3.19.93 \(\int \sqrt {a+\frac {b}{x^2}} x \, dx\) [1893]

Optimal. Leaf size=47 \[ \frac {1}{2} \sqrt {a+\frac {b}{x^2}} x^2+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 43, 65, 214} \begin {gather*} \frac {1}{2} x^2 \sqrt {a+\frac {b}{x^2}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^2]*x,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 272

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

Rubi steps

\begin {align*} \int \sqrt {a+\frac {b}{x^2}} x \, dx &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {1}{2} \sqrt {a+\frac {b}{x^2}} x^2-\frac {1}{4} b \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {1}{2} \sqrt {a+\frac {b}{x^2}} x^2-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right )\\ &=\frac {1}{2} \sqrt {a+\frac {b}{x^2}} x^2+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2 \sqrt {a}}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 58, normalized size = 1.23 \begin {gather*} \frac {1}{2} \sqrt {a+\frac {b}{x^2}} x \left (x-\frac {b \log \left (-\sqrt {a} x+\sqrt {b+a x^2}\right )}{\sqrt {a} \sqrt {b+a x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^2]*x,x]

[Out]

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

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Maple [A]
time = 0.02, size = 62, normalized size = 1.32

method	result	size
default	\(\frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x \left (x \sqrt {a \,x^{2}+b}\, \sqrt {a}+b \ln \left (x \sqrt {a}+\sqrt {a \,x^{2}+b}\right )\right )}{2 \sqrt {a \,x^{2}+b}\, \sqrt {a}}\)	\(62\)
risch	\(\frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x^{2}}{2}+\frac {b \ln \left (x \sqrt {a}+\sqrt {a \,x^{2}+b}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{2 \sqrt {a}\, \sqrt {a \,x^{2}+b}}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 53, normalized size = 1.13 \begin {gather*} \frac {1}{2} \, \sqrt {a + \frac {b}{x^{2}}} x^{2} - \frac {b \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{4 \, \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.38, size = 127, normalized size = 2.70 \begin {gather*} \left [\frac {2 \, a x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} + \sqrt {a} b \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right )}{4 \, a}, \frac {a x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - \sqrt {-a} b \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{2 \, a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*a*x^2*sqrt((a*x^2 + b)/x^2) + sqrt(a)*b*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + b)/x^2) - b))/a, 1/
2*(a*x^2*sqrt((a*x^2 + b)/x^2) - sqrt(-a)*b*arctan(sqrt(-a)*x^2*sqrt((a*x^2 + b)/x^2)/(a*x^2 + b)))/a]

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Sympy [A]
time = 0.94, size = 41, normalized size = 0.87 \begin {gather*} \frac {\sqrt {b} x \sqrt {\frac {a x^{2}}{b} + 1}}{2} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{2 \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.83, size = 52, normalized size = 1.11 \begin {gather*} \frac {b \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{4 \, \sqrt {a}} + \frac {1}{2} \, {\left (\sqrt {a x^{2} + b} x - \frac {b \log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right )}{\sqrt {a}}\right )} \mathrm {sgn}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*log(abs(b))*sgn(x)/sqrt(a) + 1/2*(sqrt(a*x^2 + b)*x - b*log(abs(-sqrt(a)*x + sqrt(a*x^2 + b)))/sqrt(a))*
sgn(x)

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Mupad [B]
time = 1.38, size = 35, normalized size = 0.74 \begin {gather*} \frac {x^2\,\sqrt {a+\frac {b}{x^2}}}{2}+\frac {b\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2\,\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b/x^2)^(1/2),x)

[Out]

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

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