\(\int \frac {1}{(a+\frac {b}{x^2})^{3/2} x} \, dx\) [1932]

Optimal result

Integrand size = 15, antiderivative size = 41 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x} \, dx=-\frac {1}{a \sqrt {a+\frac {b}{x^2}}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 53, 65, 214} \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a \sqrt {a+\frac {b}{x^2}}} \]

[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 53

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 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 {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 {\text {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 {\text {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] (verified)

Time = 0.13 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.73 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x} \, dx=\frac {-\sqrt {a} x+2 \sqrt {b+a x^2} \text {arctanh}\left (\frac {\sqrt {a} x}{-\sqrt {b}+\sqrt {b+a x^2}}\right )}{a^{3/2} \sqrt {a+\frac {b}{x^2}} x} \]

[In]

Integrate[1/((a + b/x^2)^(3/2)*x),x]

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.54

method	result	size
default	\(-\frac {\left (a \,x^{2}+b \right ) \left (x \,a^{\frac {3}{2}}-\ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right ) a \sqrt {a \,x^{2}+b}\right )}{\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} x^{3} a^{\frac {5}{2}}}\)	\(63\)

[In]

int(1/(a+b/x^2)^(3/2)/x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (33) = 66\).

Time = 0.32 (sec) , antiderivative size = 163, normalized size of antiderivative = 3.98 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x} \, dx=\left [-\frac {2 \, a x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - {\left (a x^{2} + b\right )} \sqrt {a} \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right )}{2 \, {\left (a^{3} x^{2} + a^{2} b\right )}}, -\frac {a x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} + {\left (a x^{2} + b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{a^{3} x^{2} + a^{2} b}\right ] \]

[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)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (34) = 68\).

Time = 1.02 (sec) , antiderivative size = 187, normalized size of antiderivative = 4.56 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x} \, dx=- \frac {2 a^{3} x^{2} \sqrt {1 + \frac {b}{a x^{2}}}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} - \frac {a^{3} x^{2} \log {\left (\frac {b}{a x^{2}} \right )}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} + \frac {2 a^{3} x^{2} \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} - \frac {a^{2} b \log {\left (\frac {b}{a x^{2}} \right )}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} + \frac {2 a^{2} b \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x} \, dx=-\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{2 \, a^{\frac {3}{2}}} - \frac {1}{\sqrt {a + \frac {b}{x^{2}}} a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x} \, dx=\frac {\log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{2 \, a^{\frac {3}{2}}} - \frac {x}{\sqrt {a x^{2} + b} a \mathrm {sgn}\left (x\right )} - \frac {\log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a\,\sqrt {a+\frac {b}{x^2}}} \]

[In]

int(1/(x*(a + b/x^2)^(3/2)),x)

[Out]

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