\(\int \frac {x^2}{\sqrt {a+\frac {b}{x^2}}} \, dx\) [1923]

Optimal result

Integrand size = 15, antiderivative size = 42 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x^2}}} \, dx=-\frac {2 b \sqrt {a+\frac {b}{x^2}} x}{3 a^2}+\frac {\sqrt {a+\frac {b}{x^2}} x^3}{3 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 197} \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {x^3 \sqrt {a+\frac {b}{x^2}}}{3 a}-\frac {2 b x \sqrt {a+\frac {b}{x^2}}}{3 a^2} \]

[In]

Int[x^2/Sqrt[a + b/x^2],x]

[Out]

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

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 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 {\sqrt {a+\frac {b}{x^2}} x^3}{3 a}-\frac {(2 b) \int \frac {1}{\sqrt {a+\frac {b}{x^2}}} \, dx}{3 a} \\ & = -\frac {2 b \sqrt {a+\frac {b}{x^2}} x}{3 a^2}+\frac {\sqrt {a+\frac {b}{x^2}} x^3}{3 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.67 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {\sqrt {a+\frac {b}{x^2}} x \left (-2 b+a x^2\right )}{3 a^2} \]

[In]

Integrate[x^2/Sqrt[a + b/x^2],x]

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.79

method	result	size
trager	\(\frac {\left (a \,x^{2}-2 b \right ) x \sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{3 a^{2}}\)	\(33\)
gosper	\(\frac {\left (a \,x^{2}+b \right ) \left (a \,x^{2}-2 b \right )}{3 a^{2} x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(38\)
default	\(\frac {\left (a \,x^{2}+b \right ) \left (a \,x^{2}-2 b \right )}{3 a^{2} x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(38\)
risch	\(\frac {\left (a \,x^{2}+b \right ) \left (a \,x^{2}-2 b \right )}{3 a^{2} x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(38\)

[In]

int(x^2/(a+b/x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.67 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {{\left (a x^{3} - 2 \, b x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, a^{2}} \]

[In]

integrate(x^2/(a+b/x^2)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.10 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {\sqrt {b} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a} - \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{2}} \]

[In]

integrate(x**2/(a+b/x**2)**(1/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.76 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} x^{3} - 3 \, \sqrt {a + \frac {b}{x^{2}}} b x}{3 \, a^{2}} \]

[In]

integrate(x^2/(a+b/x^2)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.14 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {2 \, b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{3 \, a^{2}} + \frac {{\left (a x^{2} + b\right )}^{\frac {3}{2}}}{3 \, a^{2} \mathrm {sgn}\left (x\right )} - \frac {\sqrt {a x^{2} + b} b}{a^{2} \mathrm {sgn}\left (x\right )} \]

[In]

integrate(x^2/(a+b/x^2)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.44 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.60 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {x^3\,\sqrt {a+\frac {b}{x^2}}\,\left (a-\frac {2\,b}{x^2}\right )}{3\,a^2} \]

[In]

int(x^2/(a + b/x^2)^(1/2),x)

[Out]

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