Optimal. Leaf size=19 \[ x^2 \left (5+x^4\right ) \left (x+\log \left (\frac {1}{x^2}+2 x\right )\right ) \]
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Rubi [B] time = 0.66, antiderivative size = 129, normalized size of antiderivative = 6.79, number of steps used = 54, number of rules used = 19, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6725, 292, 31, 634, 617, 204, 628, 260, 321, 266, 43, 302, 200, 2528, 2525, 12, 459, 446, 77} \begin {gather*} x^7+5 x^3+5 x^2 \log \left (\frac {2 x^3+1}{x^2}\right )+x^6 \log \left (\frac {2 x^3+1}{x^2}\right )-\frac {5 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{2^{2/3}}+\frac {5 \sqrt [3]{2} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {5 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 43
Rule 77
Rule 200
Rule 204
Rule 260
Rule 266
Rule 292
Rule 302
Rule 321
Rule 446
Rule 459
Rule 617
Rule 628
Rule 634
Rule 2525
Rule 2528
Rule 6725
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {10 x}{1+2 x^3}+\frac {15 x^2}{1+2 x^3}+\frac {10 x^4}{1+2 x^3}+\frac {28 x^5}{1+2 x^3}+\frac {7 x^6}{1+2 x^3}+\frac {2 x^8}{1+2 x^3}+\frac {14 x^9}{1+2 x^3}+2 x \left (5+3 x^4\right ) \log \left (\frac {1+2 x^3}{x^2}\right )\right ) \, dx\\ &=2 \int \frac {x^8}{1+2 x^3} \, dx+2 \int x \left (5+3 x^4\right ) \log \left (\frac {1+2 x^3}{x^2}\right ) \, dx+7 \int \frac {x^6}{1+2 x^3} \, dx-10 \int \frac {x}{1+2 x^3} \, dx+10 \int \frac {x^4}{1+2 x^3} \, dx+14 \int \frac {x^9}{1+2 x^3} \, dx+15 \int \frac {x^2}{1+2 x^3} \, dx+28 \int \frac {x^5}{1+2 x^3} \, dx\\ &=\frac {5 x^2}{2}+\frac {5}{2} \log \left (1+2 x^3\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {x^2}{1+2 x} \, dx,x,x^3\right )+2 \int \left (5 x \log \left (\frac {1+2 x^3}{x^2}\right )+3 x^5 \log \left (\frac {1+2 x^3}{x^2}\right )\right ) \, dx-5 \int \frac {x}{1+2 x^3} \, dx+7 \int \left (-\frac {1}{4}+\frac {x^3}{2}+\frac {1}{4 \left (1+2 x^3\right )}\right ) \, dx+\frac {28}{3} \operatorname {Subst}\left (\int \frac {x}{1+2 x} \, dx,x,x^3\right )+14 \int \left (\frac {1}{8}-\frac {x^3}{4}+\frac {x^6}{2}-\frac {1}{8 \left (1+2 x^3\right )}\right ) \, dx+\frac {1}{3} \left (5\ 2^{2/3}\right ) \int \frac {1}{1+\sqrt [3]{2} x} \, dx-\frac {1}{3} \left (5\ 2^{2/3}\right ) \int \frac {1+\sqrt [3]{2} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx\\ &=\frac {5 x^2}{2}+x^7+\frac {5}{3} \sqrt [3]{2} \log \left (1+\sqrt [3]{2} x\right )+\frac {5}{2} \log \left (1+2 x^3\right )+\frac {2}{3} \operatorname {Subst}\left (\int \left (-\frac {1}{4}+\frac {x}{2}+\frac {1}{4 (1+2 x)}\right ) \, dx,x,x^3\right )+6 \int x^5 \log \left (\frac {1+2 x^3}{x^2}\right ) \, dx+\frac {28}{3} \operatorname {Subst}\left (\int \left (\frac {1}{2}-\frac {1}{2 (1+2 x)}\right ) \, dx,x,x^3\right )+10 \int x \log \left (\frac {1+2 x^3}{x^2}\right ) \, dx-\frac {5 \int \frac {-\sqrt [3]{2}+2\ 2^{2/3} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx}{3\ 2^{2/3}}+\frac {5 \int \frac {1}{1+\sqrt [3]{2} x} \, dx}{3 \sqrt [3]{2}}-\frac {5 \int \frac {1+\sqrt [3]{2} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx}{3 \sqrt [3]{2}}-\frac {5 \int \frac {1}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx}{\sqrt [3]{2}}\\ &=\frac {5 x^2}{2}+\frac {9 x^3}{2}+\frac {x^6}{6}+x^7+\frac {5 \log \left (1+\sqrt [3]{2} x\right )}{3\ 2^{2/3}}+\frac {5}{3} \sqrt [3]{2} \log \left (1+\sqrt [3]{2} x\right )-\frac {5 \log \left (1-\sqrt [3]{2} x+2^{2/3} x^2\right )}{3\ 2^{2/3}}+\frac {1}{4} \log \left (1+2 x^3\right )+5 x^2 \log \left (\frac {1+2 x^3}{x^2}\right )+x^6 \log \left (\frac {1+2 x^3}{x^2}\right )-5 \int \frac {2 x \left (-1+x^3\right )}{1+2 x^3} \, dx-\frac {5 \int \frac {-\sqrt [3]{2}+2\ 2^{2/3} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx}{6\ 2^{2/3}}-\frac {5 \int \frac {1}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx}{2 \sqrt [3]{2}}-\left (5 \sqrt [3]{2}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{2} x\right )-\int \frac {2 x^5 \left (-1+x^3\right )}{1+2 x^3} \, dx\\ &=\frac {5 x^2}{2}+\frac {9 x^3}{2}+\frac {x^6}{6}+x^7+\frac {5 \sqrt [3]{2} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {5 \log \left (1+\sqrt [3]{2} x\right )}{3\ 2^{2/3}}+\frac {5}{3} \sqrt [3]{2} \log \left (1+\sqrt [3]{2} x\right )-\frac {5 \log \left (1-\sqrt [3]{2} x+2^{2/3} x^2\right )}{2\ 2^{2/3}}+\frac {1}{4} \log \left (1+2 x^3\right )+5 x^2 \log \left (\frac {1+2 x^3}{x^2}\right )+x^6 \log \left (\frac {1+2 x^3}{x^2}\right )-2 \int \frac {x^5 \left (-1+x^3\right )}{1+2 x^3} \, dx-10 \int \frac {x \left (-1+x^3\right )}{1+2 x^3} \, dx-\frac {5 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{2} x\right )}{2^{2/3}}\\ &=\frac {9 x^3}{2}+\frac {x^6}{6}+x^7+\frac {5 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {5 \sqrt [3]{2} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {5 \log \left (1+\sqrt [3]{2} x\right )}{3\ 2^{2/3}}+\frac {5}{3} \sqrt [3]{2} \log \left (1+\sqrt [3]{2} x\right )-\frac {5 \log \left (1-\sqrt [3]{2} x+2^{2/3} x^2\right )}{2\ 2^{2/3}}+\frac {1}{4} \log \left (1+2 x^3\right )+5 x^2 \log \left (\frac {1+2 x^3}{x^2}\right )+x^6 \log \left (\frac {1+2 x^3}{x^2}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {(-1+x) x}{1+2 x} \, dx,x,x^3\right )+15 \int \frac {x}{1+2 x^3} \, dx\\ &=\frac {9 x^3}{2}+\frac {x^6}{6}+x^7+\frac {5 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {5 \sqrt [3]{2} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {5 \log \left (1+\sqrt [3]{2} x\right )}{3\ 2^{2/3}}+\frac {5}{3} \sqrt [3]{2} \log \left (1+\sqrt [3]{2} x\right )-\frac {5 \log \left (1-\sqrt [3]{2} x+2^{2/3} x^2\right )}{2\ 2^{2/3}}+\frac {1}{4} \log \left (1+2 x^3\right )+5 x^2 \log \left (\frac {1+2 x^3}{x^2}\right )+x^6 \log \left (\frac {1+2 x^3}{x^2}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \left (-\frac {3}{4}+\frac {x}{2}+\frac {3}{4 (1+2 x)}\right ) \, dx,x,x^3\right )-\frac {5 \int \frac {1}{1+\sqrt [3]{2} x} \, dx}{\sqrt [3]{2}}+\frac {5 \int \frac {1+\sqrt [3]{2} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx}{\sqrt [3]{2}}\\ &=5 x^3+x^7+\frac {5 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {5 \sqrt [3]{2} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {5 \log \left (1-\sqrt [3]{2} x+2^{2/3} x^2\right )}{2\ 2^{2/3}}+5 x^2 \log \left (\frac {1+2 x^3}{x^2}\right )+x^6 \log \left (\frac {1+2 x^3}{x^2}\right )+\frac {5 \int \frac {-\sqrt [3]{2}+2\ 