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 = 0.0321963, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, 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 1960
Rule 288
Rule 204
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*}
Mathematica [A] time = 0.0465422, size = 104, normalized size = 1.44 \[ \frac{\sqrt{\frac{5-7 x^2}{5 x^2+7}} \sqrt{5 x^2+7} \left (35 \sqrt{5 x^2+7} \left (7 x^2-5\right )+74 \sqrt{35} \sqrt{5-7 x^2} \sin ^{-1}\left (\sqrt{\frac{5}{74}} \sqrt{5-7 x^2}\right )\right )}{350 \left (7 x^2-5\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 78, normalized size = 1.1 \begin{align*}{\frac{5\,{x}^{2}+7}{350}\sqrt{-{\frac{7\,{x}^{2}-5}{5\,{x}^{2}+7}}} \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 ){\frac{1}{\sqrt{- \left ( 7\,{x}^{2}-5 \right ) \left ( 5\,{x}^{2}+7 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67384, size = 103, normalized size = 1.43 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54251, size = 197, normalized size = 2.74 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 126.062, size = 66, normalized size = 0.92 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17914, size = 41, normalized size = 0.57 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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