Integrand size = 13, antiderivative size = 62 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x \, dx=-b \sqrt {a+\frac {b}{x^2}}+\frac {1}{2} a \sqrt {a+\frac {b}{x^2}} x^2+\frac {3}{2} \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right ) \] Output:
-b*(a+b/x^2)^(1/2)+1/2*a*(a+b/x^2)^(1/2)*x^2+3/2*a^(1/2)*b*arctanh((a+b/x^ 2)^(1/2)/a^(1/2))
Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.39 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x \, dx=\frac {\sqrt {a+\frac {b}{x^2}} \left (\left (-2 b+a x^2\right ) \sqrt {b+a x^2}+6 \sqrt {a} b x \text {arctanh}\left (\frac {\sqrt {a} x}{-\sqrt {b}+\sqrt {b+a x^2}}\right )\right )}{2 \sqrt {b+a x^2}} \] Input:
Integrate[(a + b/x^2)^(3/2)*x,x]
Output:
(Sqrt[a + b/x^2]*((-2*b + a*x^2)*Sqrt[b + a*x^2] + 6*Sqrt[a]*b*x*ArcTanh[( Sqrt[a]*x)/(-Sqrt[b] + Sqrt[b + a*x^2])]))/(2*Sqrt[b + a*x^2])
Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {798, 51, 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.
\(\displaystyle \int x \left (a+\frac {b}{x^2}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\frac {1}{2} \int \left (a+\frac {b}{x^2}\right )^{3/2} x^4d\frac {1}{x^2}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{2} \left (x^2 \left (a+\frac {b}{x^2}\right )^{3/2}-\frac {3}{2} b \int \sqrt {a+\frac {b}{x^2}} x^2d\frac {1}{x^2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (x^2 \left (a+\frac {b}{x^2}\right )^{3/2}-\frac {3}{2} b \left (a \int \frac {x^2}{\sqrt {a+\frac {b}{x^2}}}d\frac {1}{x^2}+2 \sqrt {a+\frac {b}{x^2}}\right )\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (x^2 \left (a+\frac {b}{x^2}\right )^{3/2}-\frac {3}{2} b \left (\frac {2 a \int \frac {1}{\frac {1}{b x^4}-\frac {a}{b}}d\sqrt {a+\frac {b}{x^2}}}{b}+2 \sqrt {a+\frac {b}{x^2}}\right )\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (x^2 \left (a+\frac {b}{x^2}\right )^{3/2}-\frac {3}{2} b \left (2 \sqrt {a+\frac {b}{x^2}}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )\right )\right )\) |
Input:
Int[(a + b/x^2)^(3/2)*x,x]
Output:
((a + b/x^2)^(3/2)*x^2 - (3*b*(2*Sqrt[a + b/x^2] - 2*Sqrt[a]*ArcTanh[Sqrt[ a + b/x^2]/Sqrt[a]]))/2)/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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 = 1.15
method | result | size |
risch | \(\frac {\left (a \,x^{2}-2 b \right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{2}+\frac {3 \sqrt {a}\, b \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{2 \sqrt {a \,x^{2}+b}}\) | \(71\) |
default | \(\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} x^{2} \left (2 a^{\frac {3}{2}} \left (a \,x^{2}+b \right )^{\frac {3}{2}} x^{2}+3 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b}\, b \,x^{2}-2 \left (a \,x^{2}+b \right )^{\frac {5}{2}} \sqrt {a}+3 \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right ) a \,b^{2} x \right )}{2 \left (a \,x^{2}+b \right )^{\frac {3}{2}} b \sqrt {a}}\) | \(107\) |
Input:
int((a+b/x^2)^(3/2)*x,x,method=_RETURNVERBOSE)
Output:
1/2*(a*x^2-2*b)*((a*x^2+b)/x^2)^(1/2)+3/2*a^(1/2)*b*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 = 0.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.08 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x \, dx=\left [\frac {3}{4} \, \sqrt {a} b \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) + \frac {1}{2} \, {\left (a x^{2} - 2 \, b\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}, -\frac {3}{2} \, \sqrt {-a} b \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + \frac {1}{2} \, {\left (a x^{2} - 2 \, b\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}\right ] \] Input:
integrate((a+b/x^2)^(3/2)*x,x, algorithm="fricas")
Output:
[3/4*sqrt(a)*b*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + b)/x^2) - b) + 1 /2*(a*x^2 - 2*b)*sqrt((a*x^2 + b)/x^2), -3/2*sqrt(-a)*b*arctan(sqrt(-a)*x^ 2*sqrt((a*x^2 + b)/x^2)/(a*x^2 + b)) + 1/2*(a*x^2 - 2*b)*sqrt((a*x^2 + b)/ x^2)]
Time = 1.60 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.42 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x \, dx=\frac {3 \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{2} + \frac {a^{2} x^{3}}{2 \sqrt {b} \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {a \sqrt {b} x}{2 \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {b^{\frac {3}{2}}}{x \sqrt {\frac {a x^{2}}{b} + 1}} \] Input:
integrate((a+b/x**2)**(3/2)*x,x)
Output:
3*sqrt(a)*b*asinh(sqrt(a)*x/sqrt(b))/2 + a**2*x**3/(2*sqrt(b)*sqrt(a*x**2/ b + 1)) - a*sqrt(b)*x/(2*sqrt(a*x**2/b + 1)) - b**(3/2)/(x*sqrt(a*x**2/b + 1))
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.06 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x \, dx=\frac {1}{2} \, \sqrt {a + \frac {b}{x^{2}}} a x^{2} - \frac {3}{4} \, \sqrt {a} b \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right ) - \sqrt {a + \frac {b}{x^{2}}} b \] Input:
integrate((a+b/x^2)^(3/2)*x,x, algorithm="maxima")
Output:
1/2*sqrt(a + b/x^2)*a*x^2 - 3/4*sqrt(a)*b*log((sqrt(a + b/x^2) - sqrt(a))/ (sqrt(a + b/x^2) + sqrt(a))) - sqrt(a + b/x^2)*b
Time = 0.16 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.27 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x \, dx=\frac {1}{2} \, \sqrt {a x^{2} + b} a x \mathrm {sgn}\left (x\right ) - \frac {3}{4} \, \sqrt {a} b \log \left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2}\right ) \mathrm {sgn}\left (x\right ) + \frac {2 \, \sqrt {a} b^{2} \mathrm {sgn}\left (x\right )}{{\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b} \] Input:
integrate((a+b/x^2)^(3/2)*x,x, algorithm="giac")
Output:
1/2*sqrt(a*x^2 + b)*a*x*sgn(x) - 3/4*sqrt(a)*b*log((sqrt(a)*x - sqrt(a*x^2 + b))^2)*sgn(x) + 2*sqrt(a)*b^2*sgn(x)/((sqrt(a)*x - sqrt(a*x^2 + b))^2 - b)
Time = 0.88 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.77 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x \, dx=\frac {a\,x^2\,\sqrt {a+\frac {b}{x^2}}}{2}-b\,\sqrt {a+\frac {b}{x^2}}+\frac {3\,\sqrt {a}\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2} \] Input:
int(x*(a + b/x^2)^(3/2),x)
Output:
(a*x^2*(a + b/x^2)^(1/2))/2 - b*(a + b/x^2)^(1/2) + (3*a^(1/2)*b*atanh((a + b/x^2)^(1/2)/a^(1/2)))/2
Time = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x \, dx=\frac {4 \sqrt {a \,x^{2}+b}\, a \,x^{2}-8 \sqrt {a \,x^{2}+b}\, b +12 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{2}+b}+\sqrt {a}\, x}{\sqrt {b}}\right ) b x -9 \sqrt {a}\, b x}{8 x} \] Input:
int((a+b/x^2)^(3/2)*x,x)
Output:
(4*sqrt(a*x**2 + b)*a*x**2 - 8*sqrt(a*x**2 + b)*b + 12*sqrt(a)*log((sqrt(a *x**2 + b) + sqrt(a)*x)/sqrt(b))*b*x - 9*sqrt(a)*b*x)/(8*x)