Optimal. Leaf size=94 \[ \frac {4}{11} \sqrt {x} \left (x^{3/2}+x\right )^{3/2}+\frac {64 \left (x^{3/2}+x\right )^{3/2}}{231 \sqrt {x}}-\frac {256 \left (x^{3/2}+x\right )^{3/2}}{1155 x}+\frac {512 \left (x^{3/2}+x\right )^{3/2}}{3465 x^{3/2}}-\frac {32}{99} \left (x^{3/2}+x\right )^{3/2} \]
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Rubi [A] time = 0.09, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2016, 2002, 2014} \[ \frac {4}{11} \sqrt {x} \left (x^{3/2}+x\right )^{3/2}+\frac {64 \left (x^{3/2}+x\right )^{3/2}}{231 \sqrt {x}}-\frac {256 \left (x^{3/2}+x\right )^{3/2}}{1155 x}+\frac {512 \left (x^{3/2}+x\right )^{3/2}}{3465 x^{3/2}}-\frac {32}{99} \left (x^{3/2}+x\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int x \sqrt {x+x^{3/2}} \, dx &=\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*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.54 \[ \frac {4 \left (\sqrt {x}+1\right ) \sqrt {x^{3/2}+x} \left (-280 x^{3/2}+315 x^2+240 x-192 \sqrt {x}+128\right )}{3465 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 40, normalized size = 0.43 \[ \frac {4 \, {\left (315 \, x^{3} - 40 \, x^{2} + {\left (35 \, x^{2} + 48 \, x + 128\right )} \sqrt {x} - 64 \, x\right )} \sqrt {x^{\frac {3}{2}} + x}}{3465 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 51, normalized size = 0.54 \[ \frac {4}{3465} \, {\left (315 \, {\left (\sqrt {x} + 1\right )}^{\frac {11}{2}} - 1540 \, {\left (\sqrt {x} + 1\right )}^{\frac {9}{2}} + 2970 \, {\left (\sqrt {x} + 1\right )}^{\frac {7}{2}} - 2772 \, {\left (\sqrt {x} + 1\right )}^{\frac {5}{2}} + 1155 \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} - 128\right )} \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 38, normalized size = 0.40 \[ \frac {4 \sqrt {x^{\frac {3}{2}}+x}\, \left (\sqrt {x}+1\right ) \left (315 x^{2}-280 x^{\frac {3}{2}}+240 x -192 \sqrt {x}+128\right )}{3465 \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x^{\frac {3}{2}} + x} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\sqrt {x+x^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {x^{\frac {3}{2}} + x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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