Optimal. Leaf size=222 \[ \frac{\log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{\sqrt{10 \left (2+\sqrt{35}\right )}}-\frac{\log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{\sqrt{10 \left (2+\sqrt{35}\right )}}-\sqrt{\frac{2}{5 \left (\sqrt{35}-2\right )}} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\sqrt{\frac{2}{5 \left (\sqrt{35}-2\right )}} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]
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Rubi [A] time = 0.247047, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {699, 1127, 1161, 618, 204, 1164, 628} \[ \frac{\log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{\sqrt{10 \left (2+\sqrt{35}\right )}}-\frac{\log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{\sqrt{10 \left (2+\sqrt{35}\right )}}-\sqrt{\frac{2}{5 \left (\sqrt{35}-2\right )}} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\sqrt{\frac{2}{5 \left (\sqrt{35}-2\right )}} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 699
Rule 1127
Rule 1161
Rule 618
Rule 204
Rule 1164
Rule 628
Rubi steps
\begin{align*} \int \frac{\sqrt{1+2 x}}{2+3 x+5 x^2} \, dx &=4 \operatorname{Subst}\left (\int \frac{x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt{1+2 x}\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{\sqrt{\frac{7}{5}}-x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt{1+2 x}\right )\right )+2 \operatorname{Subst}\left (\int \frac{\sqrt{\frac{7}{5}}+x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt{1+2 x}\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{\sqrt{\frac{7}{5}}-\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )} x+x^2} \, dx,x,\sqrt{1+2 x}\right )+\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{\sqrt{\frac{7}{5}}+\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )} x+x^2} \, dx,x,\sqrt{1+2 x}\right )+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+2 x}{-\sqrt{\frac{7}{5}}-\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )} x-x^2} \, dx,x,\sqrt{1+2 x}\right )}{\sqrt{10 \left (2+\sqrt{35}\right )}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}-2 x}{-\sqrt{\frac{7}{5}}+\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )} x-x^2} \, dx,x,\sqrt{1+2 x}\right )}{\sqrt{10 \left (2+\sqrt{35}\right )}}\\ &=\frac{\log \left (\sqrt{35}-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{1+2 x}+5 (1+2 x)\right )}{\sqrt{10 \left (2+\sqrt{35}\right )}}-\frac{\log \left (\sqrt{35}+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{1+2 x}+5 (1+2 x)\right )}{\sqrt{10 \left (2+\sqrt{35}\right )}}-\frac{2}{5} \operatorname{Subst}\left (\int \frac{1}{\frac{2}{5} \left (2-\sqrt{35}\right )-x^2} \, dx,x,-\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+2 \sqrt{1+2 x}\right )-\frac{2}{5} \operatorname{Subst}\left (\int \frac{1}{\frac{2}{5} \left (2-\sqrt{35}\right )-x^2} \, dx,x,\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+2 \sqrt{1+2 x}\right )\\ &=-\sqrt{\frac{2}{5 \left (-2+\sqrt{35}\right )}} \tan ^{-1}\left (\sqrt{\frac{5}{2 \left (-2+\sqrt{35}\right )}} \left (\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}-2 \sqrt{1+2 x}\right )\right )+\sqrt{\frac{2}{5 \left (-2+\sqrt{35}\right )}} \tan ^{-1}\left (\sqrt{\frac{5}{2 \left (-2+\sqrt{35}\right )}} \left (\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+2 \sqrt{1+2 x}\right )\right )+\frac{\log \left (\sqrt{35}-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{1+2 x}+5 (1+2 x)\right )}{\sqrt{10 \left (2+\sqrt{35}\right )}}-\frac{\log \left (\sqrt{35}+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{1+2 x}+5 (1+2 x)\right )}{\sqrt{10 \left (2+\sqrt{35}\right )}}\\ \end{align*}
