Optimal. Leaf size=259 \[ \frac {\sqrt {1-c_0} \tan ^{-1}\left (\frac {\sqrt {1-c_0} \sqrt {-1+c_1} \sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}}}{-1+c_0}\right )}{\sqrt {-1+c_1}}+\frac {\sqrt {-1-c_0} \tan ^{-1}\left (\frac {\sqrt {-1-c_0} \sqrt {1+c_1} \sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}}}{1+c_0}\right )}{3 \sqrt {1+c_1}}-\frac {4 \sqrt {1-2 c_0} \tan ^{-1}\left (\frac {\sqrt {1-2 c_0} \sqrt {-1+2 c_1} \sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}}}{-1+2 c_0}\right )}{3 \sqrt {-1+2 c_1}} \]
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Rubi [F] time = 41.72, 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^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{\left (x^3+2 x^5 c_3+2 c_4\right ) \left (-x^6+x^{10} c_3{}^2+2 x^5 c_3 c_4+c_4{}^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{\left (x^3+2 x^5 c_3+2 c_4\right ) \left (-x^6+x^{10} c_3{}^2+2 x^5 c_3 c_4+c_4{}^2\right )} \, dx &=\frac {\left (\sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}} \sqrt {x^3 c_1+x^5 c_3+c_4}\right ) \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {x^3 c_0+x^5 c_3+c_4}}{\sqrt {x^3 c_1+x^5 c_3+c_4} \left (x^3+2 x^5 c_3+2 c_4\right ) \left (-x^6+x^{10} c_3{}^2+2 x^5 c_3 c_4+c_4{}^2\right )} \, dx}{\sqrt {x^3 c_0+x^5 c_3+c_4}}\\ &=\frac {\left (\sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}} \sqrt {x^3 c_1+x^5 c_3+c_4}\right ) \int \left (\frac {x^2 \left (-3+5 x^2 c_3\right ) \sqrt {x^3 c_0+x^5 c_3+c_4}}{6 \left (-x^3+x^5 c_3+c_4\right ) \sqrt {x^3 c_1+x^5 c_3+c_4}}+\frac {x^2 \left (3+5 x^2 c_3\right ) \sqrt {x^3 c_0+x^5 c_3+c_4}}{2 \left (x^3+x^5 c_3+c_4\right ) \sqrt {x^3 c_1+x^5 c_3+c_4}}-\frac {2 x^2 \left (3+10 x^2 c_3\right ) \sqrt {x^3 c_0+x^5 c_3+c_4}}{3 \sqrt {x^3 c_1+x^5 c_3+c_4} \left (x^3+2 x^5 c_3+2 c_4\right )}\right ) \, dx}{\sqrt {x^3 c_0+x^5 c_3+c_4}}\\ &=\frac {\left (\sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}} \sqrt {x^3 c_1+x^5 c_3+c_4}\right ) \int \frac {x^2 \left (-3+5 x^2 c_3\right ) \sqrt {x^3 c_0+x^5 c_3+c_4}}{\left (-x^3+x^5 c_3+c_4\right ) \sqrt {x^3 c_1+x^5 c_3+c_4}} \, dx}{6 \sqrt {x^3 c_0+x^5 c_3+c_4}}+\frac {\left (\sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}} \sqrt {x^3 c_1+x^5 c_3+c_4}\right ) \int \frac {x^2 \left (3+5 x^2 c_3\right ) \sqrt {x^3 c_0+x^5 c_3+c_4}}{\left (x^3+x^5 c_3+c_4\right ) \sqrt {x^3 c_1+x^5 c_3+c_4}} \, dx}{2 \sqrt {x^3 c_0+x^5 c_3+c_4}}-\frac {\left (2 \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}} \sqrt {x^3 c_1+x^5 c_3+c_4}\right ) \int \frac {x^2 \left (3+10 x^2 c_3\right ) \sqrt {x^3 c_0+x^5 c_3+c_4}}{\sqrt {x^3 c_1+x^5 c_3+c_4} \left (x^3+2 x^5 c_3+2 c_4\right )} \, dx}{3 \sqrt {x^3 c_0+x^5 c_3+c_4}}\\ &=\frac {\left (\sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}} \sqrt {x^3 c_1+x^5 c_3+c_4}\right ) \int \left (-\frac {3 x^2 \sqrt {x^3 c_0+x^5 c_3+c_4}}{\left (-x^3+x^5 c_3+c_4\right ) \sqrt {x^3 c_1+x^5 c_3+c_4}}+\frac {5 x^4 c_3 \sqrt {x^3 c_0+x^5 c_3+c_4}}{\left (-x^3+x^5 c_3+c_4\right ) \sqrt {x^3 c_1+x^5 c_3+c_4}}\right ) \, dx}{6 \sqrt {x^3 c_0+x^5 c_3+c_4}}+\frac {\left (\sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}} \sqrt {x^3 c_1+x^5 c_3+c_4}\right ) \int \left (\frac {3 x^2 \sqrt {x^3 c_0+x^5 c_3+c_4}}{\left (x^3+x^5 c_3+c_4\right ) \sqrt {x^3 c_1+x^5 c_3+c_4}}+\frac {5 x^4 c_3 \sqrt {x^3 c_0+x^5 c_3+c_4}}{\left (x^3+x^5 c_3+c_4\right ) \sqrt {x^3 c_1+x^5 c_3+c_4}}\right ) \, dx}{2 \sqrt {x^3 c_0+x^5 c_3+c_4}}-\frac {\left (2 \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}} \sqrt {x^3 c_1+x^5 c_3+c_4}\right ) \int \left (\frac {3 x^2 \sqrt {x^3 c_0+x^5 c_3+c_4}}{\sqrt {x^3 c_1+x^5 c_3+c_4} \left (x^3+2 x^5 c_3+2 c_4\right )}+\frac {10 x^4 c_3 \sqrt {x^3 c_0+x^5 c_3+c_4}}{\sqrt {x^3 c_1+x^5 c_3+c_4} \left (x^3+2 x^5 c_3+2 c_4\right )}\right ) \, dx}{3 \sqrt {x^3 c_0+x^5 c_3+c_4}}\\ &=-\frac {\left (\sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}} \sqrt {x^3 c_1+x^5 c_3+c_4}\right ) \int \frac {x^2 \sqrt {x^3 c_0+x^5 c_3+c_4}}{\left (-x^3+x^5 c_3+c_4\right ) \sqrt {x^3 c_1+x^5 c_3+c_4}} \, dx}{2 \sqrt {x^3 c_0+x^5 c_3+c_4}}+\frac {\left (3 \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}} \sqrt {x^3 c_1+x^5 c_3+c_4}\right ) \int \frac {x^2 \sqrt {x^3 c_0+x^5 c_3+c_4}}{\left (x^3+x^5 c_3+c_4\right ) \sqrt {x^3 c_1+x^5 c_3+c_4}} \, dx}{2 \sqrt {x^3 c_0+x^5 c_3+c_4}}-\frac {\left (2 \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}} \sqrt {x^3 c_1+x^5 c_3+c_4}\right ) \int \frac {x^2 \sqrt {x^3 c_0+x^5 c_3+c_4}}{\sqrt {x^3 c_1+x^5 c_3+c_4} \left (x^3+2 x^5 c_3+2 c_4\right )} \, dx}{\sqrt {x^3 c_0+x^5 c_3+c_4}}+\frac {\left (5 c_3 \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}} \sqrt {x^3 c_1+x^5 c_3+c_4}\right ) \int \frac {x^4 \sqrt {x^3 c_0+x^5 c_3+c_4}}{\left (-x^3+x^5 c_3+c_4\right ) \sqrt {x^3 c_1+x^5 c_3+c_4}} \, dx}{6 \sqrt {x^3 c_0+x^5 c_3+c_4}}+\frac {\left (5 c_3 \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}} \sqrt {x^3 c_1+x^5 c_3+c_4}\right ) \int \frac {x^4 \sqrt {x^3 c_0+x^5 c_3+c_4}}{\left (x^3+x^5 c_3+c_4\right ) \sqrt {x^3 c_1+x^5 c_3+c_4}} \, dx}{2 \sqrt {x^3 c_0+x^5 c_3+c_4}}-\frac {\left (20 c_3 \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}} \sqrt {x^3 c_1+x^5 c_3+c_4}\right ) \int \frac {x^4 \sqrt {x^3 c_0+x^5 c_3+c_4}}{\sqrt {x^3 c_1+x^5 c_3+c_4} \left (x^3+2 x^5 c_3+2 c_4\right )} \, dx}{3 \sqrt {x^3 c_0+x^5 c_3+c_4}}\\ \end {align*}
