Integrand size = 22, antiderivative size = 61 \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^2} \, dx=-\frac {b \sqrt {c+\frac {d}{x^2}}}{2 d x}+\frac {(b c-2 a d) \text {arctanh}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )}{2 d^{3/2}} \] Output:
-1/2*b*(c+d/x^2)^(1/2)/d/x+1/2*(-2*a*d+b*c)*arctanh(d^(1/2)/(c+d/x^2)^(1/2 )/x)/d^(3/2)
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.31 \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^2} \, dx=\frac {-b \sqrt {d} \left (d+c x^2\right )+(b c-2 a d) x^2 \sqrt {d+c x^2} \text {arctanh}\left (\frac {\sqrt {d+c x^2}}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {c+\frac {d}{x^2}} x^3} \] Input:
Integrate[(a + b/x^2)/(Sqrt[c + d/x^2]*x^2),x]
Output:
(-(b*Sqrt[d]*(d + c*x^2)) + (b*c - 2*a*d)*x^2*Sqrt[d + c*x^2]*ArcTanh[Sqrt [d + c*x^2]/Sqrt[d]])/(2*d^(3/2)*Sqrt[c + d/x^2]*x^3)
Time = 0.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {959, 858, 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.
\(\displaystyle \int \frac {a+\frac {b}{x^2}}{x^2 \sqrt {c+\frac {d}{x^2}}} \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle -\frac {(b c-2 a d) \int \frac {1}{\sqrt {c+\frac {d}{x^2}} x^2}dx}{2 d}-\frac {b \sqrt {c+\frac {d}{x^2}}}{2 d x}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {(b c-2 a d) \int \frac {1}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}}{2 d}-\frac {b \sqrt {c+\frac {d}{x^2}}}{2 d x}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {(b c-2 a d) \int \frac {1}{1-\frac {d}{x^2}}d\frac {1}{\sqrt {c+\frac {d}{x^2}} x}}{2 d}-\frac {b \sqrt {c+\frac {d}{x^2}}}{2 d x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {(b c-2 a d) \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{2 d^{3/2}}-\frac {b \sqrt {c+\frac {d}{x^2}}}{2 d x}\) |
Input:
Int[(a + b/x^2)/(Sqrt[c + d/x^2]*x^2),x]
Output:
-1/2*(b*Sqrt[c + d/x^2])/(d*x) + ((b*c - 2*a*d)*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/(2*d^(3/2))
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[(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]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.52
method | result | size |
risch | \(-\frac {b \left (c \,x^{2}+d \right )}{2 d \,x^{3} \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}-\frac {\left (2 a d -c b \right ) \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) \sqrt {c \,x^{2}+d}}{2 d^{\frac {3}{2}} \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}\) | \(93\) |
default | \(-\frac {\sqrt {c \,x^{2}+d}\, \left (2 a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) d^{2} x^{2}-\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) b c d \,x^{2}+d^{\frac {3}{2}} \sqrt {c \,x^{2}+d}\, b \right )}{2 \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x^{3} d^{\frac {5}{2}}}\) | \(105\) |
Input:
int((a+b/x^2)/(c+d/x^2)^(1/2)/x^2,x,method=_RETURNVERBOSE)
Output:
-1/2/d*b*(c*x^2+d)/x^3/((c*x^2+d)/x^2)^(1/2)-1/2*(2*a*d-b*c)/d^(3/2)*ln((2 *d+2*d^(1/2)*(c*x^2+d)^(1/2))/x)/((c*x^2+d)/x^2)^(1/2)/x*(c*x^2+d)^(1/2)
Time = 0.11 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.26 \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^2} \, dx=\left [-\frac {{\left (b c - 2 \, a d\right )} \sqrt {d} x \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, b d \sqrt {\frac {c x^{2} + d}{x^{2}}}}{4 \, d^{2} x}, -\frac {{\left (b c - 2 \, a d\right )} \sqrt {-d} x \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) + b d \sqrt {\frac {c x^{2} + d}{x^{2}}}}{2 \, d^{2} x}\right ] \] Input:
integrate((a+b/x^2)/(c+d/x^2)^(1/2)/x^2,x, algorithm="fricas")
Output:
[-1/4*((b*c - 2*a*d)*sqrt(d)*x*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x^2 + d)/ x^2) + 2*d)/x^2) + 2*b*d*sqrt((c*x^2 + d)/x^2))/(d^2*x), -1/2*((b*c - 2*a* d)*sqrt(-d)*x*arctan(sqrt(-d)*x*sqrt((c*x^2 + d)/x^2)/d) + b*d*sqrt((c*x^2 + d)/x^2))/(d^2*x)]
Time = 3.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.08 \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^2} \, dx=- \frac {a \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{\sqrt {d}} - \frac {b \sqrt {c} \sqrt {1 + \frac {d}{c x^{2}}}}{2 d x} + \frac {b c \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{2 d^{\frac {3}{2}}} \] Input:
integrate((a+b/x**2)/(c+d/x**2)**(1/2)/x**2,x)
Output:
-a*asinh(sqrt(d)/(sqrt(c)*x))/sqrt(d) - b*sqrt(c)*sqrt(1 + d/(c*x**2))/(2* d*x) + b*c*asinh(sqrt(d)/(sqrt(c)*x))/(2*d**(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (49) = 98\).
