3.1951 \(\int \frac{x^2}{(a+\frac{b}{x^2})^{5/2}} \, dx\)

Optimal. Leaf size=82 \[ -\frac{16 b x \sqrt{a+\frac{b}{x^2}}}{3 a^4}+\frac{8 b x}{3 a^3 \sqrt{a+\frac{b}{x^2}}}+\frac{2 b x}{3 a^2 \left (a+\frac{b}{x^2}\right )^{3/2}}+\frac{x^3}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]

[Out]

(2*b*x)/(3*a^2*(a + b/x^2)^(3/2)) + (8*b*x)/(3*a^3*Sqrt[a + b/x^2]) - (16*b*Sqrt[a + b/x^2]*x)/(3*a^4) + x^3/(
3*a*(a + b/x^2)^(3/2))

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Rubi [A]  time = 0.0221335, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {271, 192, 191} \[ -\frac{16 b x \sqrt{a+\frac{b}{x^2}}}{3 a^4}+\frac{8 b x}{3 a^3 \sqrt{a+\frac{b}{x^2}}}+\frac{2 b x}{3 a^2 \left (a+\frac{b}{x^2}\right )^{3/2}}+\frac{x^3}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x^2)^(5/2),x]

[Out]

(2*b*x)/(3*a^2*(a + b/x^2)^(3/2)) + (8*b*x)/(3*a^3*Sqrt[a + b/x^2]) - (16*b*Sqrt[a + b/x^2]*x)/(3*a^4) + x^3/(
3*a*(a + b/x^2)^(3/2))

Rule 271

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

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0257678, size = 61, normalized size = 0.74 \[ \frac{-6 a^2 b x^4+a^3 x^6-24 a b^2 x^2-16 b^3}{3 a^4 x \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.005, size = 60, normalized size = 0.7 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) \left ({a}^{3}{x}^{6}-6\,{a}^{2}b{x}^{4}-24\,a{b}^{2}{x}^{2}-16\,{b}^{3} \right ) }{3\,{a}^{4}{x}^{5}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+1/x^2*b)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.02417, size = 96, normalized size = 1.17 \begin{align*} \frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{3} - 9 \, \sqrt{a + \frac{b}{x^{2}}} b x}{3 \, a^{4}} - \frac{9 \,{\left (a + \frac{b}{x^{2}}\right )} b^{2} x^{2} - b^{3}}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b/x^2)^(5/2),x, algorithm="maxima")

[Out]

1/3*((a + b/x^2)^(3/2)*x^3 - 9*sqrt(a + b/x^2)*b*x)/a^4 - 1/3*(9*(a + b/x^2)*b^2*x^2 - b^3)/((a + b/x^2)^(3/2)
*a^4*x^3)

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Fricas [A]  time = 1.47798, size = 150, normalized size = 1.83 \begin{align*} \frac{{\left (a^{3} x^{7} - 6 \, a^{2} b x^{5} - 24 \, a b^{2} x^{3} - 16 \, b^{3} x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a^3*x^7 - 6*a^2*b*x^5 - 24*a*b^2*x^3 - 16*b^3*x)*sqrt((a*x^2 + b)/x^2)/(a^6*x^4 + 2*a^5*b*x^2 + a^4*b^2)

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Sympy [B]  time = 2.2906, size = 337, normalized size = 4.11 \begin{align*} \frac{a^{4} b^{\frac{19}{2}} x^{8} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} - \frac{5 a^{3} b^{\frac{21}{2}} x^{6} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} - \frac{30 a^{2} b^{\frac{23}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} - \frac{40 a b^{\frac{25}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} - \frac{16 b^{\frac{27}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+b/x**2)**(5/2),x)

[Out]

a**4*b**(19/2)*x**8*sqrt(a*x**2/b + 1)/(3*a**7*b**9*x**6 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**2 + 3*a**4*b**1
2) - 5*a**3*b**(21/2)*x**6*sqrt(a*x**2/b + 1)/(3*a**7*b**9*x**6 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**2 + 3*a*
*4*b**12) - 30*a**2*b**(23/2)*x**4*sqrt(a*x**2/b + 1)/(3*a**7*b**9*x**6 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**
2 + 3*a**4*b**12) - 40*a*b**(25/2)*x**2*sqrt(a*x**2/b + 1)/(3*a**7*b**9*x**6 + 9*a**6*b**10*x**4 + 9*a**5*b**1
1*x**2 + 3*a**4*b**12) - 16*b**(27/2)*sqrt(a*x**2/b + 1)/(3*a**7*b**9*x**6 + 9*a**6*b**10*x**4 + 9*a**5*b**11*
x**2 + 3*a**4*b**12)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b/x^2)^(5/2),x, algorithm="giac")

[Out]

integrate(x^2/(a + b/x^2)^(5/2), x)