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Mathematica [F] time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{\left (x^3+2 x^5 c_3+2 c_4\right ) \left (-x^6+x^{10} c_3{}^2+2 x^5 c_3 c_4+c_4{}^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.94, size = 259, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {1-c_0} \sqrt {-1+c_1} \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{-1+c_0}\right ) \sqrt {1-c_0}}{\sqrt {-1+c_1}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-1-c_0} \sqrt {1+c_1} \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{1+c_0}\right ) \sqrt {-1-c_0}}{3 \sqrt {1+c_1}}-\frac {4 \tan ^{-1}\left (\frac {\sqrt {1-2 c_0} \sqrt {-1+2 c_1} \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{-1+2 c_0}\right ) \sqrt {1-2 c_0}}{3 \sqrt {-1+2 c_1}} \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(-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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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (2 \textit {\_C3} \,x^{5}-3 \textit {\_C4} \right ) \sqrt {\frac {\textit {\_C3} \,x^{5}+\textit {\_C0} \,x^{3}+\textit {\_C4}}{\textit {\_C3} \,x^{5}+\textit {\_C1} \,x^{3}+\textit {\_C4}}}}{\left (2 \textit {\_C3} \,x^{5}+x^{3}+2 \textit {\_C4} \right ) \left (\textit {\_C3}^{2} x^{10}+2 \textit {\_C3} \textit {\_C4} \,x^{5}-x^{6}+\textit {\_C4}^{2}\right )}\, dx\]
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 \, \_{C_{3}} x^{5} - 3 \, \_{C_{4}}\right )} x^{5} \sqrt {\frac {\_{C_{3}} x^{5} + \_{C_{0}} x^{3} + \_{C_{4}}}{\_{C_{3}} x^{5} + \_{C_{1}} x^{3} + \_{C_{4}}}}}{{\left (\_{C_{3}}^{2} x^{10} + 2 \, \_{C_{3}} \_{C_{4}} x^{5} - x^{6} + \_{C_{4}}^{2}\right )} {\left (2 \, \_{C_{3}} x^{5} + x^{3} + 2 \, \_{C_{4}}\right )}}\,{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 = -0.00 \begin {gather*} \int -\frac {x^5\,\sqrt {\frac {_{\mathrm {C3}}\,x^5+_{\mathrm {C0}}\,x^3+_{\mathrm {C4}}}{_{\mathrm {C3}}\,x^5+_{\mathrm {C1}}\,x^3+_{\mathrm {C4}}}}\,\left (3\,_{\mathrm {C4}}-2\,_{\mathrm {C3}}\,x^5\right )}{\left (2\,_{\mathrm {C3}}\,x^5+x^3+2\,_{\mathrm {C4}}\right )\,\left ({_{\mathrm {C3}}}^2\,x^{10}+2\,_{\mathrm {C3}}\,_{\mathrm {C4}}\,x^5+{_{\mathrm {C4}}}^2-x^6\right )} \,d x \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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