Optimal. Leaf size=250 \[ -\frac {\sqrt [3]{(1-2 x)^2} \left (7^{2/3} \sqrt [3]{2 x-1}+7\right ) \left (\sqrt [3]{7} (2 x-1)^{2/3}-7^{2/3} \sqrt [3]{2 x-1}+7\right )^2 \left (\frac {3 (2 x-1)^{2/3}}{2^{2/3}}+\sqrt [3]{2} 7^{2/3} \log \left (7^{2/3} \sqrt [3]{2 x-1}+7\right )-\left (\frac {7}{2}\right )^{2/3} \log \left (-\sqrt [3]{7} (2 x-1)^{2/3}+7^{2/3} \sqrt [3]{2 x-1}-7\right )+\sqrt [3]{2} \sqrt {3} 7^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{2 x-1}}{\sqrt {3} \sqrt [3]{7}}\right )\right )}{14 (x+3) \sqrt [3]{2 x-1} \left (-2 \sqrt [3]{7} x+(14 x-7)^{2/3}-7 \sqrt [3]{2 x-1}+\sqrt [3]{7}\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 179, normalized size of antiderivative = 0.72, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {646, 50, 56, 617, 204, 31} \begin {gather*} \frac {3 \sqrt [3]{4 x^2-4 x+1}}{2^{2/3}}-\frac {\left (\frac {7}{2}\right )^{2/3} \sqrt [3]{4 x^2-4 x+1} \log (x+3)}{(2 x-1)^{2/3}}+\frac {3 \left (\frac {7}{2}\right )^{2/3} \sqrt [3]{4 x^2-4 x+1} \log \left (\sqrt [3]{8 x-4}+2^{2/3} \sqrt [3]{7}\right )}{(2 x-1)^{2/3}}+\frac {\sqrt [3]{2} \sqrt {3} 7^{2/3} \sqrt [3]{4 x^2-4 x+1} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{2 x-1}}{\sqrt {3} \sqrt [3]{7}}\right )}{(2 x-1)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 56
Rule 204
Rule 617
Rule 646
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{2-8 x+8 x^2}}{3+x} \, dx &=\frac {\sqrt [3]{2-8 x+8 x^2} \int \frac {(-4+8 x)^{2/3}}{3+x} \, dx}{(-4+8 x)^{2/3}}\\ &=\frac {3 \sqrt [3]{1-4 x+4 x^2}}{2^{2/3}}-\frac {\left (28 \sqrt [3]{2-8 x+8 x^2}\right ) \int \frac {1}{(3+x) \sqrt [3]{-4+8 x}} \, dx}{(-4+8 x)^{2/3}}\\ &=\frac {3 \sqrt [3]{1-4 x+4 x^2}}{2^{2/3}}-\frac {\left (\frac {7}{2}\right )^{2/3} \sqrt [3]{1-4 x+4 x^2} \log (3+x)}{(-1+2 x)^{2/3}}-\frac {\left (42 \sqrt [3]{2-8 x+8 x^2}\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 )}{(-4+8 x)^{2/3}}+\frac {\left (3 \sqrt [3]{2} 7^{2/3} \sqrt [3]{2-8 x+8 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{2^{2/3} \sqrt [3]{7}+x} \, dx,x,\sqrt [3]{-4+8 x}\right )}{(-4+8 x)^{2/3}}\\ &=\frac {3 \sqrt [3]{1-4 x+4 x^2}}{2^{2/3}}-\frac {\left (\frac {7}{2}\right )^{2/3} \sqrt [3]{1-4 x+4 x^2} \log (3+x)}{(-1+2 x)^{2/3}}+\frac {3 \left (\frac {7}{2}\right )^{2/3} \sqrt [3]{1-4 x+4 x^2} \log \left (2^{2/3} \sqrt [3]{7}+\sqrt [3]{-4+8 x}\right )}{(-1+2 x)^{2/3}}-\frac {\left (6 \sqrt [3]{2} 7^{2/3} \sqrt [3]{2-8 x+8 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\sqrt [3]{\frac {2}{7}} \sqrt [3]{-4+8 x}\right )}{(-4+8 x)^{2/3}}\\ &=\frac {3 \sqrt [3]{1-4 x+4 x^2}}{2^{2/3}}+\frac {\sqrt [3]{2} \sqrt {3} 7^{2/3} \sqrt [3]{1-4 x+4 x^2} \tan ^{-1}\left (\frac {7-2\ 7^{2/3} \sqrt [3]{-1+2 x}}{7 \sqrt {3}}\right )}{(-1+2 x)^{2/3}}-\frac {\left (\frac {7}{2}\right )^{2/3} \sqrt [3]{1-4 x+4 x^2} \log (3+x)}{(-1+2 x)^{2/3}}+\frac {3 \left (\frac {7}{2}\right )^{2/3} \sqrt [3]{1-4 x+4 x^2} \log \left (2^{2/3} \sqrt [3]{7}+\sqrt [3]{-4+8 x}\right )}{(-1+2 x)^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 37, normalized size = 0.15 \begin {gather*} -\frac {3 \sqrt [3]{(1-2 