Optimal. Leaf size=30 \[ 2 \sqrt {-1+x}+\frac {4}{3} (-1+x)^{3/2}-\frac {4 x^{3/2}}{3} \]
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Rubi [A]
time = 0.05, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6821, 45}
\begin {gather*} -\frac {4 x^{3/2}}{3}+\frac {4}{3} (x-1)^{3/2}+2 \sqrt {x-1} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6821
Rubi steps
\begin {align*} \int \frac {1}{\left (\sqrt {-1+x}+\sqrt {x}\right )^2 \sqrt {-1+x}} \, dx &=\int \left (-\frac {1}{\sqrt {-1+x}}-2 \sqrt {x}+\frac {2 x}{\sqrt {-1+x}}\right ) \, dx\\ &=-2 \sqrt {-1+x}-\frac {4 x^{3/2}}{3}+2 \int \frac {x}{\sqrt {-1+x}} \, dx\\ &=-2 \sqrt {-1+x}-\frac {4 x^{3/2}}{3}+2 \int \left (\frac {1}{\sqrt {-1+x}}+\sqrt {-1+x}\right ) \, dx\\ &=2 \sqrt {-1+x}+\frac {4}{3} (-1+x)^{3/2}-\frac {4 x^{3/2}}{3}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 26, normalized size = 0.87 \begin {gather*} -\frac {4 x^{3/2}}{3}+\frac {2}{3} \sqrt {-1+x} (1+2 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 21, normalized size = 0.70
method | result | size |
default | \(\frac {4 \left (-1+x \right )^{\frac {3}{2}}}{3}-\frac {4 x^{\frac {3}{2}}}{3}+2 \sqrt {-1+x}\) | \(21\) |
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 = 0.81, size = 18, normalized size = 0.60 \begin {gather*} \frac {2}{3} \, {\left (2 \, x + 1\right )} \sqrt {x - 1} - \frac {4}{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. 53 vs.
\(2 (26) = 52\).
time = 0.27, size = 53, normalized size = 1.77 \begin {gather*} - \frac {4 \sqrt {x}}{6 \sqrt {x} \sqrt {x - 1} + 6 x - 3} - \frac {2 \sqrt {x - 1}}{6 \sqrt {x} \sqrt {x - 1} + 6 x - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 20, normalized size = 0.67 \begin {gather*} \frac {4}{3} \, {\left (x - 1\right )}^{\frac {3}{2}} - \frac {4}{3} \, x^{\frac {3}{2}} + 2 \, \sqrt {x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 21, normalized size = 0.70 \begin {gather*} \frac {4\,x\,\sqrt {x-1}}{3}+\frac {2\,\sqrt {x-1}}{3}-\frac {4\,x^{3/2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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