\(\int \frac {(a+\frac {b}{x^2})^{3/2}}{x} \, dx\) [1903]

Optimal result

Integrand size = 15, antiderivative size = 54 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=-a \sqrt {a+\frac {b}{x^2}}-\frac {1}{3} \left (a+\frac {b}{x^2}\right )^{3/2}+a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 52, 65, 214} \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )-a \sqrt {a+\frac {b}{x^2}}-\frac {1}{3} \left (a+\frac {b}{x^2}\right )^{3/2} \]

[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 52

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 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*} \text {integral}& = -\left (\frac {1}{2} \text {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 \text {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 \text {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 \text {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 [A] (verified)

Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=\frac {\sqrt {a+\frac {b}{x^2}} \left (-b-4 a x^2+\frac {6 a^{3/2} x^3 \text {arctanh}\left (\frac {\sqrt {a} x}{-\sqrt {b}+\sqrt {b+a x^2}}\right )}{\sqrt {b+a x^2}}\right )}{3 x^2} \]

[In]

Integrate[(a + b/x^2)^(3/2)/x,x]

[Out]

(Sqrt[a + b/x^2]*(-b - 4*a*x^2 + (6*a^(3/2)*x^3*ArcTanh[(Sqrt[a]*x)/(-Sqrt[b] + Sqrt[b + a*x^2])])/Sqrt[b + a*
x^2]))/(3*x^2)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.31

method	result	size
risch	\(-\frac {\left (4 a \,x^{2}+b \right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{3 x^{2}}+\frac {a^{\frac {3}{2}} \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{\sqrt {a \,x^{2}+b}}\)	\(71\)
default	\(\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} \left (2 a^{\frac {5}{2}} \left (a \,x^{2}+b \right )^{\frac {3}{2}} x^{4}+3 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b}\, b \,x^{4}-2 a^{\frac {3}{2}} \left (a \,x^{2}+b \right )^{\frac {5}{2}} x^{2}+3 \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right ) a^{2} b^{2} x^{3}-\left (a \,x^{2}+b \right )^{\frac {5}{2}} b \sqrt {a}\right )}{3 \left (a \,x^{2}+b \right )^{\frac {3}{2}} b^{2} \sqrt {a}}\)	\(126\)

[In]

int((a+b/x^2)^(3/2)/x,x,method=_RETURNVERBOSE)

[Out]

-1/3*(4*a*x^2+b)/x^2*((a*x^2+b)/x^2)^(1/2)+a^(3/2)*ln(a^(1/2)*x+(a*x^2+b)^(1/2))*((a*x^2+b)/x^2)^(1/2)*x/(a*x^
2+b)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.61 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=\left [\frac {3 \, a^{\frac {3}{2}} x^{2} \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) - 2 \, {\left (4 \, a x^{2} + b\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{6 \, x^{2}}, -\frac {3 \, \sqrt {-a} a x^{2} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (4 \, a x^{2} + b\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, x^{2}}\right ] \]

[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]

Sympy [A] (verification not implemented)

Time = 1.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=- \frac {4 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{2}}}}{3} - \frac {a^{\frac {3}{2}} \log {\left (\frac {b}{a x^{2}} \right )}}{2} + a^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )} - \frac {\sqrt {a} b \sqrt {1 + \frac {b}{a x^{2}}}}{3 x^{2}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=-\frac {1}{2} \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right ) - \frac {1}{3} \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} - \sqrt {a + \frac {b}{x^{2}}} a \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (42) = 84\).

Time = 0.47 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.26 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=-\frac {1}{2} \, a^{\frac {3}{2}} \log \left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2}\right ) \mathrm {sgn}\left (x\right ) + \frac {4 \, {\left (3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{4} a^{\frac {3}{2}} b \mathrm {sgn}\left (x\right ) - 3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} a^{\frac {3}{2}} b^{2} \mathrm {sgn}\left (x\right ) + 2 \, a^{\frac {3}{2}} b^{3} \mathrm {sgn}\left (x\right )\right )}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b\right )}^{3}} \]

[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

Mupad [B] (verification not implemented)

Time = 6.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )-a\,\sqrt {a+\frac {b}{x^2}}-\frac {{\left (a+\frac {b}{x^2}\right )}^{3/2}}{3} \]

[In]

int((a + b/x^2)^(3/2)/x,x)

[Out]

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