Optimal. Leaf size=183 \[ \frac{4}{5} (2 x+1)^{5/2}-12 \sqrt{2 x+1}-\frac{3 \sqrt [4]{3} \log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\frac{3 \sqrt [4]{3} \log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}-3 \sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )+3 \sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right ) \]
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Rubi [A] time = 0.17134, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {692, 694, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{4}{5} (2 x+1)^{5/2}-12 \sqrt{2 x+1}-\frac{3 \sqrt [4]{3} \log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\frac{3 \sqrt [4]{3} \log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}-3 \sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )+3 \sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right ) \]
Antiderivative was successfully verified.
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Rule 692
Rule 694
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(1+2 x)^{7/2}}{1+x+x^2} \, dx &=\frac{4}{5} (1+2 x)^{5/2}-3 \int \frac{(1+2 x)^{3/2}}{1+x+x^2} \, dx\\ &=-12 \sqrt{1+2 x}+\frac{4}{5} (1+2 x)^{5/2}+9 \int \frac{1}{\sqrt{1+2 x} \left (1+x+x^2\right )} \, dx\\ &=-12 \sqrt{1+2 x}+\frac{4}{5} (1+2 x)^{5/2}+\frac{9}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (\frac{3}{4}+\frac{x^2}{4}\right )} \, dx,x,1+2 x\right )\\ &=-12 \sqrt{1+2 x}+\frac{4}{5} (1+2 x)^{5/2}+9 \operatorname{Subst}\left (\int \frac{1}{\frac{3}{4}+\frac{x^4}{4}} \, dx,x,\sqrt{1+2 x}\right )\\ &=-12 \sqrt{1+2 x}+\frac{4}{5} (1+2 x)^{5/2}+\frac{1}{2} \left (3 \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{3}-x^2}{\frac{3}{4}+\frac{x^4}{4}} \, dx,x,\sqrt{1+2 x}\right )+\frac{1}{2} \left (3 \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{3}+x^2}{\frac{3}{4}+\frac{x^4}{4}} \, dx,x,\sqrt{1+2 x}\right )\\ &=-12 \sqrt{1+2 x}+\frac{4}{5} (1+2 x)^{5/2}-\frac{\left (3 \sqrt [4]{3}\right ) \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{\left (3 \sqrt [4]{3}\right ) \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}}+\left (3 \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{3}-\sqrt{2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt{1+2 x}\right )+\left (3 \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{3}+\sqrt{2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt{1+2 x}\right )\\ &=-12 \sqrt{1+2 x}+\frac{4}{5} (1+2 x)^{5/2}-\frac{3 \sqrt [4]{3} \log \left (1+\sqrt{3}+2 x-\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{\sqrt{2}}+\frac{3 \sqrt [4]{3} \log \left (1+\sqrt{3}+2 x+\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{\sqrt{2}}+\left (3 \sqrt{2} \sqrt [4]{3}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2+4 x}}{\sqrt [4]{3}}\right )-\left (3 \sqrt{2} \sqrt [4]{3}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2+4 x}}{\sqrt [4]{3}}\right )\\ &=-12 \sqrt{1+2 x}+\frac{4}{5} (1+2 x)^{5/2}-3 \sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{1+2 x}}{\sqrt [4]{3}}\right )+3 \sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{1+2 x}}{\sqrt [4]{3}}\right )-\frac{3 \sqrt [4]{3} \log \left (1+\sqrt{3}+2 x-\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{\sqrt{2}}+\frac{3 \sqrt [4]{3} \log \left (1+\sqrt{3}+2 x+\sqrt{2} \sqrt [4]{3} \sqrt{1+2 x}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0929894, size = 182, normalized size = 0.99 \[ \frac{16}{5} \sqrt{2 x+1} x^2+\frac{16}{5} \sqrt{2 x+1} x-\frac{56}{5} \sqrt{2 x+1}-\frac{3 \sqrt [4]{3} \log \left (2 x-\sqrt [4]{3} \sqrt{4 x+2}+\sqrt{3}+1\right )}{\sqrt{2}}+\frac{3 \sqrt [4]{3} \log \left (2 x+\sqrt [4]{3} \sqrt{4 x+2}+\sqrt{3}+1\right )}{\sqrt{2}}-3 \sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{4 x+2}}{\sqrt [4]{3}}\right )+3 \sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{4 x+2}}{\sqrt [4]{3}}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 129, normalized size = 0.7 \begin{align*}{\frac{4}{5} \left ( 1+2\,x \right ) ^{{\frac{5}{2}}}}-12\,\sqrt{1+2\,x}+3\,\sqrt [4]{3}\arctan \left ( 1+1/3\,\sqrt{2}\sqrt{1+2\,x}{3}^{3/4} \right ) \sqrt{2}+3\,\sqrt [4]{3}\arctan \left ( -1+1/3\,\sqrt{2}\sqrt{1+2\,x}{3}^{3/4} \right ) \sqrt{2}+{\frac{3\,\sqrt [4]{3}\sqrt{2}}{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 = 1.80035, size = 203, normalized size = 1.11 \begin{align*} \frac{4}{5} \,{\left (2 \, x + 1\right )}^{\frac{5}{2}} + 3 \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 ) + 3 \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{3}{2} \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{3}{2} \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 ) - 12 \, \sqrt{2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70997, size = 651, normalized size = 3.56 \begin{align*} -6 \cdot 3^{\frac{1}{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 ) - 6 \cdot 3^{\frac{1}{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{3}{2} \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{3}{2} \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{8}{5} \,{\left (2 \, x^{2} + 2 \, x - 7\right )} \sqrt{2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 54.7249, size = 180, normalized size = 0.98 \begin{align*} \frac{4 \left (2 x + 1\right )^{\frac{5}{2}}}{5} - 12 \sqrt{2 x + 1} - \frac{3 \sqrt{2} \sqrt [4]{3} \log{\left (2 x - \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{2} + \frac{3 \sqrt{2} \sqrt [4]{3} \log{\left (2 x + \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{2} + 3 \sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} - 1 \right )} + 3 \sqrt{2} \sqrt [4]{3} \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 = 1.18237, size = 186, normalized size = 1.02 \begin{align*} \frac{4}{5} \,{\left (2 \, x + 1\right )}^{\frac{5}{2}} + 3 \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 ) + 3 \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{3}{2} \cdot 12^{\frac{1}{4}} \log \left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - \frac{3}{2} \cdot 12^{\frac{1}{4}} \log \left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - 12 \, \sqrt{2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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