3.39 \(\int \left (a^2-x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=84 \[ \frac{5}{24} a^2 x \left (a^2-x^2\right )^{3/2}+\frac{1}{6} x \left (a^2-x^2\right )^{5/2}+\frac{5}{16} a^6 \tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right )+\frac{5}{16} a^4 x \sqrt{a^2-x^2} \]

[Out]

(5*a^4*x*Sqrt[a^2 - x^2])/16 + (5*a^2*x*(a^2 - x^2)^(3/2))/24 + (x*(a^2 - x^2)^(
5/2))/6 + (5*a^6*ArcTan[x/Sqrt[a^2 - x^2]])/16

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Rubi [A]  time = 0.036184, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{5}{24} a^2 x \left (a^2-x^2\right )^{3/2}+\frac{1}{6} x \left (a^2-x^2\right )^{5/2}+\frac{5}{16} a^6 \tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right )+\frac{5}{16} a^4 x \sqrt{a^2-x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 - x^2)^(5/2),x]

[Out]

(5*a^4*x*Sqrt[a^2 - x^2])/16 + (5*a^2*x*(a^2 - x^2)^(3/2))/24 + (x*(a^2 - x^2)^(
5/2))/6 + (5*a^6*ArcTan[x/Sqrt[a^2 - x^2]])/16

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Rubi in Sympy [A]  time = 2.68722, size = 70, normalized size = 0.83 \[ \frac{5 a^{6} \operatorname{atan}{\left (\frac{x}{\sqrt{a^{2} - x^{2}}} \right )}}{16} + \frac{5 a^{4} x \sqrt{a^{2} - x^{2}}}{16} + \frac{5 a^{2} x \left (a^{2} - x^{2}\right )^{\frac{3}{2}}}{24} + \frac{x \left (a^{2} - x^{2}\right )^{\frac{5}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**6*atan(x/sqrt(a**2 - x**2))/16 + 5*a**4*x*sqrt(a**2 - x**2)/16 + 5*a**2*x*(
a**2 - x**2)**(3/2)/24 + x*(a**2 - x**2)**(5/2)/6

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Mathematica [A]  time = 0.0612947, size = 60, normalized size = 0.71 \[ \frac{1}{48} \left (15 a^6 \tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right )+x \sqrt{a^2-x^2} \left (33 a^4-26 a^2 x^2+8 x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 - x^2)^(5/2),x]

[Out]

(x*Sqrt[a^2 - x^2]*(33*a^4 - 26*a^2*x^2 + 8*x^4) + 15*a^6*ArcTan[x/Sqrt[a^2 - x^
2]])/48

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Maple [A]  time = 0.028, size = 69, normalized size = 0.8 \[{\frac{5\,{a}^{2}x}{24} \left ({a}^{2}-{x}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{x}{6} \left ({a}^{2}-{x}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{a}^{6}}{16}\arctan \left ({x{\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}}} \right ) }+{\frac{5\,{a}^{4}x}{16}\sqrt{{a}^{2}-{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a^2-x^2)^(5/2),x)

[Out]

5/24*a^2*x*(a^2-x^2)^(3/2)+1/6*x*(a^2-x^2)^(5/2)+5/16*a^6*arctan(x/(a^2-x^2)^(1/
2))+5/16*a^4*x*(a^2-x^2)^(1/2)

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Maxima [A]  time = 1.50729, size = 84, normalized size = 1. \[ \frac{5}{16} \, a^{6} \arcsin \left (\frac{x}{\sqrt{a^{2}}}\right ) + \frac{5}{16} \, \sqrt{a^{2} - x^{2}} a^{4} x + \frac{5}{24} \,{\left (a^{2} - x^{2}\right )}^{\frac{3}{2}} a^{2} x + \frac{1}{6} \,{\left (a^{2} - x^{2}\right )}^{\frac{5}{2}} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/16*a^6*arcsin(x/sqrt(a^2)) + 5/16*sqrt(a^2 - x^2)*a^4*x + 5/24*(a^2 - x^2)^(3/
2)*a^2*x + 1/6*(a^2 - x^2)^(5/2)*x

