Optimal. Leaf size=29 \[ \frac {2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{\sqrt {x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {271}
\begin {gather*} \frac {2 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 271
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}} \, dx &=\frac {2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{\sqrt {x}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(146\) vs. \(2(29)=58\).
time = 0.88, size = 146, normalized size = 5.03 \begin {gather*} \frac {\left (-1+\sqrt {-1+\sqrt {x}}\right ) \left (\sqrt {3}-\sqrt {1+\sqrt {x}}\right ) \left (-2+\sqrt {-1+\sqrt {x}}+\sqrt {3} \sqrt {1+\sqrt {x}}-\sqrt {x}\right )}{\left (3+2 \sqrt {-1+\sqrt {x}}-2 \sqrt {3} \sqrt {1+\sqrt {x}}-\sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}+2 \sqrt {x}\right ) \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.33, size = 20, normalized size = 0.69
method | result | size |
derivativedivides | \(\frac {2 \sqrt {-1+\sqrt {x}}\, \sqrt {\sqrt {x}+1}}{\sqrt {x}}\) | \(20\) |
default | \(\frac {2 \sqrt {-1+\sqrt {x}}\, \sqrt {\sqrt {x}+1}}{\sqrt {x}}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 10, normalized size = 0.34 \begin {gather*} \frac {2 \, \sqrt {x - 1}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.10, size = 25, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (\sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + x\right )}}{x} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{\frac {3}{2}} \sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.74, size = 25, normalized size = 0.86 \begin {gather*} \frac {16}{{\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.56, size = 19, normalized size = 0.66 \begin {gather*} \frac {2\,\sqrt {\sqrt {x}-1}\,\sqrt {\sqrt {x}+1}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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