Integrand size = 24, antiderivative size = 167 \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {a^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {3 a^2 b x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {3 a b^2 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {b^3 x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )} \]
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Time = 0.02 (sec) , antiderivative size = 167, 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 = {1369, 276} \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {3 a b^2 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {3 a^2 b x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {b^3 x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {a^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]
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Rule 276
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int x \left (a b+b^2 x^3\right )^3 \, dx}{b^2 \left (a b+b^2 x^3\right )} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (a^3 b^3 x+3 a^2 b^4 x^4+3 a b^5 x^7+b^6 x^{10}\right ) \, dx}{b^2 \left (a b+b^2 x^3\right )} \\ & = \frac {a^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {3 a^2 b x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {3 a b^2 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {b^3 x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.37 \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {x^2 \sqrt {\left (a+b x^3\right )^2} \left (220 a^3+264 a^2 b x^3+165 a b^2 x^6+40 b^3 x^9\right )}{440 \left (a+b x^3\right )} \]
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Time = 1.86 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.35
method | result | size |
gosper | \(\frac {x^{2} \left (40 b^{3} x^{9}+165 b^{2} x^{6} a +264 a^{2} b \,x^{3}+220 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{440 \left (b \,x^{3}+a \right )^{3}}\) | \(58\) |
default | \(\frac {x^{2} \left (40 b^{3} x^{9}+165 b^{2} x^{6} a +264 a^{2} b \,x^{3}+220 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{440 \left (b \,x^{3}+a \right )^{3}}\) | \(58\) |
risch | \(\frac {a^{3} x^{2} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 b \,x^{3}+2 a}+\frac {3 a^{2} b \,x^{5} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{5 \left (b \,x^{3}+a \right )}+\frac {3 a \,b^{2} x^{8} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{8 \left (b \,x^{3}+a \right )}+\frac {b^{3} x^{11} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{11 b \,x^{3}+11 a}\) | \(116\) |
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.21 \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {1}{11} \, b^{3} x^{11} + \frac {3}{8} \, a b^{2} x^{8} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{2} \, a^{3} x^{2} \]
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\[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\int x \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.21 \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {1}{11} \, b^{3} x^{11} + \frac {3}{8} \, a b^{2} x^{8} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{2} \, a^{3} x^{2} \]
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Time = 0.31 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.40 \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {1}{11} \, b^{3} x^{11} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{8} \, a b^{2} x^{8} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{5} \, a^{2} b x^{5} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{2} \, a^{3} x^{2} \mathrm {sgn}\left (b x^{3} + a\right ) \]
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Timed out. \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\int x\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2} \,d x \]
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