Optimal. Leaf size=210 \[ \frac {\sqrt [3]{9 x^3-30 x^2+66 x-19}}{2 x-3}+\frac {1}{3} \sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{9 x^3-30 x^2+66 x-19}-4 x+6\right )-\frac {\log \left (8 x^2+\sqrt [3]{2} \left (9 x^3-30 x^2+66 x-19\right )^{2/3}+\left (2\ 2^{2/3} x-3\ 2^{2/3}\right ) \sqrt [3]{9 x^3-30 x^2+66 x-19}-24 x+18\right )}{3\ 2^{2/3}}+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {2 \sqrt {3} x-3 \sqrt {3}}{2^{2/3} \sqrt [3]{9 x^3-30 x^2+66 x-19}+2 x-3}\right )}{\sqrt {3}} \]
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Rubi [F] time = 1.51, 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 = {} \begin {gather*} \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx &=\int \left (\frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{21 (-5+x)}-\frac {2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2}+\frac {2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{7 (-3+2 x)}+\frac {(5-4 x) \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{21 \left (1-x+x^2\right )}\right ) \, dx\\ &=\frac {1}{21} \int \frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-5+x} \, dx+\frac {1}{21} \int \frac {(5-4 x) \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{1-x+x^2} \, dx+\frac {2}{7} \int \frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-3+2 x} \, dx-2 \int \frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2} \, dx\\ &=\frac {1}{21} \int \left (\frac {\left (-4-2 i \sqrt {3}\right ) \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-1-i \sqrt {3}+2 x}+\frac {\left (-4+2 i \sqrt {3}\right ) \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-1+i \sqrt {3}+2 x}\right ) \, dx+\frac {1}{21} \operatorname {Subst}\left (\int \frac {\sqrt [3]{\frac {2401}{81}+\frac {98 x}{3}+9 x^3}}{-\frac {35}{9}+x} \, dx,x,-\frac {10}{9}+x\right )+\frac {2}{7} \operatorname {Subst}\left (\int \frac {\sqrt [3]{\frac {2401}{81}+\frac {98 x}{3}+9 x^3}}{-\frac {7}{9}+2 x} \, dx,x,-\frac {10}{9}+x\right )-2 \operatorname {Subst}\left (\int \frac {\sqrt [3]{\frac {2401}{81}+\frac {98 x}{3}+9 x^3}}{\left (-\frac {7}{9}+2 x\right )^2} \, dx,x,-\frac {10}{9}+x\right )\\ &=-\left (\frac {1}{21} \left (2 \left (2-i \sqrt {3}\right )\right ) \int \frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-1+i \sqrt {3}+2 x} \, dx\right )-\frac {1}{21} \left (2 \left (2+i \sqrt {3}\right )\right ) \int \frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-1-i \sqrt {3}+2 x} \, dx+\frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{-\frac {35}{9}+x} \, dx,x,-\frac {10}{9}+x\right )}{63 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}+\frac {\left (2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{-\frac {7}{9}+2 x} \, dx,x,-\frac {10}{9}+x\right )}{21 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}-\frac {\left (2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{\left (-\frac {7}{9}+2 x\right )^2} \, dx,x,-\frac {10}{9}+x\right )}{3 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}\\ &=-\left (\frac {1}{21} \left (2 \left (2-i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{\frac {2401}{81}+\frac {98 x}{3}+9 x^3}}{\frac {1}{27} \left (60+27 \left (-1+i \sqrt {3}\right )\right )+2 x} \, dx,x,-\frac {10}{9}+x\right )\right )-\frac {1}{21} \left (2 \left (2+i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{\frac {2401}{81}+\frac {98 x}{3}+9 x^3}}{\frac {1}{27} \left (60+27 \left (-1-i \sqrt {3}\right )\right )+2 x} \, dx,x,-\frac {10}{9}+x\right )+\frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{-\frac {35}{9}+x} \, dx,x,-\frac {10}{9}+x\right )}{63 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}+\frac {\left (2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{-\frac {7}{9}+2 x} \, dx,x,-\frac {10}{9}+x\right )}{21 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}-\frac {\left (2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{\left (-\frac {7}{9}+2 x\right )^2} \, dx,x,-\frac {10}{9}+x\right )}{3 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}\\ &=\frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{-\frac {35}{9}+x} \, dx,x,-\frac {10}{9}+x\right )}{63 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}+\frac {\left (2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{-\frac {7}{9}+2 x} \, dx,x,-\frac {10}{9}+x\right )}{21 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}-\frac {\left (2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{\left (-\frac {7}{9}+2 x\right )^2} \, dx,x,-\frac {10}{9}+x\right )}{3 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}-\frac {\left (2 \left (2-i \sqrt {3}\right ) \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{\frac {1}{27} \left (60+27 \left (-1+i \sqrt {3}\right )\right )+2 x} \, dx,x,-\frac {10}{9}+x\right )}{63 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}-\frac {\left (2 \left (2+i \sqrt {3}\right ) \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{\frac {1}{27} \left (60+27 \left (-1-i \sqrt {3}\right )\right )+2 x} \, dx,x,-\frac {10}{9}+x\right )}{63 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}\\ \end {align*}
