Optimal. Leaf size=63 \[ \frac {2 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{3 x^{3/2}}+\frac {4 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{3 \sqrt {x}} \]
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Rubi [A] time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {272, 265} \[ \frac {2 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{3 x^{3/2}}+\frac {4 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{3 \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 265
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}} \, dx &=\frac {2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{3 x^{3/2}}+\frac {2}{3} \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}}}{3 x^{3/2}}+\frac {4 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{3 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 0.57 \[ \frac {2 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} (2 x+1)}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 34, normalized size = 0.54 \[ \frac {2 \, {\left ({\left (2 \, x + 1\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + 2 \, x^{2}\right )}}{3 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 48, normalized size = 0.76 \[ \frac {128 \, {\left (3 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 4\right )}}{3 \, {\left ({\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 4\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 25, normalized size = 0.40 \[ \frac {2 \sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (2 x +1\right )}{3 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 21, normalized size = 0.33 \[ \frac {4 \, \sqrt {x - 1}}{3 \, \sqrt {x}} + \frac {2 \, \sqrt {x - 1}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.50, size = 33, normalized size = 0.52 \[ \frac {\sqrt {\sqrt {x}-1}\,\left (\frac {4\,x}{3}+\frac {2\,\sqrt {x}}{3}+\frac {4\,x^{3/2}}{3}+\frac {2}{3}\right )}{x^{3/2}\,\sqrt {\sqrt {x}+1}} \]
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 \[ \int \frac {1}{x^{\frac {5}{2}} \sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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