Integrand size = 20, antiderivative size = 32 \[ \int \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 b} \]
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Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {623} \[ \int \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 b} \]
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Rule 623
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {x \sqrt {(a+b x)^2} (2 a+b x)}{2 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.59
method | result | size |
default | \(\frac {\operatorname {csgn}\left (b x +a \right ) \left (b x +a \right )^{2}}{2 b}\) | \(19\) |
gosper | \(\frac {x \left (b x +2 a \right ) \sqrt {\left (b x +a \right )^{2}}}{2 b x +2 a}\) | \(27\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, a x}{b x +a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, b \,x^{2}}{2 b x +2 a}\) | \(43\) |
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none
Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.31 \[ \int \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{2} \, b x^{2} + a x \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (19) = 38\).
Time = 0.49 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.75 \[ \int \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\begin {cases} \left (\frac {a}{2 b} + \frac {x}{2}\right ) \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} & \text {for}\: b^{2} \neq 0 \\\frac {\left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3 a b} & \text {for}\: a b \neq 0 \\x \sqrt {a^{2}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).
Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44 \[ \int \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} x + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a}{2 \, b} \]
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none
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \mathrm {sgn}\left (b x + a\right ) + \frac {a^{2} \mathrm {sgn}\left (b x + a\right )}{2 \, b} \]
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Time = 9.94 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.59 \[ \int \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )}{2\,b} \]
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