Optimal. Leaf size=31 \[ \frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{3 x^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 31, 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 = {265} \begin {gather*} \frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 265
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{5/2}} \, dx &=\frac {2 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{3 x^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 31, normalized size = 1.00 \begin {gather*} \frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 2.38, size = 512, normalized size = 16.52 \begin {gather*} \frac {\left (\frac {\left (\sqrt {\sqrt {x}-1}-1\right )^2}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^2}+1\right ) \left (\frac {9 \left (\sqrt {\sqrt {x}-1}-1\right )^8}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^8}+\frac {42 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^7}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^7}+\frac {256 \left (\sqrt {\sqrt {x}-1}-1\right )^6}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^6}+\frac {294 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^5}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^5}+\frac {638 \left (\sqrt {\sqrt {x}-1}-1\right )^4}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^4}+\frac {294 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^3}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^3}+\frac {256 \left (\sqrt {\sqrt {x}-1}-1\right )^2}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^2}+\frac {42 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )}{\sqrt {3}-\sqrt {\sqrt {x}+1}}+9\right ) \left (\frac {1}{384}-\frac {\sqrt {\sqrt {x}-1}}{384}\right ) \left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{12}}{\left (\sqrt {\sqrt {x}+1}-\sqrt {3}\right ) \left (-2 x-2 \sqrt {\sqrt {x}-1} \sqrt {x}+\sqrt {3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}+2 \sqrt {3} \sqrt {\sqrt {x}+1} \sqrt {x}-3 \sqrt {x}\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 30, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left ({\left (x - 1\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + x^{2}\right )}}{3 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.23, size = 48, normalized size = 1.55 \begin {gather*} \frac {16 \, {\left (3 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{8} + 16\right )}}{3 \, {\left ({\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 4\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 23, normalized size = 0.74 \begin {gather*} \frac {2 \sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (x -1\right )}{3 x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.32, size = 10, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (x - 1\right )}^{\frac {3}{2}}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.26, size = 31, normalized size = 1.00 \begin {gather*} \frac {\sqrt {\sqrt {x}-1}\,\left (\frac {2\,x\,\sqrt {\sqrt {x}+1}}{3}-\frac {2\,\sqrt {\sqrt {x}+1}}{3}\right )}{x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}{x^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________