Optimal. Leaf size=42 \[ -\frac {1}{15} \left (-x^4-2 x^3-x^2+1\right )^{3/2} \left (3 x^4+6 x^3+3 x^2+2\right ) \]
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Rubi [A] time = 0.24, antiderivative size = 59, normalized size of antiderivative = 1.40, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1593, 1680, 12, 1247, 692, 629} \[ -\frac {1}{5} x^2 \left (-x^4-2 x^3-x^2+1\right )^{3/2} (x+1)^2-\frac {2}{15} \left (-x^4-2 x^3-x^2+1\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 629
Rule 692
Rule 1247
Rule 1593
Rule 1680
Rubi steps
\begin {align*} \int (1+2 x) \left (x+x^2\right )^3 \sqrt {1-\left (x+x^2\right )^2} \, dx &=\int x^3 (1+x)^3 (1+2 x) \sqrt {1-\left (x+x^2\right )^2} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1}{128} x \left (-1+4 x^2\right )^3 \sqrt {15+8 x^2-16 x^4} \, dx,x,\frac {1}{2}+x\right )\\ &=\frac {1}{128} \operatorname {Subst}\left (\int x \left (-1+4 x^2\right )^3 \sqrt {15+8 x^2-16 x^4} \, dx,x,\frac {1}{2}+x\right )\\ &=\frac {1}{256} \operatorname {Subst}\left (\int (-1+4 x)^3 \sqrt {15+8 x-16 x^2} \, dx,x,\left (\frac {1}{2}+x\right )^2\right )\\ &=-\frac {1}{5} x^2 (1+x)^2 \left (1-x^2-2 x^3-x^4\right )^{3/2}+\frac {1}{40} \operatorname {Subst}\left (\int (-1+4 x) \sqrt {15+8 x-16 x^2} \, dx,x,\left (\frac {1}{2}+x\right )^2\right )\\ &=-\frac {2}{15} \left (1-x^2-2 x^3-x^4\right )^{3/2}-\frac {1}{5} x^2 (1+x)^2 \left (1-x^2-2 x^3-x^4\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 62, normalized size = 1.48 \[ \frac {1}{15} \sqrt {-x^4-2 x^3-x^2+1} \left (3 x^8+12 x^7+18 x^6+12 x^5+2 x^4-2 x^3-x^2-2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 58, normalized size = 1.38 \[ \frac {1}{15} \, {\left (3 \, x^{8} + 12 \, x^{7} + 18 \, x^{6} + 12 \, x^{5} + 2 \, x^{4} - 2 \, x^{3} - x^{2} - 2\right )} \sqrt {-x^{4} - 2 \, x^{3} - x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 58, normalized size = 1.38 \[ \frac {1}{5} \, {\left (x^{4} + 2 \, x^{3} + x^{2} - 1\right )}^{2} \sqrt {-x^{4} - 2 \, x^{3} - x^{2} + 1} - \frac {1}{3} \, {\left (-x^{4} - 2 \, x^{3} - x^{2} + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 51, normalized size = 1.21 \[ \frac {\left (x^{2}+x +1\right ) \left (x^{2}+x -1\right ) \left (3 x^{4}+6 x^{3}+3 x^{2}+2\right ) \sqrt {-x^{4}-2 x^{3}-x^{2}+1}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 59, normalized size = 1.40 \[ \frac {1}{15} \, {\left (3 \, x^{8} + 12 \, x^{7} + 18 \, x^{6} + 12 \, x^{5} + 2 \, x^{4} - 2 \, x^{3} - x^{2} - 2\right )} \sqrt {x^{2} + x + 1} \sqrt {-x^{2} - x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.42, size = 51, normalized size = 1.21 \[ \sqrt {1-{\left (x^2+x\right )}^2}\,\left (\frac {x^8}{5}+\frac {4\,x^7}{5}+\frac {6\,x^6}{5}+\frac {4\,x^5}{5}+\frac {2\,x^4}{15}-\frac {2\,x^3}{15}-\frac {x^2}{15}-\frac {2}{15}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 10.18, size = 182, normalized size = 4.33 \[ \frac {x^{8} \sqrt {- x^{4} - 2 x^{3} - x^{2} + 1}}{5} + \frac {4 x^{7} \sqrt {- x^{4} - 2 x^{3} - x^{2} + 1}}{5} + \frac {6 x^{6} \sqrt {- x^{4} - 2 x^{3} - x^{2} + 1}}{5} + \frac {4 x^{5} \sqrt {- x^{4} - 2 x^{3} - x^{2} + 1}}{5} + \frac {2 x^{4} \sqrt {- x^{4} - 2 x^{3} - x^{2} + 1}}{15} - \frac {2 x^{3} \sqrt {- x^{4} - 2 x^{3} - x^{2} + 1}}{15} - \frac {x^{2} \sqrt {- x^{4} - 2 x^{3} - x^{2} + 1}}{15} - \frac {2 \sqrt {- x^{4} - 2 x^{3} - x^{2} + 1}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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