Optimal. Leaf size=101 \[ \frac {x (a+b x)}{6 \left (x^3+1\right )^2}+\frac {x (5 a+4 b x)}{18 \left (x^3+1\right )}-\frac {1}{54} (5 a-2 b) \log \left (x^2-x+1\right )+\frac {1}{27} (5 a-2 b) \log (x+1)-\frac {(5 a+2 b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{9 \sqrt {3}} \]
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Rubi [A] time = 0.10, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {809, 1855, 1860, 31, 634, 618, 204, 628} \begin {gather*} \frac {x (a+b x)}{6 \left (x^3+1\right )^2}+\frac {x (5 a+4 b x)}{18 \left (x^3+1\right )}-\frac {1}{54} (5 a-2 b) \log \left (x^2-x+1\right )+\frac {1}{27} (5 a-2 b) \log (x+1)-\frac {(5 a+2 b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{9 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 618
Rule 628
Rule 634
Rule 809
Rule 1855
Rule 1860
Rubi steps
\begin {align*} \int \frac {a+b x}{(1+x)^3 \left (1-x+x^2\right )^3} \, dx &=\int \frac {a+b x}{\left (1+x^3\right )^3} \, dx\\ &=\frac {x (a+b x)}{6 \left (1+x^3\right )^2}-\frac {1}{6} \int \frac {-5 a-4 b x}{\left (1+x^3\right )^2} \, dx\\ &=\frac {x (a+b x)}{6 \left (1+x^3\right )^2}+\frac {x (5 a+4 b x)}{18 \left (1+x^3\right )}+\frac {1}{18} \int \frac {10 a+4 b x}{1+x^3} \, dx\\ &=\frac {x (a+b x)}{6 \left (1+x^3\right )^2}+\frac {x (5 a+4 b x)}{18 \left (1+x^3\right )}+\frac {1}{54} \int \frac {20 a+4 b+(-10 a+4 b) x}{1-x+x^2} \, dx+\frac {1}{27} (5 a-2 b) \int \frac {1}{1+x} \, dx\\ &=\frac {x (a+b x)}{6 \left (1+x^3\right )^2}+\frac {x (5 a+4 b x)}{18 \left (1+x^3\right )}+\frac {1}{27} (5 a-2 b) \log (1+x)+\frac {1}{54} (-5 a+2 b) \int \frac {-1+2 x}{1-x+x^2} \, dx+\frac {1}{18} (5 a+2 b) \int \frac {1}{1-x+x^2} \, dx\\ &=\frac {x (a+b x)}{6 \left (1+x^3\right )^2}+\frac {x (5 a+4 b x)}{18 \left (1+x^3\right )}+\frac {1}{27} (5 a-2 b) \log (1+x)-\frac {1}{54} (5 a-2 b) \log \left (1-x+x^2\right )+\frac {1}{9} (-5 a-2 b) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=\frac {x (a+b x)}{6 \left (1+x^3\right )^2}+\frac {x (5 a+4 b x)}{18 \left (1+x^3\right )}-\frac {(5 a+2 b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {1}{27} (5 a-2 b) \log (1+x)-\frac {1}{54} (5 a-2 b) \log \left (1-x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 94, normalized size = 0.93 \begin {gather*} \frac {1}{54} \left (\frac {9 x (a+b x)}{\left (x^3+1\right )^2}+\frac {3 x (5 a+4 b x)}{x^3+1}+(2 b-5 a) \log \left (x^2-x+1\right )+2 (5 a-2 b) \log (x+1)+2 \sqrt {3} (5 a+2 b) \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x}{(1+x)^3 \left (1-x+x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 160, normalized size = 1.58 \begin {gather*} \frac {12 \, b x^{5} + 15 \, a x^{4} + 21 \, b x^{2} + 2 \, \sqrt {3} {\left ({\left (5 \, a + 2 \, b\right )} x^{6} + 2 \, {\left (5 \, a + 2 \, b\right )} x^{3} + 5 \, a + 2 \, b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 24 \, a x - {\left ({\left (5 \, a - 2 \, b\right )} x^{6} + 2 \, {\left (5 \, a - 2 \, b\right )} x^{3} + 5 \, a - 2 \, b\right )} \log \left (x^{2} - x + 1\right ) + 2 \, {\left ({\left (5 \, a - 2 \, b\right )} x^{6} + 2 \, {\left (5 \, a - 2 \, b\right )} x^{3} + 5 \, a - 2 \, b\right )} \log \left (x + 1\right )}{54 \, {\left (x^{6} + 2 \, x^{3} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 88, normalized size = 0.87 \begin {gather*} \frac {1}{27} \, \sqrt {3} {\left (5 \, a + 2 \, b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{54} \, {\left (5 \, a - 2 \, b\right )} \log \left (x^{2} - x + 1\right ) + \frac {1}{27} \, {\left (5 \, a - 2 \, b\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac {4 \, b x^{5} + 5 \, a x^{4} + 7 \, b x^{2} + 8 \, a x}{18 \, {\left (x^{3} + 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 154, normalized size = 1.52 \begin {gather*} \frac {5 \sqrt {3}\, a \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{27}+\frac {5 a \ln \left (x +1\right )}{27}-\frac {5 a \ln \left (x^{2}-x +1\right )}{54}+\frac {2 \sqrt {3}\, b \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{27}-\frac {2 b \ln \left (x +1\right )}{27}+\frac {b \ln \left (x^{2}-x +1\right )}{27}-\frac {a}{54 \left (x +1\right )^{2}}-\frac {a}{9 \left (x +1\right )}+\frac {b}{54 \left (x +1\right )^{2}}+\frac {2 b}{27 \left (x +1\right )}-\frac {\left (-3 a -4 b \right ) x^{3}+\left (a +\frac {13 b}{2}\right ) x^{2}-\frac {7 a}{2}+\frac {5 b}{2}+\left (-a -8 b \right ) x}{27 \left (x^{2}-x +1\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 92, normalized size = 0.91 \begin {gather*} \frac {1}{27} \, \sqrt {3} {\left (5 \, a + 2 \, b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{54} \, {\left (5 \, a - 2 \, b\right )} \log \left (x^{2} - x + 1\right ) + \frac {1}{27} \, {\left (5 \, a - 2 \, b\right )} \log \left (x + 1\right ) + \frac {4 \, b x^{5} + 5 \, a x^{4} + 7 \, b x^{2} + 8 \, a x}{18 \, {\left (x^{6} + 2 \, x^{3} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 114, normalized size = 1.13 \begin {gather*} \ln \left (x+1\right )\,\left (\frac {5\,a}{27}-\frac {2\,b}{27}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {b}{27}-\frac {5\,a}{54}+\frac {\sqrt {3}\,a\,5{}\mathrm {i}}{54}+\frac {\sqrt {3}\,b\,1{}\mathrm {i}}{27}\right )-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {5\,a}{54}-\frac {b}{27}+\frac {\sqrt {3}\,a\,5{}\mathrm {i}}{54}+\frac {\sqrt {3}\,b\,1{}\mathrm {i}}{27}\right )+\frac {\frac {2\,b\,x^5}{9}+\frac {5\,a\,x^4}{18}+\frac {7\,b\,x^2}{18}+\frac {4\,a\,x}{9}}{x^6+2\,x^3+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.71, size = 292, normalized size = 2.89 \begin {gather*} \frac {\left (5 a - 2 b\right ) \log {\left (x + \frac {25 a^{2} \left (5 a - 2 b\right ) + 40 a b^{2} + 2 b \left (5 a - 2 b\right )^{2}}{125 a^{3} + 8 b^{3}} \right )}}{27} + \left (- \frac {5 a}{54} + \frac {b}{27} - \frac {\sqrt {3} i \left (5 a + 2 b\right )}{54}\right ) \log {\left (x + \frac {675 a^{2} \left (- \frac {5 a}{54} + \frac {b}{27} - \frac {\sqrt {3} i \left (5 a + 2 b\right )}{54}\right ) + 40 a b^{2} + 1458 b \left (- \frac {5 a}{54} + \frac {b}{27} - \frac {\sqrt {3} i \left (5 a + 2 b\right )}{54}\right )^{2}}{125 a^{3} + 8 b^{3}} \right )} + \left (- \frac {5 a}{54} + \frac {b}{27} + \frac {\sqrt {3} i \left (5 a + 2 b\right )}{54}\right ) \log {\left (x + \frac {675 a^{2} \left (- \frac {5 a}{54} + \frac {b}{27} + \frac {\sqrt {3} i \left (5 a + 2 b\right )}{54}\right ) + 40 a b^{2} + 1458 b \left (- \frac {5 a}{54} + \frac {b}{27} + \frac {\sqrt {3} i \left (5 a + 2 b\right )}{54}\right )^{2}}{125 a^{3} + 8 b^{3}} \right )} + \frac {5 a x^{4} + 8 a x + 4 b x^{5} + 7 b x^{2}}{18 x^{6} + 36 x^{3} + 18} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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