x)^2} \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {1}{7} (1-2 x)\right )-1\right )}{2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 5.42, size = 250, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [3]{(1-2 x)^2} \left (7+7^{2/3} \sqrt [3]{-1+2 x}\right ) \left (7-7^{2/3} \sqrt [3]{-1+2 x}+\sqrt [3]{7} (-1+2 x)^{2/3}\right )^2 \left (\frac {3 (-1+2 x)^{2/3}}{2^{2/3}}+\sqrt [3]{2} \sqrt {3} 7^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+2 x}}{\sqrt {3} \sqrt [3]{7}}\right )+\sqrt [3]{2} 7^{2/3} \log \left (7+7^{2/3} \sqrt [3]{-1+2 x}\right )-\left (\frac {7}{2}\right )^{2/3} \log \left (-7+7^{2/3} \sqrt [3]{-1+2 x}-\sqrt [3]{7} (-1+2 x)^{2/3}\right )\right )}{14 (3+x) \sqrt [3]{-1+2 x} \left (\sqrt [3]{7}-2 \sqrt [3]{7} x-7 \sqrt [3]{-1+2 x}+(-7+14 x)^{2/3}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 169, normalized size = 0.68 \begin {gather*} 98^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {98^{\frac {2}{3}} \sqrt {3} {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}} - 7 \, \sqrt {3} {\left (2 \, x - 1\right )}}{21 \, {\left (2 \, x - 1\right )}}\right ) - \frac {1}{2} \cdot 98^{\frac {1}{3}} \log \left (\frac {98^{\frac {2}{3}} {\left (4 \, x^{2} - 4 \, x + 1\right )} - 7 \cdot 98^{\frac {1}{3}} {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}} {\left (2 \, x - 1\right )} + 49 \, {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {2}{3}}}{4 \, x^{2} - 4 \, x + 1}\right ) + 98^{\frac {1}{3}} \log \left (\frac {98^{\frac {1}{3}} {\left (2 \, x - 1\right )} + 7 \, {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}}}{2 \, x - 1}\right ) + \frac {3}{2} \, {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}} \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 {{\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}}}{x + 3}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.56, size = 110, normalized size = 0.44
method | result | size |
risch | \(\frac {3 \,2^{\frac {1}{3}} \left (\left (-1+2 x \right )^{2}\right )^{\frac {1}{3}}}{2}+\frac {\left (7^{\frac {2}{3}} \ln \left (\left (-1+2 x \right )^{\frac {1}{3}}+7^{\frac {1}{3}}\right )-\frac {7^{\frac {2}{3}} \ln \left (\left (-1+2 x \right )^{\frac {2}{3}}-7^{\frac {1}{3}} \left (-1+2 x \right )^{\frac {1}{3}}+7^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, 7^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,7^{\frac {2}{3}} \left (-1+2 x \right )^{\frac {1}{3}}}{7}-1\right )}{3}\right )\right ) 2^{\frac {1}{3}} \left (\left (-1+2 x \right )^{2}\right )^{\frac {1}{3}}}{\left (-1+2 x \right )^{\frac {2}{3}}}\) | \(110\) |
trager | \(\text {Expression too large to display}\) | \(1484\) |
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 {{\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}}}{x + 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 = -0.00 \begin {gather*} \int \frac {{\left (8\,x^2-8\,x+2\right )}^{1/3}}{x+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 {\sqrt [3]{4 x^{2} - 4 x + 1}}{x + 3}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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