Optimal. Leaf size=180 \[ -\frac{4}{9 (2 x+1)^{3/2}}+\frac{\log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{3 \sqrt{2} 3^{3/4}}-\frac{\log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{3 \sqrt{2} 3^{3/4}}+\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right )}{3\ 3^{3/4}} \]
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Rubi [A] time = 0.13279, antiderivative size = 180, 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 = {693, 694, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{4}{9 (2 x+1)^{3/2}}+\frac{\log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{3 \sqrt{2} 3^{3/4}}-\frac{\log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{3 \sqrt{2} 3^{3/4}}+\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right )}{3\ 3^{3/4}} \]
Antiderivative was successfully verified.
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Rule 693
Rule 694
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx &=-\frac{4}{9 (1+2 x)^{3/2}}-\frac{1}{3} \int \frac{1}{\sqrt{1+2 x} \left (1+x+x^2\right )} \, dx\\ &=-\frac{4}{9 (1+2 x)^{3/2}}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (\frac{3}{4}+\frac{x^2}{4}\right )} \, dx,x,1+2 x\right )\\ &=-\frac{4}{9 (1+2 x)^{3/2}}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\frac{3}{4}+\frac{x^4}{4}} \, dx,x,\sqrt{1+2 x}\right )\\ &=-\frac{4}{9 (1+2 x)^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{3}-x^2}{\frac{3}{4}+\frac{x^4}{4}} \, dx,x,\sqrt{1+2 x}\right )}{6 \sqrt{3}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{3}+x^2}{\frac{3}{4}+\frac{x^4}{4}} \, dx,x,\sqrt{1+2 x}\right )}{6 \sqrt{3}}\\ &=-\frac{4}{9 (1+2 x)^{3/2}}+\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 )}{3 \sqrt{2} 3^{3/4}}+\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 )}{3 \sqrt{2} 3^{3/4}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{3}-\sqrt{2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt{1+2 x}\right )}{3 \sqrt{3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{3}+\sqrt{2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt{1+2 x}\right )}{3 \sqrt{3}}\\ &=-\frac{4}{9 (1+2 x)^{3/2}}+\frac{\log \left (1+\sqrt{3}+2 x-\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{3 \sqrt{2} 3^{3/4}}-\frac{\log \left (1+\sqrt{3}+2 x+\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{3 \sqrt{2} 3^{3/4}}-\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2+4 x}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}+\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2+4 x}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}\\ &=-\frac{4}{9 (1+2 x)^{3/2}}+\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{1+2 x}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}-\frac{\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{1+2 x}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}+\frac{\log \left (1+\sqrt{3}+2 x-\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{3 \sqrt{2} 3^{3/4}}-\frac{\log \left (1+\sqrt{3}+2 x+\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{3 \sqrt{2} 3^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.008174, size = 32, normalized size = 0.18 \[ -\frac{4 \, _2F_1\left (-\frac{3}{4},1;\frac{1}{4};-\frac{1}{3} (2 x+1)^2\right )}{9 (2 x+1)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 120, normalized size = 0.7 \begin{align*} -{\frac{\sqrt [4]{3}\sqrt{2}}{9}\arctan \left ( 1+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\sqrt{1+2\,x}} \right ) }-{\frac{\sqrt [4]{3}\sqrt{2}}{9}\arctan \left ( -1+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\sqrt{1+2\,x}} \right ) }-{\frac{\sqrt [4]{3}\sqrt{2}}{18}\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 ) }-{\frac{4}{9} \left ( 1+2\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.01312, size = 190, normalized size = 1.06 \begin{align*} -\frac{1}{9} \cdot 3^{\frac{1}{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}{9} \cdot 3^{\frac{1}{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}{18} \cdot 3^{\frac{1}{4}} \sqrt{2} \log \left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + \frac{1}{18} \cdot 3^{\frac{1}{4}} \sqrt{2} \log \left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - \frac{4}{9 \,{\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 [B] time = 1.68823, size = 811, normalized size = 4.51 \begin{align*} \frac{4 \cdot 27^{\frac{3}{4}} \sqrt{2}{\left (4 \, x^{2} + 4 \, x + 1\right )} \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 ) + 4 \cdot 27^{\frac{3}{4}} \sqrt{2}{\left (4 \, x^{2} + 4 \, x + 1\right )} \arctan \left (\frac{1}{54} \cdot 27^{\frac{1}{4}} \sqrt{2} \sqrt{-36 \cdot 27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 648 \, x + 324 \, \sqrt{3} + 324} - \frac{1}{3} \cdot 27^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 1\right ) - 27^{\frac{3}{4}} \sqrt{2}{\left (4 \, x^{2} + 4 \, x + 1\right )} \log \left (36 \cdot 27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 648 \, x + 324 \, \sqrt{3} + 324\right ) + 27^{\frac{3}{4}} \sqrt{2}{\left (4 \, x^{2} + 4 \, x + 1\right )} \log \left (-36 \cdot 27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 648 \, x + 324 \, \sqrt{3} + 324\right ) - 72 \, \sqrt{2 \, x + 1}}{162 \,{\left (4 \, x^{2} + 4 \, 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 x + 1\right )^{\frac{5}{2}} \left (x^{2} + x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17082, size = 174, normalized size = 0.97 \begin{align*} -\frac{1}{9} \cdot 12^{\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}{9} \cdot 12^{\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}{18} \cdot 12^{\frac{1}{4}} \log \left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + \frac{1}{18} \cdot 12^{\frac{1}{4}} \log \left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - \frac{4}{9 \,{\left (2 \, x + 1\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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