Mathematica [C] time = 0.202769, size = 112, normalized size = 0.5 \[ \frac{2 \left (\sqrt{-2+i \sqrt{31}} \left (\sqrt{31}-2 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )+\sqrt{-2-i \sqrt{31}} \left (\sqrt{31}+2 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )\right )}{5 \sqrt{217}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.087, size = 486, normalized size = 2.2 \begin{align*}{\frac{\sqrt{5}\sqrt{2\,\sqrt{5}\sqrt{7}+4}}{155}\ln \left ( \sqrt{5}\sqrt{7}+10\,x+5+\sqrt{2\,\sqrt{5}\sqrt{7}+4}\sqrt{5}\sqrt{1+2\,x} \right ) }-{\frac{4\,\sqrt{5}\sqrt{7}+8}{31\,\sqrt{10\,\sqrt{5}\sqrt{7}-20}}\arctan \left ({\frac{1}{\sqrt{10\,\sqrt{5}\sqrt{7}-20}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{5}\sqrt{2\,\sqrt{5}\sqrt{7}+4} \right ) } \right ) }-{\frac{\sqrt{7}\sqrt{2\,\sqrt{5}\sqrt{7}+4}}{62}\ln \left ( \sqrt{5}\sqrt{7}+10\,x+5+\sqrt{2\,\sqrt{5}\sqrt{7}+4}\sqrt{5}\sqrt{1+2\,x} \right ) }+{\frac{\sqrt{5} \left ( 2\,\sqrt{5}\sqrt{7}+4 \right ) \sqrt{7}}{31\,\sqrt{10\,\sqrt{5}\sqrt{7}-20}}\arctan \left ({\frac{1}{\sqrt{10\,\sqrt{5}\sqrt{7}-20}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{5}\sqrt{2\,\sqrt{5}\sqrt{7}+4} \right ) } \right ) }-{\frac{\sqrt{5}\sqrt{2\,\sqrt{5}\sqrt{7}+4}}{155}\ln \left ( -\sqrt{2\,\sqrt{5}\sqrt{7}+4}\sqrt{5}\sqrt{1+2\,x}+\sqrt{5}\sqrt{7}+10\,x+5 \right ) }-{\frac{4\,\sqrt{5}\sqrt{7}+8}{31\,\sqrt{10\,\sqrt{5}\sqrt{7}-20}}\arctan \left ({\frac{1}{\sqrt{10\,\sqrt{5}\sqrt{7}-20}} \left ( -\sqrt{5}\sqrt{2\,\sqrt{5}\sqrt{7}+4}+10\,\sqrt{1+2\,x} \right ) } \right ) }+{\frac{\sqrt{7}\sqrt{2\,\sqrt{5}\sqrt{7}+4}}{62}\ln \left ( -\sqrt{2\,\sqrt{5}\sqrt{7}+4}\sqrt{5}\sqrt{1+2\,x}+\sqrt{5}\sqrt{7}+10\,x+5 \right ) }+{\frac{\sqrt{5} \left ( 2\,\sqrt{5}\sqrt{7}+4 \right ) \sqrt{7}}{31\,\sqrt{10\,\sqrt{5}\sqrt{7}-20}}\arctan \left ({\frac{1}{\sqrt{10\,\sqrt{5}\sqrt{7}-20}} \left ( -\sqrt{5}\sqrt{2\,\sqrt{5}\sqrt{7}+4}+10\,\sqrt{1+2\,x} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 \, x + 1}}{5 \, x^{2} + 3 \, x + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.65576, size = 1442, normalized size = 6.5 \begin{align*} \frac{1}{336350} \, \sqrt{155} 35^{\frac{1}{4}}{\left (2 \, \sqrt{35} \sqrt{31} - 35 \, \sqrt{31}\right )} \sqrt{4 \, \sqrt{35} + 70} \log \left (\frac{124}{7} \, \sqrt{155} 35^{\frac{3}{4}} \sqrt{31} \sqrt{2 \, x + 1} \sqrt{4 \, \sqrt{35} + 70} + 192200 \, x + 19220 \, \sqrt{35} + 96100\right ) - \frac{1}{336350} \, \sqrt{155} 35^{\frac{1}{4}}{\left (2 \, \sqrt{35} \sqrt{31} - 35 \, \sqrt{31}\right )} \sqrt{4 \, \sqrt{35} + 70} \log \left (-\frac{124}{7} \, \sqrt{155} 35^{\frac{3}{4}} \sqrt{31} \sqrt{2 \, x + 1} \sqrt{4 \, \sqrt{35} + 70} + 192200 \, x + 19220 \, \sqrt{35} + 96100\right ) - \frac{2}{5425} \, \sqrt{155} 35^{\frac{3}{4}} \sqrt{4 \, \sqrt{35} + 70} \arctan \left (\frac{1}{1177225} \, \sqrt{155} 35^{\frac{3}{4}} \sqrt{31} \sqrt{7} \sqrt{\sqrt{155} 35^{\frac{3}{4}} \sqrt{31} \sqrt{2 \, x + 1} \sqrt{4 \, \sqrt{35} + 70} + 10850 \, x + 1085 \, \sqrt{35} + 5425} \sqrt{4 \, \sqrt{35} + 70} - \frac{1}{1085} \, \sqrt{155} 35^{\frac{3}{4}} \sqrt{2 \, x + 1} \sqrt{4 \, \sqrt{35} + 70} - \frac{1}{31} \, \sqrt{35} \sqrt{31} - \frac{2}{31} \, \sqrt{31}\right ) - \frac{2}{5425} \, \sqrt{155} 35^{\frac{3}{4}} \sqrt{4 \, \sqrt{35} + 70} \arctan \left (\frac{1}{2354450} \, \sqrt{155} 35^{\frac{3}{4}} \sqrt{7} \sqrt{-124 \, \sqrt{155} 35^{\frac{3}{4}} \sqrt{31} \sqrt{2 \, x + 1} \sqrt{4 \, \sqrt{35} + 70} + 1345400 \, x + 134540 \, \sqrt{35} + 672700} \sqrt{4 \, \sqrt{35} + 70} - \frac{1}{1085} \, \sqrt{155} 35^{\frac{3}{4}} \sqrt{2 \, x + 1} \sqrt{4 \, \sqrt{35} + 70} + \frac{1}{31} \, \sqrt{35} \sqrt{31} + \frac{2}{31} \, \sqrt{31}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.5134, size = 32, normalized size = 0.14 \begin{align*} 4 \operatorname{RootSum}{\left (1230080 t^{4} + 1984 t^{2} + 7, \left ( t \mapsto t \log{\left (9920 t^{3} + 8 t + \sqrt{2 x + 1} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 \, x + 1}}{5 \, x^{2} + 3 \, x + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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