Optimal. Leaf size=142 \[ -\frac {1}{4} \tanh ^{-1}\left (\frac {(1-3 y) \sqrt {-5 y^2-5 y+1}}{(1-5 y) \sqrt {-y^2-y+1}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {(3 y+4) \sqrt {-5 y^2-5 y+1}}{(5 y+6) \sqrt {-y^2-y+1}}\right )+\frac {9}{4} \tanh ^{-1}\left (\frac {(7 y+11) \sqrt {-5 y^2-5 y+1}}{3 (5 y+7) \sqrt {-y^2-y+1}}\right ) \]
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Rubi [F] time = 3.32, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(1+2 y) \sqrt {1-5 y-5 y^2}}{y (1+y) (2+y) \sqrt {1-y-y^2}} \, dy \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(1+2 y) \sqrt {1-5 y-5 y^2}}{y (1+y) (2+y) \sqrt {1-y-y^2}} \, dy &=\int \left (\frac {\sqrt {1-5 y-5 y^2}}{2 y \sqrt {1-y-y^2}}+\frac {\sqrt {1-5 y-5 y^2}}{(1+y) \sqrt {1-y-y^2}}-\frac {3 \sqrt {1-5 y-5 y^2}}{2 (2+y) \sqrt {1-y-y^2}}\right ) \, dy\\ &=\frac {1}{2} \int \frac {\sqrt {1-5 y-5 y^2}}{y \sqrt {1-y-y^2}} \, dy-\frac {3}{2} \int \frac {\sqrt {1-5 y-5 y^2}}{(2+y) \sqrt {1-y-y^2}} \, dy+\int \frac {\sqrt {1-5 y-5 y^2}}{(1+y) \sqrt {1-y-y^2}} \, dy\\ \end {align*}
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Mathematica [C] time = 1.57, size = 630, normalized size = 4.44 \[ \frac {\left (-1-\frac {2}{\sqrt {5}}\right ) \left (2 y+\sqrt {5}+1\right )^2 \sqrt {\frac {10 y+3 \sqrt {5}+5}{10 y+5 \sqrt {5}+5}} \left (20 \left (\sqrt {5} \sqrt {\frac {-10 y+3 \sqrt {5}-5}{2 y+\sqrt {5}+1}} \sqrt {\frac {-2 y+\sqrt {5}-1}{2 y+\sqrt {5}+1}}-4 \sqrt {\frac {-10 y+3 \sqrt {5}-5}{2 y+\sqrt {5}+1}} \sqrt {\frac {-2 y+\sqrt {5}-1}{2 y+\sqrt {5}+1}}-2 \sqrt {5} \sqrt {-\frac {2 \sqrt {5} y+\sqrt {5}-5}{2 y+\sqrt {5}+1}} \sqrt {-\frac {2 \sqrt {5} y+\sqrt {5}-3}{2 y+\sqrt {5}+1}}+5 \sqrt {-\frac {2 \sqrt {5} y+\sqrt {5}-5}{2 y+\sqrt {5}+1}} \sqrt {-\frac {2 \sqrt {5} y+\sqrt {5}-3}{2 y+\sqrt {5}+1}}\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {2 \sqrt {\frac {10 y+3 \sqrt {5}+5}{2 y+\sqrt {5}+1}}}{\sqrt {15}}\right ),\frac {15}{16}\right )+\sqrt {\frac {-10 y+3 \sqrt {5}-5}{2 y+\sqrt {5}+1}} \sqrt {\frac {-2 y+\sqrt {5}-1}{2 y+\sqrt {5}+1}} \left (9 \sqrt {5} \Pi \left (\frac {5}{8}-\frac {\sqrt {5}}{8};\sin ^{-1}\left (\frac {2 \sqrt {\frac {10 y+3 \sqrt {5}+5}{2 y+\sqrt {5}+1}}}{\sqrt {15}}\right )|\frac {15}{16}\right )+\left (9 \sqrt {5}-20\right ) \Pi \left (-\frac {3}{8} \left (-5+\sqrt {5}\right );\sin ^{-1}\left (\frac {2 \sqrt {\frac {10 y+3 \sqrt {5}+5}{2 y+\sqrt {5}+1}}}{\sqrt {15}}\right )|\frac {15}{16}\right )+2 \sqrt {5} \Pi \left (\frac {3}{8} \left (5+\sqrt {5}\right );\sin ^{-1}\left (\frac {2 \sqrt {\frac {10 y+3 \sqrt {5}+5}{2 y+\sqrt {5}+1}}}{\sqrt {15}}\right )|\frac {15}{16}\right )\right )\right )}{16 \sqrt {-5 y^2-5 y+1} \sqrt {-y^2-y+1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.56, size = 223, normalized size = 1.57 \[ \frac {9}{8} \, \log \left (-\frac {235 \, y^{4} + 935 \, y^{3} - 3 \, {\left (35 \, y^{2} + 104 \, y + 77\right )} \sqrt {-y^{2} - y + 1} \sqrt {-5 \, y^{2} - 5 \, y + 1} + 1086 \, y^{2} + 131 \, y - 281}{y^{4} + 8 \, y^{3} + 24 \, y^{2} + 32 \, y + 16}\right ) + \frac {1}{4} \, \log \left (\frac {35 \, y^{4} + 125 \, y^{3} + {\left (15 \, y^{2} + 38 \, y + 24\right )} \sqrt {-y^{2} - y + 1} \sqrt {-5 \, y^{2} - 5 \, y + 1} + 131 \, y^{2} + 16 \, y - 26}{y^{4} + 4 \, y^{3} + 6 \, y^{2} + 4 \, y + 1}\right ) + \frac {1}{8} \, \log \left (\frac {35 \, y^{4} + 15 \, y^{3} + {\left (15 \, y^{2} - 8 \, y + 1\right )} \sqrt {-y^{2} - y + 1} \sqrt {-5 \, y^{2} - 5 \, y + 1} - 34 \, y^{2} + 11 \, y - 1}{y^{4}}\right ) \]
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 \[ \int \frac {\sqrt {-5 \, y^{2} - 5 \, y + 1} {\left (2 \, y + 1\right )}}{\sqrt {-y^{2} - y + 1} {\left (y + 2\right )} {\left (y + 1\right )} y}\,{d y} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 352, normalized size = 2.48 \[ -\frac {300 \sqrt {-5 y^{2}-5 y +1}\, \sqrt {-y^{2}-y +1}\, \sqrt {-\frac {10 y +5+3 \sqrt {5}}{-10 y -5+3 \sqrt {5}}}\, \left (-10 y -5+3 \sqrt {5}\right )^{2} \sqrt {\frac {-2 y -1+\sqrt {5}}{-10 y -5+3 \sqrt {5}}}\, \sqrt {5}\, \sqrt {\frac {2 y +1+\sqrt {5}}{-10 y -5+3 \sqrt {5}}}\, \left (-3 \EllipticPi \left (2 \sqrt {-\frac {10 y +5+3 \sqrt {5}}{-10 y -5+3 \sqrt {5}}}, -\frac {5+\sqrt {5}}{4 \left (\sqrt {5}-5\right )}, \frac {1}{4}\right )+2 \EllipticPi \left (2 \sqrt {-\frac {10 y +5+3 \sqrt {5}}{-10 y -5+3 \sqrt {5}}}, -\frac {5+3 \sqrt {5}}{4 \left (3 \sqrt {5}-5\right )}, \frac {1}{4}\right )+\EllipticPi \left (2 \sqrt {-\frac {10 y +5+3 \sqrt {5}}{-10 y -5+3 \sqrt {5}}}, -\frac {3 \sqrt {5}-5}{4 \left (5+3 \sqrt {5}\right )}, \frac {1}{4}\right )\right )}{\sqrt {5 y^{4}+10 y^{3}-y^{2}-6 y +1}\, \sqrt {\left (10 y +5+3 \sqrt {5}\right ) \left (-10 y -5+3 \sqrt {5}\right ) \left (-2 y -1+\sqrt {5}\right ) \left (2 y +1+\sqrt {5}\right )}\, \left (5+3 \sqrt {5}\right ) \left (3 \sqrt {5}-5\right ) \left (5+\sqrt {5}\right ) \left (\sqrt {5}-5\right )} \]
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 \[ \int \frac {\sqrt {-5 \, y^{2} - 5 \, y + 1} {\left (2 \, y + 1\right )}}{\sqrt {-y^{2} - y + 1} {\left (y + 2\right )} {\left (y + 1\right )} y}\,{d y} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (2\,y+1\right )\,\sqrt {-5\,y^2-5\,y+1}}{y\,\left (y+1\right )\,\left (y+2\right )\,\sqrt {-y^2-y+1}} \,d y \]
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 \[ \int \frac {\left (2 y + 1\right ) \sqrt {- 5 y^{2} - 5 y + 1}}{y \left (y + 1\right ) \left (y + 2\right ) \sqrt {- y^{2} - y + 1}}\, dy \]
Verification of antiderivative is not currently implemented for this CAS.
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