3.476 \(\int \frac{(25-x^2)^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=40 \[ -\frac{\left (25-x^2\right )^{3/2}}{3 x^3}+\frac{\sqrt{25-x^2}}{x}+\sin ^{-1}\left (\frac{x}{5}\right ) \]

[Out]

Sqrt[25 - x^2]/x - (25 - x^2)^(3/2)/(3*x^3) + ArcSin[x/5]

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Rubi [A]  time = 0.0085835, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {277, 216} \[ -\frac{\left (25-x^2\right )^{3/2}}{3 x^3}+\frac{\sqrt{25-x^2}}{x}+\sin ^{-1}\left (\frac{x}{5}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(25 - x^2)^(3/2)/x^4,x]

[Out]

Sqrt[25 - x^2]/x - (25 - x^2)^(3/2)/(3*x^3) + ArcSin[x/5]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 216

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

Rubi steps

\begin{align*} \int \frac{\left (25-x^2\right )^{3/2}}{x^4} \, dx &=-\frac{\left (25-x^2\right )^{3/2}}{3 x^3}-\int \frac{\sqrt{25-x^2}}{x^2} \, dx\\ &=\frac{\sqrt{25-x^2}}{x}-\frac{\left (25-x^2\right )^{3/2}}{3 x^3}+\int \frac{1}{\sqrt{25-x^2}} \, dx\\ &=\frac{\sqrt{25-x^2}}{x}-\frac{\left (25-x^2\right )^{3/2}}{3 x^3}+\sin ^{-1}\left (\frac{x}{5}\right )\\ \end{align*}

Mathematica [C]  time = 0.0031677, size = 24, normalized size = 0.6 \[ -\frac{125 \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{x^2}{25}\right )}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(25 - x^2)^(3/2)/x^4,x]

[Out]

(-125*Hypergeometric2F1[-3/2, -3/2, -1/2, x^2/25])/(3*x^3)

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Maple [A]  time = 0.005, size = 58, normalized size = 1.5 \begin{align*} -{\frac{1}{75\,{x}^{3}} \left ( -{x}^{2}+25 \right ) ^{{\frac{5}{2}}}}+{\frac{2}{1875\,x} \left ( -{x}^{2}+25 \right ) ^{{\frac{5}{2}}}}+{\frac{2\,x}{1875} \left ( -{x}^{2}+25 \right ) ^{{\frac{3}{2}}}}+{\frac{x}{25}\sqrt{-{x}^{2}+25}}+\arcsin \left ({\frac{x}{5}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+25)^(3/2)/x^4,x)

[Out]

-1/75/x^3*(-x^2+25)^(5/2)+2/1875/x*(-x^2+25)^(5/2)+2/1875*x*(-x^2+25)^(3/2)+1/25*x*(-x^2+25)^(1/2)+arcsin(1/5*
x)

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Maxima [A]  time = 1.42977, size = 61, normalized size = 1.52 \begin{align*} \frac{1}{25} \, \sqrt{-x^{2} + 25} x + \frac{2 \,{\left (-x^{2} + 25\right )}^{\frac{3}{2}}}{75 \, x} - \frac{{\left (-x^{2} + 25\right )}^{\frac{5}{2}}}{75 \, x^{3}} + \arcsin \left (\frac{1}{5} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+25)^(3/2)/x^4,x, algorithm="maxima")

[Out]

1/25*sqrt(-x^2 + 25)*x + 2/75*(-x^2 + 25)^(3/2)/x - 1/75*(-x^2 + 25)^(5/2)/x^3 + arcsin(1/5*x)

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Fricas [A]  time = 2.01717, size = 109, normalized size = 2.72 \begin{align*} -\frac{6 \, x^{3} \arctan \left (\frac{\sqrt{-x^{2} + 25} - 5}{x}\right ) -{\left (4 \, x^{2} - 25\right )} \sqrt{-x^{2} + 25}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+25)^(3/2)/x^4,x, algorithm="fricas")

[Out]

-1/3*(6*x^3*arctan((sqrt(-x^2 + 25) - 5)/x) - (4*x^2 - 25)*sqrt(-x^2 + 25))/x^3

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Sympy [A]  time = 2.56946, size = 32, normalized size = 0.8 \begin{align*} \operatorname{asin}{\left (\frac{x}{5} \right )} + \frac{4 \sqrt{25 - x^{2}}}{3 x} - \frac{25 \sqrt{25 - x^{2}}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+25)**(3/2)/x**4,x)

[Out]

asin(x/5) + 4*sqrt(25 - x**2)/(3*x) - 25*sqrt(25 - x**2)/(3*x**3)

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Giac [B]  time = 1.0877, size = 104, normalized size = 2.6 \begin{align*} -\frac{x^{3}{\left (\frac{15 \,{\left (\sqrt{-x^{2} + 25} - 5\right )}^{2}}{x^{2}} - 1\right )}}{24 \,{\left (\sqrt{-x^{2} + 25} - 5\right )}^{3}} + \frac{5 \,{\left (\sqrt{-x^{2} + 25} - 5\right )}}{8 \, x} - \frac{{\left (\sqrt{-x^{2} + 25} - 5\right )}^{3}}{24 \, x^{3}} + \arcsin \left (\frac{1}{5} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+25)^(3/2)/x^4,x, algorithm="giac")

[Out]

-1/24*x^3*(15*(sqrt(-x^2 + 25) - 5)^2/x^2 - 1)/(sqrt(-x^2 + 25) - 5)^3 + 5/8*(sqrt(-x^2 + 25) - 5)/x - 1/24*(s
qrt(-x^2 + 25) - 5)^3/x^3 + arcsin(1/5*x)