Integrand size = 15, antiderivative size = 74 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{x^4} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}-\frac {a \sqrt {a+\frac {b}{x^2}}}{8 b x}+\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{8 b^{3/2}} \] Output:
-1/4*(a+b/x^2)^(1/2)/x^3-1/8*a*(a+b/x^2)^(1/2)/b/x+1/8*a^2*arctanh(b^(1/2) /(a+b/x^2)^(1/2)/x)/b^(3/2)
Time = 0.09 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{x^4} \, dx=\frac {\sqrt {a+\frac {b}{x^2}} \left (-\sqrt {b} \left (2 b+a x^2\right )+\frac {a^2 x^4 \text {arctanh}\left (\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )}{\sqrt {b+a x^2}}\right )}{8 b^{3/2} x^3} \] Input:
Integrate[Sqrt[a + b/x^2]/x^4,x]
Output:
(Sqrt[a + b/x^2]*(-(Sqrt[b]*(2*b + a*x^2)) + (a^2*x^4*ArcTanh[Sqrt[b + a*x ^2]/Sqrt[b]])/Sqrt[b + a*x^2]))/(8*b^(3/2)*x^3)
Time = 0.32 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {858, 248, 262, 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.
\(\displaystyle \int \frac {\sqrt {a+\frac {b}{x^2}}}{x^4} \, dx\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\int \frac {\sqrt {a+\frac {b}{x^2}}}{x^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 248 |
\(\displaystyle -\frac {1}{4} a \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2}d\frac {1}{x}-\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {1}{4} a \left (\frac {\sqrt {a+\frac {b}{x^2}}}{2 b x}-\frac {a \int \frac {1}{\sqrt {a+\frac {b}{x^2}}}d\frac {1}{x}}{2 b}\right )-\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -\frac {1}{4} a \left (\frac {\sqrt {a+\frac {b}{x^2}}}{2 b x}-\frac {a \int \frac {1}{1-\frac {b}{x^2}}d\frac {1}{\sqrt {a+\frac {b}{x^2}} x}}{2 b}\right )-\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {1}{4} a \left (\frac {\sqrt {a+\frac {b}{x^2}}}{2 b x}-\frac {a \text {arctanh}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{2 b^{3/2}}\right )-\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}\) |
Input:
Int[Sqrt[a + b/x^2]/x^4,x]
Output:
-1/4*Sqrt[a + b/x^2]/x^3 - (a*(Sqrt[a + b/x^2]/(2*b*x) - (a*ArcTanh[Sqrt[b ]/(Sqrt[a + b/x^2]*x)])/(2*b^(3/2))))/4
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 + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 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 = 0.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.16
method | result | size |
risch | \(-\frac {\left (a \,x^{2}+2 b \right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{8 x^{3} b}+\frac {a^{2} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{8 b^{\frac {3}{2}} \sqrt {a \,x^{2}+b}}\) | \(86\) |
default | \(\frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, \left (\sqrt {b}\, \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) a^{2} x^{4}-\sqrt {a \,x^{2}+b}\, a^{2} x^{4}+\left (a \,x^{2}+b \right )^{\frac {3}{2}} a \,x^{2}-2 \left (a \,x^{2}+b \right )^{\frac {3}{2}} b \right )}{8 x^{3} \sqrt {a \,x^{2}+b}\, b^{2}}\) | \(106\) |
Input:
int((a+b/x^2)^(1/2)/x^4,x,method=_RETURNVERBOSE)
Output:
-1/8*(a*x^2+2*b)/x^3/b*((a*x^2+b)/x^2)^(1/2)+1/8/b^(3/2)*a^2*ln((2*b+2*b^( 1/2)*(a*x^2+b)^(1/2))/x)*((a*x^2+b)/x^2)^(1/2)*x/(a*x^2+b)^(1/2)
