Optimal. Leaf size=170 \[ \frac{4}{3} (2 x+1)^{3/2}-\frac{3^{3/4} \log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\frac{3^{3/4} \log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\sqrt{2} 3^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )-\sqrt{2} 3^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right ) \]
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Rubi [A] time = 0.136914, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {692, 694, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{4}{3} (2 x+1)^{3/2}-\frac{3^{3/4} \log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\frac{3^{3/4} \log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\sqrt{2} 3^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )-\sqrt{2} 3^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right ) \]
Antiderivative was successfully verified.
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Rule 692
Rule 694
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(1+2 x)^{5/2}}{1+x+x^2} \, dx &=\frac{4}{3} (1+2 x)^{3/2}-3 \int \frac{\sqrt{1+2 x}}{1+x+x^2} \, dx\\ &=\frac{4}{3} (1+2 x)^{3/2}-\frac{3}{2} \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\frac{3}{4}+\frac{x^2}{4}} \, dx,x,1+2 x\right )\\ &=\frac{4}{3} (1+2 x)^{3/2}-3 \operatorname{Subst}\left (\int \frac{x^2}{\frac{3}{4}+\frac{x^4}{4}} \, dx,x,\sqrt{1+2 x}\right )\\ &=\frac{4}{3} (1+2 x)^{3/2}+\frac{3}{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{3}{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{4}{3} (1+2 x)^{3/2}-3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{3}-\sqrt{2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt{1+2 x}\right )-3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{3}+\sqrt{2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt{1+2 x}\right )-\frac{3^{3/4} \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}}-\frac{3^{3/4} \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}}\\ &=\frac{4}{3} (1+2 x)^{3/2}-\frac{3^{3/4} \log \left (1+\sqrt{3}+2 x-\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{\sqrt{2}}+\frac{3^{3/4} \log \left (1+\sqrt{3}+2 x+\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{\sqrt{2}}-\left (\sqrt{2} 3^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2+4 x}}{\sqrt [4]{3}}\right )+\left (\sqrt{2} 3^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2+4 x}}{\sqrt [4]{3}}\right )\\ &=\frac{4}{3} (1+2 x)^{3/2}+\sqrt{2} 3^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{1+2 x}}{\sqrt [4]{3}}\right )-\sqrt{2} 3^{3/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{1+2 x}}{\sqrt [4]{3}}\right )-\frac{3^{3/4} \log \left (1+\sqrt{3}+2 x-\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{\sqrt{2}}+\frac{3^{3/4} \log \left (1+\sqrt{3}+2 x+\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0104773, size = 34, normalized size = 0.2 \[ -\frac{4}{3} (2 x+1)^{3/2} \left (\, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\frac{1}{3} (2 x+1)^2\right )-1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 120, normalized size = 0.7 \begin{align*}{\frac{4}{3} \left ( 1+2\,x \right ) ^{{\frac{3}{2}}}}-{3}^{{\frac{3}{4}}}\arctan \left ( 1+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\sqrt{1+2\,x}} \right ) \sqrt{2}-{3}^{{\frac{3}{4}}}\arctan \left ( -1+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\sqrt{1+2\,x}} \right ) \sqrt{2}-{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{2}\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 = 1.91176, size = 190, normalized size = 1.12 \begin{align*} -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 ) - 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}{2} \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}{2} \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{4}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67823, size = 682, normalized size = 4.01 \begin{align*} 2 \cdot 27^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{1}{9} \cdot 27^{\frac{1}{4}} \sqrt{2} \sqrt{27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 18 \, x + 9 \, \sqrt{3} + 9} - \frac{1}{3} \cdot 27^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} - 1\right ) + 2 \cdot 27^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{1}{27} \cdot 27^{\frac{1}{4}} \sqrt{2} \sqrt{-9 \cdot 27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 162 \, x + 81 \, \sqrt{3} + 81} - \frac{1}{3} \cdot 27^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 1\right ) + \frac{1}{2} \cdot 27^{\frac{1}{4}} \sqrt{2} \log \left (9 \cdot 27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 162 \, x + 81 \, \sqrt{3} + 81\right ) - \frac{1}{2} \cdot 27^{\frac{1}{4}} \sqrt{2} \log \left (-9 \cdot 27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 162 \, x + 81 \, \sqrt{3} + 81\right ) + \frac{4}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 32.9937, size = 163, normalized size = 0.96 \begin{align*} \frac{4 \left (2 x + 1\right )^{\frac{3}{2}}}{3} - \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 )}}{2} + \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 )}}{2} - \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 )} - \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19452, size = 174, normalized size = 1.02 \begin{align*} \frac{4}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} - 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 ) - 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}{2} \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}{2} \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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