Optimal. Leaf size=35 \[ -\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}+\frac {4}{5} \left (1-\sqrt {x}\right )^{5/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {196, 45}
\begin {gather*} \frac {4}{5} \left (1-\sqrt {x}\right )^{5/2}-\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 196
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=2 \text {Subst}\left (\int \sqrt {1-x} x \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (\sqrt {1-x}-(1-x)^{3/2}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}+\frac {4}{5} \left (1-\sqrt {x}\right )^{5/2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.74 \begin {gather*} -\frac {4}{15} \left (1-\sqrt {x}\right )^{3/2} \left (2+3 \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 24, normalized size = 0.69
method | result | size |
derivativedivides | \(-\frac {4 \left (1-\sqrt {x}\right )^{\frac {3}{2}}}{3}+\frac {4 \left (1-\sqrt {x}\right )^{\frac {5}{2}}}{5}\) | \(24\) |
default | \(-\frac {4 \left (1-\sqrt {x}\right )^{\frac {3}{2}}}{3}+\frac {4 \left (1-\sqrt {x}\right )^{\frac {5}{2}}}{5}\) | \(24\) |
meijerg | \(-\frac {-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-\sqrt {x}\right )^{\frac {3}{2}} \left (3 \sqrt {x}+2\right )}{15}}{\sqrt {\pi }}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 23, normalized size = 0.66 \begin {gather*} \frac {4}{5} \, {\left (-\sqrt {x} + 1\right )}^{\frac {5}{2}} - \frac {4}{3} \, {\left (-\sqrt {x} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.58, size = 21, normalized size = 0.60 \begin {gather*} \frac {4}{15} \, {\left (3 \, x - \sqrt {x} - 2\right )} \sqrt {-\sqrt {x} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.60, size = 306, normalized size = 8.74 \begin {gather*} \begin {cases} - \frac {12 i x^{\frac {7}{2}} \sqrt {\sqrt {x} - 1}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} + \frac {4 i x^{\frac {5}{2}} \sqrt {\sqrt {x} - 1}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} - \frac {8 x^{\frac {5}{2}}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} + \frac {16 i x^{3} \sqrt {\sqrt {x} - 1}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} - \frac {8 i x^{2} \sqrt {\sqrt {x} - 1}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} + \frac {8 x^{2}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} & \text {for}\: \left |{\sqrt {x}}\right | > 1 \\- \frac {12 x^{\frac {7}{2}} \sqrt {1 - \sqrt {x}}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} + \frac {4 x^{\frac {5}{2}} \sqrt {1 - \sqrt {x}}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} - \frac {8 x^{\frac {5}{2}}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} + \frac {16 x^{3} \sqrt {1 - \sqrt {x}}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} - \frac {8 x^{2} \sqrt {1 - \sqrt {x}}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} + \frac {8 x^{2}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 30, normalized size = 0.86 \begin {gather*} \frac {4}{5} \, {\left (\sqrt {x} - 1\right )}^{2} \sqrt {-\sqrt {x} + 1} - \frac {4}{3} \, {\left (-\sqrt {x} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 10, normalized size = 0.29 \begin {gather*} x\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},2;\ 3;\ \sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 11, normalized size = 0.31 \begin {gather*} -\frac {4 \left (1-\sqrt {x}\right )^{\frac {5}{2}}}{5} \end {gather*}
Warning: Unable to verify antiderivative.
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