Optimal. Leaf size=38 \[ \frac {x^4}{2}+\frac {2}{5} \left (1-x^2\right )^{5/2}-\frac {2}{3} \left (1-x^2\right )^{3/2} \]
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Rubi [A] time = 0.11, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6742, 266, 43} \[ \frac {x^4}{2}+\frac {2}{5} \left (1-x^2\right )^{5/2}-\frac {2}{3} \left (1-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 6742
Rubi steps
\begin {align*} \int x^3 \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx &=\int \left (2 x^3+2 x^3 \sqrt {1-x^2}\right ) \, dx\\ &=\frac {x^4}{2}+2 \int x^3 \sqrt {1-x^2} \, dx\\ &=\frac {x^4}{2}+\operatorname {Subst}\left (\int \sqrt {1-x} x \, dx,x,x^2\right )\\ &=\frac {x^4}{2}+\operatorname {Subst}\left (\int \left (\sqrt {1-x}-(1-x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=\frac {x^4}{2}-\frac {2}{3} \left (1-x^2\right )^{3/2}+\frac {2}{5} \left (1-x^2\right )^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 38, normalized size = 1.00 \[ \frac {x^4}{2}+\frac {2}{5} \left (1-x^2\right )^{5/2}-\frac {2}{3} \left (1-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 32, normalized size = 0.84 \[ \frac {1}{2} \, x^{4} + \frac {2}{15} \, {\left (3 \, x^{4} - x^{2} - 2\right )} \sqrt {x + 1} \sqrt {-x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 77, normalized size = 2.03 \[ \frac {1}{2} \, x^{4} + \frac {1}{60} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 33, normalized size = 0.87 \[ \frac {x^{4}}{2}+\frac {2 \sqrt {x +1}\, \sqrt {-x +1}\, \left (x^{2}-1\right ) \left (3 x^{2}+2\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 31, normalized size = 0.82 \[ \frac {1}{2} \, x^{4} - \frac {2}{5} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} - \frac {4}{15} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.02, size = 45, normalized size = 1.18 \[ \frac {x^4}{2}-\frac {\sqrt {1-x}\,\left (-\frac {2\,x^5}{5}-\frac {2\,x^4}{5}+\frac {2\,x^3}{15}+\frac {2\,x^2}{15}+\frac {4\,x}{15}+\frac {4}{15}\right )}{\sqrt {x+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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