Optimal. Leaf size=23 \[ \log \left (x^2+\sqrt {x^4+2 x^3+x^2+3}+x\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 11, normalized size of antiderivative = 0.48, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1680, 12, 1107, 619, 215} \begin {gather*} \sinh ^{-1}\left (\frac {x (x+1)}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 215
Rule 619
Rule 1107
Rule 1680
Rubi steps
\begin {align*} \int \frac {1+2 x}{\sqrt {3+x^2+2 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {8 x}{\sqrt {49-8 x^2+16 x^4}} \, dx,x,\frac {1}{2}+x\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x}{\sqrt {49-8 x^2+16 x^4}} \, dx,x,\frac {1}{2}+x\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {49-8 x+16 x^2}} \, dx,x,\left (\frac {1}{2}+x\right )^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3072}}} \, dx,x,32 x (1+x)\right )}{32 \sqrt {3}}\\ &=\sinh ^{-1}\left (\frac {x (1+x)}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [C] time = 0.59, size = 305, normalized size = 13.26 \begin {gather*} -\frac {(-1)^{2/3} \sqrt {2} \sqrt [4]{3} \sqrt {\frac {2 i x+\sqrt {3}+3 i}{x+(-1)^{2/3}}} \left (x+(-1)^{2/3}\right )^2 \sqrt {\frac {2 \sqrt {3} x+3 \sqrt {3}+3 i}{\left (\sqrt {3}-2 i\right ) \left (x+(-1)^{2/3}\right )}} \sqrt {\frac {\left (4-2 i \sqrt {3}\right ) x-i \sqrt {3}-5}{x+(-1)^{2/3}}} \left (\left (\sqrt {3}+2 i\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {2 \left (2 i+\sqrt {3}\right ) x+\sqrt {3}-5 i}{-2 i x+\sqrt {3}+i}}}{\sqrt {2}}\right )|\frac {4}{7}\right )-2 \sqrt {3} \Pi \left (\frac {2 i}{2 i+\sqrt {3}};\sin ^{-1}\left (\frac {\sqrt {-\frac {2 \left (2 i+\sqrt {3}\right ) x+\sqrt {3}-5 i}{-2 i x+\sqrt {3}+i}}}{\sqrt {2}}\right )|\frac {4}{7}\right )\right )}{\left (\sqrt {3}+9 i\right ) \sqrt {x^4+2 x^3+x^2+3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.07, size = 23, normalized size = 1.00 \begin {gather*} \log \left (x+x^2+\sqrt {3+x^2+2 x^3+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 21, normalized size = 0.91 \begin {gather*} \log \left (x^{2} + x + \sqrt {x^{4} + 2 \, x^{3} + x^{2} + 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 23, normalized size = 1.00 \begin {gather*} -\log \left (-x^{2} - x + \sqrt {{\left (x^{2} + x\right )}^{2} + 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 28, normalized size = 1.22
method | result | size |
trager | \(-\ln \left (-x^{2}+\sqrt {x^{4}+2 x^{3}+x^{2}+3}-x \right )\) | \(28\) |
default | \(\frac {i \left (2-i \sqrt {3}\right ) \sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {2}\, \sqrt {-\frac {i \sqrt {3}\, \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}{x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \EllipticF \left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )}{6 \sqrt {\left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {i \left (2-i \sqrt {3}\right ) \sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {2}\, \sqrt {-\frac {i \sqrt {3}\, \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}{x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \left (\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )-i \sqrt {3}\, \EllipticPi \left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, 1-\frac {i \sqrt {3}}{2}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )\right )}{3 \sqrt {\left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(528\) |
elliptic | \(\frac {i \left (2-i \sqrt {3}\right ) \sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {2}\, \sqrt {-\frac {i \sqrt {3}\, \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}{x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \EllipticF \left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )}{6 \sqrt {\left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {i \left (2-i \sqrt {3}\right ) \sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {2}\, \sqrt {-\frac {i \sqrt {3}\, \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}{x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \left (\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )-i \sqrt {3}\, \EllipticPi \left (\sqrt {-\frac {2 \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (-2+i \sqrt {3}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, 1-\frac {i \sqrt {3}}{2}, \frac {\sqrt {\left (2+i \sqrt {3}\right ) \left (2-i \sqrt {3}\right )}}{2}\right )\right )}{3 \sqrt {\left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(528\) |
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 {2 \, x + 1}{\sqrt {x^{4} + 2 \, x^{3} + x^{2} + 3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {2\,x+1}{\sqrt {x^4+2\,x^3+x^2+3}} \,d x \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 {2 x + 1}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + 3 x + 3\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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