Optimal. Leaf size=72 \[ \frac {1}{10} \sqrt {\frac {5-7 x^2}{5 x^2+7}} \left (5 x^2+7\right )-\frac {37 \tan ^{-1}\left (\sqrt {\frac {5}{7}} \sqrt {\frac {5-7 x^2}{5 x^2+7}}\right )}{5 \sqrt {35}} \]
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Rubi [A] time = 0.03, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1960, 288, 204} \[ \frac {1}{10} \sqrt {\frac {5-7 x^2}{5 x^2+7}} \left (5 x^2+7\right )-\frac {37 \tan ^{-1}\left (\sqrt {\frac {5}{7}} \sqrt {\frac {5-7 x^2}{5 x^2+7}}\right )}{5 \sqrt {35}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 288
Rule 1960
Rubi steps
\begin {align*} \int x \sqrt {\frac {5-7 x^2}{7+5 x^2}} \, dx &=-\left (74 \operatorname {Subst}\left (\int \frac {x^2}{\left (-7-5 x^2\right )^2} \, dx,x,\sqrt {\frac {5-7 x^2}{7+5 x^2}}\right )\right )\\ &=\frac {1}{10} \sqrt {\frac {5-7 x^2}{7+5 x^2}} \left (7+5 x^2\right )+\frac {37}{5} \operatorname {Subst}\left (\int \frac {1}{-7-5 x^2} \, dx,x,\sqrt {\frac {5-7 x^2}{7+5 x^2}}\right )\\ &=\frac {1}{10} \sqrt {\frac {5-7 x^2}{7+5 x^2}} \left (7+5 x^2\right )-\frac {37 \tan ^{-1}\left (\sqrt {\frac {5}{7}} \sqrt {\frac {5-7 x^2}{7+5 x^2}}\right )}{5 \sqrt {35}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 104, normalized size = 1.44 \[ \frac {\sqrt {\frac {5-7 x^2}{5 x^2+7}} \sqrt {5 x^2+7} \left (35 \sqrt {5 x^2+7} \left (7 x^2-5\right )-74 \sqrt {35} \sqrt {7 x^2-5} \sinh ^{-1}\left (\sqrt {\frac {5}{74}} \sqrt {7 x^2-5}\right )\right )}{350 \left (7 x^2-5\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 77, normalized size = 1.07 \[ \frac {1}{10} \, {\left (5 \, x^{2} + 7\right )} \sqrt {-\frac {7 \, x^{2} - 5}{5 \, x^{2} + 7}} - \frac {37}{350} \, \sqrt {35} \arctan \left (\frac {\sqrt {35} {\left (35 \, x^{2} + 12\right )} \sqrt {-\frac {7 \, x^{2} - 5}{5 \, x^{2} + 7}}}{35 \, {\left (7 \, x^{2} - 5\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 30, normalized size = 0.42 \[ \frac {37}{350} \, \sqrt {35} \arcsin \left (\frac {35}{37} \, x^{2} + \frac {12}{37}\right ) + \frac {1}{10} \, \sqrt {-35 \, x^{4} - 24 \, x^{2} + 35} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 78, normalized size = 1.08 \[ \frac {\sqrt {-\frac {7 x^{2}-5}{5 x^{2}+7}}\, \left (5 x^{2}+7\right ) \left (37 \sqrt {35}\, \arcsin \left (\frac {35 x^{2}}{37}+\frac {12}{37}\right )+35 \sqrt {-35 x^{4}-24 x^{2}+35}\right )}{350 \sqrt {-\left (7 x^{2}-5\right ) \left (5 x^{2}+7\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.05, size = 76, normalized size = 1.06 \[ -\frac {37}{175} \, \sqrt {35} \arctan \left (\frac {1}{7} \, \sqrt {35} \sqrt {-\frac {7 \, x^{2} - 5}{5 \, x^{2} + 7}}\right ) - \frac {37 \, \sqrt {-\frac {7 \, x^{2} - 5}{5 \, x^{2} + 7}}}{5 \, {\left (\frac {5 \, {\left (7 \, x^{2} - 5\right )}}{5 \, x^{2} + 7} - 7\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 88, normalized size = 1.22 \[ -\frac {37\,\sqrt {35}\,\mathrm {atan}\left (\frac {\sqrt {5}\,\sqrt {7}\,\sqrt {-\frac {7\,x^2-5}{5\,x^2+7}}}{7}\right )}{175}-\frac {37\,\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,\sqrt {-\frac {7\,x^2-5}{5\,x^2+7}}}{1225\,\left (\frac {5\,x^2-\frac {25}{7}}{5\,x^2+7}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 66.86, size = 66, normalized size = 0.92 \[ \begin {cases} \frac {5 \sqrt {35} \left (\frac {\sqrt {25 - 35 x^{2}} \sqrt {35 x^{2} + 49}}{125} - \frac {74 \operatorname {asin}{\left (\frac {\sqrt {74} \sqrt {25 - 35 x^{2}}}{74} \right )}}{125}\right )}{14} & \text {for}\: x > - \frac {\sqrt {35}}{7} \wedge x < \frac {\sqrt {35}}{7} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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