Integrand size = 11, antiderivative size = 43 \[ \int \frac {(a+b x)^3}{x^7} \, dx=-\frac {a^3}{6 x^6}-\frac {3 a^2 b}{5 x^5}-\frac {3 a b^2}{4 x^4}-\frac {b^3}{3 x^3} \]
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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} \[ \int \frac {(a+b x)^3}{x^7} \, dx=-\frac {a^3}{6 x^6}-\frac {3 a^2 b}{5 x^5}-\frac {3 a b^2}{4 x^4}-\frac {b^3}{3 x^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3}{x^7}+\frac {3 a^2 b}{x^6}+\frac {3 a b^2}{x^5}+\frac {b^3}{x^4}\right ) \, dx \\ & = -\frac {a^3}{6 x^6}-\frac {3 a^2 b}{5 x^5}-\frac {3 a b^2}{4 x^4}-\frac {b^3}{3 x^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^3}{x^7} \, dx=-\frac {a^3}{6 x^6}-\frac {3 a^2 b}{5 x^5}-\frac {3 a b^2}{4 x^4}-\frac {b^3}{3 x^3} \]
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Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81
method | result | size |
norman | \(\frac {-\frac {1}{3} b^{3} x^{3}-\frac {3}{4} a \,b^{2} x^{2}-\frac {3}{5} a^{2} b x -\frac {1}{6} a^{3}}{x^{6}}\) | \(35\) |
risch | \(\frac {-\frac {1}{3} b^{3} x^{3}-\frac {3}{4} a \,b^{2} x^{2}-\frac {3}{5} a^{2} b x -\frac {1}{6} a^{3}}{x^{6}}\) | \(35\) |
gosper | \(-\frac {20 b^{3} x^{3}+45 a \,b^{2} x^{2}+36 a^{2} b x +10 a^{3}}{60 x^{6}}\) | \(36\) |
default | \(-\frac {a^{3}}{6 x^{6}}-\frac {3 a^{2} b}{5 x^{5}}-\frac {3 a \,b^{2}}{4 x^{4}}-\frac {b^{3}}{3 x^{3}}\) | \(36\) |
parallelrisch | \(\frac {-20 b^{3} x^{3}-45 a \,b^{2} x^{2}-36 a^{2} b x -10 a^{3}}{60 x^{6}}\) | \(36\) |
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Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^3}{x^7} \, dx=-\frac {20 \, b^{3} x^{3} + 45 \, a b^{2} x^{2} + 36 \, a^{2} b x + 10 \, a^{3}}{60 \, x^{6}} \]
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Time = 0.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^3}{x^7} \, dx=\frac {- 10 a^{3} - 36 a^{2} b x - 45 a b^{2} x^{2} - 20 b^{3} x^{3}}{60 x^{6}} \]
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Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^3}{x^7} \, dx=-\frac {20 \, b^{3} x^{3} + 45 \, a b^{2} x^{2} + 36 \, a^{2} b x + 10 \, a^{3}}{60 \, x^{6}} \]
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Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^3}{x^7} \, dx=-\frac {20 \, b^{3} x^{3} + 45 \, a b^{2} x^{2} + 36 \, a^{2} b x + 10 \, a^{3}}{60 \, x^{6}} \]
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Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^3}{x^7} \, dx=-\frac {\frac {a^3}{6}+\frac {3\,a^2\,b\,x}{5}+\frac {3\,a\,b^2\,x^2}{4}+\frac {b^3\,x^3}{3}}{x^6} \]
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