Integrand size = 20, antiderivative size = 32 \[ \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b} \]
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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} \[ \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b} \]
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Rule 623
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {(a+b x) \left ((a+b x)^2\right )^{3/2}}{4 b} \]
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Time = 2.66 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62
method | result | size |
default | \(\frac {\left (b x +a \right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{4 b}\) | \(20\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (b x +a \right )^{3}}{4 b}\) | \(22\) |
gosper | \(\frac {x \left (b^{3} x^{3}+4 a \,b^{2} x^{2}+6 a^{2} b x +4 a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{4 \left (b x +a \right )^{3}}\) | \(49\) |
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Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{4} \, b^{3} x^{4} + a b^{2} x^{3} + \frac {3}{2} \, a^{2} b x^{2} + a^{3} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (29) = 58\).
Time = 0.78 (sec) , antiderivative size = 294, normalized size of antiderivative = 9.19 \[ \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=a^{2} \left (\begin {cases} \left (\frac {a}{2 b} + \frac {x}{2}\right ) \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} & \text {for}\: b^{2} \neq 0 \\\frac {\left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3 a b} & \text {for}\: a b \neq 0 \\x \sqrt {a^{2}} & \text {otherwise} \end {cases}\right ) + 2 a b \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {a^{2}}{6 b^{2}} + \frac {a x}{6 b} + \frac {x^{2}}{3}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- \frac {a^{2} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5}}{2 a^{2} b^{2}} & \text {for}\: a b \neq 0 \\\frac {x^{2} \sqrt {a^{2}}}{2} & \text {otherwise} \end {cases}\right ) + b^{2} \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (\frac {a^{3}}{12 b^{3}} - \frac {a^{2} x}{12 b^{2}} + \frac {a x^{2}}{12 b} + \frac {x^{3}}{4}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {\frac {a^{4} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} - \frac {2 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7}}{4 a^{3} b^{3}} & \text {for}\: a b \neq 0 \\\frac {x^{3} \sqrt {a^{2}}}{3} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44 \[ \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a}{4 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (28) = 56\).
Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.78 \[ \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} a^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {a^{4} \mathrm {sgn}\left (b x + a\right )}{4 \, b} + \frac {1}{4} \, {\left (b x^{2} + 2 \, a x\right )}^{2} b \mathrm {sgn}\left (b x + a\right ) \]
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Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {\left (x\,b^2+a\,b\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b^2} \]
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