3.1938 \(\int \frac{x^2}{(a+\frac{b}{x^2})^{3/2}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{8 b x \sqrt{a+\frac{b}{x^2}}}{3 a^3}+\frac{4 b x}{3 a^2 \sqrt{a+\frac{b}{x^2}}}+\frac{x^3}{3 a \sqrt{a+\frac{b}{x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.0162914, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {271, 192, 191} \[ -\frac{8 b x \sqrt{a+\frac{b}{x^2}}}{3 a^3}+\frac{4 b x}{3 a^2 \sqrt{a+\frac{b}{x^2}}}+\frac{x^3}{3 a \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x^2)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0181407, size = 41, normalized size = 0.66 \[ \frac{a^2 x^4-4 a b x^2-8 b^2}{3 a^3 x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 49, normalized size = 0.8 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) \left ({a}^{2}{x}^{4}-4\,ab{x}^{2}-8\,{b}^{2} \right ) }{3\,{a}^{3}{x}^{3}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+1/x^2*b)^(3/2),x)

[Out]

1/3*(a*x^2+b)*(a^2*x^4-4*a*b*x^2-8*b^2)/a^3/x^3/((a*x^2+b)/x^2)^(3/2)

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Maxima [A]  time = 1.02799, size = 72, normalized size = 1.16 \begin{align*} \frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{3} - 6 \, \sqrt{a + \frac{b}{x^{2}}} b x}{3 \, a^{3}} - \frac{b^{2}}{\sqrt{a + \frac{b}{x^{2}}} a^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b/x^2)^(3/2),x, algorithm="maxima")

[Out]

1/3*((a + b/x^2)^(3/2)*x^3 - 6*sqrt(a + b/x^2)*b*x)/a^3 - b^2/(sqrt(a + b/x^2)*a^3*x)

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Fricas [A]  time = 1.52303, size = 104, normalized size = 1.68 \begin{align*} \frac{{\left (a^{2} x^{5} - 4 \, a b x^{3} - 8 \, b^{2} x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{4} x^{2} + a^{3} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b/x^2)^(3/2),x, algorithm="fricas")

[Out]

1/3*(a^2*x^5 - 4*a*b*x^3 - 8*b^2*x)*sqrt((a*x^2 + b)/x^2)/(a^4*x^2 + a^3*b)

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Sympy [B]  time = 1.08123, size = 219, normalized size = 3.53 \begin{align*} \frac{a^{3} b^{\frac{9}{2}} x^{6} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} - \frac{3 a^{2} b^{\frac{11}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} - \frac{12 a b^{\frac{13}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} - \frac{8 b^{\frac{15}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+b/x**2)**(3/2),x)

[Out]

a**3*b**(9/2)*x**6*sqrt(a*x**2/b + 1)/(3*a**5*b**4*x**4 + 6*a**4*b**5*x**2 + 3*a**3*b**6) - 3*a**2*b**(11/2)*x
**4*sqrt(a*x**2/b + 1)/(3*a**5*b**4*x**4 + 6*a**4*b**5*x**2 + 3*a**3*b**6) - 12*a*b**(13/2)*x**2*sqrt(a*x**2/b
+ 1)/(3*a**5*b**4*x**4 + 6*a**4*b**5*x**2 + 3*a**3*b**6) - 8*b**(15/2)*sqrt(a*x**2/b + 1)/(3*a**5*b**4*x**4 +
6*a**4*b**5*x**2 + 3*a**3*b**6)

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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{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b/x^2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^2/(a + b/x^2)^(3/2), x)