Optimal. Leaf size=104 \[ \frac{1}{3} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}-\frac{1}{12} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{1}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{1}{8} \cosh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0485153, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {280, 323, 330, 52} \[ \frac{1}{3} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}-\frac{1}{12} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{1}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{1}{8} \cosh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 280
Rule 323
Rule 330
Rule 52
Rubi steps
\begin{align*} \int \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2} \, dx &=\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}-\frac{1}{6} \int \frac{x^{3/2}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=-\frac{1}{12} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}-\frac{1}{8} \int \frac{\sqrt{x}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=-\frac{1}{8} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}-\frac{1}{12} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}-\frac{1}{16} \int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}} \, dx\\ &=-\frac{1}{8} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}-\frac{1}{12} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}-\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{1}{8} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}-\frac{1}{12} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}-\frac{1}{8} \cosh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0371856, size = 99, normalized size = 0.95 \[ \frac{\sqrt{\sqrt{x}+1} \sqrt{x} \left (8 x^{5/2}-8 x^2-2 x^{3/2}+2 x-3 \sqrt{x}+3\right )+6 \sqrt{1-\sqrt{x}} \sin ^{-1}\left (\frac{\sqrt{1-\sqrt{x}}}{\sqrt{2}}\right )}{24 \sqrt{\sqrt{x}-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 65, normalized size = 0.6 \begin{align*} -{\frac{1}{24}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( -8\,{x}^{5/2}\sqrt{-1+x}+2\,{x}^{3/2}\sqrt{-1+x}+3\,\sqrt{x}\sqrt{-1+x}+3\,\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.932527, size = 63, normalized size = 0.61 \begin{align*} \frac{1}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} x^{\frac{3}{2}} + \frac{1}{4} \,{\left (x - 1\right )}^{\frac{3}{2}} \sqrt{x} + \frac{1}{8} \, \sqrt{x - 1} \sqrt{x} - \frac{1}{8} \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.05156, size = 184, normalized size = 1.77 \begin{align*} \frac{1}{24} \,{\left (8 \, x^{2} - 2 \, x - 3\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + \frac{1}{16} \, \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{\frac{3}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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