Optimal. Leaf size=149 \[ \frac {3 \sqrt [3]{x^8+2 x^3-1}}{x}+\sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{x^8+2 x^3-1}-2 x\right )+\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^8+2 x^3-1}+x}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^8+2 x^3-1} x+\sqrt [3]{2} \left (x^8+2 x^3-1\right )^{2/3}\right )}{2^{2/3}} \]
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Rubi [F] time = 1.59, 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 [F] time = 0.72, size = 0, normalized size = 0.00 \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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IntegrateAlgebraic [A] time = 2.84, size = 149, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x}+\sqrt [3]{2} \sqrt {3} \tan ^{-1}\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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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 (5 \, x^{8} + 3\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{{\left (x^{8} - 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 9.58, size = 2498, normalized size = 16.77 \[\text {Expression too large to display}\]
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*} \int \frac {{\left (5 \, x^{8} + 3\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{{\left (x^{8} - 1\right )} x^{2}}\,{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 (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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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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