Optimal. Leaf size=106 \[ \frac {1}{250} \sqrt {\frac {5-7 x^5}{5 x^5+7}} \left (5 x^5+7\right )^2-\frac {27}{350} \sqrt {\frac {5-7 x^5}{5 x^5+7}} \left (5 x^5+7\right )+\frac {2257 \tan ^{-1}\left (\sqrt {\frac {5}{7}} \sqrt {\frac {5-7 x^5}{5 x^5+7}}\right )}{875 \sqrt {35}} \]
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Rubi [A] time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1960, 455, 385, 204} \[ \frac {1}{250} \sqrt {\frac {5-7 x^5}{5 x^5+7}} \left (5 x^5+7\right )^2-\frac {27}{350} \sqrt {\frac {5-7 x^5}{5 x^5+7}} \left (5 x^5+7\right )+\frac {2257 \tan ^{-1}\left (\sqrt {\frac {5}{7}} \sqrt {\frac {5-7 x^5}{5 x^5+7}}\right )}{875 \sqrt {35}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 385
Rule 455
Rule 1960
Rubi steps
\begin {align*} \int x^9 \sqrt {\frac {5-7 x^5}{7+5 x^5}} \, dx &=-\left (\frac {148}{5} \operatorname {Subst}\left (\int \frac {x^2 \left (-5+7 x^2\right )}{\left (-7-5 x^2\right )^3} \, dx,x,\sqrt {\frac {5-7 x^5}{7+5 x^5}}\right )\right )\\ &=\frac {1}{250} \sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (7+5 x^5\right )^2+\frac {37}{125} \operatorname {Subst}\left (\int \frac {-74+140 x^2}{\left (-7-5 x^2\right )^2} \, dx,x,\sqrt {\frac {5-7 x^5}{7+5 x^5}}\right )\\ &=-\frac {27}{350} \sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (7+5 x^5\right )+\frac {1}{250} \sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (7+5 x^5\right )^2-\frac {2257}{875} \operatorname {Subst}\left (\int \frac {1}{-7-5 x^2} \, dx,x,\sqrt {\frac {5-7 x^5}{7+5 x^5}}\right )\\ &=-\frac {27}{350} \sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (7+5 x^5\right )+\frac {1}{250} \sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (7+5 x^5\right )^2+\frac {2257 \tan ^{-1}\left (\sqrt {\frac {5}{7}} \sqrt {\frac {5-7 x^5}{7+5 x^5}}\right )}{875 \sqrt {35}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 109, normalized size = 1.03 \[ \frac {\sqrt {\frac {5-7 x^5}{5 x^5+7}} \sqrt {5 x^5+7} \left (4514 \sqrt {35} \sqrt {7 x^5-5} \sinh ^{-1}\left (\sqrt {\frac {5}{74}} \sqrt {7 x^5-5}\right )+35 \sqrt {5 x^5+7} \left (245 x^{10}-777 x^5+430\right )\right )}{61250 \left (7 x^5-5\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 82, normalized size = 0.77 \[ \frac {1}{1750} \, {\left (175 \, x^{10} - 185 \, x^{5} - 602\right )} \sqrt {-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}} + \frac {2257}{61250} \, \sqrt {35} \arctan \left (\frac {\sqrt {35} {\left (35 \, x^{5} + 12\right )} \sqrt {-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}}}{35 \, {\left (7 \, x^{5} - 5\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 47, normalized size = 0.44 \[ \frac {1}{61250} \, {\left (35 \, \sqrt {-35 \, x^{10} - 24 \, x^{5} + 35} {\left (35 \, x^{5} - 86\right )} - 2257 \, \sqrt {35} \arcsin \left (\frac {35}{37} \, x^{5} + \frac {12}{37}\right )\right )} \mathrm {sgn}\left (5 \, x^{5} + 7\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 130, normalized size = 1.23 \[ -\frac {2257 \sqrt {-\frac {7 x^{5}-5}{5 x^{5}+7}}\, \sqrt {-\left (5 x^{5}+7\right ) \left (7 x^{5}-5\right )}\, \RootOf \left (\textit {\_Z}^{2}+35\right ) \ln \left (35 x^{5} \RootOf \left (\textit {\_Z}^{2}+35\right )+12 \RootOf \left (\textit {\_Z}^{2}+35\right )+35 \sqrt {-35 x^{10}-24 x^{5}+35}\right )}{61250 \left (7 x^{5}-5\right )}+\frac {\left (35 x^{5}-86\right ) \left (5 x^{5}+7\right ) \sqrt {-\frac {7 x^{5}-5}{5 x^{5}+7}}}{1750} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.99, size = 121, normalized size = 1.14 \[ \frac {2257}{30625} \, \sqrt {35} \arctan \left (\frac {1}{7} \, \sqrt {35} \sqrt {-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}}\right ) - \frac {37 \, {\left (675 \, \left (-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}\right )^{\frac {3}{2}} + 427 \, \sqrt {-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}}\right )}}{875 \, {\left (\frac {25 \, {\left (7 \, x^{5} - 5\right )}^{2}}{{\left (5 \, x^{5} + 7\right )}^{2}} - \frac {70 \, {\left (7 \, x^{5} - 5\right )}}{5 \, x^{5} + 7} + 49\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.99, size = 134, normalized size = 1.26 \[ \frac {2257\,\sqrt {35}\,\mathrm {atan}\left (\frac {\sqrt {5}\,\sqrt {7}\,\sqrt {-\frac {7\,x^5-5}{5\,x^5+7}}}{7}\right )}{30625}-\frac {43\,\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,\sqrt {-\frac {7\,x^5-5}{5\,x^5+7}}}{4375}-\frac {37\,\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,x^5\,\sqrt {-\frac {7\,x^5-5}{5\,x^5+7}}}{12250}+\frac {\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,x^{10}\,\sqrt {-\frac {7\,x^5-5}{5\,x^5+7}}}{350} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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