Integrand size = 11, antiderivative size = 65 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} \, dx=-\frac {b \sqrt {a+\frac {b}{x^2}}}{2 x}+a \sqrt {a+\frac {b}{x^2}} x-\frac {3}{2} a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right ) \] Output:
-1/2*b*(a+b/x^2)^(1/2)/x+a*(a+b/x^2)^(1/2)*x-3/2*a*b^(1/2)*arctanh(b^(1/2) /(a+b/x^2)^(1/2)/x)
Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.22 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}} \left (\left (b-2 a x^2\right ) \sqrt {b+a x^2}+3 a \sqrt {b} x^2 \text {arctanh}\left (\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )\right )}{2 x \sqrt {b+a x^2}} \] Input:
Integrate[(a + b/x^2)^(3/2),x]
Output:
-1/2*(Sqrt[a + b/x^2]*((b - 2*a*x^2)*Sqrt[b + a*x^2] + 3*a*Sqrt[b]*x^2*Arc Tanh[Sqrt[b + a*x^2]/Sqrt[b]]))/(x*Sqrt[b + a*x^2])
Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {773, 247, 211, 224, 219}
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.
\(\displaystyle \int \left (a+\frac {b}{x^2}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 773 |
\(\displaystyle -\int \left (a+\frac {b}{x^2}\right )^{3/2} x^2d\frac {1}{x}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle x \left (a+\frac {b}{x^2}\right )^{3/2}-3 b \int \sqrt {a+\frac {b}{x^2}}d\frac {1}{x}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle x \left (a+\frac {b}{x^2}\right )^{3/2}-3 b \left (\frac {1}{2} a \int \frac {1}{\sqrt {a+\frac {b}{x^2}}}d\frac {1}{x}+\frac {\sqrt {a+\frac {b}{x^2}}}{2 x}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle x \left (a+\frac {b}{x^2}\right )^{3/2}-3 b \left (\frac {1}{2} a \int \frac {1}{1-\frac {b}{x^2}}d\frac {1}{\sqrt {a+\frac {b}{x^2}} x}+\frac {\sqrt {a+\frac {b}{x^2}}}{2 x}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle x \left (a+\frac {b}{x^2}\right )^{3/2}-3 b \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{2 \sqrt {b}}+\frac {\sqrt {a+\frac {b}{x^2}}}{2 x}\right )\) |
Input:
Int[(a + b/x^2)^(3/2),x]
Output:
(a + b/x^2)^(3/2)*x - 3*b*(Sqrt[a + b/x^2]/(2*x) + (a*ArcTanh[Sqrt[b]/(Sqr t[a + b/x^2]*x)])/(2*Sqrt[b]))
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^ 2, x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && !IntegerQ[p]
Time = 0.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.32
method | result | size |
risch | \(-\frac {b \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{2 x}+\frac {\left (-\frac {3 \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) \sqrt {b}\, a}{2}+\sqrt {a \,x^{2}+b}\, a \right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{\sqrt {a \,x^{2}+b}}\) | \(86\) |
default | \(-\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} x \left (3 b^{\frac {3}{2}} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) a \,x^{2}-\left (a \,x^{2}+b \right )^{\frac {3}{2}} a \,x^{2}+\left (a \,x^{2}+b \right )^{\frac {5}{2}}-3 \sqrt {a \,x^{2}+b}\, a b \,x^{2}\right )}{2 \left (a \,x^{2}+b \right )^{\frac {3}{2}} b}\) | \(100\) |
Input:
int((a+b/x^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2*b*((a*x^2+b)/x^2)^(1/2)/x+(-3/2*ln((2*b+2*b^(1/2)*(a*x^2+b)^(1/2))/x) *b^(1/2)*a+(a*x^2+b)^(1/2)*a)*((a*x^2+b)/x^2)^(1/2)*x/(a*x^2+b)^(1/2)
