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] Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(19)=38\).
time = 0.42, 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 = {6857,
298, 31, 648, 631, 210, 642, 266, 327, 272, 45, 308, 206, 2608, 2605, 12, 470, 457, 78}
\begin {gather*} -\frac {5 \sqrt {3} \text {ArcTan}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{2^{2/3}}+\frac {5 \sqrt [3]{2} \text {ArcTan}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {5 \text {ArcTan}\left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 45
Rule 78
Rule 206
Rule 210
Rule 266
Rule 272
Rule 298
Rule 308
Rule 327
Rule 457
Rule 470
Rule 631
Rule 642
Rule 648
Rule 2605
Rule 2608
Rule 6857
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} \text {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} \text {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} \text {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} \text {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 ) \text {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 \text {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} \text {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} \text {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 \text {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 = 0.06, 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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(42\) vs.
\(2(19)=38\).
time = 0.12, size = 43, normalized size = 2.26
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] Leaf count of result is larger than twice the leaf count of optimal. 136 vs.
\(2 (19) = 38\).
time = 0.86, 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 \left (x\right ) - \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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Fricas [A]
time = 0.34, 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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Sympy [A]
time = 0.07, 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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Giac [A]
time = 0.42, 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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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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