Optimal. Leaf size=168 \[ 4 \sqrt {2 x+1}+\frac {\sqrt [4]{3} \log \left (2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{\sqrt {2}}-\frac {\sqrt [4]{3} \log \left (2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{\sqrt {2}}+\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )-\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {692, 694, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} 4 \sqrt {2 x+1}+\frac {\sqrt [4]{3} \log \left (2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{\sqrt {2}}-\frac {\sqrt [4]{3} \log \left (2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{\sqrt {2}}+\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )-\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 329
Rule 617
Rule 628
Rule 692
Rule 694
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(1+2 x)^{3/2}}{1+x+x^2} \, dx &=4 \sqrt {1+2 x}-3 \int \frac {1}{\sqrt {1+2 x} \left (1+x+x^2\right )} \, dx\\ &=4 \sqrt {1+2 x}-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (\frac {3}{4}+\frac {x^2}{4}\right )} \, dx,x,1+2 x\right )\\ &=4 \sqrt {1+2 x}-3 \operatorname {Subst}\left (\int \frac {1}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )\\ &=4 \sqrt {1+2 x}-\frac {1}{2} \sqrt {3} \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} \sqrt {3} \operatorname {Subst}\left (\int \frac {\sqrt {3}+x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )\\ &=4 \sqrt {1+2 x}+\frac {\sqrt [4]{3} \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 {\sqrt [4]{3} \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 {3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3}-\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt {1+2 x}\right )-\sqrt {3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3}+\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt {1+2 x}\right )\\ &=4 \sqrt {1+2 x}+\frac {\sqrt [4]{3} \log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}}-\frac {\sqrt [4]{3} \log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}}-\left (\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 (\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 )\\ &=4 \sqrt {1+2 x}+\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )-\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )+\frac {\sqrt [4]{3} \log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}}-\frac {\sqrt [4]{3} \log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 148, normalized size = 0.88 \begin {gather*} 4 \sqrt {2 x+1}+\frac {\sqrt [4]{3} \log \left (2 x-\sqrt [4]{3} \sqrt {4 x+2}+\sqrt {3}+1\right )}{\sqrt {2}}-\frac {\sqrt [4]{3} \log \left (2 x+\sqrt [4]{3} \sqrt {4 x+2}+\sqrt {3}+1\right )}{\sqrt {2}}+\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac {\sqrt {4 x+2}}{\sqrt [4]{3}}\right )-\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt {4 x+2}}{\sqrt [4]{3}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 112, normalized size = 0.67 \begin {gather*} 4 \sqrt {2 x+1}-\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (\frac {\frac {2 x+1}{\sqrt {2} \sqrt [4]{3}}-\frac {\sqrt [4]{3}}{\sqrt {2}}}{\sqrt {2 x+1}}\right )-\sqrt {2} \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt {2} 3^{3/4} \sqrt {2 x+1}}{\sqrt {3} (2 x+1)+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 190, normalized size = 1.13 \begin {gather*} 2 \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 ) + 2 \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 {1}{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 {1}{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 ) + 4 \, \sqrt {2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 129, normalized size = 0.77 \begin {gather*} -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 ) - 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}{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 {1}{2} \cdot 12^{\frac {1}{4}} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) + 4 \, \sqrt {2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 120, normalized size = 0.71 \begin {gather*} -3^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )-3^{\frac {1}{4}} \sqrt {2}\, \arctan \left (1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )-\frac {3^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {2 x +1+\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}{2 x +1+\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}\right )}{2}+4 \sqrt {2 x +1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.06, size = 141, normalized size = 0.84 \begin {gather*} -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^{\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}{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 {1}{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 ) + 4 \, \sqrt {2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 66, normalized size = 0.39 \begin {gather*} 4\,\sqrt {2\,x+1}+\sqrt {2}\,3^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-1-\mathrm {i}\right )+\sqrt {2}\,3^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-1+1{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.26, size = 162, normalized size = 0.96 \begin {gather*} 4 \sqrt {2 x + 1} + \frac {\sqrt {2} \sqrt [4]{3} \log {\left (2 x - \sqrt {2} \sqrt [4]{3} \sqrt {2 x + 1} + 1 + \sqrt {3} \right )}}{2} - \frac {\sqrt {2} \sqrt [4]{3} \log {\left (2 x + \sqrt {2} \sqrt [4]{3} \sqrt {2 x + 1} + 1 + \sqrt {3} \right )}}{2} - \sqrt {2} \sqrt [4]{3} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \sqrt {2 x + 1}}{3} - 1 \right )} - \sqrt {2} \sqrt [4]{3} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \sqrt {2 x + 1}}{3} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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