Integrand size = 26, antiderivative size = 27 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {1}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 27, 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} \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {1}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
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Rule 643
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {1}{3 b \left ((a+b x)^2\right )^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(-\frac {\operatorname {csgn}\left (b x +a \right )}{3 b \left (b x +a \right )^{3}}\) | \(19\) |
gosper | \(-\frac {\left (b x +a \right )^{2}}{3 b \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(22\) |
default | \(-\frac {\left (b x +a \right )^{2}}{3 b \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(22\) |
risch | \(-\frac {\sqrt {\left (b x +a \right )^{2}}}{3 \left (b x +a \right )^{4} b}\) | \(22\) |
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none
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {1}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (26) = 52\).
Time = 0.39 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.59 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\begin {cases} - \frac {1}{3 a^{2} b \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} + 6 a b^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} + 3 b^{3} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}} & \text {for}\: b \neq 0 \\\frac {a x}{\left (a^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {1}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b} \]
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none
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {1}{3 \, {\left (b x + a\right )}^{3} b \mathrm {sgn}\left (b x + a\right )} \]
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Time = 10.79 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,b\,{\left (a+b\,x\right )}^4} \]
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