3.1932 \(\int \frac{1}{(a+\frac{b}{x^2})^{3/2} x} \, dx\)

Optimal. Leaf size=41 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a \sqrt{a+\frac{b}{x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.0236517, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^(3/2)*x),x]

[Out]

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

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{1}{a \sqrt{a+\frac{b}{x^2}}}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )}{2 a}\\ &=-\frac{1}{a \sqrt{a+\frac{b}{x^2}}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^2}}\right )}{a b}\\ &=-\frac{1}{a \sqrt{a+\frac{b}{x^2}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0554261, size = 62, normalized size = 1.51 \[ \frac{\sqrt{b} \sqrt{\frac{a x^2}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )-\sqrt{a} x}{a^{3/2} x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^(3/2)*x),x]

[Out]

(-(Sqrt[a]*x) + Sqrt[b]*Sqrt[1 + (a*x^2)/b]*ArcSinh[(Sqrt[a]*x)/Sqrt[b]])/(a^(3/2)*Sqrt[a + b/x^2]*x)

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Maple [A]  time = 0.007, size = 63, normalized size = 1.5 \begin{align*} -{\frac{a{x}^{2}+b}{{x}^{3}} \left ( x{a}^{{\frac{3}{2}}}-\ln \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ) a\sqrt{a{x}^{2}+b} \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*x^2+b)*(x*a^(3/2)-ln(x*a^(1/2)+(a*x^2+b)^(1/2))*a*(a*x^2+b)^(1/2))/((a*x^2+b)/x^2)^(3/2)/x^3/a^(5/2)

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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(1/(a+b/x^2)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.50511, size = 359, normalized size = 8.76 \begin{align*} \left [-\frac{2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (a x^{2} + b\right )} \sqrt{a} \log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right )}{2 \,{\left (a^{3} x^{2} + a^{2} b\right )}}, -\frac{a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (a x^{2} + b\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{a^{3} x^{2} + a^{2} b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(3/2)/x,x, algorithm="fricas")

[Out]

[-1/2*(2*a*x^2*sqrt((a*x^2 + b)/x^2) - (a*x^2 + b)*sqrt(a)*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + b)/x^2)
- b))/(a^3*x^2 + a^2*b), -(a*x^2*sqrt((a*x^2 + b)/x^2) + (a*x^2 + b)*sqrt(-a)*arctan(sqrt(-a)*x^2*sqrt((a*x^2
+ b)/x^2)/(a*x^2 + b)))/(a^3*x^2 + a^2*b)]

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Sympy [B]  time = 1.8982, size = 187, normalized size = 4.56 \begin{align*} - \frac{2 a^{3} x^{2} \sqrt{1 + \frac{b}{a x^{2}}}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} - \frac{a^{3} x^{2} \log{\left (\frac{b}{a x^{2}} \right )}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} + \frac{2 a^{3} x^{2} \log{\left (\sqrt{1 + \frac{b}{a x^{2}}} + 1 \right )}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} - \frac{a^{2} b \log{\left (\frac{b}{a x^{2}} \right )}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} + \frac{2 a^{2} b \log{\left (\sqrt{1 + \frac{b}{a x^{2}}} + 1 \right )}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**(3/2)/x,x)

[Out]

-2*a**3*x**2*sqrt(1 + b/(a*x**2))/(2*a**(9/2)*x**2 + 2*a**(7/2)*b) - a**3*x**2*log(b/(a*x**2))/(2*a**(9/2)*x**
2 + 2*a**(7/2)*b) + 2*a**3*x**2*log(sqrt(1 + b/(a*x**2)) + 1)/(2*a**(9/2)*x**2 + 2*a**(7/2)*b) - a**2*b*log(b/
(a*x**2))/(2*a**(9/2)*x**2 + 2*a**(7/2)*b) + 2*a**2*b*log(sqrt(1 + b/(a*x**2)) + 1)/(2*a**(9/2)*x**2 + 2*a**(7
/2)*b)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(3/2)/x,x, algorithm="giac")

[Out]

integrate(1/((a + b/x^2)^(3/2)*x), x)