Optimal. Leaf size=170 \[ \frac {2 \sqrt [4]{2 x^5-2 x^4-x^2-2}}{x}-\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{2 x^5-2 x^4-x^2-2}}{\sqrt {2} x^2-\sqrt {2 x^5-2 x^4-x^2-2}}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{2 x^5-2 x^4-x^2-2}}{2 x^2+\sqrt {2} \sqrt {2 x^5-2 x^4-x^2-2}}\right )}{\sqrt [4]{2}} \]
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Rubi [F] time = 1.12, 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 (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+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 (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx &=\int \left (-\frac {2 \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2}+\frac {\left (-1+5 x^3\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{-2-x^2+2 x^5}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2} \, dx\right )+\int \frac {\left (-1+5 x^3\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{-2-x^2+2 x^5} \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2} \, dx\right )+\int \left (\frac {\sqrt [4]{-2-x^2-2 x^4+2 x^5}}{2+x^2-2 x^5}+\frac {5 x^3 \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{-2-x^2+2 x^5}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2} \, dx\right )+5 \int \frac {x^3 \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{-2-x^2+2 x^5} \, dx+\int \frac {\sqrt [4]{-2-x^2-2 x^4+2 x^5}}{2+x^2-2 x^5} \, dx\\ \end {align*}
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Mathematica [F] time = 0.57, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.41, size = 170, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x}-\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{\sqrt {2} x^2-\sqrt {-2-x^2-2 x^4+2 x^5}}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{2 x^2+\sqrt {2} \sqrt {-2-x^2-2 x^4+2 x^5}}\right )}{\sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 91.06, size = 1274, normalized size = 7.49
result too large to display
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 (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} {\left (x^{5} + x^{2} + 4\right )}}{{\left (2 \, x^{5} - x^{2} - 2\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{5}+x^{2}+4\right ) \left (2 x^{5}-2 x^{4}-x^{2}-2\right )^{\frac {1}{4}}}{x^{2} \left (2 x^{5}-x^{2}-2\right )}\, dx\]
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 (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} {\left (x^{5} + x^{2} + 4\right )}}{{\left (2 \, x^{5} - x^{2} - 2\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 (x^5+x^2+4\right )\,{\left (2\,x^5-2\,x^4-x^2-2\right )}^{1/4}}{x^2\,\left (-2\,x^5+x^2+2\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 (x^{5} + x^{2} + 4\right ) \sqrt [4]{2 x^{5} - 2 x^{4} - x^{2} - 2}}{x^{2} \left (2 x^{5} - x^{2} - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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