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

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 + (2257*ArcTan[Sqrt[5/7]*Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]])
/(875*Sqrt[35])

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Rubi [A]  time = 0.118541, 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 \[ \frac{1}{250} \sqrt{\frac{5-7 x^5}{5 x^5+7}} \left (5 x^5+7\right )^2-\frac{27}{350} \sqrt{\frac{5-7 x^5}{5 x^5+7}} \left (5 x^5+7\right )+\frac{2257 \tan ^{-1}\left (\sqrt{\frac{5}{7}} \sqrt{\frac{5-7 x^5}{5 x^5+7}}\right )}{875 \sqrt{35}} \]

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 + (2257*ArcTan[Sqrt[5/7]*Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]])
/(875*Sqrt[35])

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Rubi in Sympy [A]  time = 6.16264, size = 104, normalized size = 0.98 \[ - \frac{999 \sqrt{\frac{- 7 x^{5} + 5}{5 x^{5} + 7}}}{175 \left (\frac{5 \left (- 7 x^{5} + 5\right )}{5 x^{5} + 7} + 7\right )} + \frac{2738 \sqrt{\frac{- 7 x^{5} + 5}{5 x^{5} + 7}}}{125 \left (\frac{5 \left (- 7 x^{5} + 5\right )}{5 x^{5} + 7} + 7\right )^{2}} + \frac{2257 \sqrt{35} \operatorname{atan}{\left (\frac{\sqrt{35} \sqrt{\frac{- 7 x^{5} + 5}{5 x^{5} + 7}}}{7} \right )}}{30625} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9*((-7*x**5+5)/(5*x**5+7))**(1/2),x)

[Out]

-999*sqrt((-7*x**5 + 5)/(5*x**5 + 7))/(175*(5*(-7*x**5 + 5)/(5*x**5 + 7) + 7)) +
2738*sqrt((-7*x**5 + 5)/(5*x**5 + 7))/(125*(5*(-7*x**5 + 5)/(5*x**5 + 7) + 7)**
2) + 2257*sqrt(35)*atan(sqrt(35)*sqrt((-7*x**5 + 5)/(5*x**5 + 7))/7)/30625

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Mathematica [A]  time = 0.125431, size = 124, normalized size = 1.17 \[ \frac{\sqrt{\frac{5-7 x^5}{5 x^5+7}} \left (2257 \sqrt{35} \sqrt{5 x^5+7} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{7}-\frac{x^5}{5}} \left (35 x^5+12\right )}{\sqrt{5 x^5+7} \left (7 x^5-5\right )}\right )+35 \sqrt{5-7 x^5} \left (175 x^{10}-185 x^5-602\right )\right )}{61250 \sqrt{5-7 x^5}} \]

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)]*(35*Sqrt[5 - 7*x^5]*(-602 - 185*x^5 + 175*x^10) +
2257*Sqrt[35]*Sqrt[7 + 5*x^5]*ArcTan[(Sqrt[1/7 - x^5/5]*(12 + 35*x^5))/(Sqrt[7
+ 5*x^5]*(-5 + 7*x^5))]))/(61250*Sqrt[5 - 7*x^5])

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{x}^{9}\sqrt{{\frac{-7\,{x}^{5}+5}{5\,{x}^{5}+7}}}\, dx \]

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)

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Maxima [A]  time = 0.807534, size = 163, normalized size = 1.54 \[ \frac{2257}{30625} \, \sqrt{35} \arctan \left (\frac{1}{7} \, \sqrt{35} \sqrt{-\frac{7 \, x^{5} - 5}{5 \, x^{5} + 7}}\right ) - \frac{37 \,{\left (675 \, \left (-\frac{7 \, x^{5} - 5}{5 \, x^{5} + 7}\right )^{\frac{3}{2}} + 427 \, \sqrt{-\frac{7 \, x^{5} - 5}{5 \, x^{5} + 7}}\right )}}{875 \,{\left (\frac{25 \,{\left (7 \, x^{5} - 5\right )}^{2}}{{\left (5 \, x^{5} + 7\right )}^{2}} - \frac{70 \,{\left (7 \, x^{5} - 5\right )}}{5 \, x^{5} + 7} + 49\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9*sqrt(-(7*x^5 - 5)/(5*x^5 + 7)),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)

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Fricas [A]  time = 0.27461, size = 116, normalized size = 1.09 \[ \frac{1}{61250} \, \sqrt{35}{\left (\sqrt{35}{\left (175 \, x^{10} - 185 \, x^{5} - 602\right )} \sqrt{-\frac{7 \, x^{5} - 5}{5 \, x^{5} + 7}} - 2257 \, \arctan \left (\frac{\sqrt{35}{\left (35 \, x^{5} + 12\right )}}{35 \,{\left (5 \, x^{5} + 7\right )} \sqrt{-\frac{7 \, x^{5} - 5}{5 \, x^{5} + 7}}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9*sqrt(-(7*x^5 - 5)/(5*x^5 + 7)),x, algorithm="fricas")

[Out]

1/61250*sqrt(35)*(sqrt(35)*(175*x^10 - 185*x^5 - 602)*sqrt(-(7*x^5 - 5)/(5*x^5 +
7)) - 2257*arctan(1/35*sqrt(35)*(35*x^5 + 12)/((5*x^5 + 7)*sqrt(-(7*x^5 - 5)/(5
*x^5 + 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**9*((-7*x**5+5)/(5*x**5+7))**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.268315, size = 63, normalized size = 0.59 \[ \frac{1}{61250} \,{\left (35 \, \sqrt{-35 \, x^{10} - 24 \, x^{5} + 35}{\left (35 \, x^{5} - 86\right )} - 2257 \, \sqrt{35} \arcsin \left (\frac{35}{37} \, x^{5} + \frac{12}{37}\right )\right )}{\rm sign}\left (5 \, x^{5} + 7\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9*sqrt(-(7*x^5 - 5)/(5*x^5 + 7)),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))*sign(5*x^5 + 7)