Optimal. Leaf size=44 \[ -\frac {1}{2} \sqrt {1-(x+1)^2} x+\frac {3}{2} \sqrt {1-(x+1)^2}+\frac {3}{2} \sin ^{-1}(x+1) \]
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Rubi [A] time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {371, 671, 641, 216} \[ -\frac {1}{2} \sqrt {1-(x+1)^2} x+\frac {3}{2} \sqrt {1-(x+1)^2}+\frac {3}{2} \sin ^{-1}(x+1) \]
Antiderivative was successfully verified.
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Rule 216
Rule 371
Rule 641
Rule 671
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {1-(1+x)^2}} \, dx &=\operatorname {Subst}\left (\int \frac {(-1+x)^2}{\sqrt {1-x^2}} \, dx,x,1+x\right )\\ &=-\frac {1}{2} x \sqrt {1-(1+x)^2}-\frac {3}{2} \operatorname {Subst}\left (\int \frac {-1+x}{\sqrt {1-x^2}} \, dx,x,1+x\right )\\ &=\frac {3}{2} \sqrt {1-(1+x)^2}-\frac {1}{2} x \sqrt {1-(1+x)^2}+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,1+x\right )\\ &=\frac {3}{2} \sqrt {1-(1+x)^2}-\frac {1}{2} x \sqrt {1-(1+x)^2}+\frac {3}{2} \sin ^{-1}(1+x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 1.16 \[ \frac {x \left (x^2-x-6\right )+6 \sqrt {x} \sqrt {x+2} \sinh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {2}}\right )}{2 \sqrt {-x (x+2)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 35, normalized size = 0.80 \[ -\frac {1}{2} \, \sqrt {-x^{2} - 2 \, x} {\left (x - 3\right )} - 3 \, \arctan \left (\frac {\sqrt {-x^{2} - 2 \, x}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 23, normalized size = 0.52 \[ -\frac {1}{2} \, \sqrt {-{\left (x + 1\right )}^{2} + 1} {\left (x - 3\right )} + \frac {3}{2} \, \arcsin \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 0.80 \[ -\frac {\sqrt {-x^{2}-2 x}\, x}{2}+\frac {3 \arcsin \left (x +1\right )}{2}+\frac {3 \sqrt {-x^{2}-2 x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 36, normalized size = 0.82 \[ -\frac {1}{2} \, \sqrt {-x^{2} - 2 \, x} x + \frac {3}{2} \, \sqrt {-x^{2} - 2 \, x} - \frac {3}{2} \, \arcsin \left (-x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2}{\sqrt {1-{\left (x+1\right )}^2}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {- x \left (x + 2\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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