3.526 \(\int x \sqrt{\frac{5-7 x^2}{7+5 x^2}} \, dx\)

Optimal. Leaf size=72 \[ \frac{1}{10} \sqrt{\frac{5-7 x^2}{5 x^2+7}} \left (5 x^2+7\right )-\frac{37 \tan ^{-1}\left (\sqrt{\frac{5}{7}} \sqrt{\frac{5-7 x^2}{5 x^2+7}}\right )}{5 \sqrt{35}} \]

[Out]

(Sqrt[(5 - 7*x^2)/(7 + 5*x^2)]*(7 + 5*x^2))/10 - (37*ArcTan[Sqrt[5/7]*Sqrt[(5 -
7*x^2)/(7 + 5*x^2)]])/(5*Sqrt[35])

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Rubi [A]  time = 0.0653706, 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 \[ \frac{1}{10} \sqrt{\frac{5-7 x^2}{5 x^2+7}} \left (5 x^2+7\right )-\frac{37 \tan ^{-1}\left (\sqrt{\frac{5}{7}} \sqrt{\frac{5-7 x^2}{5 x^2+7}}\right )}{5 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[(5 - 7*x^2)/(7 + 5*x^2)],x]

[Out]

(Sqrt[(5 - 7*x^2)/(7 + 5*x^2)]*(7 + 5*x^2))/10 - (37*ArcTan[Sqrt[5/7]*Sqrt[(5 -
7*x^2)/(7 + 5*x^2)]])/(5*Sqrt[35])

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Rubi in Sympy [A]  time = 2.42523, size = 66, normalized size = 0.92 \[ \frac{37 \sqrt{\frac{- 7 x^{2} + 5}{5 x^{2} + 7}}}{5 \left (\frac{5 \left (- 7 x^{2} + 5\right )}{5 x^{2} + 7} + 7\right )} - \frac{37 \sqrt{35} \operatorname{atan}{\left (\frac{\sqrt{35} \sqrt{\frac{- 7 x^{2} + 5}{5 x^{2} + 7}}}{7} \right )}}{175} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*((-7*x**2+5)/(5*x**2+7))**(1/2),x)

[Out]

37*sqrt((-7*x**2 + 5)/(5*x**2 + 7))/(5*(5*(-7*x**2 + 5)/(5*x**2 + 7) + 7)) - 37*
sqrt(35)*atan(sqrt(35)*sqrt((-7*x**2 + 5)/(5*x**2 + 7))/7)/175

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Mathematica [A]  time = 0.0989119, size = 95, normalized size = 1.32 \[ \frac{\sqrt{\frac{5-7 x^2}{5 x^2+7}} \left (35 \sqrt{5-7 x^2} \left (5 x^2+7\right )+74 \sqrt{35} \sqrt{5 x^2+7} \sin ^{-1}\left (\sqrt{\frac{7}{74}} \sqrt{5 x^2+7}\right )\right )}{350 \sqrt{5-7 x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[(5 - 7*x^2)/(7 + 5*x^2)],x]

[Out]

(Sqrt[(5 - 7*x^2)/(7 + 5*x^2)]*(35*Sqrt[5 - 7*x^2]*(7 + 5*x^2) + 74*Sqrt[35]*Sqr
t[7 + 5*x^2]*ArcSin[Sqrt[7/74]*Sqrt[7 + 5*x^2]]))/(350*Sqrt[5 - 7*x^2])

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Maple [A]  time = 0.03, size = 78, normalized size = 1.1 \[{\frac{5\,{x}^{2}+7}{350}\sqrt{-{\frac{7\,{x}^{2}-5}{5\,{x}^{2}+7}}} \left ( 37\,\sqrt{35}\arcsin \left ({\frac{35\,{x}^{2}}{37}}+{\frac{12}{37}} \right ) +35\,\sqrt{-35\,{x}^{4}-24\,{x}^{2}+35} \right ){\frac{1}{\sqrt{- \left ( 7\,{x}^{2}-5 \right ) \left ( 5\,{x}^{2}+7 \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*((-7*x^2+5)/(5*x^2+7))^(1/2),x)

[Out]

1/350*(-(7*x^2-5)/(5*x^2+7))^(1/2)*(5*x^2+7)*(37*35^(1/2)*arcsin(35/37*x^2+12/37
)+35*(-35*x^4-24*x^2+35)^(1/2))/(-(7*x^2-5)*(5*x^2+7))^(1/2)

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Maxima [A]  time = 0.812918, size = 103, normalized size = 1.43 \[ -\frac{37}{175} \, \sqrt{35} \arctan \left (\frac{1}{7} \, \sqrt{35} \sqrt{-\frac{7 \, x^{2} - 5}{5 \, x^{2} + 7}}\right ) - \frac{37 \, \sqrt{-\frac{7 \, x^{2} - 5}{5 \, x^{2} + 7}}}{5 \,{\left (\frac{5 \,{\left (7 \, x^{2} - 5\right )}}{5 \, x^{2} + 7} - 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sqrt(-(7*x^2 - 5)/(5*x^2 + 7)),x, algorithm="maxima")

[Out]

-37/175*sqrt(35)*arctan(1/7*sqrt(35)*sqrt(-(7*x^2 - 5)/(5*x^2 + 7))) - 37/5*sqrt
(-(7*x^2 - 5)/(5*x^2 + 7))/(5*(7*x^2 - 5)/(5*x^2 + 7) - 7)

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Fricas [A]  time = 0.272152, size = 109, normalized size = 1.51 \[ \frac{1}{350} \, \sqrt{35}{\left (\sqrt{35}{\left (5 \, x^{2} + 7\right )} \sqrt{-\frac{7 \, x^{2} - 5}{5 \, x^{2} + 7}} + 37 \, \arctan \left (\frac{\sqrt{35}{\left (35 \, x^{2} + 12\right )}}{35 \,{\left (5 \, x^{2} + 7\right )} \sqrt{-\frac{7 \, x^{2} - 5}{5 \, x^{2} + 7}}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sqrt(-(7*x^2 - 5)/(5*x^2 + 7)),x, algorithm="fricas")

[Out]

1/350*sqrt(35)*(sqrt(35)*(5*x^2 + 7)*sqrt(-(7*x^2 - 5)/(5*x^2 + 7)) + 37*arctan(
1/35*sqrt(35)*(35*x^2 + 12)/((5*x^2 + 7)*sqrt(-(7*x^2 - 5)/(5*x^2 + 7)))))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*((-7*x**2+5)/(5*x**2+7))**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.268742, size = 41, normalized size = 0.57 \[ \frac{37}{350} \, \sqrt{35} \arcsin \left (\frac{35}{37} \, x^{2} + \frac{12}{37}\right ) + \frac{1}{10} \, \sqrt{-35 \, x^{4} - 24 \, x^{2} + 35} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sqrt(-(7*x^2 - 5)/(5*x^2 + 7)),x, algorithm="giac")

[Out]

37/350*sqrt(35)*arcsin(35/37*x^2 + 12/37) + 1/10*sqrt(-35*x^4 - 24*x^2 + 35)