3.1931 \(\int \frac{x}{(a+\frac{b}{x^2})^{3/2}} \, dx\)

Optimal. Leaf size=69 \[ \frac{3 b}{2 a^2 \sqrt{a+\frac{b}{x^2}}}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{x^2}{2 a \sqrt{a+\frac{b}{x^2}}} \]

[Out]

(3*b)/(2*a^2*Sqrt[a + b/x^2]) + x^2/(2*a*Sqrt[a + b/x^2]) - (3*b*ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]])/(2*a^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0330882, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 208} \[ \frac{3 x^2 \sqrt{a+\frac{b}{x^2}}}{2 a^2}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{x^2}{a \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b/x^2)^(3/2),x]

[Out]

-(x^2/(a*Sqrt[a + b/x^2])) + (3*Sqrt[a + b/x^2]*x^2)/(2*a^2) - (3*b*ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]])/(2*a^(5/
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{x}{\left (a+\frac{b}{x^2}\right )^{3/2}} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{3/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{x^2}{a \sqrt{a+\frac{b}{x^2}}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )}{2 a}\\ &=-\frac{x^2}{a \sqrt{a+\frac{b}{x^2}}}+\frac{3 \sqrt{a+\frac{b}{x^2}} x^2}{2 a^2}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )}{4 a^2}\\ &=-\frac{x^2}{a \sqrt{a+\frac{b}{x^2}}}+\frac{3 \sqrt{a+\frac{b}{x^2}} x^2}{2 a^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^2}}\right )}{2 a^2}\\ &=-\frac{x^2}{a \sqrt{a+\frac{b}{x^2}}}+\frac{3 \sqrt{a+\frac{b}{x^2}} x^2}{2 a^2}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 a^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0371454, size = 74, normalized size = 1.07 \[ \frac{\sqrt{a} x \left (a x^2+3 b\right )-3 b^{3/2} \sqrt{\frac{a x^2}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{5/2} x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b/x^2)^(3/2),x]

[Out]

(Sqrt[a]*x*(3*b + a*x^2) - 3*b^(3/2)*Sqrt[1 + (a*x^2)/b]*ArcSinh[(Sqrt[a]*x)/Sqrt[b]])/(2*a^(5/2)*Sqrt[a + b/x
^2]*x)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 73, normalized size = 1.1 \begin{align*}{\frac{a{x}^{2}+b}{2\,{x}^{3}} \left ({x}^{3}{a}^{{\frac{5}{2}}}+3\,{a}^{3/2}xb-3\,\ln \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ) \sqrt{a{x}^{2}+b}ab \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*x^2+b)*(x^3*a^(5/2)+3*a^(3/2)*x*b-3*ln(x*a^(1/2)+(a*x^2+b)^(1/2))*(a*x^2+b)^(1/2)*a*b)/((a*x^2+b)/x^2)^
(3/2)/x^3/a^(7/2)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.53647, size = 421, normalized size = 6.1 \begin{align*} \left [\frac{3 \,{\left (a b x^{2} + b^{2}\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^{2} x^{4} + 3 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{4 \,{\left (a^{4} x^{2} + a^{3} b\right )}}, \frac{3 \,{\left (a b x^{2} + b^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) +{\left (a^{2} x^{4} + 3 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{2 \,{\left (a^{4} x^{2} + a^{3} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(3*(a*b*x^2 + b^2)*sqrt(a)*log(-2*a*x^2 + 2*sqrt(a)*x^2*sqrt((a*x^2 + b)/x^2) - b) + 2*(a^2*x^4 + 3*a*b*x
^2)*sqrt((a*x^2 + b)/x^2))/(a^4*x^2 + a^3*b), 1/2*(3*(a*b*x^2 + b^2)*sqrt(-a)*arctan(sqrt(-a)*x^2*sqrt((a*x^2
+ b)/x^2)/(a*x^2 + b)) + (a^2*x^4 + 3*a*b*x^2)*sqrt((a*x^2 + b)/x^2))/(a^4*x^2 + a^3*b)]

________________________________________________________________________________________

Sympy [A]  time = 3.1326, size = 71, normalized size = 1.03 \begin{align*} \frac{x^{3}}{2 a \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{3 \sqrt{b} x}{2 a^{2} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 a^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3/(2*a*sqrt(b)*sqrt(a*x**2/b + 1)) + 3*sqrt(b)*x/(2*a**2*sqrt(a*x**2/b + 1)) - 3*b*asinh(sqrt(a)*x/sqrt(b))
/(2*a**(5/2))

________________________________________________________________________________________

Giac [A]  time = 1.30483, size = 131, normalized size = 1.9 \begin{align*} \frac{1}{2} \, b{\left (\frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{2 \, a - \frac{3 \,{\left (a x^{2} + b\right )}}{x^{2}}}{{\left (a \sqrt{\frac{a x^{2} + b}{x^{2}}} - \frac{{\left (a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{x^{2}}\right )} a^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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