Optimal. Leaf size=157 \[ \frac {2}{3} \sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{2 x^3+1}+2 x\right )-\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{2 x^3+1}-x}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{2 x^3+1} x-\sqrt [3]{2} \left (2 x^3+1\right )^{2/3}-2 x^2\right )+\frac {\sqrt [3]{2 x^3+1} \left (-58 x^6-9 x^3-4\right )}{28 x^7} \]
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Rubi [A] time = 0.21, antiderivative size = 176, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {580, 583, 12, 494, 292, 31, 634, 617, 204, 628} \begin {gather*} -\frac {29 \sqrt [3]{2 x^3+1}}{14 x}+\frac {2}{3} \sqrt [3]{2} \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{2 x^3+1}}+1\right )+\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{2 x^3+1}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (2 x^3+1\right )^{4/3}}{7 x^7}-\frac {\sqrt [3]{2 x^3+1}}{28 x^4}-\frac {1}{3} \sqrt [3]{2} \log \left (-\frac {\sqrt [3]{2} x}{\sqrt [3]{2 x^3+1}}+\frac {2^{2/3} x^2}{\left (2 x^3+1\right )^{2/3}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 204
Rule 292
Rule 494
Rule 580
Rule 583
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\left (1+2 x^3\right )^{4/3} \left (1+3 x^3\right )}{x^8 \left (1+4 x^3\right )} \, dx &=-\frac {\left (1+2 x^3\right )^{4/3}}{7 x^7}+\frac {1}{7} \int \frac {\sqrt [3]{1+2 x^3} \left (1+18 x^3\right )}{x^5 \left (1+4 x^3\right )} \, dx\\ &=-\frac {\sqrt [3]{1+2 x^3}}{28 x^4}-\frac {\left (1+2 x^3\right )^{4/3}}{7 x^7}+\frac {1}{28} \int \frac {58+120 x^3}{x^2 \left (1+2 x^3\right )^{2/3} \left (1+4 x^3\right )} \, dx\\ &=-\frac {\sqrt [3]{1+2 x^3}}{28 x^4}-\frac {29 \sqrt [3]{1+2 x^3}}{14 x}-\frac {\left (1+2 x^3\right )^{4/3}}{7 x^7}-\frac {1}{28} \int \frac {112 x}{\left (1+2 x^3\right )^{2/3} \left (1+4 x^3\right )} \, dx\\ &=-\frac {\sqrt [3]{1+2 x^3}}{28 x^4}-\frac {29 \sqrt [3]{1+2 x^3}}{14 x}-\frac {\left (1+2 x^3\right )^{4/3}}{7 x^7}-4 \int \frac {x}{\left (1+2 x^3\right )^{2/3} \left (1+4 x^3\right )} \, dx\\ &=-\frac {\sqrt [3]{1+2 x^3}}{28 x^4}-\frac {29 \sqrt [3]{1+2 x^3}}{14 x}-\frac {\left (1+2 x^3\right )^{4/3}}{7 x^7}-4 \operatorname {Subst}\left (\int \frac {x}{1+2 x^3} \, dx,x,\frac {x}{\sqrt [3]{1+2 x^3}}\right )\\ &=-\frac {\sqrt [3]{1+2 x^3}}{28 x^4}-\frac {29 \sqrt [3]{1+2 x^3}}{14 x}-\frac {\left (1+2 x^3\right )^{4/3}}{7 x^7}+\frac {1}{3} \left (2\ 2^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x} \, dx,x,\frac {x}{\sqrt [3]{1+2 x^3}}\right )-\frac {1}{3} \left (2\ 2^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt [3]{2} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+2 x^3}}\right )\\ &=-\frac {\sqrt [3]{1+2 x^3}}{28 x^4}-\frac {29 \sqrt [3]{1+2 x^3}}{14 x}-\frac {\left (1+2 x^3\right )^{4/3}}{7 x^7}+\frac {2}{3} \sqrt [3]{2} \log \left (1+\frac {\sqrt [3]{2} x}{\sqrt [3]{1+2 x^3}}\right )-\frac {1}{3} \sqrt [3]{2} \operatorname {Subst}\left (\int \frac {-\sqrt [3]{2}+2\ 2^{2/3} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+2 x^3}}\right )-2^{2/3} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+2 x^3}}\right )\\ &=-\frac {\sqrt [3]{1+2 x^3}}{28 x^4}-\frac {29 \sqrt [3]{1+2 x^3}}{14 x}-\frac {\left (1+2 x^3\right )^{4/3}}{7 x^7}-\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {2^{2/3} x^2}{\left (1+2 x^3\right )^{2/3}}-\frac {\sqrt [3]{2} x}{\sqrt [3]{1+2 x^3}}\right )+\frac {2}{3} \sqrt [3]{2} \log \left (1+\frac {\sqrt [3]{2} x}{\sqrt [3]{1+2 x^3}}\right )-\left (2 \sqrt [3]{2}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{\sqrt [3]{\frac {1}{2}+x^3}}\right )\\ &=-\frac {\sqrt [3]{1+2 x^3}}{28 x^4}-\frac {29 \sqrt [3]{1+2 x^3}}{14 x}-\frac {\left (1+2 x^3\right )^{4/3}}{7 x^7}+\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1+2 x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {2^{2/3} x^2}{\left (1+2 x^3\right )^{2/3}}-\frac {\sqrt [3]{2} x}{\sqrt [3]{1+2 x^3}}\right )+\frac {2}{3} \sqrt [3]{2} \log \left (1+\frac {\sqrt [3]{2} x}{\sqrt [3]{1+2 x^3}}\right )\\ \end {align*}
