3.19.92 \(\int \sqrt {a+\frac {b}{x^2}} x^2 \, dx\) [1892]

Optimal. Leaf size=21 \[ \frac {\left (a+\frac {b}{x^2}\right )^{3/2} x^3}{3 a} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \begin {gather*} \frac {x^3 \left (a+\frac {b}{x^2}\right )^{3/2}}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^2]*x^2,x]

[Out]

((a + b/x^2)^(3/2)*x^3)/(3*a)

Rule 270

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 26, normalized size = 1.24 \begin {gather*} \frac {\sqrt {a+\frac {b}{x^2}} x \left (b+a x^2\right )}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^2]*x^2,x]

[Out]

(Sqrt[a + b/x^2]*x*(b + a*x^2))/(3*a)

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Maple [A]
time = 0.03, size = 27, normalized size = 1.29

method	result	size
gosper	\(\frac {\left (a \,x^{2}+b \right ) x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{3 a}\)	\(27\)
default	\(\frac {\left (a \,x^{2}+b \right ) x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{3 a}\)	\(27\)
risch	\(\frac {\left (a \,x^{2}+b \right ) x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{3 a}\)	\(27\)
trager	\(\frac {x \left (a \,x^{2}+b \right ) \sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{3 a}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 17, normalized size = 0.81 \begin {gather*} \frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} x^{3}}{3 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a + b/x^2)^(3/2)*x^3/a

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Fricas [A]
time = 0.38, size = 27, normalized size = 1.29 \begin {gather*} \frac {{\left (a x^{3} + b x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (15) = 30\).
time = 0.39, size = 41, normalized size = 1.95 \begin {gather*} \frac {\sqrt {b} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{3} + \frac {b^{\frac {3}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.83, size = 27, normalized size = 1.29 \begin {gather*} \frac {{\left (a x^{2} + b\right )}^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{3 \, a} - \frac {b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{3 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.24, size = 23, normalized size = 1.10 \begin {gather*} \sqrt {a+\frac {b}{x^2}}\,\left (\frac {x^3}{3}+\frac {b\,x}{3\,a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b/x^2)^(1/2),x)

[Out]

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

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