}-\frac {21}{10} a^{2} b^{5} x^{5}-\frac {7}{9} a \,b^{6} x^{6}-\frac {1}{8} b^{7} x^{7}}{x^{15}}\) | \(79\) |
gosper | \(-\frac {6435 b^{7} x^{7}+40040 a \,b^{6} x^{6}+108108 a^{2} b^{5} x^{5}+163800 a^{3} b^{4} x^{4}+150150 a^{4} b^{3} x^{3}+83160 a^{5} b^{2} x^{2}+25740 a^{6} b x +3432 a^{7}}{51480 x^{15}}\) | \(80\) |
default | \(-\frac {a^{7}}{15 x^{15}}-\frac {a^{6} b}{2 x^{14}}-\frac {21 a^{5} b^{2}}{13 x^{13}}-\frac {35 a^{4} b^{3}}{12 x^{12}}-\frac {35 a^{3} b^{4}}{11 x^{11}}-\frac {21 a^{2} b^{5}}{10 x^{10}}-\frac {7 a \,b^{6}}{9 x^{9}}-\frac {b^{7}}{8 x^{8}}\) | \(80\) |
parallelrisch | \(\frac {-6435 b^{7} x^{7}-40040 a \,b^{6} x^{6}-108108 a^{2} b^{5} x^{5}-163800 a^{3} b^{4} x^{4}-150150 a^{4} b^{3} x^{3}-83160 a^{5} b^{2} x^{2}-25740 a^{6} b x -3432 a^{7}}{51480 x^{15}}\) | \(80\) |
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Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^7}{x^{16}} \, dx=-\frac {6435 \, b^{7} x^{7} + 40040 \, a b^{6} x^{6} + 108108 \, a^{2} b^{5} x^{5} + 163800 \, a^{3} b^{4} x^{4} + 150150 \, a^{4} b^{3} x^{3} + 83160 \, a^{5} b^{2} x^{2} + 25740 \, a^{6} b x + 3432 \, a^{7}}{51480 \, x^{15}} \]
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Time = 0.37 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^7}{x^{16}} \, dx=\frac {- 3432 a^{7} - 25740 a^{6} b x - 83160 a^{5} b^{2} x^{2} - 150150 a^{4} b^{3} x^{3} - 163800 a^{3} b^{4} x^{4} - 108108 a^{2} b^{5} x^{5} - 40040 a b^{6} x^{6} - 6435 b^{7} x^{7}}{51480 x^{15}} \]
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Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^7}{x^{16}} \, dx=-\frac {6435 \, b^{7} x^{7} + 40040 \, a b^{6} x^{6} + 108108 \, a^{2} b^{5} x^{5} + 163800 \, a^{3} b^{4} x^{4} + 150150 \, a^{4} b^{3} x^{3} + 83160 \, a^{5} b^{2} x^{2} + 25740 \, a^{6} b x + 3432 \, a^{7}}{51480 \, x^{15}} \]
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Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^7}{x^{16}} \, dx=-\frac {6435 \, b^{7} x^{7} + 40040 \, a b^{6} x^{6} + 108108 \, a^{2} b^{5} x^{5} + 163800 \, a^{3} b^{4} x^{4} + 150150 \, a^{4} b^{3} x^{3} + 83160 \, a^{5} b^{2} x^{2} + 25740 \, a^{6} b x + 3432 \, a^{7}}{51480 \, x^{15}} \]
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Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^7}{x^{16}} \, dx=-\frac {\frac {a^7}{15}+\frac {a^6\,b\,x}{2}+\frac {21\,a^5\,b^2\,x^2}{13}+\frac {35\,a^4\,b^3\,x^3}{12}+\frac {35\,a^3\,b^4\,x^4}{11}+\frac {21\,a^2\,b^5\,x^5}{10}+\frac {7\,a\,b^6\,x^6}{9}+\frac {b^7\,x^7}{8}}{x^{15}} \]
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