Optimal. Leaf size=218 \[ -\sqrt {\frac {2}{217} \left (2+\sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\sqrt {\frac {2}{217} \left (2+\sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {\log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{\sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{\sqrt {14 \left (2+\sqrt {35}\right )}} \]
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Rubi [A]
time = 0.19, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {721, 1108,
648, 632, 210, 642} \begin {gather*} -\sqrt {\frac {2}{217} \left (2+\sqrt {35}\right )} \text {ArcTan}\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}{217} \left (2+\sqrt {35}\right )} \text {ArcTan}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {\log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{\sqrt {14 \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 {14 \left (2+\sqrt {35}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 721
Rule 1108
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx &=4 \text {Subst}\left (\int \frac {1}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=\sqrt {\frac {2}{7 \left (2+\sqrt {35}\right )}} \text {Subst}\left (\int \frac {\sqrt {\frac {2}{5} \left (2+\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 )+\sqrt {\frac {2}{7 \left (2+\sqrt {35}\right )}} \text {Subst}\left (\int \frac {\sqrt {\frac {2}{5} \left (2+\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 )\\ &=\frac {\text {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 )}{\sqrt {35}}+\frac {\text {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 )}{\sqrt {35}}-\frac {\text {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 {14 \left (2+\sqrt {35}\right )}}+\frac {\text {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 {14 \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 {14 \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 {14 \left (2+\sqrt {35}\right )}}-\frac {2 \text {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 {35}}-\frac {2 \text {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 {35}}\\ &=-\sqrt {\frac {2}{7 \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}{7 \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 {14 \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 {14 \left (2+\sqrt {35}\right )}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.22, size = 100, normalized size = 0.46 \begin {gather*} \frac {2 \left (\sqrt {2+i \sqrt {31}} \tan ^{-1}\left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+\sqrt {2-i \sqrt {31}} \tan ^{-1}\left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{\sqrt {217}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(379\) vs.
\(2(157)=314\).
time = 1.79, size = 380, normalized size = 1.74
method | result | size |
derivativedivides | \(-\frac {\left (35 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}-10 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+\sqrt {5}\, \sqrt {7}+10 x +5\right )}{2170}-\frac {2 \left (-62 \sqrt {5}\, \sqrt {7}+\frac {\left (35 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}-10 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}}{10}\right ) \arctan \left (\frac {-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{217 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (35 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}-10 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+10 x +5+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}\right )}{2170}+\frac {2 \left (62 \sqrt {5}\, \sqrt {7}-\frac {\left (35 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}-10 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}}{10}\right ) \arctan \left (\frac {10 \sqrt {2 x +1}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{217 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(380\) |
default | \(-\frac {\left (35 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}-10 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+\sqrt {5}\, \sqrt {7}+10 x +5\right )}{2170}-\frac {2 \left (-62 \sqrt {5}\, \sqrt {7}+\frac {\left (35 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}-10 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}}{10}\right ) \arctan \left (\frac {-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{217 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (35 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}-10 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+10 x +5+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}\right )}{2170}+\frac {2 \left (62 \sqrt {5}\, \sqrt {7}-\frac {\left (35 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}-10 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}}{10}\right ) \arctan \left (\frac {10 \sqrt {2 x +1}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{217 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(380\) |
trager | \(\RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right ) \ln \left (-\frac {517979 x \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{5}-20615 x \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{3}-19096 \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{3}+651 \sqrt {2 x +1}\, \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{2}+200 \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right ) x +320 \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )+145 \sqrt {2 x +1}}{217 \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{2} x +5 x +4}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}+47089 \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{2}+868\right ) \ln \left (-\frac {73997 \RootOf \left (\textit {\_Z}^{2}+47089 \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{2}+868\right ) \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{4} x +5673 \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{2} \RootOf \left (\textit {\_Z}^{2}+47089 \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{2}+868\right ) x +2728 \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{2} \RootOf \left (\textit {\_Z}^{2}+47089 \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{2}+868\right )-20181 \sqrt {2 x +1}\, \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{2}+108 \RootOf \left (\textit {\_Z}^{2}+47089 \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{2}+868\right ) x +96 \RootOf \left (\textit {\_Z}^{2}+47089 \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{2}+868\right )+4123 \sqrt {2 x +1}}{217 \RootOf \left (6727 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+5\right )^{2} x -x -4}\right )}{217}\) | \(422\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 365 vs.