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Fricas [A]  time = 0.214446, size = 343, normalized size = 4.08 \[ -\frac{1056 \, a^{11} x - 2944 \, a^{9} x^{3} + 3174 \, a^{7} x^{5} - 1698 \, a^{5} x^{7} + 460 \, a^{3} x^{9} - 48 \, a x^{11} + 30 \,{\left (32 \, a^{12} - 48 \, a^{10} x^{2} + 18 \, a^{8} x^{4} - a^{6} x^{6} - 2 \,{\left (16 \, a^{11} - 16 \, a^{9} x^{2} + 3 \, a^{7} x^{4}\right )} \sqrt{a^{2} - x^{2}}\right )} \arctan \left (-\frac{a - \sqrt{a^{2} - x^{2}}}{x}\right ) -{\left (1056 \, a^{10} x - 2416 \, a^{8} x^{3} + 2098 \, a^{6} x^{5} - 885 \, a^{4} x^{7} + 170 \, a^{2} x^{9} - 8 \, x^{11}\right )} \sqrt{a^{2} - x^{2}}}{48 \,{\left (32 \, a^{6} - 48 \, a^{4} x^{2} + 18 \, a^{2} x^{4} - x^{6} - 2 \,{\left (16 \, a^{5} - 16 \, a^{3} x^{2} + 3 \, a x^{4}\right )} \sqrt{a^{2} - x^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(1056*a^11*x - 2944*a^9*x^3 + 3174*a^7*x^5 - 1698*a^5*x^7 + 460*a^3*x^9 -
48*a*x^11 + 30*(32*a^12 - 48*a^10*x^2 + 18*a^8*x^4 - a^6*x^6 - 2*(16*a^11 - 16*a
^9*x^2 + 3*a^7*x^4)*sqrt(a^2 - x^2))*arctan(-(a - sqrt(a^2 - x^2))/x) - (1056*a^
10*x - 2416*a^8*x^3 + 2098*a^6*x^5 - 885*a^4*x^7 + 170*a^2*x^9 - 8*x^11)*sqrt(a^
2 - x^2))/(32*a^6 - 48*a^4*x^2 + 18*a^2*x^4 - x^6 - 2*(16*a^5 - 16*a^3*x^2 + 3*a
*x^4)*sqrt(a^2 - x^2))

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Sympy [A]  time = 6.10086, size = 180, normalized size = 2.14 \[ \begin{cases} - \frac{5 i a^{6} \operatorname{acosh}{\left (\frac{x}{a} \right )}}{16} - \frac{11 i a^{5} x}{16 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{59 i a^{3} x^{3}}{48 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{17 i a x^{5}}{24 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{7}}{6 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{x^{2}}{a^{2}}}\right | > 1 \\\frac{5 a^{6} \operatorname{asin}{\left (\frac{x}{a} \right )}}{16} + \frac{11 a^{5} x \sqrt{1 - \frac{x^{2}}{a^{2}}}}{16} - \frac{13 a^{3} x^{3} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{24} + \frac{a x^{5} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{6} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a**2-x**2)**(5/2),x)

[Out]

Piecewise((-5*I*a**6*acosh(x/a)/16 - 11*I*a**5*x/(16*sqrt(-1 + x**2/a**2)) + 59*
I*a**3*x**3/(48*sqrt(-1 + x**2/a**2)) - 17*I*a*x**5/(24*sqrt(-1 + x**2/a**2)) +
I*x**7/(6*a*sqrt(-1 + x**2/a**2)), Abs(x**2/a**2) > 1), (5*a**6*asin(x/a)/16 + 1
1*a**5*x*sqrt(1 - x**2/a**2)/16 - 13*a**3*x**3*sqrt(1 - x**2/a**2)/24 + a*x**5*s
qrt(1 - x**2/a**2)/6, True))

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GIAC/XCAS [A]  time = 0.249435, size = 68, normalized size = 0.81 \[ \frac{5}{16} \, a^{6} \arcsin \left (\frac{x}{a}\right ){\rm sign}\left (a\right ) + \frac{1}{48} \,{\left (33 \, a^{4} - 2 \,{\left (13 \, a^{2} - 4 \, x^{2}\right )} x^{2}\right )} \sqrt{a^{2} - x^{2}} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/16*a^6*arcsin(x/a)*sign(a) + 1/48*(33*a^4 - 2*(13*a^2 - 4*x^2)*x^2)*sqrt(a^2 -
x^2)*x