Optimal. Leaf size=74 \[ -\frac{4 \sqrt{x} \sqrt{x+1}}{3 (1-3 x)}+\frac{8}{9 (1-3 x)}+\frac{5}{9} \log (1-3 x)-\frac{8}{9} \sinh ^{-1}\left (\sqrt{x}\right )+\frac{10}{9} \tanh ^{-1}\left (\frac{2 \sqrt{x}}{\sqrt{x+1}}\right ) \]
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Rubi [A] time = 0.0559415, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {6742, 97, 157, 54, 215, 93, 207} \[ -\frac{4 \sqrt{x} \sqrt{x+1}}{3 (1-3 x)}+\frac{8}{9 (1-3 x)}+\frac{5}{9} \log (1-3 x)-\frac{8}{9} \sinh ^{-1}\left (\sqrt{x}\right )+\frac{10}{9} \tanh ^{-1}\left (\frac{2 \sqrt{x}}{\sqrt{x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 6742
Rule 97
Rule 157
Rule 54
Rule 215
Rule 93
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{\left (2 \sqrt{x}+\sqrt{1+x}\right )^2} \, dx &=\int \left (\frac{8}{3 (-1+3 x)^2}-\frac{4 \sqrt{x} \sqrt{1+x}}{(-1+3 x)^2}+\frac{5}{3 (-1+3 x)}\right ) \, dx\\ &=\frac{8}{9 (1-3 x)}+\frac{5}{9} \log (1-3 x)-4 \int \frac{\sqrt{x} \sqrt{1+x}}{(-1+3 x)^2} \, dx\\ &=\frac{8}{9 (1-3 x)}-\frac{4 \sqrt{x} \sqrt{1+x}}{3 (1-3 x)}+\frac{5}{9} \log (1-3 x)-\frac{4}{3} \int \frac{\frac{1}{2}+x}{\sqrt{x} \sqrt{1+x} (-1+3 x)} \, dx\\ &=\frac{8}{9 (1-3 x)}-\frac{4 \sqrt{x} \sqrt{1+x}}{3 (1-3 x)}+\frac{5}{9} \log (1-3 x)-\frac{4}{9} \int \frac{1}{\sqrt{x} \sqrt{1+x}} \, dx-\frac{10}{9} \int \frac{1}{\sqrt{x} \sqrt{1+x} (-1+3 x)} \, dx\\ &=\frac{8}{9 (1-3 x)}-\frac{4 \sqrt{x} \sqrt{1+x}}{3 (1-3 x)}+\frac{5}{9} \log (1-3 x)-\frac{8}{9} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\sqrt{x}\right )-\frac{20}{9} \operatorname{Subst}\left (\int \frac{1}{-1+4 x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{1+x}}\right )\\ &=\frac{8}{9 (1-3 x)}-\frac{4 \sqrt{x} \sqrt{1+x}}{3 (1-3 x)}-\frac{8}{9} \sinh ^{-1}\left (\sqrt{x}\right )+\frac{10}{9} \tanh ^{-1}\left (\frac{2 \sqrt{x}}{\sqrt{1+x}}\right )+\frac{5}{9} \log (1-3 x)\\ \end{align*}
Mathematica [A] time = 0.132287, size = 126, normalized size = 1.7 \[ \frac{12 x^{3/2}+12 \sqrt{x}-8 \sqrt{x+1}+15 \sqrt{x+1} x \log (1-3 x)-5 \sqrt{x+1} \log (1-3 x)+10 \sqrt{-x-1} (3 x-1) \tan ^{-1}\left (\frac{2 \sqrt{x}}{\sqrt{-x-1}}\right )-8 \sqrt{x+1} (3 x-1) \sinh ^{-1}\left (\sqrt{x}\right )}{9 \sqrt{x+1} (3 x-1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 115, normalized size = 1.6 \begin{align*} -{\frac{8}{27\,x-9}}+{\frac{5\,\ln \left ( 3\,x-1 \right ) }{9}}-{\frac{1}{27\,x-9}\sqrt{x}\sqrt{1+x} \left ( 12\,\ln \left ( 1/2+x+\sqrt{x \left ( 1+x \right ) } \right ) x-15\,{\it Artanh} \left ( 1/4\,{\frac{1+5\,x}{\sqrt{x \left ( 1+x \right ) }}} \right ) x-4\,\ln \left ( 1/2+x+\sqrt{x \left ( 1+x \right ) } \right ) +5\,{\it Artanh} \left ( 1/4\,{\frac{1+5\,x}{\sqrt{x \left ( 1+x \right ) }}} \right ) -12\,\sqrt{x \left ( 1+x \right ) } \right ){\frac{1}{\sqrt{x \left ( 1+x \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\sqrt{x + 1} + 2 \, \sqrt{x}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12849, size = 309, normalized size = 4.18 \begin{align*} -\frac{5 \,{\left (3 \, x - 1\right )} \log \left (3 \, \sqrt{x + 1} \sqrt{x} - 3 \, x - 1\right ) - 4 \,{\left (3 \, x - 1\right )} \log \left (2 \, \sqrt{x + 1} \sqrt{x} - 2 \, x - 1\right ) - 5 \,{\left (3 \, x - 1\right )} \log \left (\sqrt{x + 1} \sqrt{x} - x + 1\right ) - 5 \,{\left (3 \, x - 1\right )} \log \left (3 \, x - 1\right ) - 12 \, \sqrt{x + 1} \sqrt{x} - 12 \, x + 12}{9 \,{\left (3 \, 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{1}{\left (2 \sqrt{x} + \sqrt{x + 1}\right )^{2}}\, 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: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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