Optimal. Leaf size=140 \[ \frac {3 \left (1+x^5\right )^{2/3} \left (2+5 x^3+2 x^5\right )}{10 x^5}-\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^5}}\right )}{2^{2/3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^5}\right )}{2^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^5}+2^{2/3} \left (1+x^5\right )^{2/3}\right )}{2\ 2^{2/3}} \]
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Rubi [F]
time = 0.87, 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 {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right ) \left (2+x^3+2 x^5\right )}{x^6 \left (2-x^3+2 x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right ) \left (2+x^3+2 x^5\right )}{x^6 \left (2-x^3+2 x^5\right )} \, dx &=\int \left (-\frac {3 \left (1+x^5\right )^{2/3}}{x^6}-\frac {3 \left (1+x^5\right )^{2/3}}{x^3}+\frac {2 \left (1+x^5\right )^{2/3}}{x}+\frac {\left (-3+10 x^2\right ) \left (1+x^5\right )^{2/3}}{2-x^3+2 x^5}\right ) \, dx\\ &=2 \int \frac {\left (1+x^5\right )^{2/3}}{x} \, dx-3 \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx-3 \int \frac {\left (1+x^5\right )^{2/3}}{x^3} \, dx+\int \frac {\left (-3+10 x^2\right ) \left (1+x^5\right )^{2/3}}{2-x^3+2 x^5} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {2}{5};\frac {3}{5};-x^5\right )}{2 x^2}+\frac {2}{5} \text {Subst}\left (\int \frac {(1+x)^{2/3}}{x} \, dx,x,x^5\right )-\frac {3}{5} \text {Subst}\left (\int \frac {(1+x)^{2/3}}{x^2} \, dx,x,x^5\right )+\int \left (-\frac {3 \left (1+x^5\right )^{2/3}}{2-x^3+2 x^5}+\frac {10 x^2 \left (1+x^5\right )^{2/3}}{2-x^3+2 x^5}\right ) \, dx\\ &=\frac {3}{5} \left (1+x^5\right )^{2/3}+\frac {3 \left (1+x^5\right )^{2/3}}{5 x^5}+\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {2}{5};\frac {3}{5};-x^5\right )}{2 x^2}-3 \int \frac {\left (1+x^5\right )^{2/3}}{2-x^3+2 x^5} \, dx+10 \int \frac {x^2 \left (1+x^5\right )^{2/3}}{2-x^3+2 x^5} \, dx\\ \end {align*}
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Mathematica [A]
time = 2.19, size = 140, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+x^5\right )^{2/3} \left (2+5 x^3+2 x^5\right )}{10 x^5}-\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^5}}\right )}{2^{2/3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^5}\right )}{2^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^5}+2^{2/3} \left (1+x^5\right )^{2/3}\right )}{2\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 206.37, size = 873, normalized size = 6.24
method | result | size |
risch | \(\text {Expression too large to display}\) | \(873\) |
trager | \(\text {Expression too large to display}\) | \(1490\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 399 vs.
\(2 (109) = 218\).
time = 65.29, size = 399, normalized size = 2.85 \begin {gather*} -\frac {20 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{11} + x^{9} - x^{7} + 4 \, x^{6} + x^{4} + 2 \, x\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (8 \, x^{15} + 60 \, x^{13} + 24 \, x^{11} + 24 \, x^{10} - x^{9} + 120 \, x^{8} + 24 \, x^{6} + 24 \, x^{5} + 60 \, x^{3} + 8\right )} + 12 \, {\left (4 \, x^{12} + 14 \, x^{10} + x^{8} + 8 \, x^{7} + 14 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (8 \, x^{15} - 12 \, x^{13} - 48 \, x^{11} + 24 \, x^{10} - x^{9} - 24 \, x^{8} - 48 \, x^{6} + 24 \, x^{5} - 12 \, x^{3} + 8\right )}}\right ) - 10 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{5} + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (2 \, x^{5} - x^{3} + 2\right )} - 12 \, {\left (x^{5} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{5} - x^{3} + 2}\right ) + 5 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{6} + x^{4} + x\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (4 \, x^{10} + 14 \, x^{8} + x^{6} + 8 \, x^{5} + 14 \, x^{3} + 4\right )} + 6 \, {\left (4 \, x^{7} + x^{5} + 4 \, x^{2}\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}}{4 \, x^{10} - 4 \, x^{8} + x^{6} + 8 \, x^{5} - 4 \, x^{3} + 4}\right ) - 36 \, {\left (2 \, x^{5} + 5 \, x^{3} + 2\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}}}{120 \, x^{5}} \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 (\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )\right )^{\frac {2}{3}} \cdot \left (2 x^{5} - 3\right ) \left (2 x^{5} + x^{3} + 2\right )}{x^{6} \cdot \left (2 x^{5} - x^{3} + 2\right )}\, dx \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*} \text {could not integrate} \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 (x^5+1\right )}^{2/3}\,\left (2\,x^5-3\right )\,\left (2\,x^5+x^3+2\right )}{x^6\,\left (2\,x^5-x^3+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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