3.462 \(\int \frac{\left (-10+x^2\right )^{5/2}}{x} \, dx\)

Optimal. Leaf size=61 \[ \frac{1}{5} \left (x^2-10\right )^{5/2}-\frac{10}{3} \left (x^2-10\right )^{3/2}+100 \sqrt{x^2-10}-100 \sqrt{10} \tan ^{-1}\left (\frac{\sqrt{x^2-10}}{\sqrt{10}}\right ) \]

[Out]

100*Sqrt[-10 + x^2] - (10*(-10 + x^2)^(3/2))/3 + (-10 + x^2)^(5/2)/5 - 100*Sqrt[
10]*ArcTan[Sqrt[-10 + x^2]/Sqrt[10]]

_______________________________________________________________________________________

Rubi [A]  time = 0.0689656, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{1}{5} \left (x^2-10\right )^{5/2}-\frac{10}{3} \left (x^2-10\right )^{3/2}+100 \sqrt{x^2-10}-100 \sqrt{10} \tan ^{-1}\left (\frac{\sqrt{x^2-10}}{\sqrt{10}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-10 + x^2)^(5/2)/x,x]

[Out]

100*Sqrt[-10 + x^2] - (10*(-10 + x^2)^(3/2))/3 + (-10 + x^2)^(5/2)/5 - 100*Sqrt[
10]*ArcTan[Sqrt[-10 + x^2]/Sqrt[10]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.16264, size = 54, normalized size = 0.89 \[ \frac{\left (x^{2} - 10\right )^{\frac{5}{2}}}{5} - \frac{10 \left (x^{2} - 10\right )^{\frac{3}{2}}}{3} + 100 \sqrt{x^{2} - 10} - 100 \sqrt{10} \operatorname{atan}{\left (\frac{\sqrt{10} \sqrt{x^{2} - 10}}{10} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2-10)**(5/2)/x,x)

[Out]

(x**2 - 10)**(5/2)/5 - 10*(x**2 - 10)**(3/2)/3 + 100*sqrt(x**2 - 10) - 100*sqrt(
10)*atan(sqrt(10)*sqrt(x**2 - 10)/10)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0319282, size = 47, normalized size = 0.77 \[ 100 \sqrt{10} \tan ^{-1}\left (\frac{1}{\sqrt{\frac{x^2}{10}-1}}\right )+\frac{1}{15} \sqrt{x^2-10} \left (3 x^4-110 x^2+2300\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-10 + x^2)^(5/2)/x,x]

[Out]

(Sqrt[-10 + x^2]*(2300 - 110*x^2 + 3*x^4))/15 + 100*Sqrt[10]*ArcTan[1/Sqrt[-1 +
x^2/10]]

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 46, normalized size = 0.8 \[{\frac{1}{5} \left ({x}^{2}-10 \right ) ^{{\frac{5}{2}}}}-{\frac{10}{3} \left ({x}^{2}-10 \right ) ^{{\frac{3}{2}}}}+100\,\sqrt{{x}^{2}-10}+100\,\sqrt{10}\arctan \left ({\frac{\sqrt{10}}{\sqrt{{x}^{2}-10}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-10)^(5/2)/x,x)

[Out]

1/5*(x^2-10)^(5/2)-10/3*(x^2-10)^(3/2)+100*(x^2-10)^(1/2)+100*10^(1/2)*arctan(10
^(1/2)/(x^2-10)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 1.49045, size = 57, normalized size = 0.93 \[ \frac{1}{5} \,{\left (x^{2} - 10\right )}^{\frac{5}{2}} - \frac{10}{3} \,{\left (x^{2} - 10\right )}^{\frac{3}{2}} + 100 \, \sqrt{10} \arcsin \left (\frac{\sqrt{10}}{{\left | x \right |}}\right ) + 100 \, \sqrt{x^{2} - 10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 10)^(5/2)/x,x, algorithm="maxima")

[Out]

1/5*(x^2 - 10)^(5/2) - 10/3*(x^2 - 10)^(3/2) + 100*sqrt(10)*arcsin(sqrt(10)/abs(
x)) + 100*sqrt(x^2 - 10)

