Integrand size = 26, antiderivative size = 27 \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b} \]
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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} \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b} \]
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Rule 643
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {(a+b x)^4 \sqrt {(a+b x)^2}}{5 b} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(\frac {\operatorname {csgn}\left (b x +a \right ) \left (b x +a \right )^{5}}{5 b}\) | \(19\) |
default | \(\frac {\left (b x +a \right )^{2} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{5 b}\) | \(22\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (b x +a \right )^{4}}{5 b}\) | \(22\) |
gosper | \(\frac {x \left (x^{4} b^{4}+5 a \,b^{3} x^{3}+10 x^{2} b^{2} a^{2}+10 b \,a^{3} x +5 a^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{5 \left (b x +a \right )^{3}}\) | \(60\) |
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none
Time = 0.31 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{5} \, b^{4} x^{5} + a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{3} + 2 \, a^{3} b x^{2} + a^{4} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (22) = 44\).
Time = 0.16 (sec) , antiderivative size = 158, normalized size of antiderivative = 5.85 \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\begin {cases} \frac {a^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{5 b} + \frac {4 a^{3} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{5} + \frac {6 a^{2} b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{5} + \frac {4 a b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{5} + \frac {b^{3} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{5} & \text {for}\: b \neq 0 \\a x \left (a^{2}\right )^{\frac {3}{2}} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}{5 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.19 \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{5} \, b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + a b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{4} x \mathrm {sgn}\left (b x + a\right ) + \frac {a^{5} \mathrm {sgn}\left (b x + a\right )}{5 \, b} \]
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Time = 10.75 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {{\left (a+b\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b} \]
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