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Mathematica [F] time = 1.90, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.06, size = 210, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-3+2 x}+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {-3 \sqrt {3}+2 \sqrt {3} x}{-3+2 x+2^{2/3} \sqrt [3]{-19+66 x-30 x^2+9 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt [3]{2} \log \left (6-4 x+2^{2/3} \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right )-\frac {\log \left (18-24 x+8 x^2+\left (-3 2^{2/3}+2\ 2^{2/3} x\right ) \sqrt [3]{-19+66 x-30 x^2+9 x^3}+\sqrt [3]{2} \left (-19+66 x-30 x^2+9 x^3\right )^{2/3}\right )}{3\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 23.16, size = 532, normalized size = 2.53 \begin {gather*} \frac {2 \, \sqrt {3} 2^{\frac {1}{3}} {\left (2 \, x - 3\right )} \arctan \left (-\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (5380 \, x^{8} - 59100 \, x^{7} + 301161 \, x^{6} - 909412 \, x^{5} + 1740060 \, x^{4} - 2110416 \, x^{3} + 1545376 \, x^{2} - 606864 \, x + 94131\right )} {\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {1}{3}} - 42 \, \sqrt {3} 2^{\frac {1}{3}} {\left (82 \, x^{7} - 963 \, x^{6} + 4404 \, x^{5} - 10852 \, x^{4} + 15852 \, x^{3} - 14316 \, x^{2} + 7786 \, x - 1905\right )} {\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {2}{3}} + \sqrt {3} {\left (43721 \, x^{9} - 510066 \, x^{8} + 2889414 \, x^{7} - 10065027 \, x^{6} + 23187528 \, x^{5} - 35703864 \, x^{4} + 35637567 \, x^{3} - 21385926 \, x^{2} + 6711858 \, x - 806653\right )}}{3 \, {\left (62551 \, x^{9} - 773406 \, x^{8} + 4465170 \, x^{7} - 15587817 \, x^{6} + 35620200 \, x^{5} - 54275256 \, x^{4} + 54133401 \, x^{3} - 33459498 \, x^{2} + 11334294 \, x - 1538783\right )}}\right ) - 2^{\frac {1}{3}} {\left (2 \, x - 3\right )} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (82 \, x^{4} - 471 \, x^{3} + 1086 \, x^{2} - 1100 \, x + 381\right )} {\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (1345 \, x^{6} - 10740 \, x^{5} + 40044 \, x^{4} - 83056 \, x^{3} + 95748 \, x^{2} - 53484 \, x + 10459\right )} + 12 \, {\left (68 \, x^{5} - 468 \, x^{4} + 1425 \, x^{3} - 2218 \, x^{2} + 1632 \, x - 414\right )} {\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {1}{3}}}{x^{6} - 12 \, x^{5} + 48 \, x^{4} - 82 \, x^{3} + 96 \, x^{2} - 60 \, x + 25}\right ) + 2 \cdot 2^{\frac {1}{3}} {\left (2 \, x - 3\right )} \log \left (\frac {7 \cdot 2^{\frac {2}{3}} {\left (x^{3} - 6 \, x^{2} + 6 \, x - 5\right )} - 6 \cdot 2^{\frac {1}{3}} {\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {1}{3}} {\left (4 \, x^{2} - 12 \, x + 9\right )} + 6 \, {\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {2}{3}} {\left (2 \, x - 3\right )}}{x^{3} - 6 \, x^{2} + 6 \, x - 5}\right ) + 18 \, {\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {1}{3}}}{18 \, {\left (2 \, x - 3\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 {{\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {1}{3}} {\left (x + 2\right )}^{2}}{{\left (x^{3} - 6 \, x^{2} + 6 \, x - 5\right )} {\left (2 \, x - 3\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 38.34, size = 2403, normalized size = 11.44
method | result | size |
trager | \(\text {Expression too large to display}\) | \(2403\) |
risch | \(\text {Expression too large to display}\) | \(3445\) |
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 (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {1}{3}} {\left (x + 2\right )}^{2}}{{\left (x^{3} - 6 \, x^{2} + 6 \, x - 5\right )} {\left (2 \, x - 3\right )}^{2}}\,{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 (x+2\right )}^2\,{\left (9\,x^3-30\,x^2+66\,x-19\right )}^{1/3}}{{\left (2\,x-3\right )}^2\,\left (x^3-6\,x^2+6\,x-5\right )} \,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 {\sqrt [3]{\left (3 x - 1\right ) \left (3 x^{2} - 9 x + 19\right )} \left (x + 2\right )^{2}}{\left (x - 5\right ) \left (2 x - 3\right )^{2} \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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