3.1903 \(\int \frac{(a+\frac{b}{x^2})^{3/2}}{x} \, dx\)

Optimal. Leaf size=54 \[ a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )-a \sqrt{a+\frac{b}{x^2}}-\frac{1}{3} \left (a+\frac{b}{x^2}\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0302899, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )-a \sqrt{a+\frac{b}{x^2}}-\frac{1}{3} \left (a+\frac{b}{x^2}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^(3/2)/x,x]

[Out]

-(a*Sqrt[a + b/x^2]) - (a + b/x^2)^(3/2)/3 + a^(3/2)*ArcTanh[Sqrt[a + b/x^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 50

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

Mathematica [C]  time = 0.0130098, size = 52, normalized size = 0.96 \[ -\frac{b \sqrt{a+\frac{b}{x^2}} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{a x^2}{b}\right )}{3 x^2 \sqrt{\frac{a x^2}{b}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^(3/2)/x,x]

[Out]

-(b*Sqrt[a + b/x^2]*Hypergeometric2F1[-3/2, -3/2, -1/2, -((a*x^2)/b)])/(3*x^2*Sqrt[1 + (a*x^2)/b])

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Maple [B]  time = 0.007, size = 125, normalized size = 2.3 \begin{align*} -{\frac{1}{3\,{b}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( -2\, \left ( a{x}^{2}+b \right ) ^{3/2}{a}^{5/2}{x}^{4}+2\, \left ( a{x}^{2}+b \right ) ^{5/2}{a}^{3/2}{x}^{2}-3\,\sqrt{a{x}^{2}+b}{a}^{5/2}{x}^{4}b-3\,\ln \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ){x}^{3}{a}^{2}{b}^{2}+ \left ( a{x}^{2}+b \right ) ^{{\frac{5}{2}}}b\sqrt{a} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*((a*x^2+b)/x^2)^(3/2)*(-2*(a*x^2+b)^(3/2)*a^(5/2)*x^4+2*(a*x^2+b)^(5/2)*a^(3/2)*x^2-3*(a*x^2+b)^(1/2)*a^(
5/2)*x^4*b-3*ln(x*a^(1/2)+(a*x^2+b)^(1/2))*x^3*a^2*b^2+(a*x^2+b)^(5/2)*b*a^(1/2))/(a*x^2+b)^(3/2)/b^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)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.574, size = 333, normalized size = 6.17 \begin{align*} \left [\frac{3 \, a^{\frac{3}{2}} x^{2} \log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right ) - 2 \,{\left (4 \, a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{6 \, x^{2}}, -\frac{3 \, \sqrt{-a} a x^{2} \arctan \left (\frac{\sqrt{-a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) +{\left (4 \, a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \, x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*a^(3/2)*x^2*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + b)/x^2) - b) - 2*(4*a*x^2 + b)*sqrt((a*x^2 + b)
/x^2))/x^2, -1/3*(3*sqrt(-a)*a*x^2*arctan(sqrt(-a)*x^2*sqrt((a*x^2 + b)/x^2)/(a*x^2 + b)) + (4*a*x^2 + b)*sqrt
((a*x^2 + b)/x^2))/x^2]

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Sympy [A]  time = 2.13007, size = 78, normalized size = 1.44 \begin{align*} - \frac{4 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{3} - \frac{a^{\frac{3}{2}} \log{\left (\frac{b}{a x^{2}} \right )}}{2} + a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b}{a x^{2}}} + 1 \right )} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x^{2}}}}{3 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*a**(3/2)*sqrt(1 + b/(a*x**2))/3 - a**(3/2)*log(b/(a*x**2))/2 + a**(3/2)*log(sqrt(1 + b/(a*x**2)) + 1) - sqr
t(a)*b*sqrt(1 + b/(a*x**2))/(3*x**2)

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Giac [B]  time = 1.49469, size = 165, normalized size = 3.06 \begin{align*} -\frac{1}{2} \, a^{\frac{3}{2}} \log \left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2}\right ) \mathrm{sgn}\left (x\right ) + \frac{4 \,{\left (3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{4} a^{\frac{3}{2}} b \mathrm{sgn}\left (x\right ) - 3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} a^{\frac{3}{2}} b^{2} \mathrm{sgn}\left (x\right ) + 2 \, a^{\frac{3}{2}} b^{3} \mathrm{sgn}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^(3/2)*log((sqrt(a)*x - sqrt(a*x^2 + b))^2)*sgn(x) + 4/3*(3*(sqrt(a)*x - sqrt(a*x^2 + b))^4*a^(3/2)*b*sg
n(x) - 3*(sqrt(a)*x - sqrt(a*x^2 + b))^2*a^(3/2)*b^2*sgn(x) + 2*a^(3/2)*b^3*sgn(x))/((sqrt(a)*x - sqrt(a*x^2 +
b))^2 - b)^3