3.1893 \(\int \sqrt{a+\frac{b}{x^2}} x \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0224933, antiderivative size = 47, normalized size of antiderivative = 1., 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 = {266, 47, 63, 208} \[ \frac{1}{2} x^2 \sqrt{a+\frac{b}{x^2}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

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 266

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]]

Rule 47

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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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 208

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

Rubi steps

\begin{align*} \int \sqrt{a+\frac{b}{x^2}} x \, dx &=-\left (\frac{1}{2} \operatorname{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 \operatorname{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} \operatorname{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*}

Mathematica [A]  time = 0.0263612, size = 58, normalized size = 1.23 \[ \frac{1}{2} x \sqrt{a+\frac{b}{x^2}} \left (\frac{b \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{\sqrt{a} \sqrt{a x^2+b}}+x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^2]*x,x]

[Out]

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

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Maple [A]  time = 0.001, size = 62, normalized size = 1.3 \begin{align*}{\frac{x}{2}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ( x\sqrt{a{x}^{2}+b}\sqrt{a}+b\ln \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ) \right ){\frac{1}{\sqrt{a{x}^{2}+b}}}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51477, size = 292, normalized size = 6.21 \begin{align*} \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{align*}

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 = 1.93149, size = 41, normalized size = 0.87 \begin{align*} \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{align*}

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 = 1.71551, size = 70, normalized size = 1.49 \begin{align*} \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{align*}

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)