Integrand size = 24, antiderivative size = 71 \[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {a x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {b x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]
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Time = 0.01 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {660, 45} \[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {b x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {a x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^2 \left (a b+b^2 x\right ) \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b x^2+b^2 x^3\right ) \, dx}{a b+b^2 x} \\ & = \frac {a x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {b x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.14 \[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {x^3 (4 a+3 b x) \left (\sqrt {a^2} b x+a \left (\sqrt {a^2}-\sqrt {(a+b x)^2}\right )\right )}{12 \left (-a^2-a b x+\sqrt {a^2} \sqrt {(a+b x)^2}\right )} \]
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Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.42
method | result | size |
gosper | \(\frac {x^{3} \left (3 b x +4 a \right ) \sqrt {\left (b x +a \right )^{2}}}{12 b x +12 a}\) | \(30\) |
default | \(\frac {\operatorname {csgn}\left (b x +a \right ) \left (b x +a \right )^{2} \left (3 b^{2} x^{2}-2 a b x +a^{2}\right )}{12 b^{3}}\) | \(36\) |
risch | \(\frac {a \,x^{3} \sqrt {\left (b x +a \right )^{2}}}{3 b x +3 a}+\frac {b \,x^{4} \sqrt {\left (b x +a \right )^{2}}}{4 b x +4 a}\) | \(46\) |
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Time = 0.37 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.18 \[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (46) = 92\).
Time = 1.01 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.80 \[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (\frac {a^{3}}{12 b^{3}} - \frac {a^{2} x}{12 b^{2}} + \frac {a x^{2}}{12 b} + \frac {x^{3}}{4}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {\frac {a^{4} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} - \frac {2 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7}}{4 a^{3} b^{3}} & \text {for}\: a b \neq 0 \\\frac {x^{3} \sqrt {a^{2}}}{3} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (45) = 90\).
Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.44 \[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} x}{2 \, b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}}{2 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} x}{4 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a}{12 \, b^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.55 \[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{4} \, b x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {a^{4} \mathrm {sgn}\left (b x + a\right )}{12 \, b^{3}} \]
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Time = 9.18 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{12\,b^3} \]
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