Optimal. Leaf size=52 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {x^5-x}}\right )}{a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {x^5-x}}\right )}{a^{3/4}} \]
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Rubi [F] time = 1.46, 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 {x+3 x^5}{\sqrt {-x+x^5} \left (1-a x^2-2 x^4+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x+3 x^5}{\sqrt {-x+x^5} \left (1-a x^2-2 x^4+x^8\right )} \, dx &=\int \frac {x \left (1+3 x^4\right )}{\sqrt {-x+x^5} \left (1-a x^2-2 x^4+x^8\right )} \, dx\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^4}\right ) \int \frac {\sqrt {x} \left (1+3 x^4\right )}{\sqrt {-1+x^4} \left (1-a x^2-2 x^4+x^8\right )} \, dx}{\sqrt {-x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (1+3 x^8\right )}{\sqrt {-1+x^8} \left (1-a x^4-2 x^8+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {x^2}{\sqrt {-1+x^8} \left (1-a x^4-2 x^8+x^{16}\right )}+\frac {3 x^{10}}{\sqrt {-1+x^8} \left (1-a x^4-2 x^8+x^{16}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^8} \left (1-a x^4-2 x^8+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}+\frac {\left (6 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\sqrt {-1+x^8} \left (1-a x^4-2 x^8+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}\\ \end {align*}
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Mathematica [F] time = 0.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x+3 x^5}{\sqrt {-x+x^5} \left (1-a x^2-2 x^4+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.17, size = 52, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {-x+x^5}}\right )}{a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {-x+x^5}}\right )}{a^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 231, normalized size = 4.44 \begin {gather*} \frac {1}{a^{3}}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {x^{5} - x} a \frac {1}{a^{3}}^{\frac {1}{4}}}{x^{4} - 1}\right ) - \frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {x^{8} - 2 \, x^{4} + a x^{2} + 2 \, \sqrt {x^{5} - x} {\left (a^{3} \frac {1}{a^{3}}^{\frac {3}{4}} x + {\left (a x^{4} - a\right )} \frac {1}{a^{3}}^{\frac {1}{4}}\right )} + 2 \, {\left (a^{2} x^{5} - a^{2} x\right )} \sqrt {\frac {1}{a^{3}}} + 1}{x^{8} - 2 \, x^{4} - a x^{2} + 1}\right ) + \frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {x^{8} - 2 \, x^{4} + a x^{2} - 2 \, \sqrt {x^{5} - x} {\left (a^{3} \frac {1}{a^{3}}^{\frac {3}{4}} x + {\left (a x^{4} - a\right )} \frac {1}{a^{3}}^{\frac {1}{4}}\right )} + 2 \, {\left (a^{2} x^{5} - a^{2} x\right )} \sqrt {\frac {1}{a^{3}}} + 1}{x^{8} - 2 \, x^{4} - a x^{2} + 1}\right ) \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 {3 \, x^{5} + x}{{\left (x^{8} - 2 \, x^{4} - a x^{2} + 1\right )} \sqrt {x^{5} - x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {3 x^{5}+x}{\sqrt {x^{5}-x}\, \left (x^{8}-2 x^{4}-a \,x^{2}+1\right )}\, dx\]
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 {3 \, x^{5} + x}{{\left (x^{8} - 2 \, x^{4} - a x^{2} + 1\right )} \sqrt {x^{5} - x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.01, size = 134, normalized size = 2.58 \begin {gather*} \frac {\ln \left (\frac {a^2+2\,\sqrt {x^5-x}\,{\left (a^3\right )}^{3/4}-a^2\,x^4-a\,x\,\sqrt {a^3}}{a-a\,x^4+x\,\sqrt {a^3}}\right )}{2\,{\left (a^3\right )}^{1/4}}+\frac {\ln \left (\frac {a^2\,1{}\mathrm {i}-2\,\sqrt {x^5-x}\,{\left (a^3\right )}^{3/4}-a^2\,x^4\,1{}\mathrm {i}+a\,x\,\sqrt {a^3}\,1{}\mathrm {i}}{a\,x^4-a+x\,\sqrt {a^3}}\right )\,1{}\mathrm {i}}{2\,{\left (a^3\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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