Optimal. Leaf size=49 \[ -\frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {x^5+1}}\right )}{a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {x^5+1}}\right )}{a^{3/4}} \]
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Rubi [F] time = 0.55, 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 {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx &=\int \left (-\frac {2 \sqrt {1+x^5}}{a-x^4+2 a x^5+a x^{10}}+\frac {3 x^5 \sqrt {1+x^5}}{a-x^4+2 a x^5+a x^{10}}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1+x^5}}{a-x^4+2 a x^5+a x^{10}} \, dx\right )+3 \int \frac {x^5 \sqrt {1+x^5}}{a-x^4+2 a x^5+a x^{10}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 7.28, size = 49, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {1+x^5}}\right )}{a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {1+x^5}}\right )}{a^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 238, normalized size = 4.86 \begin {gather*} -\frac {1}{a^{3}}^{\frac {1}{4}} \arctan \left (\frac {a^{2} \frac {1}{a^{3}}^{\frac {3}{4}} x}{\sqrt {x^{5} + 1}}\right ) - \frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {a x^{10} + 2 \, a x^{5} + x^{4} + 2 \, \sqrt {x^{5} + 1} {\left (a \frac {1}{a^{3}}^{\frac {1}{4}} x^{3} + {\left (a^{3} x^{6} + a^{3} x\right )} \frac {1}{a^{3}}^{\frac {3}{4}}\right )} + 2 \, {\left (a^{2} x^{7} + a^{2} x^{2}\right )} \sqrt {\frac {1}{a^{3}}} + a}{a x^{10} + 2 \, a x^{5} - x^{4} + a}\right ) + \frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {a x^{10} + 2 \, a x^{5} + x^{4} - 2 \, \sqrt {x^{5} + 1} {\left (a \frac {1}{a^{3}}^{\frac {1}{4}} x^{3} + {\left (a^{3} x^{6} + a^{3} x\right )} \frac {1}{a^{3}}^{\frac {3}{4}}\right )} + 2 \, {\left (a^{2} x^{7} + a^{2} x^{2}\right )} \sqrt {\frac {1}{a^{3}}} + a}{a x^{10} + 2 \, a x^{5} - x^{4} + a}\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 {{\left (3 \, x^{5} - 2\right )} \sqrt {x^{5} + 1}}{a x^{10} + 2 \, a x^{5} - x^{4} + a}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{5}+1}\, \left (3 x^{5}-2\right )}{a \,x^{10}+2 a \,x^{5}-x^{4}+a}\, dx\]
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 (3 \, x^{5} - 2\right )} \sqrt {x^{5} + 1}}{a x^{10} + 2 \, a x^{5} - x^{4} + a}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.59, size = 309, normalized size = 6.31 \begin {gather*} \frac {\ln \left (\frac {\left (x\,\sqrt {a^3}+a^2\,x^4-2\,a\,\sqrt {x^5+1}\,{\left (a^3\right )}^{1/4}\right )\,\left (a\,x^3-2\,x^6\,\sqrt {a^3}-3\,x\,\sqrt {a^3}+a^2\,x^4+a^2\,x^9+2\,a\,\sqrt {x^5+1}\,{\left (a^3\right )}^{1/4}\right )}{\left (a^2-x^2\,\sqrt {a^3}+a^2\,x^5\right )\,\left (4\,\sqrt {a^3}+2\,x^5\,\sqrt {a^3}-a\,x^2-a^2\,x^8\right )}\right )}{2\,{\left (a^3\right )}^{1/4}}+\frac {\ln \left (\frac {\left (2\,a\,\sqrt {x^5+1}\,{\left (a^3\right )}^{1/4}+x\,\sqrt {a^3}\,1{}\mathrm {i}-a^2\,x^4\,1{}\mathrm {i}\right )\,\left (2\,a\,\sqrt {x^5+1}\,{\left (a^3\right )}^{1/4}+x^6\,\sqrt {a^3}\,2{}\mathrm {i}+a\,x^3\,1{}\mathrm {i}+x\,\sqrt {a^3}\,3{}\mathrm {i}+a^2\,x^4\,1{}\mathrm {i}+a^2\,x^9\,1{}\mathrm {i}\right )}{\left (x^2\,\sqrt {a^3}+a^2+a^2\,x^5\right )\,\left (4\,\sqrt {a^3}+2\,x^5\,\sqrt {a^3}+a\,x^2+a^2\,x^8\right )}\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (3 x^{5} - 2\right )}{a x^{10} + 2 a x^{5} + a - x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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