Optimal. Leaf size=157 \[ \frac{\log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2} \sqrt [4]{3}}-\frac{\log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2} \sqrt [4]{3}}-\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}+\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt [4]{3}} \]
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Rubi [A] time = 0.120514, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {694, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{\log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2} \sqrt [4]{3}}-\frac{\log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2} \sqrt [4]{3}}-\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}+\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt [4]{3}} \]
Antiderivative was successfully verified.
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Rule 694
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{\sqrt{1+2 x}}{1+x+x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\frac{3}{4}+\frac{x^2}{4}} \, dx,x,1+2 x\right )\\ &=\operatorname{Subst}\left (\int \frac{x^2}{\frac{3}{4}+\frac{x^4}{4}} \, dx,x,\sqrt{1+2 x}\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{3}-x^2}{\frac{3}{4}+\frac{x^4}{4}} \, dx,x,\sqrt{1+2 x}\right )\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{3}+x^2}{\frac{3}{4}+\frac{x^4}{4}} \, dx,x,\sqrt{1+2 x}\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt [4]{3}+2 x}{-\sqrt{3}-\sqrt{2} \sqrt [4]{3} x-x^2} \, dx,x,\sqrt{1+2 x}\right )}{\sqrt{2} \sqrt [4]{3}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt [4]{3}-2 x}{-\sqrt{3}+\sqrt{2} \sqrt [4]{3} x-x^2} \, dx,x,\sqrt{1+2 x}\right )}{\sqrt{2} \sqrt [4]{3}}+\operatorname{Subst}\left (\int \frac{1}{\sqrt{3}-\sqrt{2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt{1+2 x}\right )+\operatorname{Subst}\left (\int \frac{1}{\sqrt{3}+\sqrt{2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt{1+2 x}\right )\\ &=\frac{\log \left (1+\sqrt{3}+2 x-\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{\sqrt{2} \sqrt [4]{3}}-\frac{\log \left (1+\sqrt{3}+2 x+\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{\sqrt{2} \sqrt [4]{3}}+\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2+4 x}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}-\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2+4 x}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}\\ &=-\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{1+2 x}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}+\frac{\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{1+2 x}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}+\frac{\log \left (1+\sqrt{3}+2 x-\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{\sqrt{2} \sqrt [4]{3}}-\frac{\log \left (1+\sqrt{3}+2 x+\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{\sqrt{2} \sqrt [4]{3}}\\ \end{align*}
Mathematica [C] time = 0.0058424, size = 32, normalized size = 0.2 \[ \frac{4}{9} (2 x+1)^{3/2} \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\frac{1}{3} (2 x+1)^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 111, normalized size = 0.7 \begin{align*}{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\arctan \left ( 1+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\sqrt{1+2\,x}} \right ) }+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\arctan \left ( -1+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\sqrt{1+2\,x}} \right ) }+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{6}\ln \left ({ \left ( 1+2\,x+\sqrt{3}-\sqrt [4]{3}\sqrt{2}\sqrt{1+2\,x} \right ) \left ( 1+2\,x+\sqrt{3}+\sqrt [4]{3}\sqrt{2}\sqrt{1+2\,x} \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.18092, size = 178, normalized size = 1.13 \begin{align*} \frac{1}{3} \cdot 3^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{2 \, x + 1}\right )}\right ) + \frac{1}{3} \cdot 3^{\frac{3}{4}} \sqrt{2} \arctan \left (-\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{2 \, x + 1}\right )}\right ) - \frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + \frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59657, size = 621, normalized size = 3.96 \begin{align*} -\frac{2}{3} \cdot 3^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{3} \cdot 3^{\frac{3}{4}} \sqrt{2} \sqrt{3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1} - \frac{1}{3} \cdot 3^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} - 1\right ) - \frac{2}{3} \cdot 3^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2} \sqrt{-4 \cdot 3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 8 \, x + 4 \, \sqrt{3} + 4} - \frac{1}{3} \cdot 3^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 1\right ) - \frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (4 \cdot 3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 8 \, x + 4 \, \sqrt{3} + 4\right ) + \frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (-4 \cdot 3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 8 \, x + 4 \, \sqrt{3} + 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.76604, size = 155, normalized size = 0.99 \begin{align*} \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x - \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{6} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x + \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{6} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} - 1 \right )}}{3} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} + 1 \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16337, size = 162, normalized size = 1.03 \begin{align*} \frac{1}{3} \cdot 108^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{2 \, x + 1}\right )}\right ) + \frac{1}{3} \cdot 108^{\frac{1}{4}} \arctan \left (-\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{2 \, x + 1}\right )}\right ) - \frac{1}{6} \cdot 108^{\frac{1}{4}} \log \left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + \frac{1}{6} \cdot 108^{\frac{1}{4}} \log \left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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