Integrand size = 13, antiderivative size = 30 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{x^4} \, dx=-\frac {b^2}{5 x^5}-\frac {a b}{2 x^4}-\frac {a^2}{3 x^3} \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 45} \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{x^4} \, dx=-\frac {a^2}{3 x^3}-\frac {a b}{2 x^4}-\frac {b^2}{5 x^5} \]
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Rule 45
Rule 269
Rubi steps \begin{align*} \text {integral}& = \int \frac {(b+a x)^2}{x^6} \, dx \\ & = \int \left (\frac {b^2}{x^6}+\frac {2 a b}{x^5}+\frac {a^2}{x^4}\right ) \, dx \\ & = -\frac {b^2}{5 x^5}-\frac {a b}{2 x^4}-\frac {a^2}{3 x^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{x^4} \, dx=-\frac {b^2}{5 x^5}-\frac {a b}{2 x^4}-\frac {a^2}{3 x^3} \]
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Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80
method | result | size |
norman | \(\frac {-\frac {1}{3} a^{2} x^{2}-\frac {1}{2} a b x -\frac {1}{5} b^{2}}{x^{5}}\) | \(24\) |
risch | \(\frac {-\frac {1}{3} a^{2} x^{2}-\frac {1}{2} a b x -\frac {1}{5} b^{2}}{x^{5}}\) | \(24\) |
gosper | \(-\frac {10 a^{2} x^{2}+15 a b x +6 b^{2}}{30 x^{5}}\) | \(25\) |
default | \(-\frac {b^{2}}{5 x^{5}}-\frac {a b}{2 x^{4}}-\frac {a^{2}}{3 x^{3}}\) | \(25\) |
parallelrisch | \(\frac {-10 a^{2} x^{2}-15 a b x -6 b^{2}}{30 x^{5}}\) | \(25\) |
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Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{x^4} \, dx=-\frac {10 \, a^{2} x^{2} + 15 \, a b x + 6 \, b^{2}}{30 \, x^{5}} \]
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Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{x^4} \, dx=\frac {- 10 a^{2} x^{2} - 15 a b x - 6 b^{2}}{30 x^{5}} \]
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Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{x^4} \, dx=-\frac {10 \, a^{2} x^{2} + 15 \, a b x + 6 \, b^{2}}{30 \, x^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{x^4} \, dx=-\frac {10 \, a^{2} x^{2} + 15 \, a b x + 6 \, b^{2}}{30 \, x^{5}} \]
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Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{x^4} \, dx=-\frac {\frac {a^2\,x^2}{3}+\frac {a\,b\,x}{2}+\frac {b^2}{5}}{x^5} \]
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