Time = 0.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{x^4} \, dx=\left [\frac {a^{2} \sqrt {b} x^{3} \log \left (-\frac {a x^{2} + 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) - 2 \, {\left (a b x^{2} + 2 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{16 \, b^{2} x^{3}}, -\frac {a^{2} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{b}\right ) + {\left (a b x^{2} + 2 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{8 \, b^{2} x^{3}}\right ] \] Input:
integrate((a+b/x^2)^(1/2)/x^4,x, algorithm="fricas")
Output:
[1/16*(a^2*sqrt(b)*x^3*log(-(a*x^2 + 2*sqrt(b)*x*sqrt((a*x^2 + b)/x^2) + 2 *b)/x^2) - 2*(a*b*x^2 + 2*b^2)*sqrt((a*x^2 + b)/x^2))/(b^2*x^3), -1/8*(a^2 *sqrt(-b)*x^3*arctan(sqrt(-b)*x*sqrt((a*x^2 + b)/x^2)/b) + (a*b*x^2 + 2*b^ 2)*sqrt((a*x^2 + b)/x^2))/(b^2*x^3)]
Time = 2.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{x^4} \, dx=- \frac {a^{\frac {3}{2}}}{8 b x \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {3 \sqrt {a}}{8 x^{3} \sqrt {1 + \frac {b}{a x^{2}}}} + \frac {a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x} \right )}}{8 b^{\frac {3}{2}}} - \frac {b}{4 \sqrt {a} x^{5} \sqrt {1 + \frac {b}{a x^{2}}}} \] Input:
integrate((a+b/x**2)**(1/2)/x**4,x)
Output:
-a**(3/2)/(8*b*x*sqrt(1 + b/(a*x**2))) - 3*sqrt(a)/(8*x**3*sqrt(1 + b/(a*x **2))) + a**2*asinh(sqrt(b)/(sqrt(a)*x))/(8*b**(3/2)) - b/(4*sqrt(a)*x**5* sqrt(1 + b/(a*x**2)))
Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{x^4} \, dx=-\frac {a^{2} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right )}{16 \, b^{\frac {3}{2}}} - \frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{2} x^{3} + \sqrt {a + \frac {b}{x^{2}}} a^{2} b x}{8 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{2} b x^{4} - 2 \, {\left (a + \frac {b}{x^{2}}\right )} b^{2} x^{2} + b^{3}\right )}} \] Input:
integrate((a+b/x^2)^(1/2)/x^4,x, algorithm="maxima")
Output:
-1/16*a^2*log((sqrt(a + b/x^2)*x - sqrt(b))/(sqrt(a + b/x^2)*x + sqrt(b))) /b^(3/2) - 1/8*((a + b/x^2)^(3/2)*a^2*x^3 + sqrt(a + b/x^2)*a^2*b*x)/((a + b/x^2)^2*b*x^4 - 2*(a + b/x^2)*b^2*x^2 + b^3)
Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{x^4} \, dx=-\frac {\frac {a^{3} \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b} b} + \frac {{\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{3} \mathrm {sgn}\left (x\right ) + \sqrt {a x^{2} + b} a^{3} b \mathrm {sgn}\left (x\right )}{a^{2} b x^{4}}}{8 \, a} \] Input:
integrate((a+b/x^2)^(1/2)/x^4,x, algorithm="giac")
Output:
-1/8*(a^3*arctan(sqrt(a*x^2 + b)/sqrt(-b))*sgn(x)/(sqrt(-b)*b) + ((a*x^2 + b)^(3/2)*a^3*sgn(x) + sqrt(a*x^2 + b)*a^3*b*sgn(x))/(a^2*b*x^4))/a
Timed out. \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{x^4} \, dx=\int \frac {\sqrt {a+\frac {b}{x^2}}}{x^4} \,d x \] Input:
int((a + b/x^2)^(1/2)/x^4,x)
Output:
int((a + b/x^2)^(1/2)/x^4, x)
Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.35 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{x^4} \, dx=\frac {-\sqrt {a \,x^{2}+b}\, a b \,x^{2}-2 \sqrt {a \,x^{2}+b}\, b^{2}-\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{2}+b}+\sqrt {a}\, x -\sqrt {b}}{\sqrt {b}}\right ) a^{2} x^{4}+\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{2}+b}+\sqrt {a}\, x +\sqrt {b}}{\sqrt {b}}\right ) a^{2} x^{4}}{8 b^{2} x^{4}} \] Input:
int((a+b/x^2)^(1/2)/x^4,x)
Output:
( - sqrt(a*x**2 + b)*a*b*x**2 - 2*sqrt(a*x**2 + b)*b**2 - sqrt(b)*log((sqr t(a*x**2 + b) + sqrt(a)*x - sqrt(b))/sqrt(b))*a**2*x**4 + sqrt(b)*log((sqr t(a*x**2 + b) + sqrt(a)*x + sqrt(b))/sqrt(b))*a**2*x**4)/(8*b**2*x**4)