Optimal. Leaf size=279 \[ -\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]
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Rubi [A]
time = 0.43, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {717, 838, 840,
1183, 648, 632, 210, 642} \begin {gather*} \frac {1}{125} \sqrt {\frac {2}{155} \left (42875 \sqrt {35}-168698\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 )-\frac {1}{125} \sqrt {\frac {2}{155} \left (42875 \sqrt {35}-168698\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 {4}{25} (2 x+1)^{5/2}+\frac {16}{75} (2 x+1)^{3/2}-\frac {76}{125} \sqrt {2 x+1}-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 717
Rule 838
Rule 840
Rule 1183
Rubi steps
\begin {align*} \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx &=\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{5} \int \frac {(1+2 x)^{3/2} (-3+8 x)}{2+3 x+5 x^2} \, dx\\ &=\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{25} \int \frac {(-47-38 x) \sqrt {1+2 x}}{2+3 x+5 x^2} \, dx\\ &=-\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{125} \int \frac {-83-432 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=-\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {2}{125} \text {Subst}\left (\int \frac {266-432 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {\text {Subst}\left (\int \frac {266 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (266+432 \sqrt {\frac {7}{5}}\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 )}{125 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {266 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (266+432 \sqrt {\frac {7}{5}}\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 )}{125 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{625} \left (-216+19 \sqrt {35}\right ) \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 )+\frac {1}{625} \left (-216+19 \sqrt {35}\right ) \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 )-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \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 )+\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \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 )\\ &=-\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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}{625} \left (2 \left (216-19 \sqrt {35}\right )\right ) \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 )+\frac {1}{625} \left (2 \left (216-19 \sqrt {35}\right )\right ) \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 )\\ &=-\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \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}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \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}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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] Result contains complex when optimal does not.
time = 0.55, size = 124, normalized size = 0.44 \begin {gather*} \frac {1240 \sqrt {1+2 x} \left (-11+50 x+30 x^2\right )-6 \sqrt {155 \left (-168698-34021 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-6 \sqrt {155 \left (-168698+34021 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{58125} \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. \(406\) vs.
\(2(185)=370\).
time = 2.25, size = 407, normalized size = 1.46 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*} \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. 468 vs.
\(2 (188) = 376\).
time = 2.37, size = 468, normalized size = 1.68 \begin {gather*} \frac {1}{1752203762968750} \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (168698 \, \sqrt {35} \sqrt {31} + 1500625 \, \sqrt {31}\right )} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} \log \left (\frac {26582500}{34021} \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (216 \, \sqrt {35} \sqrt {31} + 665 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} + 353314653125000 \, x + 35331465312500 \, \sqrt {35} + 176657326562500\right ) - \frac {1}{1752203762968750} \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (168698 \, \sqrt {35} \sqrt {31} + 1500625 \, \sqrt {31}\right )} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} \log \left (-\frac {26582500}{34021} \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (216 \, \sqrt {35} \sqrt {31} + 665 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} + 353314653125000 \, x + 35331465312500 \, \sqrt {35} + 176657326562500\right ) + \frac {2}{830703125} \cdot 42875^{\frac {1}{4}} \sqrt {155} \sqrt {35} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} \arctan \left (\frac {1}{2044682140744622640625} \cdot 42875^{\frac {3}{4}} \sqrt {34021} \sqrt {217} \sqrt {155} \sqrt {42875^{\frac {1}{4}} \sqrt {155} {\left (216 \, \sqrt {35} \sqrt {31} + 665 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} + 452181616250 \, x + 45218161625 \, \sqrt {35} + 226090808125} {\left (19 \, \sqrt {35} + 216\right )} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} - \frac {1}{1582635656875} \cdot 42875^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} {\left (19 \, \sqrt {35} + 216\right )} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + \frac {2}{830703125} \cdot 42875^{\frac {1}{4}} \sqrt {155} \sqrt {35} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} \arctan \left (\frac {1}{715638749260617924218750} \cdot 42875^{\frac {3}{4}} \sqrt {34021} \sqrt {155} \sqrt {-26582500 \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (216 \, \sqrt {35} \sqrt {31} + 665 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} + 12020117813965625000 \, x + 1202011781396562500 \, \sqrt {35} + 6010058906982812500} {\left (19 \, \sqrt {35} + 216\right )} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} - \frac {1}{1582635656875} \cdot 42875^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} {\left (19 \, \sqrt {35} + 216\right )} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + \frac {8}{375} \, {\left (30 \, x^{2} + 50 \, x - 11\right )} \sqrt {2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 38.99, size = 292, normalized size = 1.05 \begin {gather*} \frac {4 \left (2 x + 1\right )^{\frac {5}{2}}}{25} + \frac {16 \left (2 x + 1\right )^{\frac {3}{2}}}{75} - \frac {76 \sqrt {2 x + 1}}{125} + 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 )} - \frac {408 \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 )}}{25} - \frac {84 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} - \frac {12 \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 )}}{5} + \frac {112 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{125} + \frac {504 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{25} + \frac {976 \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 )}}{125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 614 vs.
\(2 (188) = 376\).
time = 1.72, size = 614, normalized size = 2.20 \begin {gather*} \frac {4}{25} \, {\left (2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1}{1162984375} \, \sqrt {31} {\left (11340 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 54 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 108 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 22680 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 162925 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} - 325850 \, \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}{1162984375} \, \sqrt {31} {\left (11340 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 54 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 108 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 22680 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 162925 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} - 325850 \, \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}{2325968750} \, \sqrt {31} {\left (54 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 11340 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 22680 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 108 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} - 162925 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} + 325850 \, \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}{2325968750} \, \sqrt {31} {\left (54 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 11340 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 22680 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 108 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} - 162925 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} + 325850 \, \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 {16}{75} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} - \frac {76}{125} \, \sqrt {2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.04, size = 200, normalized size = 0.72 \begin {gather*} \frac {16\,{\left (2\,x+1\right )}^{3/2}}{75}-\frac {76\,\sqrt {2\,x+1}}{125}+\frac {4\,{\left (2\,x+1\right )}^{5/2}}{25}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}\,4354688{}\mathrm {i}}{6103515625\,\left (-\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}+\frac {8709376\,\sqrt {31}\,\sqrt {155}\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}}{189208984375\,\left (-\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}\right )\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,2{}\mathrm {i}}{19375}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}\,4354688{}\mathrm {i}}{6103515625\,\left (\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}-\frac {8709376\,\sqrt {31}\,\sqrt {155}\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}}{189208984375\,\left (\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}\right )\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,2{}\mathrm {i}}{19375} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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