Optimal. Leaf size=34 \[ \frac {2}{7} (x+1)^{7/2}-\frac {4}{5} (x+1)^{5/2}+\frac {2}{3} (x+1)^{3/2} \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \[ \frac {2}{7} (x+1)^{7/2}-\frac {4}{5} (x+1)^{5/2}+\frac {2}{3} (x+1)^{3/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int x^2 \sqrt {1+x} \, dx &=\int \left (\sqrt {1+x}-2 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx\\ &=\frac {2}{3} (1+x)^{3/2}-\frac {4}{5} (1+x)^{5/2}+\frac {2}{7} (1+x)^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.62 \[ \frac {2}{105} (x+1)^{3/2} \left (15 x^2-12 x+8\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 22, normalized size = 0.65 \[ \frac {2}{105} \, {\left (15 \, x^{3} + 3 \, x^{2} - 4 \, x + 8\right )} \sqrt {x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 22, normalized size = 0.65 \[ \frac {2}{7} \, {\left (x + 1\right )}^{\frac {7}{2}} - \frac {4}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} + \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.53 \[ \frac {2 \left (x +1\right )^{\frac {3}{2}} \left (15 x^{2}-12 x +8\right )}{105} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 22, normalized size = 0.65 \[ \frac {2}{7} \, {\left (x + 1\right )}^{\frac {7}{2}} - \frac {4}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} + \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 19, normalized size = 0.56 \[ -\frac {2\,{\left (x+1\right )}^{3/2}\,\left (42\,x-15\,{\left (x+1\right )}^2+7\right )}{105} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.39, size = 48, normalized size = 1.41 \[ \frac {2 x^{3} \sqrt {x + 1}}{7} + \frac {2 x^{2} \sqrt {x + 1}}{35} - \frac {8 x \sqrt {x + 1}}{105} + \frac {16 \sqrt {x + 1}}{105} \]
Verification of antiderivative is not currently implemented for this CAS.
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