Integrand size = 26, antiderivative size = 25 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {1}{b \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {643} \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {1}{b \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 643
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{b \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.64 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {1}{b \sqrt {(a+b x)^2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(-\frac {\operatorname {csgn}\left (b x +a \right )}{b \left (b x +a \right )}\) | \(19\) |
gosper | \(-\frac {\left (b x +a \right )^{2}}{b \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(22\) |
default | \(-\frac {\left (b x +a \right )^{2}}{b \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(22\) |
risch | \(-\frac {\sqrt {\left (b x +a \right )^{2}}}{\left (b x +a \right )^{2} b}\) | \(22\) |
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none
Time = 0.31 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.52 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {1}{b^{2} x + a b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (14) = 28\).
Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\begin {cases} - \frac {1}{b \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}} & \text {for}\: b \neq 0 \\\frac {a x}{\left (a^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {1}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b} \]
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none
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {1}{{\left (b x + a\right )} b \mathrm {sgn}\left (b x + a\right )} \]
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Time = 10.85 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {1}{b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}} \]
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