\(2 (160) = 320\).
time = 2.70, size = 365, normalized size = 1.67 \begin {gather*} -\frac {1}{470890} \, \sqrt {217} 35^{\frac {1}{4}} {\left (2 \, \sqrt {35} \sqrt {31} - 35 \, \sqrt {31}\right )} \sqrt {4 \, \sqrt {35} + 70} \log \left (4340 \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {31} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} + 9417800 \, x + 941780 \, \sqrt {35} + 4708900\right ) + \frac {1}{470890} \, \sqrt {217} 35^{\frac {1}{4}} {\left (2 \, \sqrt {35} \sqrt {31} - 35 \, \sqrt {31}\right )} \sqrt {4 \, \sqrt {35} + 70} \log \left (-4340 \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {31} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} + 9417800 \, x + 941780 \, \sqrt {35} + 4708900\right ) - \frac {2}{7595} \, \sqrt {217} 35^{\frac {3}{4}} \sqrt {4 \, \sqrt {35} + 70} \arctan \left (\frac {1}{235445} \, \sqrt {1085} \sqrt {217} 35^{\frac {1}{4}} \sqrt {\sqrt {217} 35^{\frac {1}{4}} \sqrt {31} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} + 2170 \, x + 217 \, \sqrt {35} + 1085} \sqrt {4 \, \sqrt {35} + 70} - \frac {1}{217} \, \sqrt {217} 35^{\frac {1}{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}{7595} \, \sqrt {217} 35^{\frac {3}{4}} \sqrt {4 \, \sqrt {35} + 70} \arctan \left (\frac {1}{470890} \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {-4340 \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {31} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} + 9417800 \, x + 941780 \, \sqrt {35} + 4708900} \sqrt {4 \, \sqrt {35} + 70} - \frac {1}{217} \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) \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} \cdot \left (5 x^{2} + 3 x + 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.01, size = 279, normalized size = 1.28 \begin {gather*} \frac {1}{7595} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450}\right )} \arctan \left (\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) + \frac {1}{7595} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450}\right )} \arctan \left (-\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} - \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) + \frac {1}{15190} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450}\right )} \log \left (2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) - \frac {1}{15190} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450}\right )} \log \left (-2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.02, size = 167, normalized size = 0.77 \begin {gather*} \frac {\sqrt {217}\,\mathrm {atan}\left (\frac {256\,\sqrt {7}\,\sqrt {-2-\sqrt {31}\,1{}\mathrm {i}}\,\sqrt {2\,x+1}}{6125\,\left (\frac {256}{875}+\frac {\sqrt {31}\,128{}\mathrm {i}}{875}\right )}+\frac {\sqrt {217}\,\sqrt {-2-\sqrt {31}\,1{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{6125\,\left (\frac {256}{875}+\frac {\sqrt {31}\,128{}\mathrm {i}}{875}\right )}\right )\,\sqrt {-2-\sqrt {31}\,1{}\mathrm {i}}\,2{}\mathrm {i}}{217}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {256\,\sqrt {7}\,\sqrt {-2+\sqrt {31}\,1{}\mathrm {i}}\,\sqrt {2\,x+1}}{6125\,\left (-\frac {256}{875}+\frac {\sqrt {31}\,128{}\mathrm {i}}{875}\right )}-\frac {\sqrt {217}\,\sqrt {-2+\sqrt {31}\,1{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{6125\,\left (-\frac {256}{875}+\frac {\sqrt {31}\,128{}\mathrm {i}}{875}\right )}\right )\,\sqrt {-2+\sqrt {31}\,1{}\mathrm {i}}\,2{}\mathrm {i}}{217} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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