Optimal. Leaf size=270 \[ \frac {\sqrt {2 x+1} (20 x+37)}{217 \left (5 x^2+3 x+2\right )}-\frac {1}{217} \sqrt {\frac {1}{434} \left (10325 \sqrt {35}-32678\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{217} \sqrt {\frac {1}{434} \left (10325 \sqrt {35}-32678\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{217} \sqrt {\frac {2}{217} \left (32678+10325 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{217} \sqrt {\frac {2}{217} \left (32678+10325 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]
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Rubi [A] time = 0.32, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {740, 826, 1169, 634, 618, 204, 628} \begin {gather*} \frac {\sqrt {2 x+1} (20 x+37)}{217 \left (5 x^2+3 x+2\right )}-\frac {1}{217} \sqrt {\frac {1}{434} \left (10325 \sqrt {35}-32678\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{217} \sqrt {\frac {1}{434} \left (10325 \sqrt {35}-32678\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{217} \sqrt {\frac {2}{217} \left (32678+10325 \sqrt {35}\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 )+\frac {1}{217} \sqrt {\frac {2}{217} \left (32678+10325 \sqrt {35}\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 740
Rule 826
Rule 1169
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx &=\frac {\sqrt {1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}+\frac {1}{217} \int \frac {107+20 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}+\frac {2}{217} \operatorname {Subst}\left (\int \frac {194+20 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {194 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (194-4 \sqrt {35}\right ) x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{217 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {194 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (194-4 \sqrt {35}\right ) x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{217 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}+\frac {\left (70+97 \sqrt {35}\right ) \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 )}{7595}+\frac {\left (70+97 \sqrt {35}\right ) \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 )}{7595}-\frac {1}{217} \sqrt {\frac {1}{434} \left (-32678+10325 \sqrt {35}\right )} \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 )+\frac {1}{217} \sqrt {\frac {1}{434} \left (-32678+10325 \sqrt {35}\right )} \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 )\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}-\frac {1}{217} \sqrt {\frac {1}{434} \left (-32678+10325 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{217} \sqrt {\frac {1}{434} \left (-32678+10325 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )-\frac {\left (2 \left (70+97 \sqrt {35}\right )\right ) \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 )}{7595}-\frac {\left (2 \left (70+97 \sqrt {35}\right )\right ) \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 )}{7595}\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}-\frac {1}{217} \sqrt {\frac {2}{217} \left (32678+10325 \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 {1}{217} \sqrt {\frac {2}{217} \left (32678+10325 \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 {1}{217} \sqrt {\frac {1}{434} \left (-32678+10325 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{217} \sqrt {\frac {1}{434} \left (-32678+10325 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )\\ \end {align*}
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Mathematica [C] time = 0.32, size = 166, normalized size = 0.61 \begin {gather*} \frac {1}{217} \left (\frac {\sqrt {2 x+1} (20 x+37)}{5 x^2+3 x+2}+\frac {2 \sqrt {10-5 i \sqrt {31}} \left (101 \sqrt {31}-62 i\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )}{31 \left (\sqrt {31}+2 i\right )}+\frac {2 \sqrt {10+5 i \sqrt {31}} \left (101 \sqrt {31}+62 i\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )}{31 \left (\sqrt {31}-2 i\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.75, size = 149, normalized size = 0.55 \begin {gather*} \frac {4 \sqrt {2 x+1} (10 (2 x+1)+27)}{217 \left (5 (2 x+1)^2-4 (2 x+1)+7\right )}+\frac {2}{217} \sqrt {\frac {1}{217} \left (32678+9269 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}-\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right )+\frac {2}{217} \sqrt {\frac {1}{217} \left (32678-9269 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}+\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 572, normalized size = 2.12 \begin {gather*} -\frac {1149356 \cdot 5969915^{\frac {1}{4}} \sqrt {826} \sqrt {217} \sqrt {35} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {32678 \, \sqrt {35} + 361375} \arctan \left (\frac {1}{229833696387700298075} \cdot 5969915^{\frac {3}{4}} \sqrt {826} \sqrt {299} \sqrt {295} \sqrt {217} \sqrt {5969915^{\frac {1}{4}} \sqrt {826} \sqrt {217} {\left (2 \, \sqrt {35} \sqrt {31} - 97 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {32678 \, \sqrt {35} + 361375} + 8306970490 \, x + 830697049 \, \sqrt {35} + 4153485245} \sqrt {32678 \, \sqrt {35} + 361375} {\left (97 \, \sqrt {35} - 70\right )} - \frac {1}{12007725843295} \cdot 5969915^{\frac {3}{4}} \sqrt {826} \sqrt {217} \sqrt {2 \, x + 1} \sqrt {32678 \, \sqrt {35} + 361375} {\left (97 \, \sqrt {35} - 70\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 1149356 \cdot 5969915^{\frac {1}{4}} \sqrt {826} \sqrt {217} \sqrt {35} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {32678 \, \sqrt {35} + 361375} \arctan \left (\frac {1}{80441793735695104326250} \cdot 5969915^{\frac {3}{4}} \sqrt {826} \sqrt {299} \sqrt {217} \sqrt {-36137500 \cdot 5969915^{\frac {1}{4}} \sqrt {826} \sqrt {217} {\left (2 \, \sqrt {35} \sqrt {31} - 97 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {32678 \, \sqrt {35} + 361375} + 300193146082375000 \, x + 30019314608237500 \, \sqrt {35} + 150096573041187500} \sqrt {32678 \, \sqrt {35} + 361375} {\left (97 \, \sqrt {35} - 70\right )} - \frac {1}{12007725843295} \cdot 5969915^{\frac {3}{4}} \sqrt {826} \sqrt {217} \sqrt {2 \, x + 1} \sqrt {32678 \, \sqrt {35} + 361375} {\left (97 \, \sqrt {35} - 70\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) - 5969915^{\frac {1}{4}} \sqrt {826} \sqrt {217} {\left (32678 \, \sqrt {35} \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} - 361375 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )}\right )} \sqrt {32678 \, \sqrt {35} + 361375} \log \left (\frac {36137500}{299} \cdot 5969915^{\frac {1}{4}} \sqrt {826} \sqrt {217} {\left (2 \, \sqrt {35} \sqrt {31} - 97 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {32678 \, \sqrt {35} + 361375} + 1003990455125000 \, x + 100399045512500 \, \sqrt {35} + 501995227562500\right ) + 5969915^{\frac {1}{4}} \sqrt {826} \sqrt {217} {\left (32678 \, \sqrt {35} \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} - 361375 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )}\right )} \sqrt {32678 \, \sqrt {35} + 361375} \log \left (-\frac {36137500}{299} \cdot 5969915^{\frac {1}{4}} \sqrt {826} \sqrt {217} {\left (2 \, \sqrt {35} \sqrt {31} - 97 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {32678 \, \sqrt {35} + 361375} + 1003990455125000 \, x + 100399045512500 \, \sqrt {35} + 501995227562500\right ) - 1802612596330 \, {\left (20 \, x + 37\right )} \sqrt {2 \, x + 1}}{391166933403610 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.91, size = 622, normalized size = 2.30
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.93, size = 968, normalized size = 3.59
result too large to display
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*} \int \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2} \sqrt {2 \, x + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 207, normalized size = 0.77 \begin {gather*} \frac {\frac {108\,\sqrt {2\,x+1}}{1085}+\frac {8\,{\left (2\,x+1\right )}^{3/2}}{217}}{{\left (2\,x+1\right )}^2-\frac {8\,x}{5}+\frac {3}{5}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-32678-\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}\,38272{}\mathrm {i}}{2018940875\,\left (\frac {10103808}{288420125}+\frac {\sqrt {31}\,3712384{}\mathrm {i}}{288420125}\right )}+\frac {76544\,\sqrt {31}\,\sqrt {217}\,\sqrt {-32678-\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}}{62587167125\,\left (\frac {10103808}{288420125}+\frac {\sqrt {31}\,3712384{}\mathrm {i}}{288420125}\right )}\right )\,\sqrt {-32678-\sqrt {31}\,9269{}\mathrm {i}}\,2{}\mathrm {i}}{47089}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-32678+\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}\,38272{}\mathrm {i}}{2018940875\,\left (-\frac {10103808}{288420125}+\frac {\sqrt {31}\,3712384{}\mathrm {i}}{288420125}\right )}-\frac {76544\,\sqrt {31}\,\sqrt {217}\,\sqrt {-32678+\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}}{62587167125\,\left (-\frac {10103808}{288420125}+\frac {\sqrt {31}\,3712384{}\mathrm {i}}{288420125}\right )}\right )\,\sqrt {-32678+\sqrt {31}\,9269{}\mathrm {i}}\,2{}\mathrm {i}}{47089} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {2 x + 1} \left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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