Integrand size = 15, antiderivative size = 54 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=-a \sqrt {a+\frac {b}{x^2}}-\frac {1}{3} \left (a+\frac {b}{x^2}\right )^{3/2}+a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right ) \] Output:
-a*(a+b/x^2)^(1/2)-1/3*(a+b/x^2)^(3/2)+a^(3/2)*arctanh((a+b/x^2)^(1/2)/a^( 1/2))
Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=\frac {\sqrt {a+\frac {b}{x^2}} \left (-b-4 a x^2+\frac {6 a^{3/2} x^3 \text {arctanh}\left (\frac {\sqrt {a} x}{-\sqrt {b}+\sqrt {b+a x^2}}\right )}{\sqrt {b+a x^2}}\right )}{3 x^2} \] Input:
Integrate[(a + b/x^2)^(3/2)/x,x]
Output:
(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)
Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 60, 60, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\frac {1}{2} \int \left (a+\frac {b}{x^2}\right )^{3/2} x^2d\frac {1}{x^2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (-a \int \sqrt {a+\frac {b}{x^2}} x^2d\frac {1}{x^2}-\frac {2}{3} \left (a+\frac {b}{x^2}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (-a \left (a \int \frac {x^2}{\sqrt {a+\frac {b}{x^2}}}d\frac {1}{x^2}+2 \sqrt {a+\frac {b}{x^2}}\right )-\frac {2}{3} \left (a+\frac {b}{x^2}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (-a \left (\frac {2 a \int \frac {1}{\frac {1}{b x^4}-\frac {a}{b}}d\sqrt {a+\frac {b}{x^2}}}{b}+2 \sqrt {a+\frac {b}{x^2}}\right )-\frac {2}{3} \left (a+\frac {b}{x^2}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (-a \left (2 \sqrt {a+\frac {b}{x^2}}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )\right )-\frac {2}{3} \left (a+\frac {b}{x^2}\right )^{3/2}\right )\) |
Input:
Int[(a + b/x^2)^(3/2)/x,x]
Output:
((-2*(a + b/x^2)^(3/2))/3 - a*(2*Sqrt[a + b/x^2] - 2*Sqrt[a]*ArcTanh[Sqrt[ a + b/x^2]/Sqrt[a]]))/2
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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Time = 0.08 (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 | \(\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} \left (2 a^{\frac {5}{2}} \left (a \,x^{2}+b \right )^{\frac {3}{2}} x^{4}+3 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b}\, b \,x^{4}-2 a^{\frac {3}{2}} \left (a \,x^{2}+b \right )^{\frac {5}{2}} x^{2}+3 \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right ) a^{2} b^{2} x^{3}-\left (a \,x^{2}+b \right )^{\frac {5}{2}} b \sqrt {a}\right )}{3 \left (a \,x^{2}+b \right )^{\frac {3}{2}} b^{2} \sqrt {a}}\) | \(126\) |
Input:
int((a+b/x^2)^(3/2)/x,x,method=_RETURNVERBOSE)
Output:
-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)
Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.61 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=\left [\frac {3 \, a^{\frac {3}{2}} x^{2} \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) - 2 \, {\left (4 \, a x^{2} + b\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{6 \, x^{2}}, -\frac {3 \, \sqrt {-a} a x^{2} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (4 \, a x^{2} + b\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, x^{2}}\right ] \] Input:
integrate((a+b/x^2)^(3/2)/x,x, algorithm="fricas")
Output:
[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*arc tan(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]
Time = 1.62 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=- \frac {4 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{2}}}}{3} - \frac {a^{\frac {3}{2}} \log {\left (\frac {b}{a x^{2}} \right )}}{2} + a^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )} - \frac {\sqrt {a} b \sqrt {1 + \frac {b}{a x^{2}}}}{3 x^{2}} \] Input:
integrate((a+b/x**2)**(3/2)/x,x)
Output:
-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) - sqrt(a)*b*sqrt(1 + b/(a*x**2))/(3*x**2)
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=-\frac {1}{2} \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right ) - \frac {1}{3} \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} - \sqrt {a + \frac {b}{x^{2}}} a \] Input:
integrate((a+b/x^2)^(3/2)/x,x, algorithm="maxima")
Output:
-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
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (42) = 84\).
Time = 0.35 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.26 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=-\frac {1}{2} \, a^{\frac {3}{2}} \log \left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2}\right ) \mathrm {sgn}\left (x\right ) + \frac {4 \, {\left (3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{4} a^{\frac {3}{2}} b \mathrm {sgn}\left (x\right ) - 3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} a^{\frac {3}{2}} b^{2} \mathrm {sgn}\left (x\right ) + 2 \, a^{\frac {3}{2}} b^{3} \mathrm {sgn}\left (x\right )\right )}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b\right )}^{3}} \] Input:
integrate((a+b/x^2)^(3/2)/x,x, algorithm="giac")
Output:
-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*sgn(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
Time = 0.66 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )-a\,\sqrt {a+\frac {b}{x^2}}-\frac {{\left (a+\frac {b}{x^2}\right )}^{3/2}}{3} \] Input:
int((a + b/x^2)^(3/2)/x,x)
Output:
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
Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x} \, dx=\frac {-4 \sqrt {a \,x^{2}+b}\, a \,x^{2}-\sqrt {a \,x^{2}+b}\, b +3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{2}+b}+\sqrt {a}\, x}{\sqrt {b}}\right ) a \,x^{3}}{3 x^{3}} \] Input:
int((a+b/x^2)^(3/2)/x,x)
Output:
( - 4*sqrt(a*x**2 + b)*a*x**2 - sqrt(a*x**2 + b)*b + 3*sqrt(a)*log((sqrt(a *x**2 + b) + sqrt(a)*x)/sqrt(b))*a*x**3)/(3*x**3)