3.293 \(\int x^9 \sqrt{\frac{5-7 x^5}{7+5 x^5}} \, dx\)

Optimal. Leaf size=106 \[ \frac{1}{250} \sqrt{\frac{5-7 x^5}{5 x^5+7}} \left (5 x^5+7\right )^2-\frac{27}{350} \sqrt{\frac{5-7 x^5}{5 x^5+7}} \left (5 x^5+7\right )+\frac{2257 \tan ^{-1}\left (\sqrt{\frac{5}{7}} \sqrt{\frac{5-7 x^5}{5 x^5+7}}\right )}{875 \sqrt{35}} \]

[Out]

(-27*Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]*(7 + 5*x^5))/350 + (Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]*(7 + 5*x^5)^2)/250 + (225
7*ArcTan[Sqrt[5/7]*Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]])/(875*Sqrt[35])

________________________________________________________________________________________

Rubi [A]  time = 0.0557393, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, 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.

[In]

Int[x^9*Sqrt[(5 - 7*x^5)/(7 + 5*x^5)],x]

[Out]

(-27*Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]*(7 + 5*x^5))/350 + (Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]*(7 + 5*x^5)^2)/250 + (225
7*ArcTan[Sqrt[5/7]*Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]])/(875*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 455

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

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^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*}

Mathematica [A]  time = 0.055097, size = 109, normalized size = 1.03 \[ \frac{\sqrt{\frac{5-7 x^5}{5 x^5+7}} \sqrt{5 x^5+7} \left (35 \sqrt{5 x^5+7} \left (245 x^{10}-777 x^5+430\right )-4514 \sqrt{35} \sqrt{5-7 x^5} \sin ^{-1}\left (\sqrt{\frac{5}{74}} \sqrt{5-7 x^5}\right )\right )}{61250 \left (7 x^5-5\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9*Sqrt[(5 - 7*x^5)/(7 + 5*x^5)],x]

[Out]

(Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]*Sqrt[7 + 5*x^5]*(35*Sqrt[7 + 5*x^5]*(430 - 777*x^5 + 245*x^10) - 4514*Sqrt[35]*
Sqrt[5 - 7*x^5]*ArcSin[Sqrt[5/74]*Sqrt[5 - 7*x^5]]))/(61250*(-5 + 7*x^5))

________________________________________________________________________________________

Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{x}^{9}\sqrt{{\frac{-7\,{x}^{5}+5}{5\,{x}^{5}+7}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*((-7*x^5+5)/(5*x^5+7))^(1/2),x)

[Out]

int(x^9*((-7*x^5+5)/(5*x^5+7))^(1/2),x)

________________________________________________________________________________________

Maxima [A]  time = 1.52384, size = 163, normalized size = 1.54 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*((-7*x^5+5)/(5*x^5+7))^(1/2),x, algorithm="maxima")

[Out]

2257/30625*sqrt(35)*arctan(1/7*sqrt(35)*sqrt(-(7*x^5 - 5)/(5*x^5 + 7))) - 37/875*(675*(-(7*x^5 - 5)/(5*x^5 + 7
))^(3/2) + 427*sqrt(-(7*x^5 - 5)/(5*x^5 + 7)))/(25*(7*x^5 - 5)^2/(5*x^5 + 7)^2 - 70*(7*x^5 - 5)/(5*x^5 + 7) +
49)

________________________________________________________________________________________

Fricas [A]  time = 1.5163, size = 225, normalized size = 2.12 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*((-7*x^5+5)/(5*x^5+7))^(1/2),x, algorithm="fricas")

[Out]

1/1750*(175*x^10 - 185*x^5 - 602)*sqrt(-(7*x^5 - 5)/(5*x^5 + 7)) + 2257/61250*sqrt(35)*arctan(1/35*sqrt(35)*(3
5*x^5 + 12)*sqrt(-(7*x^5 - 5)/(5*x^5 + 7))/(7*x^5 - 5))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9*((-7*x**5+5)/(5*x**5+7))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.18099, size = 63, normalized size = 0.59 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*((-7*x^5+5)/(5*x^5+7))^(1/2),x, algorithm="giac")

[Out]

1/61250*(35*sqrt(-35*x^10 - 24*x^5 + 35)*(35*x^5 - 86) - 2257*sqrt(35)*arcsin(35/37*x^5 + 12/37))*sgn(5*x^5 +
7)