Integrand size = 11, antiderivative size = 69 \[ \int \frac {(a+b x)^5}{x^{12}} \, dx=-\frac {a^5}{11 x^{11}}-\frac {a^4 b}{2 x^{10}}-\frac {10 a^3 b^2}{9 x^9}-\frac {5 a^2 b^3}{4 x^8}-\frac {5 a b^4}{7 x^7}-\frac {b^5}{6 x^6} \]
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Time = 0.02 (sec) , antiderivative size = 69, 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} \[ \int \frac {(a+b x)^5}{x^{12}} \, dx=-\frac {a^5}{11 x^{11}}-\frac {a^4 b}{2 x^{10}}-\frac {10 a^3 b^2}{9 x^9}-\frac {5 a^2 b^3}{4 x^8}-\frac {5 a b^4}{7 x^7}-\frac {b^5}{6 x^6} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^5}{x^{12}}+\frac {5 a^4 b}{x^{11}}+\frac {10 a^3 b^2}{x^{10}}+\frac {10 a^2 b^3}{x^9}+\frac {5 a b^4}{x^8}+\frac {b^5}{x^7}\right ) \, dx \\ & = -\frac {a^5}{11 x^{11}}-\frac {a^4 b}{2 x^{10}}-\frac {10 a^3 b^2}{9 x^9}-\frac {5 a^2 b^3}{4 x^8}-\frac {5 a b^4}{7 x^7}-\frac {b^5}{6 x^6} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{x^{12}} \, dx=-\frac {a^5}{11 x^{11}}-\frac {a^4 b}{2 x^{10}}-\frac {10 a^3 b^2}{9 x^9}-\frac {5 a^2 b^3}{4 x^8}-\frac {5 a b^4}{7 x^7}-\frac {b^5}{6 x^6} \]
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Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83
method | result | size |
norman | \(\frac {-\frac {1}{6} b^{5} x^{5}-\frac {5}{7} a \,b^{4} x^{4}-\frac {5}{4} a^{2} b^{3} x^{3}-\frac {10}{9} a^{3} b^{2} x^{2}-\frac {1}{2} a^{4} b x -\frac {1}{11} a^{5}}{x^{11}}\) | \(57\) |
risch | \(\frac {-\frac {1}{6} b^{5} x^{5}-\frac {5}{7} a \,b^{4} x^{4}-\frac {5}{4} a^{2} b^{3} x^{3}-\frac {10}{9} a^{3} b^{2} x^{2}-\frac {1}{2} a^{4} b x -\frac {1}{11} a^{5}}{x^{11}}\) | \(57\) |
gosper | \(-\frac {462 b^{5} x^{5}+1980 a \,b^{4} x^{4}+3465 a^{2} b^{3} x^{3}+3080 a^{3} b^{2} x^{2}+1386 a^{4} b x +252 a^{5}}{2772 x^{11}}\) | \(58\) |
default | \(-\frac {a^{5}}{11 x^{11}}-\frac {a^{4} b}{2 x^{10}}-\frac {10 a^{3} b^{2}}{9 x^{9}}-\frac {5 a^{2} b^{3}}{4 x^{8}}-\frac {5 a \,b^{4}}{7 x^{7}}-\frac {b^{5}}{6 x^{6}}\) | \(58\) |
parallelrisch | \(\frac {-462 b^{5} x^{5}-1980 a \,b^{4} x^{4}-3465 a^{2} b^{3} x^{3}-3080 a^{3} b^{2} x^{2}-1386 a^{4} b x -252 a^{5}}{2772 x^{11}}\) | \(58\) |
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Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^5}{x^{12}} \, dx=-\frac {462 \, b^{5} x^{5} + 1980 \, a b^{4} x^{4} + 3465 \, a^{2} b^{3} x^{3} + 3080 \, a^{3} b^{2} x^{2} + 1386 \, a^{4} b x + 252 \, a^{5}}{2772 \, x^{11}} \]
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Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^5}{x^{12}} \, dx=\frac {- 252 a^{5} - 1386 a^{4} b x - 3080 a^{3} b^{2} x^{2} - 3465 a^{2} b^{3} x^{3} - 1980 a b^{4} x^{4} - 462 b^{5} x^{5}}{2772 x^{11}} \]
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Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^5}{x^{12}} \, dx=-\frac {462 \, b^{5} x^{5} + 1980 \, a b^{4} x^{4} + 3465 \, a^{2} b^{3} x^{3} + 3080 \, a^{3} b^{2} x^{2} + 1386 \, a^{4} b x + 252 \, a^{5}}{2772 \, x^{11}} \]
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Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^5}{x^{12}} \, dx=-\frac {462 \, b^{5} x^{5} + 1980 \, a b^{4} x^{4} + 3465 \, a^{2} b^{3} x^{3} + 3080 \, a^{3} b^{2} x^{2} + 1386 \, a^{4} b x + 252 \, a^{5}}{2772 \, x^{11}} \]
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Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^5}{x^{12}} \, dx=-\frac {\frac {a^5}{11}+\frac {a^4\,b\,x}{2}+\frac {10\,a^3\,b^2\,x^2}{9}+\frac {5\,a^2\,b^3\,x^3}{4}+\frac {5\,a\,b^4\,x^4}{7}+\frac {b^5\,x^5}{6}}{x^{11}} \]
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