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Mathematica [C] time = 0.10, size = 71, normalized size = 0.45 \begin {gather*} -\frac {2 x^2 \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {2 x^3}{4 x^3+1}\right )}{\left (4 x^3+1\right )^{2/3}}-\frac {\sqrt [3]{2 x^3+1} \left (58 x^6+9 x^3+4\right )}{28 x^7} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.36, size = 157, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{1+2 x^3} \left (-4-9 x^3-58 x^6\right )}{28 x^7}-\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1+2 x^3}}\right )}{\sqrt {3}}+\frac {2}{3} \sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{1+2 x^3}\right )-\frac {1}{3} \sqrt [3]{2} \log \left (-2 x^2+2^{2/3} x \sqrt [3]{1+2 x^3}-\sqrt [3]{2} \left (1+2 x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.13, size = 291, normalized size = 1.85 \begin {gather*} -\frac {56 \, \sqrt {3} 2^{\frac {1}{3}} x^{7} \arctan \left (-\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (20 \, x^{8} + 10 \, x^{5} - x^{2}\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}} - 6 \, \sqrt {3} 2^{\frac {1}{3}} {\left (8 \, x^{7} - 2 \, x^{4} - x\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} + \sqrt {3} {\left (8 \, x^{9} + 60 \, x^{6} + 24 \, x^{3} - 1\right )}}{3 \, {\left (152 \, x^{9} + 60 \, x^{6} - 12 \, x^{3} - 1\right )}}\right ) - 56 \cdot 2^{\frac {1}{3}} x^{7} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (4 \, x^{3} + 1\right )} + 6 \, {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} x}{4 \, x^{3} + 1}\right ) + 28 \cdot 2^{\frac {1}{3}} x^{7} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (2 \, x^{4} - x\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (20 \, x^{6} + 10 \, x^{3} - 1\right )} + 12 \, {\left (x^{5} + x^{2}\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}}}{16 \, x^{6} + 8 \, x^{3} + 1}\right ) + 9 \, {\left (58 \, x^{6} + 9 \, x^{3} + 4\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}}}{252 \, x^{7}} \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 (3 \, x^{3} + 1\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {4}{3}}}{{\left (4 \, x^{3} + 1\right )} x^{8}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 16.63, size = 1174, normalized size = 7.48
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1174\) |
risch | \(\text {Expression too large to display}\) | \(1409\) |
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 (3 \, x^{3} + 1\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {4}{3}}}{{\left (4 \, x^{3} + 1\right )} x^{8}}\,{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 = -0.01 \begin {gather*} \int \frac {{\left (2\,x^3+1\right )}^{4/3}\,\left (3\,x^3+1\right )}{x^8\,\left (4\,x^3+1\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 {\left (2 x^{3} + 1\right )^{\frac {4}{3}} \left (3 x^{3} + 1\right )}{x^{8} \left (4 x^{3} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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