Optimal. Leaf size=73 \[ \frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{3}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\frac{3}{4} \cosh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0324438, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {323, 330, 52} \[ \frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{3}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\frac{3}{4} \cosh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 323
Rule 330
Rule 52
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx &=\frac{1}{2} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{3}{4} \int \frac{\sqrt{x}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=\frac{3}{4} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}+\frac{1}{2} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{3}{8} \int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}} \, dx\\ &=\frac{3}{4} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}+\frac{1}{2} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,\sqrt{x}\right )\\ &=\frac{3}{4} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}+\frac{1}{2} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{3}{4} \cosh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0254039, size = 62, normalized size = 0.85 \[ \frac{1}{4} \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x} (2 x+3)+6 \tanh ^{-1}\left (\sqrt{\frac{\sqrt{x}-1}{\sqrt{x}+1}}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.01, size = 55, normalized size = 0.8 \begin{align*}{\frac{1}{4}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( 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 = 0.924397, size = 50, normalized size = 0.68 \begin{align*} \frac{1}{2} \, \sqrt{x - 1} x^{\frac{3}{2}} + \frac{3}{4} \, \sqrt{x - 1} \sqrt{x} + \frac{3}{4} \, \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 = 0.950867, size = 170, normalized size = 2.33 \begin{align*} \frac{1}{4} \,{\left (2 \, x + 3\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - \frac{3}{8} \, \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 \frac{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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