Optimal. Leaf size=21 \[ \frac {2}{3} (-1+x)^{3/2}+\frac {2 x^{3/2}}{3} \]
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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2131, 30, 32}
\begin {gather*} \frac {2 x^{3/2}}{3}+\frac {2}{3} (x-1)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 32
Rule 2131
Rubi steps
\begin {align*} \int \frac {1}{-\sqrt {-1+x}+\sqrt {x}} \, dx &=\int \sqrt {-1+x} \, dx+\int \sqrt {x} \, dx\\ &=\frac {2}{3} (-1+x)^{3/2}+\frac {2 x^{3/2}}{3}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 17, normalized size = 0.81 \begin {gather*} \frac {2}{3} \left ((-1+x)^{3/2}+x^{3/2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.26, size = 30, normalized size = 1.43 \begin {gather*} \frac {2 \left (-1-\sqrt {x} \sqrt {-1+x}+2 x\right )}{3 \sqrt {x}-3 \sqrt {-1+x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 14, normalized size = 0.67
method | result | size |
default | \(\frac {2 \left (-1+x \right )^{\frac {3}{2}}}{3}+\frac {2 x^{\frac {3}{2}}}{3}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{3} \, {\left (x - 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (17) = 34\)
time = 0.22, size = 63, normalized size = 3.00 \begin {gather*} \frac {2 \sqrt {x} \sqrt {x - 1}}{- 3 \sqrt {x} + 3 \sqrt {x - 1}} - \frac {4 x}{- 3 \sqrt {x} + 3 \sqrt {x - 1}} + \frac {2}{- 3 \sqrt {x} + 3 \sqrt {x - 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 43, normalized size = 2.05 \begin {gather*} 2 \left (\frac {1}{3} \sqrt {x-1} \left (x-1\right )+2 \left (\frac {1}{6} \sqrt {x-1} \sqrt {x-1}+\frac 1{6}\right ) \sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 21, normalized size = 1.00 \begin {gather*} \frac {2\,x\,\sqrt {x-1}}{3}-\frac {2\,\sqrt {x-1}}{3}+\frac {2\,x^{3/2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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