Optimal. Leaf size=214 \[ -\frac {3 \tan ^{-1}\left (\frac {\sqrt {2 \left (2+\sqrt {13}\right )}-2 \sqrt {2+3 x}}{\sqrt {2 \left (-2+\sqrt {13}\right )}}\right )}{\sqrt {2 \left (-2+\sqrt {13}\right )}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2 \left (2+\sqrt {13}\right )}+2 \sqrt {2+3 x}}{\sqrt {2 \left (-2+\sqrt {13}\right )}}\right )}{\sqrt {2 \left (-2+\sqrt {13}\right )}}+\frac {3 \log \left (2+\sqrt {13}+3 x-\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {2+3 x}\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}-\frac {3 \log \left (2+\sqrt {13}+3 x+\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {2+3 x}\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}} \]
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Rubi [A]
time = 0.29, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {714, 1141,
1175, 632, 210, 1178, 642} \begin {gather*} -\frac {3 \text {ArcTan}\left (\frac {\sqrt {2 \left (2+\sqrt {13}\right )}-2 \sqrt {3 x+2}}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )}{\sqrt {2 \left (\sqrt {13}-2\right )}}+\frac {3 \text {ArcTan}\left (\frac {2 \sqrt {3 x+2}+\sqrt {2 \left (2+\sqrt {13}\right )}}{\sqrt {2 \left (\sqrt {13}-2\right )}}\right )}{\sqrt {2 \left (\sqrt {13}-2\right )}}+\frac {3 \log \left (3 x-\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {3 x+2}+\sqrt {13}+2\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}-\frac {3 \log \left (3 x+\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {3 x+2}+\sqrt {13}+2\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 714
Rule 1141
Rule 1175
Rule 1178
Rubi steps
\begin {align*} \int \frac {\sqrt {2+3 x}}{1+x^2} \, dx &=6 \text {Subst}\left (\int \frac {x^2}{13-4 x^2+x^4} \, dx,x,\sqrt {2+3 x}\right )\\ &=-\left (3 \text {Subst}\left (\int \frac {\sqrt {13}-x^2}{13-4 x^2+x^4} \, dx,x,\sqrt {2+3 x}\right )\right )+3 \text {Subst}\left (\int \frac {\sqrt {13}+x^2}{13-4 x^2+x^4} \, dx,x,\sqrt {2+3 x}\right )\\ &=\frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt {13}-\sqrt {2 \left (2+\sqrt {13}\right )} x+x^2} \, dx,x,\sqrt {2+3 x}\right )+\frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt {13}+\sqrt {2 \left (2+\sqrt {13}\right )} x+x^2} \, dx,x,\sqrt {2+3 x}\right )+\frac {3 \text {Subst}\left (\int \frac {\sqrt {2 \left (2+\sqrt {13}\right )}+2 x}{-\sqrt {13}-\sqrt {2 \left (2+\sqrt {13}\right )} x-x^2} \, dx,x,\sqrt {2+3 x}\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {2 \left (2+\sqrt {13}\right )}-2 x}{-\sqrt {13}+\sqrt {2 \left (2+\sqrt {13}\right )} x-x^2} \, dx,x,\sqrt {2+3 x}\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}\\ &=\frac {3 \log \left (2+\sqrt {13}+3 x-\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {2+3 x}\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}-\frac {3 \log \left (2+\sqrt {13}+3 x+\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {2+3 x}\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}-3 \text {Subst}\left (\int \frac {1}{2 \left (2-\sqrt {13}\right )-x^2} \, dx,x,-\sqrt {2 \left (2+\sqrt {13}\right )}+2 \sqrt {2+3 x}\right )-3 \text {Subst}\left (\int \frac {1}{2 \left (2-\sqrt {13}\right )-x^2} \, dx,x,\sqrt {2 \left (2+\sqrt {13}\right )}+2 \sqrt {2+3 x}\right )\\ &=-\frac {3 \tan ^{-1}\left (\frac {\sqrt {2 \left (2+\sqrt {13}\right )}-2 \sqrt {2+3 x}}{\sqrt {2 \left (-2+\sqrt {13}\right )}}\right )}{\sqrt {2 \left (-2+\sqrt {13}\right )}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2 \left (2+\sqrt {13}\right )}+2 \sqrt {2+3 x}}{\sqrt {2 \left (-2+\sqrt {13}\right )}}\right )}{\sqrt {2 \left (-2+\sqrt {13}\right )}}+\frac {3 \log \left (2+\sqrt {13}+3 x-\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {2+3 x}\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}-\frac {3 \log \left (2+\sqrt {13}+3 x+\sqrt {2 \left (2+\sqrt {13}\right )} \sqrt {2+3 x}\right )}{2 \sqrt {2 \left (2+\sqrt {13}\right )}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.08, size = 61, normalized size = 0.29 \begin {gather*} \sqrt {2-3 i} \tan ^{-1}\left (\sqrt {-\frac {2}{13}-\frac {3 i}{13}} \sqrt {2+3 x}\right )+\sqrt {2+3 i} \tan ^{-1}\left (\sqrt {-\frac {2}{13}+\frac {3 i}{13}} \sqrt {2+3 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.03, size = 195, normalized size = 0.91
method | result | size |
derivativedivides | \(-\frac {\sqrt {4+2 \sqrt {13}}\, \left (-2+\sqrt {13}\right ) \left (-\frac {\ln \left (2+3 x +\sqrt {13}-\sqrt {2+3 x}\, \sqrt {4+2 \sqrt {13}}\right )}{2}-\frac {\sqrt {4+2 \sqrt {13}}\, \arctan \left (\frac {2 \sqrt {2+3 x}-\sqrt {4+2 \sqrt {13}}}{\sqrt {-4+2 \sqrt {13}}}\right )}{\sqrt {-4+2 \sqrt {13}}}\right )}{6}-\frac {\sqrt {4+2 \sqrt {13}}\, \left (-2+\sqrt {13}\right ) \left (\frac {\ln \left (2+3 x +\sqrt {13}+\sqrt {2+3 x}\, \sqrt {4+2 \sqrt {13}}\right )}{2}-\frac {\sqrt {4+2 \sqrt {13}}\, \arctan \left (\frac {2 \sqrt {2+3 x}+\sqrt {4+2 \sqrt {13}}}{\sqrt {-4+2 \sqrt {13}}}\right )}{\sqrt {-4+2 \sqrt {13}}}\right )}{6}\) | \(195\) |
default | \(-\frac {\sqrt {4+2 \sqrt {13}}\, \left (-2+\sqrt {13}\right ) \left (-\frac {\ln \left (2+3 x +\sqrt {13}-\sqrt {2+3 x}\, \sqrt {4+2 \sqrt {13}}\right )}{2}-\frac {\sqrt {4+2 \sqrt {13}}\, \arctan \left (\frac {2 \sqrt {2+3 x}-\sqrt {4+2 \sqrt {13}}}{\sqrt {-4+2 \sqrt {13}}}\right )}{\sqrt {-4+2 \sqrt {13}}}\right )}{6}-\frac {\sqrt {4+2 \sqrt {13}}\, \left (-2+\sqrt {13}\right ) \left (\frac {\ln \left (2+3 x +\sqrt {13}+\sqrt {2+3 x}\, \sqrt {4+2 \sqrt {13}}\right )}{2}-\frac {\sqrt {4+2 \sqrt {13}}\, \arctan \left (\frac {2 \sqrt {2+3 x}+\sqrt {4+2 \sqrt {13}}}{\sqrt {-4+2 \sqrt {13}}}\right )}{\sqrt {-4+2 \sqrt {13}}}\right )}{6}\) | \(195\) |
trager | \(-\RootOf \left (\RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+1\right ) \ln \left (\frac {816 \RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{4} x \RootOf \left (\RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+1\right )-216 \RootOf \left (\RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+1\right ) \RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2} x -1700 \RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2} \RootOf \left (\RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+1\right )-816 \sqrt {2+3 x}\, \RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}-96 \RootOf \left (\RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+1\right ) x -400 \RootOf \left (\RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+1\right )-1167 \sqrt {2+3 x}}{3+4 \RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2} x +2 x}\right )+\RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right ) \ln \left (-\frac {816 \RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{5} x +1848 \RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{3} x +1700 \RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{3}-816 \sqrt {2+3 x}\, \RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2}+936 \RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right ) x +1300 \RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )+351 \sqrt {2+3 x}}{4 \RootOf \left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+13\right )^{2} x +2 x -3}\right )\) | \(409\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.56, size = 282, normalized size = 1.32 \begin {gather*} \frac {1}{156} \cdot 13^{\frac {1}{4}} \sqrt {4 \, \sqrt {13} + 26} {\left (2 \, \sqrt {13} - 13\right )} \log \left (\frac {1}{13} \cdot 13^{\frac {3}{4}} \sqrt {3 \, x + 2} \sqrt {4 \, \sqrt {13} + 26} + 3 \, x + \sqrt {13} + 2\right ) - \frac {1}{156} \cdot 13^{\frac {1}{4}} \sqrt {4 \, \sqrt {13} + 26} {\left (2 \, \sqrt {13} - 13\right )} \log \left (-\frac {1}{13} \cdot 13^{\frac {3}{4}} \sqrt {3 \, x + 2} \sqrt {4 \, \sqrt {13} + 26} + 3 \, x + \sqrt {13} + 2\right ) - \frac {1}{13} \cdot 13^{\frac {3}{4}} \sqrt {4 \, \sqrt {13} + 26} \arctan \left (-\frac {1}{39} \cdot 13^{\frac {3}{4}} \sqrt {3 \, x + 2} \sqrt {4 \, \sqrt {13} + 26} + \frac {1}{39} \cdot 13^{\frac {1}{4}} \sqrt {13^{\frac {3}{4}} \sqrt {3 \, x + 2} \sqrt {4 \, \sqrt {13} + 26} + 39 \, x + 13 \, \sqrt {13} + 26} \sqrt {4 \, \sqrt {13} + 26} - \frac {1}{3} \, \sqrt {13} - \frac {2}{3}\right ) - \frac {1}{13} \cdot 13^{\frac {3}{4}} \sqrt {4 \, \sqrt {13} + 26} \arctan \left (-\frac {1}{39} \cdot 13^{\frac {3}{4}} \sqrt {3 \, x + 2} \sqrt {4 \, \sqrt {13} + 26} + \frac {1}{39} \cdot 13^{\frac {1}{4}} \sqrt {-13^{\frac {3}{4}} \sqrt {3 \, x + 2} \sqrt {4 \, \sqrt {13} + 26} + 39 \, x + 13 \, \sqrt {13} + 26} \sqrt {4 \, \sqrt {13} + 26} + \frac {1}{3} \, \sqrt {13} + \frac {2}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.87, size = 32, normalized size = 0.15 \begin {gather*} 6 \operatorname {RootSum} {\left (20736 t^{4} + 576 t^{2} + 13, \left ( t \mapsto t \log {\left (576 t^{3} + 8 t + \sqrt {3 x + 2} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.82, size = 179, normalized size = 0.84 \begin {gather*} \frac {1}{2} \, \sqrt {2 \, \sqrt {13} + 4} \arctan \left (\frac {13^{\frac {3}{4}} {\left (13^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + \sqrt {3 \, x + 2}\right )}}{13 \, \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {13} + 4} \arctan \left (-\frac {13^{\frac {3}{4}} {\left (13^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} - \sqrt {3 \, x + 2}\right )}}{13 \, \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {13} - 4} \log \left (2 \cdot 13^{\frac {1}{4}} \sqrt {3 \, x + 2} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + 3 \, x + \sqrt {13} + 2\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {13} - 4} \log \left (-2 \cdot 13^{\frac {1}{4}} \sqrt {3 \, x + 2} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + 3 \, x + \sqrt {13} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 179, normalized size = 0.84 \begin {gather*} -\mathrm {atanh}\left (-\frac {\left (1152\,\sqrt {3\,x+2}\,{\left (\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}-\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )}^2-720\,\sqrt {3\,x+2}\right )\,\left (\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}-\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )}{2808}\right )\,\left (2\,\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}-2\,\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )-\mathrm {atanh}\left (\frac {\left (720\,\sqrt {3\,x+2}-1152\,\sqrt {3\,x+2}\,{\left (\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}+\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )}^2\right )\,\left (\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}+\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right )}{2808}\right )\,\left (2\,\sqrt {-\frac {\sqrt {13}}{8}-\frac {1}{4}}+2\,\sqrt {\frac {\sqrt {13}}{8}-\frac {1}{4}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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