Integrand size = 11, antiderivative size = 76 \[ \int \frac {(a+b x)^7}{x^{12}} \, dx=-\frac {(a+b x)^8}{11 a x^{11}}+\frac {3 b (a+b x)^8}{110 a^2 x^{10}}-\frac {b^2 (a+b x)^8}{165 a^3 x^9}+\frac {b^3 (a+b x)^8}{1320 a^4 x^8} \]
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Time = 0.01 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {(a+b x)^7}{x^{12}} \, dx=\frac {b^3 (a+b x)^8}{1320 a^4 x^8}-\frac {b^2 (a+b x)^8}{165 a^3 x^9}+\frac {3 b (a+b x)^8}{110 a^2 x^{10}}-\frac {(a+b x)^8}{11 a x^{11}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^8}{11 a x^{11}}-\frac {(3 b) \int \frac {(a+b x)^7}{x^{11}} \, dx}{11 a} \\ & = -\frac {(a+b x)^8}{11 a x^{11}}+\frac {3 b (a+b x)^8}{110 a^2 x^{10}}+\frac {\left (3 b^2\right ) \int \frac {(a+b x)^7}{x^{10}} \, dx}{55 a^2} \\ & = -\frac {(a+b x)^8}{11 a x^{11}}+\frac {3 b (a+b x)^8}{110 a^2 x^{10}}-\frac {b^2 (a+b x)^8}{165 a^3 x^9}-\frac {b^3 \int \frac {(a+b x)^7}{x^9} \, dx}{165 a^3} \\ & = -\frac {(a+b x)^8}{11 a x^{11}}+\frac {3 b (a+b x)^8}{110 a^2 x^{10}}-\frac {b^2 (a+b x)^8}{165 a^3 x^9}+\frac {b^3 (a+b x)^8}{1320 a^4 x^8} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.22 \[ \int \frac {(a+b x)^7}{x^{12}} \, dx=-\frac {a^7}{11 x^{11}}-\frac {7 a^6 b}{10 x^{10}}-\frac {7 a^5 b^2}{3 x^9}-\frac {35 a^4 b^3}{8 x^8}-\frac {5 a^3 b^4}{x^7}-\frac {7 a^2 b^5}{2 x^6}-\frac {7 a b^6}{5 x^5}-\frac {b^7}{4 x^4} \]
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Time = 0.17 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04
method | result | size |
norman | \(\frac {-\frac {1}{4} b^{7} x^{7}-\frac {7}{5} a \,b^{6} x^{6}-\frac {7}{2} a^{2} b^{5} x^{5}-5 a^{3} b^{4} x^{4}-\frac {35}{8} a^{4} b^{3} x^{3}-\frac {7}{3} a^{5} b^{2} x^{2}-\frac {7}{10} a^{6} b x -\frac {1}{11} a^{7}}{x^{11}}\) | \(79\) |
risch | \(\frac {-\frac {1}{4} b^{7} x^{7}-\frac {7}{5} a \,b^{6} x^{6}-\frac {7}{2} a^{2} b^{5} x^{5}-5 a^{3} b^{4} x^{4}-\frac {35}{8} a^{4} b^{3} x^{3}-\frac {7}{3} a^{5} b^{2} x^{2}-\frac {7}{10} a^{6} b x -\frac {1}{11} a^{7}}{x^{11}}\) | \(79\) |
gosper | \(-\frac {330 b^{7} x^{7}+1848 a \,b^{6} x^{6}+4620 a^{2} b^{5} x^{5}+6600 a^{3} b^{4} x^{4}+5775 a^{4} b^{3} x^{3}+3080 a^{5} b^{2} x^{2}+924 a^{6} b x +120 a^{7}}{1320 x^{11}}\) | \(80\) |
default | \(-\frac {7 a^{6} b}{10 x^{10}}-\frac {7 a^{2} b^{5}}{2 x^{6}}-\frac {5 a^{3} b^{4}}{x^{7}}-\frac {7 a^{5} b^{2}}{3 x^{9}}-\frac {a^{7}}{11 x^{11}}-\frac {b^{7}}{4 x^{4}}-\frac {7 a \,b^{6}}{5 x^{5}}-\frac {35 a^{4} b^{3}}{8 x^{8}}\) | \(80\) |
parallelrisch | \(\frac {-330 b^{7} x^{7}-1848 a \,b^{6} x^{6}-4620 a^{2} b^{5} x^{5}-6600 a^{3} b^{4} x^{4}-5775 a^{4} b^{3} x^{3}-3080 a^{5} b^{2} x^{2}-924 a^{6} b x -120 a^{7}}{1320 x^{11}}\) | \(80\) |
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Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x)^7}{x^{12}} \, dx=-\frac {330 \, b^{7} x^{7} + 1848 \, a b^{6} x^{6} + 4620 \, a^{2} b^{5} x^{5} + 6600 \, a^{3} b^{4} x^{4} + 5775 \, a^{4} b^{3} x^{3} + 3080 \, a^{5} b^{2} x^{2} + 924 \, a^{6} b x + 120 \, a^{7}}{1320 \, x^{11}} \]
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Time = 0.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^7}{x^{12}} \, dx=\frac {- 120 a^{7} - 924 a^{6} b x - 3080 a^{5} b^{2} x^{2} - 5775 a^{4} b^{3} x^{3} - 6600 a^{3} b^{4} x^{4} - 4620 a^{2} b^{5} x^{5} - 1848 a b^{6} x^{6} - 330 b^{7} x^{7}}{1320 x^{11}} \]
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Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x)^7}{x^{12}} \, dx=-\frac {330 \, b^{7} x^{7} + 1848 \, a b^{6} x^{6} + 4620 \, a^{2} b^{5} x^{5} + 6600 \, a^{3} b^{4} x^{4} + 5775 \, a^{4} b^{3} x^{3} + 3080 \, a^{5} b^{2} x^{2} + 924 \, a^{6} b x + 120 \, a^{7}}{1320 \, x^{11}} \]
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Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x)^7}{x^{12}} \, dx=-\frac {330 \, b^{7} x^{7} + 1848 \, a b^{6} x^{6} + 4620 \, a^{2} b^{5} x^{5} + 6600 \, a^{3} b^{4} x^{4} + 5775 \, a^{4} b^{3} x^{3} + 3080 \, a^{5} b^{2} x^{2} + 924 \, a^{6} b x + 120 \, a^{7}}{1320 \, x^{11}} \]
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Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x)^7}{x^{12}} \, dx=-\frac {\frac {a^7}{11}+\frac {7\,a^6\,b\,x}{10}+\frac {7\,a^5\,b^2\,x^2}{3}+\frac {35\,a^4\,b^3\,x^3}{8}+5\,a^3\,b^4\,x^4+\frac {7\,a^2\,b^5\,x^5}{2}+\frac {7\,a\,b^6\,x^6}{5}+\frac {b^7\,x^7}{4}}{x^{11}} \]
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