2^{2/3} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx}{2\ 2^{2/3}}+\frac {15 \int \frac {1}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx}{2 \sqrt [3]{2}}\\ &=5 x^3+x^7+\frac {5 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {5 \sqrt [3]{2} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{\sqrt {3}}+5 x^2 \log \left (\frac {1+2 x^3}{x^2}\right )+x^6 \log \left (\frac {1+2 x^3}{x^2}\right )+\frac {15 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{2} x\right )}{2^{2/3}}\\ &=5 x^3+x^7+\frac {5 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {5 \sqrt [3]{2} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {5 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{2^{2/3}}+5 x^2 \log \left (\frac {1+2 x^3}{x^2}\right )+x^6 \log \left (\frac {1+2 x^3}{x^2}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.09, size = 34, normalized size = 1.79 \begin {gather*} 5 x^3+x^7+5 x^2 \log \left (\frac {1}{x^2}+2 x\right )+x^6 \log \left (\frac {1}{x^2}+2 x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 31, normalized size = 1.63 \begin {gather*} x^{7} + 5 \, x^{3} + {\left (x^{6} + 5 \, x^{2}\right )} \log \left (\frac {2 \, x^{3} + 1}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 31, normalized size = 1.63 \begin {gather*} x^{7} + 5 \, x^{3} + {\left (x^{6} + 5 \, x^{2}\right )} \log \left (\frac {2 \, x^{3} + 1}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 32, normalized size = 1.68
method | result | size |
risch | \(\left (x^{6}+5 x^{2}\right ) \ln \left (\frac {2 x^{3}+1}{x^{2}}\right )+x^{7}+5 x^{3}\) | \(32\) |
default | \(x^{7}+5 x^{3}+5 x^{2} \ln \left (\frac {2 x^{3}+1}{x^{2}}\right )+\ln \left (\frac {2 x^{3}+1}{x^{2}}\right ) x^{6}\) | \(43\) |
norman | \(x^{7}+5 x^{3}+5 x^{2} \ln \left (\frac {2 x^{3}+1}{x^{2}}\right )+\ln \left (\frac {2 x^{3}+1}{x^{2}}\right ) x^{6}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.15, size = 136, normalized size = 7.16 \begin {gather*} x^{7} + 5 \, x^{3} + {\left (x^{6} + 5 \, x^{2}\right )} \log \left (2 \, x^{3} + 1\right ) + \frac {1}{4} \, {\left (5 \cdot 2^{\frac {1}{3}} - 1\right )} \log \left (2^{\frac {2}{3}} x^{2} - 2^{\frac {1}{3}} x + 1\right ) - \frac {1}{4} \, {\left (10 \cdot 2^{\frac {1}{3}} + 1\right )} \log \left (\frac {1}{2} \cdot 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} x + 1\right )}\right ) - 2 \, {\left (x^{6} + 5 \, x^{2}\right )} \log \relax (x) - \frac {5}{4} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} x^{2} - 2^{\frac {1}{3}} x + 1\right ) + \frac {5}{2} \cdot 2^{\frac {1}{3}} \log \left (\frac {1}{2} \cdot 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} x + 1\right )}\right ) + \frac {1}{4} \, \log \left (2 \, x^{3} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 23, normalized size = 1.21 \begin {gather*} x^2\,\left (x+\ln \left (\frac {2\,x^3+1}{x^2}\right )\right )\,\left (x^4+5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 27, normalized size = 1.42 \begin {gather*} x^{7} + 5 x^{3} + \left (x^{6} + 5 x^{2}\right ) \log {\left (\frac {2 x^{3} + 1}{x^{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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