_______________________________________________________________________________________

Fricas [A]  time = 0.210487, size = 217, normalized size = 3.56 \[ -\frac{12 \, x^{10} - 650 \, x^{8} + 17875 \, x^{6} - 197500 \, x^{4} + 775000 \, x^{2} - 3000 \,{\left (\sqrt{10}{\left (4 \, x^{4} - 30 \, x^{2} + 25\right )} \sqrt{x^{2} - 10} - \sqrt{10}{\left (4 \, x^{5} - 50 \, x^{3} + 125 \, x\right )}\right )} \arctan \left (-\frac{1}{10} \, \sqrt{10}{\left (x - \sqrt{x^{2} - 10}\right )}\right ) -{\left (12 \, x^{9} - 590 \, x^{7} + 15075 \, x^{5} - 128750 \, x^{3} + 287500 \, x\right )} \sqrt{x^{2} - 10} - 575000}{15 \,{\left (4 \, x^{5} - 50 \, x^{3} -{\left (4 \, x^{4} - 30 \, x^{2} + 25\right )} \sqrt{x^{2} - 10} + 125 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 10)^(5/2)/x,x, algorithm="fricas")

[Out]

-1/15*(12*x^10 - 650*x^8 + 17875*x^6 - 197500*x^4 + 775000*x^2 - 3000*(sqrt(10)*
(4*x^4 - 30*x^2 + 25)*sqrt(x^2 - 10) - sqrt(10)*(4*x^5 - 50*x^3 + 125*x))*arctan
(-1/10*sqrt(10)*(x - sqrt(x^2 - 10))) - (12*x^9 - 590*x^7 + 15075*x^5 - 128750*x
^3 + 287500*x)*sqrt(x^2 - 10) - 575000)/(4*x^5 - 50*x^3 - (4*x^4 - 30*x^2 + 25)*
sqrt(x^2 - 10) + 125*x)

_______________________________________________________________________________________

Sympy [A]  time = 21.9911, size = 167, normalized size = 2.74 \[ \begin{cases} \frac{x^{4} \sqrt{x^{2} - 10}}{5} - \frac{22 x^{2} \sqrt{x^{2} - 10}}{3} + \frac{460 \sqrt{x^{2} - 10}}{3} - 100 \sqrt{10} i \log{\left (x \right )} + 50 \sqrt{10} i \log{\left (x^{2} \right )} + 100 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{10}}{x} \right )} & \text{for}\: \frac{\left |{x^{2}}\right |}{10} > 1 \\\frac{i x^{4} \sqrt{- x^{2} + 10}}{5} - \frac{22 i x^{2} \sqrt{- x^{2} + 10}}{3} + \frac{460 i \sqrt{- x^{2} + 10}}{3} + 50 \sqrt{10} i \log{\left (x^{2} \right )} - 100 \sqrt{10} i \log{\left (\sqrt{- \frac{x^{2}}{10} + 1} + 1 \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2-10)**(5/2)/x,x)

[Out]

Piecewise((x**4*sqrt(x**2 - 10)/5 - 22*x**2*sqrt(x**2 - 10)/3 + 460*sqrt(x**2 -
10)/3 - 100*sqrt(10)*I*log(x) + 50*sqrt(10)*I*log(x**2) + 100*sqrt(10)*asin(sqrt
(10)/x), Abs(x**2)/10 > 1), (I*x**4*sqrt(-x**2 + 10)/5 - 22*I*x**2*sqrt(-x**2 +
10)/3 + 460*I*sqrt(-x**2 + 10)/3 + 50*sqrt(10)*I*log(x**2) - 100*sqrt(10)*I*log(
sqrt(-x**2/10 + 1) + 1), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.201529, size = 62, normalized size = 1.02 \[ \frac{1}{5} \,{\left (x^{2} - 10\right )}^{\frac{5}{2}} - \frac{10}{3} \,{\left (x^{2} - 10\right )}^{\frac{3}{2}} - 100 \, \sqrt{10} \arctan \left (\frac{1}{10} \, \sqrt{10} \sqrt{x^{2} - 10}\right ) + 100 \, \sqrt{x^{2} - 10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 10)^(5/2)/x,x, algorithm="giac")

[Out]

1/5*(x^2 - 10)^(5/2) - 10/3*(x^2 - 10)^(3/2) - 100*sqrt(10)*arctan(1/10*sqrt(10)
*sqrt(x^2 - 10)) + 100*sqrt(x^2 - 10)