\(\int \frac {x^2}{(a+\frac {b}{x^2})^{3/2}} \, dx\) [1938]

Optimal result

Integrand size = 15, antiderivative size = 62 \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=\frac {4 b x}{3 a^2 \sqrt {a+\frac {b}{x^2}}}-\frac {8 b \sqrt {a+\frac {b}{x^2}} x}{3 a^3}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x^2}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {277, 198, 197} \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=-\frac {8 b x \sqrt {a+\frac {b}{x^2}}}{3 a^3}+\frac {4 b x}{3 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x^2}}} \]

[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 197

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]

Rule 198

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 277

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.66 \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=\frac {-8 b^2-4 a b x^2+a^2 x^4}{3 a^3 \sqrt {a+\frac {b}{x^2}} x} \]

[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)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.79

method	result	size
gosper	\(\frac {\left (a \,x^{2}+b \right ) \left (a^{2} x^{4}-4 a b \,x^{2}-8 b^{2}\right )}{3 a^{3} x^{3} \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}}}\)	\(49\)
default	\(\frac {\left (a \,x^{2}+b \right ) \left (a^{2} x^{4}-4 a b \,x^{2}-8 b^{2}\right )}{3 a^{3} x^{3} \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}}}\)	\(49\)
trager	\(\frac {\left (a^{2} x^{4}-4 a b \,x^{2}-8 b^{2}\right ) x \sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{3 \left (a \,x^{2}+b \right ) a^{3}}\)	\(53\)
risch	\(\frac {\left (a \,x^{2}-5 b \right ) \left (a \,x^{2}+b \right )}{3 a^{3} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}-\frac {b^{2}}{a^{3} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}\)	\(63\)

[In]

int(x^2/(a+b/x^2)^(3/2),x,method=_RETURNVERBOSE)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.82 \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=\frac {{\left (a^{2} x^{5} - 4 \, a b x^{3} - 8 \, b^{2} x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a^{4} x^{2} + a^{3} b\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (56) = 112\).

Time = 0.64 (sec) , antiderivative size = 219, normalized size of antiderivative = 3.53 \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=\frac {a^{3} b^{\frac {9}{2}} x^{6} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} - \frac {3 a^{2} b^{\frac {11}{2}} x^{4} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} - \frac {12 a b^{\frac {13}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} - \frac {8 b^{\frac {15}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=\frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} x^{3} - 6 \, \sqrt {a + \frac {b}{x^{2}}} b x}{3 \, a^{3}} - \frac {b^{2}}{\sqrt {a + \frac {b}{x^{2}}} a^{3} x} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13 \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=\frac {8 \, b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{3 \, a^{3}} - \frac {b^{2}}{\sqrt {a x^{2} + b} a^{3} \mathrm {sgn}\left (x\right )} + \frac {{\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{6} - 6 \, \sqrt {a x^{2} + b} a^{6} b}{3 \, a^{9} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

8/3*b^(3/2)*sgn(x)/a^3 - b^2/(sqrt(a*x^2 + b)*a^3*sgn(x)) + 1/3*((a*x^2 + b)^(3/2)*a^6 - 6*sqrt(a*x^2 + b)*a^6
*b)/(a^9*sgn(x))

Mupad [B] (verification not implemented)

Time = 6.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.61 \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=-\frac {-a^2\,x^4+4\,a\,b\,x^2+8\,b^2}{3\,a^3\,x\,\sqrt {a+\frac {b}{x^2}}} \]

[In]

int(x^2/(a + b/x^2)^(3/2),x)

[Out]

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