Time = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.98 \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^2} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, \sqrt {c + \frac {d}{x^{2}}} c x}{{\left (c + \frac {d}{x^{2}}\right )} d x^{2} - d^{2}} + \frac {c \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {3}{2}}}\right )} b + \frac {a \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{2 \, \sqrt {d}} \] Input:
integrate((a+b/x^2)/(c+d/x^2)^(1/2)/x^2,x, algorithm="maxima")
Output:
-1/4*(2*sqrt(c + d/x^2)*c*x/((c + d/x^2)*d*x^2 - d^2) + c*log((sqrt(c + d/ x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d)))/d^(3/2))*b + 1/2*a*log((s qrt(c + d/x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d)))/sqrt(d)
Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05 \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^2} \, dx=-\frac {c {\left (\frac {{\left (b c - 2 \, a d\right )} \arctan \left (\frac {\sqrt {c x^{2} + d}}{\sqrt {-d}}\right )}{c \sqrt {-d} d} + \frac {\sqrt {c x^{2} + d} b}{c d x^{2}}\right )}}{2 \, \mathrm {sgn}\left (x\right )} \] Input:
integrate((a+b/x^2)/(c+d/x^2)^(1/2)/x^2,x, algorithm="giac")
Output:
-1/2*c*((b*c - 2*a*d)*arctan(sqrt(c*x^2 + d)/sqrt(-d))/(c*sqrt(-d)*d) + sq rt(c*x^2 + d)*b/(c*d*x^2))/sgn(x)
Time = 4.66 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.54 \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^2} \, dx=\left \{\begin {array}{cl} -\frac {3\,a\,x^2+b}{3\,\sqrt {c}\,x^3} & \text {\ if\ \ }d=0\\ \frac {b\,c\,\ln \left (2\,\sqrt {c+\frac {d}{x^2}}+\frac {2\,\sqrt {d}}{x}\right )}{2\,d^{3/2}}-\frac {b\,\sqrt {c+\frac {d}{x^2}}}{2\,d\,x}-\frac {a\,\ln \left (\sqrt {c+\frac {d}{x^2}}+\frac {\sqrt {d}}{x}\right )}{\sqrt {d}} & \text {\ if\ \ }d\neq 0 \end {array}\right . \] Input:
int((a + b/x^2)/(x^2*(c + d/x^2)^(1/2)),x)
Output:
piecewise(d == 0, -(b + 3*a*x^2)/(3*c^(1/2)*x^3), d ~= 0, - (a*log((c + d/ x^2)^(1/2) + d^(1/2)/x))/d^(1/2) - (b*(c + d/x^2)^(1/2))/(2*d*x) + (b*c*lo g(2*(c + d/x^2)^(1/2) + (2*d^(1/2))/x))/(2*d^(3/2)))
Time = 0.20 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.36 \[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^2} \, dx=\frac {-\sqrt {c \,x^{2}+d}\, b d +2 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x -\sqrt {d}}{\sqrt {d}}\right ) a d \,x^{2}-\sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x -\sqrt {d}}{\sqrt {d}}\right ) b c \,x^{2}-2 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x +\sqrt {d}}{\sqrt {d}}\right ) a d \,x^{2}+\sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x +\sqrt {d}}{\sqrt {d}}\right ) b c \,x^{2}}{2 d^{2} x^{2}} \] Input:
int((a+b/x^2)/(c+d/x^2)^(1/2)/x^2,x)
Output:
( - sqrt(c*x**2 + d)*b*d + 2*sqrt(d)*log((sqrt(c*x**2 + d) + sqrt(c)*x - s qrt(d))/sqrt(d))*a*d*x**2 - sqrt(d)*log((sqrt(c*x**2 + d) + sqrt(c)*x - sq rt(d))/sqrt(d))*b*c*x**2 - 2*sqrt(d)*log((sqrt(c*x**2 + d) + sqrt(c)*x + s qrt(d))/sqrt(d))*a*d*x**2 + sqrt(d)*log((sqrt(c*x**2 + d) + sqrt(c)*x + sq rt(d))/sqrt(d))*b*c*x**2)/(2*d**2*x**2)