Time = 0.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.09 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} \, dx=\left [\frac {3 \, a \sqrt {b} x \log \left (-\frac {a x^{2} - 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) + 2 \, {\left (2 \, a x^{2} - b\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{4 \, x}, \frac {3 \, a \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{b}\right ) + {\left (2 \, a x^{2} - b\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{2 \, x}\right ] \] Input:
integrate((a+b/x^2)^(3/2),x, algorithm="fricas")
Output:
[1/4*(3*a*sqrt(b)*x*log(-(a*x^2 - 2*sqrt(b)*x*sqrt((a*x^2 + b)/x^2) + 2*b) /x^2) + 2*(2*a*x^2 - b)*sqrt((a*x^2 + b)/x^2))/x, 1/2*(3*a*sqrt(-b)*x*arct an(sqrt(-b)*x*sqrt((a*x^2 + b)/x^2)/b) + (2*a*x^2 - b)*sqrt((a*x^2 + b)/x^ 2))/x]
Time = 1.68 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.35 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} x}{\sqrt {1 + \frac {b}{a x^{2}}}} + \frac {\sqrt {a} b}{2 x \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {3 a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x} \right )}}{2} - \frac {b^{2}}{2 \sqrt {a} x^{3} \sqrt {1 + \frac {b}{a x^{2}}}} \] Input:
integrate((a+b/x**2)**(3/2),x)
Output:
a**(3/2)*x/sqrt(1 + b/(a*x**2)) + sqrt(a)*b/(2*x*sqrt(1 + b/(a*x**2))) - 3 *a*sqrt(b)*asinh(sqrt(b)/(sqrt(a)*x))/2 - b**2/(2*sqrt(a)*x**3*sqrt(1 + b/ (a*x**2)))
Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.32 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} \, dx=\sqrt {a + \frac {b}{x^{2}}} a x - \frac {\sqrt {a + \frac {b}{x^{2}}} a b x}{2 \, {\left ({\left (a + \frac {b}{x^{2}}\right )} x^{2} - b\right )}} + \frac {3}{4} \, a \sqrt {b} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right ) \] Input:
integrate((a+b/x^2)^(3/2),x, algorithm="maxima")
Output:
sqrt(a + b/x^2)*a*x - 1/2*sqrt(a + b/x^2)*a*b*x/((a + b/x^2)*x^2 - b) + 3/ 4*a*sqrt(b)*log((sqrt(a + b/x^2)*x - sqrt(b))/(sqrt(a + b/x^2)*x + sqrt(b) ))
Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} \, dx=\frac {1}{2} \, {\left (\frac {3 \, b \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b}} + 2 \, \sqrt {a x^{2} + b} \mathrm {sgn}\left (x\right ) - \frac {\sqrt {a x^{2} + b} b \mathrm {sgn}\left (x\right )}{a x^{2}}\right )} a \] Input:
integrate((a+b/x^2)^(3/2),x, algorithm="giac")
Output:
1/2*(3*b*arctan(sqrt(a*x^2 + b)/sqrt(-b))*sgn(x)/sqrt(-b) + 2*sqrt(a*x^2 + b)*sgn(x) - sqrt(a*x^2 + b)*b*sgn(x)/(a*x^2))*a
Time = 0.59 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.55 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} \, dx=\frac {x\,{\left (a\,x^2+b\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {b}{a\,x^2}\right )}{{\left (\frac {b}{a}+x^2\right )}^{3/2}} \] Input:
int((a + b/x^2)^(3/2),x)
Output:
(x*(b + a*x^2)^(3/2)*hypergeom([-3/2, -1/2], 1/2, -b/(a*x^2)))/(b/a + x^2) ^(3/2)
Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.40 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} \, dx=\frac {2 \sqrt {a \,x^{2}+b}\, a \,x^{2}-\sqrt {a \,x^{2}+b}\, b +3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{2}+b}+\sqrt {a}\, x -\sqrt {b}}{\sqrt {b}}\right ) a \,x^{2}-3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{2}+b}+\sqrt {a}\, x +\sqrt {b}}{\sqrt {b}}\right ) a \,x^{2}}{2 x^{2}} \] Input:
int((a+b/x^2)^(3/2),x)
Output:
(2*sqrt(a*x**2 + b)*a*x**2 - sqrt(a*x**2 + b)*b + 3*sqrt(b)*log((sqrt(a*x* *2 + b) + sqrt(a)*x - sqrt(b))/sqrt(b))*a*x**2 - 3*sqrt(b)*log((sqrt(a*x** 2 + b) + sqrt(a)*x + sqrt(b))/sqrt(b))*a*x**2)/(2*x**2)