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

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 = 0.0321963, 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.

[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])

Rule 1960

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Den
ominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^(Simplify[(m + 1)/n] - 1
))/(b*e - d*x^q)^(Simplify[(m + 1)/n] + 1), x], x, ((e*(a + b*x^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, m, n}, x] && FractionQ[p] && IntegerQ[Simplify[(m + 1)/n]]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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.

[In]

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

[Out]

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

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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.

[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 = 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.

[In]

integrate(x*((-7*x^2+5)/(5*x^2+7))^(1/2),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 = 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.

[In]

integrate(x*((-7*x^2+5)/(5*x^2+7))^(1/2),x, algorithm="fricas")

[Out]

1/10*(5*x^2 + 7)*sqrt(-(7*x^2 - 5)/(5*x^2 + 7)) - 37/350*sqrt(35)*arctan(1/35*sqrt(35)*(35*x^2 + 12)*sqrt(-(7*
x^2 - 5)/(5*x^2 + 7))/(7*x^2 - 5))

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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.

[In]

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

[Out]

Piecewise((5*sqrt(35)*(sqrt(25 - 35*x**2)*sqrt(35*x**2 + 49)/125 - 74*asin(sqrt(74)*sqrt(25 - 35*x**2)/74)/125
)/14, (x > -sqrt(35)/7) & (x < sqrt(35)/7)))

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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.

[In]

integrate(x*((-7*x^2+5)/(5*x^2+7))^(1/2),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)