Integrand size = 15, antiderivative size = 65 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^2 \, dx=\frac {4}{3} b \sqrt {a+\frac {b}{x^2}} x+\frac {1}{3} a \sqrt {a+\frac {b}{x^2}} x^3-b^{3/2} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right ) \] Output:
4/3*b*(a+b/x^2)^(1/2)*x+1/3*a*(a+b/x^2)^(1/2)*x^3-b^(3/2)*arctanh(b^(1/2)/ (a+b/x^2)^(1/2)/x)
Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.14 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^2 \, dx=\frac {\sqrt {a+\frac {b}{x^2}} x \left (\sqrt {b+a x^2} \left (4 b+a x^2\right )-3 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )\right )}{3 \sqrt {b+a x^2}} \] Input:
Integrate[(a + b/x^2)^(3/2)*x^2,x]
Output:
(Sqrt[a + b/x^2]*x*(Sqrt[b + a*x^2]*(4*b + a*x^2) - 3*b^(3/2)*ArcTanh[Sqrt [b + a*x^2]/Sqrt[b]]))/(3*Sqrt[b + a*x^2])
Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {858, 247, 247, 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.
\(\displaystyle \int x^2 \left (a+\frac {b}{x^2}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\int \left (a+\frac {b}{x^2}\right )^{3/2} x^4d\frac {1}{x}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {1}{3} x^3 \left (a+\frac {b}{x^2}\right )^{3/2}-b \int \sqrt {a+\frac {b}{x^2}} x^2d\frac {1}{x}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {1}{3} x^3 \left (a+\frac {b}{x^2}\right )^{3/2}-b \left (b \int \frac {1}{\sqrt {a+\frac {b}{x^2}}}d\frac {1}{x}-x \sqrt {a+\frac {b}{x^2}}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{3} x^3 \left (a+\frac {b}{x^2}\right )^{3/2}-b \left (b \int \frac {1}{1-\frac {b}{x^2}}d\frac {1}{\sqrt {a+\frac {b}{x^2}} x}-x \sqrt {a+\frac {b}{x^2}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} x^3 \left (a+\frac {b}{x^2}\right )^{3/2}-b \left (\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )-x \sqrt {a+\frac {b}{x^2}}\right )\) |
Input:
Int[(a + b/x^2)^(3/2)*x^2,x]
Output:
((a + b/x^2)^(3/2)*x^3)/3 - b*(-(Sqrt[a + b/x^2]*x) + Sqrt[b]*ArcTanh[Sqrt [b]/(Sqrt[a + b/x^2]*x)])
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.20
method | result | size |
default | \(-\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} x^{3} \left (3 b^{\frac {3}{2}} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right )-\left (a \,x^{2}+b \right )^{\frac {3}{2}}-3 \sqrt {a \,x^{2}+b}\, b \right )}{3 \left (a \,x^{2}+b \right )^{\frac {3}{2}}}\) | \(78\) |
Input:
int((a+b/x^2)^(3/2)*x^2,x,method=_RETURNVERBOSE)
Output:
-1/3*((a*x^2+b)/x^2)^(3/2)*x^3*(3*b^(3/2)*ln(2*(b^(1/2)*(a*x^2+b)^(1/2)+b) /x)-(a*x^2+b)^(3/2)-3*(a*x^2+b)^(1/2)*b)/(a*x^2+b)^(3/2)
Time = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.89 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^2 \, dx=\left [\frac {1}{2} \, b^{\frac {3}{2}} \log \left (-\frac {a x^{2} - 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) + \frac {1}{3} \, {\left (a x^{3} + 4 \, b x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}, \sqrt {-b} b \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{b}\right ) + \frac {1}{3} \, {\left (a x^{3} + 4 \, b x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}\right ] \] Input:
