Integrand size = 11, antiderivative size = 43 \[ \int \frac {(a+b x)^3}{x^{10}} \, dx=-\frac {a^3}{9 x^9}-\frac {3 a^2 b}{8 x^8}-\frac {3 a b^2}{7 x^7}-\frac {b^3}{6 x^6} \]
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Time = 0.01 (sec) , antiderivative size = 43, 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} \[ \int \frac {(a+b x)^3}{x^{10}} \, dx=-\frac {a^3}{9 x^9}-\frac {3 a^2 b}{8 x^8}-\frac {3 a b^2}{7 x^7}-\frac {b^3}{6 x^6} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3}{x^{10}}+\frac {3 a^2 b}{x^9}+\frac {3 a b^2}{x^8}+\frac {b^3}{x^7}\right ) \, dx \\ & = -\frac {a^3}{9 x^9}-\frac {3 a^2 b}{8 x^8}-\frac {3 a b^2}{7 x^7}-\frac {b^3}{6 x^6} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^3}{x^{10}} \, dx=-\frac {a^3}{9 x^9}-\frac {3 a^2 b}{8 x^8}-\frac {3 a b^2}{7 x^7}-\frac {b^3}{6 x^6} \]
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Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81
method | result | size |
norman | \(\frac {-\frac {1}{6} b^{3} x^{3}-\frac {3}{7} a \,b^{2} x^{2}-\frac {3}{8} a^{2} b x -\frac {1}{9} a^{3}}{x^{9}}\) | \(35\) |
risch | \(\frac {-\frac {1}{6} b^{3} x^{3}-\frac {3}{7} a \,b^{2} x^{2}-\frac {3}{8} a^{2} b x -\frac {1}{9} a^{3}}{x^{9}}\) | \(35\) |
gosper | \(-\frac {84 b^{3} x^{3}+216 a \,b^{2} x^{2}+189 a^{2} b x +56 a^{3}}{504 x^{9}}\) | \(36\) |
default | \(-\frac {a^{3}}{9 x^{9}}-\frac {3 a^{2} b}{8 x^{8}}-\frac {3 a \,b^{2}}{7 x^{7}}-\frac {b^{3}}{6 x^{6}}\) | \(36\) |
parallelrisch | \(\frac {-84 b^{3} x^{3}-216 a \,b^{2} x^{2}-189 a^{2} b x -56 a^{3}}{504 x^{9}}\) | \(36\) |
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Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^3}{x^{10}} \, dx=-\frac {84 \, b^{3} x^{3} + 216 \, a b^{2} x^{2} + 189 \, a^{2} b x + 56 \, a^{3}}{504 \, x^{9}} \]
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Time = 0.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^3}{x^{10}} \, dx=\frac {- 56 a^{3} - 189 a^{2} b x - 216 a b^{2} x^{2} - 84 b^{3} x^{3}}{504 x^{9}} \]
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none
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^3}{x^{10}} \, dx=-\frac {84 \, b^{3} x^{3} + 216 \, a b^{2} x^{2} + 189 \, a^{2} b x + 56 \, a^{3}}{504 \, x^{9}} \]
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Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^3}{x^{10}} \, dx=-\frac {84 \, b^{3} x^{3} + 216 \, a b^{2} x^{2} + 189 \, a^{2} b x + 56 \, a^{3}}{504 \, x^{9}} \]
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Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^3}{x^{10}} \, dx=-\frac {\frac {a^3}{9}+\frac {3\,a^2\,b\,x}{8}+\frac {3\,a\,b^2\,x^2}{7}+\frac {b^3\,x^3}{6}}{x^9} \]
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