Integrand size = 13, antiderivative size = 94 \[ \int x \sqrt {x+x^{3/2}} \, dx=-\frac {32}{99} \left (x+x^{3/2}\right )^{3/2}+\frac {512 \left (x+x^{3/2}\right )^{3/2}}{3465 x^{3/2}}-\frac {256 \left (x+x^{3/2}\right )^{3/2}}{1155 x}+\frac {64 \left (x+x^{3/2}\right )^{3/2}}{231 \sqrt {x}}+\frac {4}{11} \sqrt {x} \left (x+x^{3/2}\right )^{3/2} \]
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Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2041, 2027, 2039} \[ \int x \sqrt {x+x^{3/2}} \, dx=\frac {4}{11} \sqrt {x} \left (x^{3/2}+x\right )^{3/2}+\frac {64 \left (x^{3/2}+x\right )^{3/2}}{231 \sqrt {x}}-\frac {256 \left (x^{3/2}+x\right )^{3/2}}{1155 x}+\frac {512 \left (x^{3/2}+x\right )^{3/2}}{3465 x^{3/2}}-\frac {32}{99} \left (x^{3/2}+x\right )^{3/2} \]
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Rule 2027
Rule 2039
Rule 2041
Rubi steps \begin{align*} \text {integral}& = \frac {4}{11} \sqrt {x} \left (x+x^{3/2}\right )^{3/2}-\frac {8}{11} \int \sqrt {x} \sqrt {x+x^{3/2}} \, dx \\ & = -\frac {32}{99} \left (x+x^{3/2}\right )^{3/2}+\frac {4}{11} \sqrt {x} \left (x+x^{3/2}\right )^{3/2}+\frac {16}{33} \int \sqrt {x+x^{3/2}} \, dx \\ & = -\frac {32}{99} \left (x+x^{3/2}\right )^{3/2}+\frac {64 \left (x+x^{3/2}\right )^{3/2}}{231 \sqrt {x}}+\frac {4}{11} \sqrt {x} \left (x+x^{3/2}\right )^{3/2}-\frac {64}{231} \int \frac {\sqrt {x+x^{3/2}}}{\sqrt {x}} \, dx \\ & = -\frac {32}{99} \left (x+x^{3/2}\right )^{3/2}-\frac {256 \left (x+x^{3/2}\right )^{3/2}}{1155 x}+\frac {64 \left (x+x^{3/2}\right )^{3/2}}{231 \sqrt {x}}+\frac {4}{11} \sqrt {x} \left (x+x^{3/2}\right )^{3/2}+\frac {128 \int \frac {\sqrt {x+x^{3/2}}}{x} \, dx}{1155} \\ & = -\frac {32}{99} \left (x+x^{3/2}\right )^{3/2}+\frac {512 \left (x+x^{3/2}\right )^{3/2}}{3465 x^{3/2}}-\frac {256 \left (x+x^{3/2}\right )^{3/2}}{1155 x}+\frac {64 \left (x+x^{3/2}\right )^{3/2}}{231 \sqrt {x}}+\frac {4}{11} \sqrt {x} \left (x+x^{3/2}\right )^{3/2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.54 \[ \int x \sqrt {x+x^{3/2}} \, dx=\frac {4 \sqrt {x+x^{3/2}} \left (128-64 \sqrt {x}+48 x-40 x^{3/2}+35 x^2+315 x^{5/2}\right )}{3465 \sqrt {x}} \]
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Time = 1.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.40
method | result | size |
derivativedivides | \(\frac {4 \sqrt {x +x^{\frac {3}{2}}}\, \left (1+\sqrt {x}\right ) \left (315 x^{2}-280 x^{\frac {3}{2}}+240 x -192 \sqrt {x}+128\right )}{3465 \sqrt {x}}\) | \(38\) |
default | \(\frac {4 \sqrt {x +x^{\frac {3}{2}}}\, \left (1+\sqrt {x}\right ) \left (315 x^{2}-280 x^{\frac {3}{2}}+240 x -192 \sqrt {x}+128\right )}{3465 \sqrt {x}}\) | \(38\) |
meijerg | \(-\frac {\frac {512 \sqrt {\pi }}{3465}-\frac {4 \sqrt {\pi }\, \left (1+\sqrt {x}\right )^{\frac {3}{2}} \left (315 x^{2}-280 x^{\frac {3}{2}}+240 x -192 \sqrt {x}+128\right )}{3465}}{\sqrt {\pi }}\) | \(44\) |
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Time = 0.37 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43 \[ \int x \sqrt {x+x^{3/2}} \, dx=\frac {4 \, {\left (315 \, x^{3} - 40 \, x^{2} + {\left (35 \, x^{2} + 48 \, x + 128\right )} \sqrt {x} - 64 \, x\right )} \sqrt {x^{\frac {3}{2}} + x}}{3465 \, x} \]
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\[ \int x \sqrt {x+x^{3/2}} \, dx=\int x \sqrt {x^{\frac {3}{2}} + x}\, dx \]
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\[ \int x \sqrt {x+x^{3/2}} \, dx=\int { \sqrt {x^{\frac {3}{2}} + x} x \,d x } \]
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none
Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.54 \[ \int x \sqrt {x+x^{3/2}} \, dx=\frac {4}{3465} \, {\left (315 \, {\left (\sqrt {x} + 1\right )}^{\frac {11}{2}} - 1540 \, {\left (\sqrt {x} + 1\right )}^{\frac {9}{2}} + 2970 \, {\left (\sqrt {x} + 1\right )}^{\frac {7}{2}} - 2772 \, {\left (\sqrt {x} + 1\right )}^{\frac {5}{2}} + 1155 \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} - 128\right )} \mathrm {sgn}\left (x\right ) \]
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Timed out. \[ \int x \sqrt {x+x^{3/2}} \, dx=\int x\,\sqrt {x+x^{3/2}} \,d x \]
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