Optimal. Leaf size=89 \[ \frac {1}{4} \text {ArcTan}\left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )-\frac {1}{4} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x+x^2+x^3}}{1+x+x^2}\right ) \]
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Rubi [A]
time = 1.96, antiderivative size = 155, normalized size of antiderivative = 1.74, number of steps
used = 41, number of rules used = 18, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {2081, 6857,
932, 27, 6865, 1726, 1211, 1117, 1209, 1714, 1712, 212, 12, 1224, 209, 1728, 1722, 1720}
\begin {gather*} \frac {\sqrt {x^3+x^2+x} \text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt {x^2+x+1}}\right )}{4 \sqrt {x} \sqrt {x^2+x+1}}+\frac {\sqrt {x^3+x^2+x} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x^2+x+1}}\right )}{2 \sqrt {x} \sqrt {x^2+x+1}}-\frac {\sqrt {3} \sqrt {x^3+x^2+x} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x}}{\sqrt {x^2+x+1}}\right )}{4 \sqrt {x} \sqrt {x^2+x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 209
Rule 212
Rule 932
Rule 1117
Rule 1209
Rule 1211
Rule 1224
Rule 1712
Rule 1714
Rule 1720
Rule 1722
Rule 1726
Rule 1728
Rule 2081
Rule 6857
Rule 6865
Rubi steps
\begin {align*} \int \frac {\sqrt {x+x^2+x^3}}{-1+x^4} \, dx &=\frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{-1+x^4} \, dx}{\sqrt {x} \sqrt {1+x+x^2}}\\ &=\frac {\sqrt {x+x^2+x^3} \int \left (-\frac {\sqrt {x} \sqrt {1+x+x^2}}{2 \left (1-x^2\right )}-\frac {\sqrt {x} \sqrt {1+x+x^2}}{2 \left (1+x^2\right )}\right ) \, dx}{\sqrt {x} \sqrt {1+x+x^2}}\\ &=-\frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{1-x^2} \, dx}{2 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{1+x^2} \, dx}{2 \sqrt {x} \sqrt {1+x+x^2}}\\ &=-\frac {\sqrt {x+x^2+x^3} \int \left (\frac {i \sqrt {x} \sqrt {1+x+x^2}}{2 (i-x)}+\frac {i \sqrt {x} \sqrt {1+x+x^2}}{2 (i+x)}\right ) \, dx}{2 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \left (\frac {\sqrt {x} \sqrt {1+x+x^2}}{2 (1-x)}+\frac {\sqrt {x} \sqrt {1+x+x^2}}{2 (1+x)}\right ) \, dx}{2 \sqrt {x} \sqrt {1+x+x^2}}\\ &=-\frac {\left (i \sqrt {x+x^2+x^3}\right ) \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{i-x} \, dx}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (i \sqrt {x+x^2+x^3}\right ) \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{i+x} \, dx}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{1-x} \, dx}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{1+x} \, dx}{4 \sqrt {x} \sqrt {1+x+x^2}}\\ &=\frac {\left (i \sqrt {x+x^2+x^3}\right ) \int \frac {i-(2-2 i) x-(1-3 i) x^2}{\sqrt {x} (i+x) \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (i \sqrt {x+x^2+x^3}\right ) \int \frac {i+(2+2 i) x+(1+3 i) x^2}{(i-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \int \frac {1+2 x^2}{\sqrt {x} (1+x) \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {1+4 x+4 x^2}{(1-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}}\\ &=\frac {\left (i \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {i-(2-2 i) x^2-(1-3 i) x^4}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (i \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {i+(2+2 i) x^2+(1+3 i) x^4}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {(1+2 x)^2}{(1-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {1+2 x^4}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}\\ &=-\left (-\frac {\left (\left (\frac {1}{2}-\frac {i}{6}\right ) \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}\right )+\frac {\left (i \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {-3-6 i x^2}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (i \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {-3+6 i x^2}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {3}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {\left (1+2 x^2\right )^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (\left (\frac {1}{2}+\frac {i}{6}\right ) \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}\\ &=-\frac {2 \sqrt {x+x^2+x^3}}{3 (1+x)}+\frac {2 (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \sqrt {x+x^2+x^3} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{3 \sqrt {x} \left (1+x+x^2\right )}+-\frac {\left (\left (\frac {3}{4}+\frac {i}{4}\right ) \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}+-\frac {\left (\left (\frac {3}{4}-\frac {i}{4}\right ) \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}+-\frac {\left (\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1+x^2}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {5+4 x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1+x^2}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (2 \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1+x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x+x^2}}\\ &=-\frac {2 \sqrt {x+x^2+x^3}}{3 (1+x)}+\frac {\sqrt {x+x^2+x^3} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{2 \sqrt {x} \sqrt {1+x+x^2}}+\frac {2 (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \sqrt {x+x^2+x^3} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{3 \sqrt {x} \left (1+x+x^2\right )}-\frac {3 (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \sqrt {x+x^2+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{4 \sqrt {x} \left (1+x+x^2\right )}-\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{12 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (2 \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (3 \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (4 \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x+x^2}}\\ &=\frac {\sqrt {x+x^2+x^3} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{2 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (3 \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1}{1-3 x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}\\ &=\frac {\sqrt {x+x^2+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{2 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {3} \sqrt {x+x^2+x^3} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x}}{\sqrt {1+x+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 97, normalized size = 1.09 \begin {gather*} \frac {\sqrt {x} \sqrt {1+x+x^2} \left (\text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )+2 \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )-\sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x}}{\sqrt {1+x+x^2}}\right )\right )}{4 \sqrt {x \left (1+x+x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.96, size = 1788, normalized size = 20.09
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{3}+x^{2}+x}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right )^{2}}\right )}{8}-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+4 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +\RootOf \left (\textit {\_Z}^{2}-3\right )+6 \sqrt {x^{3}+x^{2}+x}}{\left (-1+x \right )^{2}}\right )}{8}+\frac {\ln \left (\frac {x^{2}+2 \sqrt {x^{3}+x^{2}+x}+2 x +1}{x^{2}+1}\right )}{4}\) | \(133\) |
elliptic | \(\text {Expression too large to display}\) | \(1500\) |
default | \(\text {Expression too large to display}\) | \(1788\) |
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 = 0.45, size = 120, normalized size = 1.35 \begin {gather*} \frac {1}{16} \, \sqrt {3} \log \left (\frac {x^{4} + 20 \, x^{3} - 4 \, \sqrt {3} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 4 \, x + 1\right )} + 30 \, x^{2} + 20 \, x + 1}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) - \frac {1}{8} \, \arctan \left (\frac {x^{2} + 1}{2 \, \sqrt {x^{3} + x^{2} + x}}\right ) + \frac {1}{4} \, \log \left (\frac {x^{2} + 2 \, x + 2 \, \sqrt {x^{3} + x^{2} + x} + 1}{x^{2} + 1}\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 {\sqrt {x \left (x^{2} + x + 1\right )}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 565, normalized size = 6.35 \begin {gather*} \frac {\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (-\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{2\,\sqrt {x^3+x^2-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,x}}+\frac {\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{2\,\sqrt {x^3+x^2-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,x}}+\frac {\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{2\,\sqrt {x^3+x^2-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,x}}-\frac {3\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{2\,\sqrt {x^3+x^2-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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