Optimal. Leaf size=135 \[ \frac{1}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{7/2}-\frac{1}{24} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}-\frac{5}{96} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{5}{64} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{5}{64} \cosh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0607087, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {280, 323, 330, 52} \[ \frac{1}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{7/2}-\frac{1}{24} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}-\frac{5}{96} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{5}{64} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{5}{64} \cosh ^{-1}\left (\sqrt{x}\right ) \]
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^{5/2} \, dx &=\frac{1}{4} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{7/2}-\frac{1}{8} \int \frac{x^{5/2}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=-\frac{1}{24} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{1}{4} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{7/2}-\frac{5}{48} \int \frac{x^{3/2}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=-\frac{5}{96} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}-\frac{1}{24} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{1}{4} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{7/2}-\frac{5}{64} \int \frac{\sqrt{x}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=-\frac{5}{64} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}-\frac{5}{96} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}-\frac{1}{24} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{1}{4} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{7/2}-\frac{5}{128} \int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}} \, dx\\ &=-\frac{5}{64} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}-\frac{5}{96} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}-\frac{1}{24} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{1}{4} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{7/2}-\frac{5}{64} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{5}{64} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}-\frac{5}{96} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}-\frac{1}{24} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{1}{4} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{7/2}-\frac{5}{64} \cosh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.057695, size = 111, normalized size = 0.82 \[ \frac{\sqrt{\sqrt{x}+1} \sqrt{x} \left (48 x^{7/2}-48 x^3-8 x^{5/2}+8 x^2-10 x^{3/2}+10 x-15 \sqrt{x}+15\right )+30 \sqrt{1-\sqrt{x}} \sin ^{-1}\left (\frac{\sqrt{1-\sqrt{x}}}{\sqrt{2}}\right )}{192 \sqrt{\sqrt{x}-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 75, normalized size = 0.6 \begin{align*} -{\frac{1}{192}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( -48\,{x}^{7/2}\sqrt{-1+x}+8\,{x}^{5/2}\sqrt{-1+x}+10\,{x}^{3/2}\sqrt{-1+x}+15\,\sqrt{x}\sqrt{-1+x}+15\,\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 = 0.960385, size = 77, normalized size = 0.57 \begin{align*} \frac{1}{4} \,{\left (x - 1\right )}^{\frac{3}{2}} x^{\frac{5}{2}} + \frac{5}{24} \,{\left (x - 1\right )}^{\frac{3}{2}} x^{\frac{3}{2}} + \frac{5}{32} \,{\left (x - 1\right )}^{\frac{3}{2}} \sqrt{x} + \frac{5}{64} \, \sqrt{x - 1} \sqrt{x} - \frac{5}{64} \, \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 = 0.925295, size = 201, normalized size = 1.49 \begin{align*} \frac{1}{192} \,{\left (48 \, x^{3} - 8 \, x^{2} - 10 \, x - 15\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + \frac{5}{128} \, \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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