Optimal. Leaf size=149 \[ \frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x}+\sqrt [3]{2} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+2 x^3+x^8}}\right )+\sqrt [3]{2} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+2 x^3+x^8}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+2 x^3+x^8}+\sqrt [3]{2} \left (-1+2 x^3+x^8\right )^{2/3}\right )}{2^{2/3}} \]
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Rubi [F]
time = 1.05, 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 {\sqrt [3]{-1+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^8\right )} \, dx &=\int \left (\frac {\sqrt [3]{-1+2 x^3+x^8}}{-1-x}+\frac {\sqrt [3]{-1+2 x^3+x^8}}{-1+x}-\frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x^2}+\frac {2 \sqrt [3]{-1+2 x^3+x^8}}{1+x^2}+\frac {4 x^2 \sqrt [3]{-1+2 x^3+x^8}}{1+x^4}\right ) \, dx\\ &=2 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{1+x^2} \, dx-3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx+4 \int \frac {x^2 \sqrt [3]{-1+2 x^3+x^8}}{1+x^4} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1-x} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1+x} \, dx\\ &=2 \int \left (\frac {i \sqrt [3]{-1+2 x^3+x^8}}{2 (i-x)}+\frac {i \sqrt [3]{-1+2 x^3+x^8}}{2 (i+x)}\right ) \, dx-3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx+4 \int \left (-\frac {\sqrt [3]{-1+2 x^3+x^8}}{2 \left (i-x^2\right )}+\frac {\sqrt [3]{-1+2 x^3+x^8}}{2 \left (i+x^2\right )}\right ) \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1-x} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1+x} \, dx\\ &=i \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i-x} \, dx+i \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i+x} \, dx-2 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i-x^2} \, dx+2 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i+x^2} \, dx-3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1-x} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1+x} \, dx\\ &=i \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i-x} \, dx+i \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i+x} \, dx-2 \int \left (-\frac {(-1)^{3/4} \sqrt [3]{-1+2 x^3+x^8}}{2 \left (\sqrt [4]{-1}-x\right )}-\frac {(-1)^{3/4} \sqrt [3]{-1+2 x^3+x^8}}{2 \left (\sqrt [4]{-1}+x\right )}\right ) \, dx+2 \int \left (-\frac {\sqrt [4]{-1} \sqrt [3]{-1+2 x^3+x^8}}{2 \left (-(-1)^{3/4}-x\right )}-\frac {\sqrt [4]{-1} \sqrt [3]{-1+2 x^3+x^8}}{2 \left (-(-1)^{3/4}+x\right )}\right ) \, dx-3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1-x} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1+x} \, dx\\ &=i \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i-x} \, dx+i \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i+x} \, dx-3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx-\sqrt [4]{-1} \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-(-1)^{3/4}-x} \, dx-\sqrt [4]{-1} \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-(-1)^{3/4}+x} \, dx+(-1)^{3/4} \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{\sqrt [4]{-1}-x} \, dx+(-1)^{3/4} \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{\sqrt [4]{-1}+x} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1-x} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1+x} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.50, size = 149, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x}+\sqrt [3]{2} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+2 x^3+x^8}}\right )+\sqrt [3]{2} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+2 x^3+x^8}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+2 x^3+x^8}+\sqrt [3]{2} \left (-1+2 x^3+x^8\right )^{2/3}\right )}{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 = 5.28, size = 2336, normalized size = 15.68 \[\text {Expression too large to display}\]
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. 369 vs.
\(2 (121) = 242\).
time = 231.98, size = 369, normalized size = 2.48 \begin {gather*} \frac {2 \, \sqrt {3} 2^{\frac {1}{3}} x \arctan \left (\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (x^{18} + 18 \, x^{13} - 2 \, x^{10} + 36 \, x^{8} - 18 \, x^{5} + x^{2}\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}} + 6 \, \sqrt {3} 2^{\frac {1}{3}} {\left (x^{17} + 6 \, x^{12} - 2 \, x^{9} - 6 \, x^{4} + x\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {2}{3}} + \sqrt {3} {\left (x^{24} + 36 \, x^{19} - 3 \, x^{16} + 180 \, x^{14} - 72 \, x^{11} + 216 \, x^{9} + 3 \, x^{8} - 180 \, x^{6} + 36 \, x^{3} - 1\right )}}{3 \, {\left (x^{24} - 3 \, x^{16} - 108 \, x^{14} - 216 \, x^{9} + 3 \, x^{8} + 108 \, x^{6} - 1\right )}}\right ) + 2 \cdot 2^{\frac {1}{3}} x \log \left (-\frac {6 \cdot 2^{\frac {1}{3}} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{8} - 1\right )} - 6 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{8} - 1}\right ) - 2^{\frac {1}{3}} x \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{9} + 6 \, x^{4} - x\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{16} + 18 \, x^{11} - 2 \, x^{8} + 36 \, x^{6} - 18 \, x^{3} + 1\right )} + 12 \, {\left (x^{10} + 3 \, x^{5} - x^{2}\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{x^{16} - 2 \, x^{8} + 1}\right ) + 18 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{6 \, 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 (5 x^{8} + 3\right ) \sqrt [3]{x^{8} + 2 x^{3} - 1}}{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\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 (5\,x^8+3\right )\,{\left (x^8+2\,x^3-1\right )}^{1/3}}{x^2\,\left (x^8-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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