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

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

[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 = 0.0145512, 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, Rules used = {195, 217, 203} \[ \frac{5}{24} a^2 x \left (a^2-x^2\right )^{3/2}+\frac{1}{6} x \left (a^2-x^2\right )^{5/2}+\frac{5}{16} a^6 \tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right )+\frac{5}{16} a^4 x \sqrt{a^2-x^2} \]

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

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

Rubi steps

\begin{align*} \int \left (a^2-x^2\right )^{5/2} \, dx &=\frac{1}{6} x \left (a^2-x^2\right )^{5/2}+\frac{1}{6} \left (5 a^2\right ) \int \left (a^2-x^2\right )^{3/2} \, dx\\ &=\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{1}{8} \left (5 a^4\right ) \int \sqrt{a^2-x^2} \, dx\\ &=\frac{5}{16} a^4 x \sqrt{a^2-x^2}+\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{1}{16} \left (5 a^6\right ) \int \frac{1}{\sqrt{a^2-x^2}} \, dx\\ &=\frac{5}{16} a^4 x \sqrt{a^2-x^2}+\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{1}{16} \left (5 a^6\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{x}{\sqrt{a^2-x^2}}\right )\\ &=\frac{5}{16} a^4 x \sqrt{a^2-x^2}+\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 )\\ \end{align*}

Mathematica [A]  time = 0.0665723, size = 62, normalized size = 0.74 \[ \frac{1}{48} \sqrt{a^2-x^2} \left (-26 a^2 x^3+\frac{15 a^5 \sin ^{-1}\left (\frac{x}{a}\right )}{\sqrt{1-\frac{x^2}{a^2}}}+33 a^4 x+8 x^5\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 - x^2)^(5/2),x]

[Out]

(Sqrt[a^2 - x^2]*(33*a^4*x - 26*a^2*x^3 + 8*x^5 + (15*a^5*ArcSin[x/a])/Sqrt[1 - x^2/a^2]))/48

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Maple [A]  time = 0.006, size = 69, normalized size = 0.8 \begin{align*}{\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}}} \end{align*}

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 = 1.43738, size = 84, normalized size = 1. \begin{align*} \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 \end{align*}

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 = 0.441388, size = 132, normalized size = 1.57 \begin{align*} -\frac{5}{8} \, a^{6} \arctan \left (-\frac{a - \sqrt{a^{2} - x^{2}}}{x}\right ) + \frac{1}{48} \,{\left (33 \, a^{4} x - 26 \, a^{2} x^{3} + 8 \, x^{5}\right )} \sqrt{a^{2} - x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/8*a^6*arctan(-(a - sqrt(a^2 - x^2))/x) + 1/48*(33*a^4*x - 26*a^2*x^3 + 8*x^5)*sqrt(a^2 - x^2)

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Sympy [A]  time = 3.84263, size = 182, normalized size = 2.17 \begin{align*} \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}\: \frac{\left |{x^{2}}\right |}{\left |{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} \end{align*}

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)/Abs(a**2) > 1),
(5*a**6*asin(x/a)/16 + 11*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*sqrt(1
- x**2/a**2)/6, True))

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Giac [A]  time = 1.10558, size = 68, normalized size = 0.81 \begin{align*} \frac{5}{16} \, a^{6} \arcsin \left (\frac{x}{a}\right ) \mathrm{sgn}\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 \end{align*}

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)*sgn(a) + 1/48*(33*a^4 - 2*(13*a^2 - 4*x^2)*x^2)*sqrt(a^2 - x^2)*x