integrate((a+b/x^2)^(3/2)*x^2,x, algorithm="fricas")
Output:
[1/2*b^(3/2)*log(-(a*x^2 - 2*sqrt(b)*x*sqrt((a*x^2 + b)/x^2) + 2*b)/x^2) + 1/3*(a*x^3 + 4*b*x)*sqrt((a*x^2 + b)/x^2), sqrt(-b)*b*arctan(sqrt(-b)*x*s qrt((a*x^2 + b)/x^2)/b) + 1/3*(a*x^3 + 4*b*x)*sqrt((a*x^2 + b)/x^2)]
Time = 1.41 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.20 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^2 \, dx=\frac {a \sqrt {b} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{3} + \frac {4 b^{\frac {3}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{3} + \frac {b^{\frac {3}{2}} \log {\left (\frac {a x^{2}}{b} \right )}}{2} - b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a x^{2}}{b} + 1} + 1 \right )} \] Input:
integrate((a+b/x**2)**(3/2)*x**2,x)
Output:
a*sqrt(b)*x**2*sqrt(a*x**2/b + 1)/3 + 4*b**(3/2)*sqrt(a*x**2/b + 1)/3 + b* *(3/2)*log(a*x**2/b)/2 - b**(3/2)*log(sqrt(a*x**2/b + 1) + 1)
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.05 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^2 \, dx=\frac {1}{3} \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} x^{3} + \sqrt {a + \frac {b}{x^{2}}} b x + \frac {1}{2} \, b^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right ) \] Input:
integrate((a+b/x^2)^(3/2)*x^2,x, algorithm="maxima")
Output:
1/3*(a + b/x^2)^(3/2)*x^3 + sqrt(a + b/x^2)*b*x + 1/2*b^(3/2)*log((sqrt(a + b/x^2)*x - sqrt(b))/(sqrt(a + b/x^2)*x + sqrt(b)))
Time = 0.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.37 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^2 \, dx=\frac {b^{2} \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b}} + \frac {1}{3} \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} \mathrm {sgn}\left (x\right ) + \sqrt {a x^{2} + b} b \mathrm {sgn}\left (x\right ) - \frac {{\left (3 \, b^{2} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 4 \, \sqrt {-b} b^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{3 \, \sqrt {-b}} \] Input:
integrate((a+b/x^2)^(3/2)*x^2,x, algorithm="giac")
Output:
b^2*arctan(sqrt(a*x^2 + b)/sqrt(-b))*sgn(x)/sqrt(-b) + 1/3*(a*x^2 + b)^(3/ 2)*sgn(x) + sqrt(a*x^2 + b)*b*sgn(x) - 1/3*(3*b^2*arctan(sqrt(b)/sqrt(-b)) + 4*sqrt(-b)*b^(3/2))*sgn(x)/sqrt(-b)
Timed out. \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^2 \, dx=\int x^2\,{\left (a+\frac {b}{x^2}\right )}^{3/2} \,d x \] Input:
int(x^2*(a + b/x^2)^(3/2),x)
Output:
int(x^2*(a + b/x^2)^(3/2), x)
Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.22 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^2 \, dx=\frac {\sqrt {a \,x^{2}+b}\, a \,x^{2}}{3}+\frac {4 \sqrt {a \,x^{2}+b}\, b}{3}+\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{2}+b}+\sqrt {a}\, x -\sqrt {b}}{\sqrt {b}}\right ) b -\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{2}+b}+\sqrt {a}\, x +\sqrt {b}}{\sqrt {b}}\right ) b \] Input:
int((a+b/x^2)^(3/2)*x^2,x)
Output:
(sqrt(a*x**2 + b)*a*x**2 + 4*sqrt(a*x**2 + b)*b + 3*sqrt(b)*log((sqrt(a*x* *2 + b) + sqrt(a)*x - sqrt(b))/sqrt(b))*b - 3*sqrt(b)*log((sqrt(a*x**2 + b ) + sqrt(a)*x + sqrt(b))/sqrt(b))*b)/3