3.83 \(\int \frac{x^4}{(1-x^2)^{5/2}} \, dx\)

Optimal. Leaf size=35 \[ \frac{x^3}{3 \left (1-x^2\right )^{3/2}}-\frac{x}{\sqrt{1-x^2}}+\sin ^{-1}(x) \]

[Out]

x^3/(3*(1 - x^2)^(3/2)) - x/Sqrt[1 - x^2] + ArcSin[x]

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Rubi [A]  time = 0.0110509, antiderivative size = 35, 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 = {288, 216} \[ \frac{x^3}{3 \left (1-x^2\right )^{3/2}}-\frac{x}{\sqrt{1-x^2}}+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[x^4/(1 - x^2)^(5/2),x]

[Out]

x^3/(3*(1 - x^2)^(3/2)) - x/Sqrt[1 - x^2] + ArcSin[x]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 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{x^4}{\left (1-x^2\right )^{5/2}} \, dx &=\frac{x^3}{3 \left (1-x^2\right )^{3/2}}-\int \frac{x^2}{\left (1-x^2\right )^{3/2}} \, dx\\ &=\frac{x^3}{3 \left (1-x^2\right )^{3/2}}-\frac{x}{\sqrt{1-x^2}}+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{x^3}{3 \left (1-x^2\right )^{3/2}}-\frac{x}{\sqrt{1-x^2}}+\sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0027465, size = 26, normalized size = 0.74 \[ \frac{x \left (4 x^2-3\right )}{3 \left (1-x^2\right )^{3/2}}+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(1 - x^2)^(5/2),x]

[Out]

(x*(-3 + 4*x^2))/(3*(1 - x^2)^(3/2)) + ArcSin[x]

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Maple [A]  time = 0., size = 30, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3} \left ( -{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}+\arcsin \left ( x \right ) -{x{\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(-x^2+1)^(5/2),x)

[Out]

1/3*x^3/(-x^2+1)^(3/2)+arcsin(x)-x/(-x^2+1)^(1/2)

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Maxima [A]  time = 1.41346, size = 59, normalized size = 1.69 \begin{align*} \frac{1}{3} \, x{\left (\frac{3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{2}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}\right )} - \frac{x}{3 \, \sqrt{-x^{2} + 1}} + \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^2+1)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.10504, size = 146, normalized size = 4.17 \begin{align*} -\frac{6 \,{\left (x^{4} - 2 \, x^{2} + 1\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) -{\left (4 \, x^{3} - 3 \, x\right )} \sqrt{-x^{2} + 1}}{3 \,{\left (x^{4} - 2 \, x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^2+1)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 2.55635, size = 105, normalized size = 3. \begin{align*} \frac{3 x^{4} \operatorname{asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} + \frac{4 x^{3} \sqrt{1 - x^{2}}}{3 x^{4} - 6 x^{2} + 3} - \frac{6 x^{2} \operatorname{asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} - \frac{3 x \sqrt{1 - x^{2}}}{3 x^{4} - 6 x^{2} + 3} + \frac{3 \operatorname{asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(-x**2+1)**(5/2),x)

[Out]

3*x**4*asin(x)/(3*x**4 - 6*x**2 + 3) + 4*x**3*sqrt(1 - x**2)/(3*x**4 - 6*x**2 + 3) - 6*x**2*asin(x)/(3*x**4 -
6*x**2 + 3) - 3*x*sqrt(1 - x**2)/(3*x**4 - 6*x**2 + 3) + 3*asin(x)/(3*x**4 - 6*x**2 + 3)

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Giac [A]  time = 1.06232, size = 39, normalized size = 1.11 \begin{align*} \frac{{\left (4 \, x^{2} - 3\right )} \sqrt{-x^{2} + 1} x}{3 \,{\left (x^{2} - 1\right )}^{2}} + \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^2+1)^(5/2),x, algorithm="giac")

[Out]

1/3*(4*x^2 - 3)*sqrt(-x^2 + 1)*x/(x^2 - 1)^2 + arcsin(x)