Integrand size = 26, antiderivative size = 27 \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b} \]
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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} \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b} \]
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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 )^{7/2}}{7 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(a+b x)^6 \sqrt {(a+b x)^2}}{7 b} \]
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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 | \(\frac {\operatorname {csgn}\left (b x +a \right ) \left (b x +a \right )^{7}}{7 b}\) | \(19\) |
default | \(\frac {\left (b x +a \right )^{2} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{7 b}\) | \(22\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (b x +a \right )^{6}}{7 b}\) | \(22\) |
gosper | \(\frac {x \left (b^{6} x^{6}+7 a \,b^{5} x^{5}+21 a^{2} b^{4} x^{4}+35 a^{3} b^{3} x^{3}+35 a^{4} b^{2} x^{2}+21 a^{5} b x +7 a^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{7 \left (b x +a \right )^{5}}\) | \(82\) |
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (23) = 46\).
Time = 0.39 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.37 \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{5} + 5 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{3} + 3 \, a^{5} b x^{2} + a^{6} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (22) = 44\).
Time = 0.31 (sec) , antiderivative size = 226, normalized size of antiderivative = 8.37 \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\begin {cases} \frac {a^{6} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7 b} + \frac {6 a^{5} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {15 a^{4} b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {20 a^{3} b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {15 a^{2} b^{3} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {6 a b^{4} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {b^{5} x^{6} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} & \text {for}\: b \neq 0 \\a x \left (a^{2}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{7 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.44 \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{7} \, b^{6} x^{7} \mathrm {sgn}\left (b x + a\right ) + a b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{6} x \mathrm {sgn}\left (b x + a\right ) + \frac {a^{7} \mathrm {sgn}\left (b x + a\right )}{7 \, b} \]
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Time = 10.95 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {{\left ({\left (a+b\,x\right )}^2\right )}^{7/2}}{7\,b} \]
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