Optimal. Leaf size=222 \[ -\frac {3 \sqrt [3]{2} \sqrt [3]{4 x^2-4 x+1}}{7 (2 x-1)}+\frac {1}{7} \sqrt [3]{\frac {2}{7}} \log \left (\left (4 x^2-4 x+1\right )^{2/3}+2 \sqrt [3]{7} x-\sqrt [3]{7}\right )-\frac {\log \left (\left (4 x^2-4 x+1\right )^{2/3}+7^{2/3} \sqrt [3]{4 x^2-4 x+1}-2 \sqrt [3]{7} x+\sqrt [3]{7}\right )}{7\ 2^{2/3} \sqrt [3]{7}}+\frac {1}{7} \sqrt [3]{\frac {2}{7}} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (4 x^2-4 x+1\right )^{2/3}}{\left (4 x^2-4 x+1\right )^{2/3}-4 \sqrt [3]{7} x+2 \sqrt [3]{7}}\right )-\frac {2}{21} \sqrt [3]{\frac {2}{7}} \log (2 x-1) \]
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Rubi [A] time = 0.11, antiderivative size = 196, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {12, 646, 51, 56, 617, 204, 31} \begin {gather*} \frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (4 x^2-4 x+1\right )^{2/3}}-\frac {(2 x-1)^{4/3} \log (x+3)}{7\ 2^{2/3} \sqrt [3]{7} \left (4 x^2-4 x+1\right )^{2/3}}+\frac {3 (2 x-1)^{4/3} \log \left (\sqrt [3]{8 x-4}+2^{2/3} \sqrt [3]{7}\right )}{7\ 2^{2/3} \sqrt [3]{7} \left (4 x^2-4 x+1\right )^{2/3}}+\frac {\sqrt [3]{\frac {2}{7}} \sqrt {3} (2 x-1)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{2 x-1}}{\sqrt {3} \sqrt [3]{7}}\right )}{7 \left (4 x^2-4 x+1\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 51
Rule 56
Rule 204
Rule 617
Rule 646
Rubi steps
\begin {align*} \int \frac {2}{(3+x) \left (2-8 x+8 x^2\right )^{2/3}} \, dx &=2 \int \frac {1}{(3+x) \left (2-8 x+8 x^2\right )^{2/3}} \, dx\\ &=\frac {\left (2 (-4+8 x)^{4/3}\right ) \int \frac {1}{(3+x) (-4+8 x)^{4/3}} \, dx}{\left (2-8 x+8 x^2\right )^{2/3}}\\ &=\frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (1-4 x+4 x^2\right )^{2/3}}-\frac {(-4+8 x)^{4/3} \int \frac {1}{(3+x) \sqrt [3]{-4+8 x}} \, dx}{14 \left (2-8 x+8 x^2\right )^{2/3}}\\ &=\frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (1-4 x+4 x^2\right )^{2/3}}-\frac {(-1+2 x)^{4/3} \log (3+x)}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}}-\frac {\left (3 (-4+8 x)^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt [3]{2} 7^{2/3}-2^{2/3} \sqrt [3]{7} x+x^2} \, dx,x,\sqrt [3]{-4+8 x}\right )}{28 \left (2-8 x+8 x^2\right )^{2/3}}+\frac {\left (3 (-4+8 x)^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{2^{2/3} \sqrt [3]{7}+x} \, dx,x,\sqrt [3]{-4+8 x}\right )}{28\ 2^{2/3} \sqrt [3]{7} \left (2-8 x+8 x^2\right )^{2/3}}\\ &=\frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (1-4 x+4 x^2\right )^{2/3}}-\frac {(-1+2 x)^{4/3} \log (3+x)}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}}+\frac {3 (-1+2 x)^{4/3} \log \left (2^{2/3} \sqrt [3]{7}+\sqrt [3]{-4+8 x}\right )}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}}-\frac {\left (3 (-4+8 x)^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\sqrt [3]{\frac {2}{7}} \sqrt [3]{-4+8 x}\right )}{14\ 2^{2/3} \sqrt [3]{7} \left (2-8 x+8 x^2\right )^{2/3}}\\ &=\frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (1-4 x+4 x^2\right )^{2/3}}+\frac {\sqrt [3]{\frac {2}{7}} \sqrt {3} (-1+2 x)^{4/3} \tan ^{-1}\left (\frac {7-2\ 7^{2/3} \sqrt [3]{-1+2 x}}{7 \sqrt {3}}\right )}{7 \left (1-4 x+4 x^2\right )^{2/3}}-\frac {(-1+2 x)^{4/3} \log (3+x)}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}}+\frac {3 (-1+2 x)^{4/3} \log \left (2^{2/3} \sqrt [3]{7}+\sqrt [3]{-4+8 x}\right )}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 40, normalized size = 0.18 \begin {gather*} -\frac {6 (2 x-1) \, _2F_1\left (-\frac {1}{3},1;\frac {2}{3};\frac {1}{7} (1-2 x)\right )}{7 \left (8 x^2-8 x+2\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.63, size = 222, normalized size = 1.00 \begin {gather*} -\frac {3 \sqrt [3]{2} \sqrt [3]{1-4 x+4 x^2}}{7 (-1+2 x)}+\frac {1}{7} \sqrt [3]{\frac {2}{7}} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-4 x+4 x^2\right )^{2/3}}{2 \sqrt [3]{7}-4 \sqrt [3]{7} x+\left (1-4 x+4 x^2\right )^{2/3}}\right )-\frac {2}{21} \sqrt [3]{\frac {2}{7}} \log (-1+2 x)+\frac {1}{7} \sqrt [3]{\frac {2}{7}} \log \left (-\sqrt [3]{7}+2 \sqrt [3]{7} x+\left (1-4 x+4 x^2\right )^{2/3}\right )-\frac {\log \left (\sqrt [3]{7}-2 \sqrt [3]{7} x+7^{2/3} \sqrt [3]{1-4 x+4 x^2}+\left (1-4 x+4 x^2\right )^{2/3}\right )}{7\ 2^{2/3} \sqrt [3]{7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 220, normalized size = 0.99 \begin {gather*} \frac {2 \cdot 7^{\frac {2}{3}} \sqrt {3} 2^{\frac {1}{3}} {\left (2 \, x - 1\right )} \arctan \left (-\frac {7^{\frac {1}{6}} \sqrt {3} {\left (7^{\frac {5}{6}} {\left (2 \, x - 1\right )} - 7 \cdot 7^{\frac {1}{6}} 2^{\frac {2}{3}} {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}}\right )}}{21 \, {\left (2 \, x - 1\right )}}\right ) - 7^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (2 \, x - 1\right )} \log \left (-\frac {7^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}} {\left (2 \, x - 1\right )} - 7^{\frac {1}{3}} 2^{\frac {2}{3}} {\left (4 \, x^{2} - 4 \, x + 1\right )} - 7 \, {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {2}{3}}}{4 \, x^{2} - 4 \, x + 1}\right ) + 2 \cdot 7^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (2 \, x - 1\right )} \log \left (\frac {7^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (2 \, x - 1\right )} + 7 \, {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}}}{2 \, x - 1}\right ) - 42 \, {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}}}{98 \, {\left (2 \, x - 1\right )}} \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*} \int \frac {2}{{\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {2}{3}} {\left (x + 3\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.09, size = 116, normalized size = 0.52
method | result | size |
risch | \(-\frac {3 \left (-1+2 x \right ) 2^{\frac {1}{3}}}{7 \left (\left (-1+2 x \right )^{2}\right )^{\frac {2}{3}}}+\frac {\left (\frac {7^{\frac {2}{3}} \ln \left (\left (-1+2 x \right )^{\frac {1}{3}}+7^{\frac {1}{3}}\right )}{49}-\frac {7^{\frac {2}{3}} \ln \left (\left (-1+2 x \right )^{\frac {2}{3}}-7^{\frac {1}{3}} \left (-1+2 x \right )^{\frac {1}{3}}+7^{\frac {2}{3}}\right )}{98}-\frac {\sqrt {3}\, 7^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,7^{\frac {2}{3}} \left (-1+2 x \right )^{\frac {1}{3}}}{7}-1\right )}{3}\right )}{49}\right ) 2^{\frac {1}{3}} \left (-1+2 x \right )^{\frac {4}{3}}}{\left (\left (-1+2 x \right )^{2}\right )^{\frac {2}{3}}}\) | \(116\) |
trager | \(-\frac {3 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}}}{7 \left (-1+2 x \right )}+\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \ln \left (\frac {-12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )^{4} x^{2}-420 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-98\right )^{3} x^{2}+6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )^{4} x +210 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-98\right )^{3} x +105 \left (8 x^{2}-8 x +2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )^{2}-64 \RootOf \left (\textit {\_Z}^{3}-98\right )^{2} x^{2}-2240 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right ) x^{2}-588 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-98\right ) x -9408 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) x +260 \RootOf \left (\textit {\_Z}^{3}-98\right )^{2} x +9100 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right ) x -1568 \left (8 x^{2}-8 x +2\right )^{\frac {2}{3}}+294 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-98\right )+4704 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right )-114 \RootOf \left (\textit {\_Z}^{3}-98\right )^{2}-3990 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )}{\left (3+x \right ) \left (-1+2 x \right )}\right )}{7}+\frac {\RootOf \left (\textit {\_Z}^{3}-98\right ) \ln \left (-\frac {12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )^{4} x^{2}-168 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-98\right )^{3} x^{2}-6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )^{4} x +84 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-98\right )^{3} x +105 \left (8 x^{2}-8 x +2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )^{2}-8 \RootOf \left (\textit {\_Z}^{3}-98\right )^{2} x^{2}+112 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right ) x^{2}+448 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-98\right ) x +12348 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) x +232 \RootOf \left (\textit {\_Z}^{3}-98\right )^{2} x -3248 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right ) x +2058 \left (8 x^{2}-8 x +2\right )^{\frac {2}{3}}-224 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-98\right )-6174 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right )-114 \RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+1596 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )}{\left (3+x \right ) \left (-1+2 x \right )}\right )}{49}\) | \(1008\) |
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*} 2 \, \int \frac {1}{{\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {2}{3}} {\left (x + 3\right )}}\,{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 = -0.00 \begin {gather*} \int \frac {2}{\left (x+3\right )\,{\left (8\,x^2-8\,x+2\right )}^{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*} \sqrt [3]{2} \int \frac {1}{x \left (4 x^{2} - 4 x + 1\right )^{\frac {2}{3}} + 3 \left (4 x^{2} - 4 x + 1\right )^{\frac {2}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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