Integrand size = 22, antiderivative size = 91 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^2} \, dx=\frac {(b c-4 a d) \sqrt {c+\frac {d}{x^2}}}{8 d x}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{4 d x}+\frac {c (b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )}{8 d^{3/2}} \] Output:
1/8*(-4*a*d+b*c)*(c+d/x^2)^(1/2)/d/x-1/4*b*(c+d/x^2)^(3/2)/d/x+1/8*c*(-4*a *d+b*c)*arctanh(d^(1/2)/(c+d/x^2)^(1/2)/x)/d^(3/2)
Time = 0.17 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^2} \, dx=\frac {\sqrt {c+\frac {d}{x^2}} \left (-\sqrt {d} \left (2 b d+b c x^2+4 a d x^2\right )+\frac {c (b c-4 a d) x^4 \text {arctanh}\left (\frac {\sqrt {d+c x^2}}{\sqrt {d}}\right )}{\sqrt {d+c x^2}}\right )}{8 d^{3/2} x^3} \] Input:
Integrate[((a + b/x^2)*Sqrt[c + d/x^2])/x^2,x]
Output:
(Sqrt[c + d/x^2]*(-(Sqrt[d]*(2*b*d + b*c*x^2 + 4*a*d*x^2)) + (c*(b*c - 4*a *d)*x^4*ArcTanh[Sqrt[d + c*x^2]/Sqrt[d]])/Sqrt[d + c*x^2]))/(8*d^(3/2)*x^3 )
Time = 0.35 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {959, 858, 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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^2} \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle -\frac {(b c-4 a d) \int \frac {\sqrt {c+\frac {d}{x^2}}}{x^2}dx}{4 d}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{4 d x}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {(b c-4 a d) \int \sqrt {c+\frac {d}{x^2}}d\frac {1}{x}}{4 d}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{4 d x}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {(b c-4 a d) \left (\frac {1}{2} c \int \frac {1}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}+\frac {\sqrt {c+\frac {d}{x^2}}}{2 x}\right )}{4 d}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{4 d x}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {(b c-4 a d) \left (\frac {1}{2} c \int \frac {1}{1-\frac {d}{x^2}}d\frac {1}{\sqrt {c+\frac {d}{x^2}} x}+\frac {\sqrt {c+\frac {d}{x^2}}}{2 x}\right )}{4 d}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{4 d x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {(b c-4 a d) \left (\frac {c \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{2 \sqrt {d}}+\frac {\sqrt {c+\frac {d}{x^2}}}{2 x}\right )}{4 d}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{4 d x}\) |
Input:
Int[((a + b/x^2)*Sqrt[c + d/x^2])/x^2,x]
Output:
-1/4*(b*(c + d/x^2)^(3/2))/(d*x) + ((b*c - 4*a*d)*(Sqrt[c + d/x^2]/(2*x) + (c*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/(2*Sqrt[d])))/(4*d)
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[(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 = 102, normalized size of antiderivative = 1.12
method | result | size |
risch | \(-\frac {\left (4 a d \,x^{2}+b c \,x^{2}+2 b d \right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{8 x^{3} d}-\frac {c \left (4 a d -c b \right ) \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}{8 d^{\frac {3}{2}} \sqrt {c \,x^{2}+d}}\) | \(102\) |
default | \(-\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (4 d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) a c \,x^{4}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) b \,c^{2} x^{4}-4 \sqrt {c \,x^{2}+d}\, a c d \,x^{4}+\sqrt {c \,x^{2}+d}\, b \,c^{2} x^{4}+4 \left (c \,x^{2}+d \right )^{\frac {3}{2}} a d \,x^{2}-\left (c \,x^{2}+d \right )^{\frac {3}{2}} b c \,x^{2}+2 \left (c \,x^{2}+d \right )^{\frac {3}{2}} b d \right )}{8 x^{3} \sqrt {c \,x^{2}+d}\, d^{2}}\) | \(175\) |
Input:
int((a+b/x^2)*(c+d/x^2)^(1/2)/x^2,x,method=_RETURNVERBOSE)
Output:
-1/8*(4*a*d*x^2+b*c*x^2+2*b*d)/x^3/d*((c*x^2+d)/x^2)^(1/2)-1/8*c*(4*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 = 188, normalized size of antiderivative = 2.07 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^2} \, dx=\left [-\frac {{\left (b c^{2} - 4 \, a c d\right )} \sqrt {d} x^{3} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (2 \, b d^{2} + {\left (b c d + 4 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{16 \, d^{2} x^{3}}, -\frac {{\left (b c^{2} - 4 \, a c d\right )} \sqrt {-d} x^{3} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) + {\left (2 \, b d^{2} + {\left (b c d + 4 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{8 \, d^{2} x^{3}}\right ] \] Input:
integrate((a+b/x^2)*(c+d/x^2)^(1/2)/x^2,x, algorithm="fricas")
Output:
[-1/16*((b*c^2 - 4*a*c*d)*sqrt(d)*x^3*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x^ 2 + d)/x^2) + 2*d)/x^2) + 2*(2*b*d^2 + (b*c*d + 4*a*d^2)*x^2)*sqrt((c*x^2 + d)/x^2))/(d^2*x^3), -1/8*((b*c^2 - 4*a*c*d)*sqrt(-d)*x^3*arctan(sqrt(-d) *x*sqrt((c*x^2 + d)/x^2)/d) + (2*b*d^2 + (b*c*d + 4*a*d^2)*x^2)*sqrt((c*x^ 2 + d)/x^2))/(d^2*x^3)]
Time = 5.85 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.58 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^2} \, dx=- \frac {a \sqrt {c} \sqrt {1 + \frac {d}{c x^{2}}}}{2 x} - \frac {a c \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{2 \sqrt {d}} - \frac {b c^{\frac {3}{2}}}{8 d x \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 b \sqrt {c}}{8 x^{3} \sqrt {1 + \frac {d}{c x^{2}}}} + \frac {b c^{2} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{8 d^{\frac {3}{2}}} - \frac {b d}{4 \sqrt {c} x^{5} \sqrt {1 + \frac {d}{c x^{2}}}} \] Input:
integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**2,x)
Output:
-a*sqrt(c)*sqrt(1 + d/(c*x**2))/(2*x) - a*c*asinh(sqrt(d)/(sqrt(c)*x))/(2* sqrt(d)) - b*c**(3/2)/(8*d*x*sqrt(1 + d/(c*x**2))) - 3*b*sqrt(c)/(8*x**3*s qrt(1 + d/(c*x**2))) + b*c**2*asinh(sqrt(d)/(sqrt(c)*x))/(8*d**(3/2)) - b* d/(4*sqrt(c)*x**5*sqrt(1 + d/(c*x**2)))
Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (75) = 150\).
Time = 0.11 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.12 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^2} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, \sqrt {c + \frac {d}{x^{2}}} c x}{{\left (c + \frac {d}{x^{2}}\right )} x^{2} - d} - \frac {c \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{\sqrt {d}}\right )} a - \frac {1}{16} \, {\left (\frac {c^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{2} x^{3} + \sqrt {c + \frac {d}{x^{2}}} c^{2} d x\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{2} d x^{4} - 2 \, {\left (c + \frac {d}{x^{2}}\right )} d^{2} x^{2} + d^{3}}\right )} b \] 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)*x^2 - d) - c*log((sqrt(c + d/x^2) *x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d)))/sqrt(d))*a - 1/16*(c^2*log((s qrt(c + d/x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d)))/d^(3/2) + 2*((c + d/x^2)^(3/2)*c^2*x^3 + sqrt(c + d/x^2)*c^2*d*x)/((c + d/x^2)^2*d*x^4 - 2*(c + d/x^2)*d^2*x^2 + d^3))*b
Time = 0.14 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^2} \, dx=-\frac {\frac {{\left (b c^{3} \mathrm {sgn}\left (x\right ) - 4 \, a c^{2} d \mathrm {sgn}\left (x\right )\right )} \arctan \left (\frac {\sqrt {c x^{2} + d}}{\sqrt {-d}}\right )}{\sqrt {-d} d} + \frac {{\left (c x^{2} + d\right )}^{\frac {3}{2}} b c^{3} \mathrm {sgn}\left (x\right ) + 4 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a c^{2} d \mathrm {sgn}\left (x\right ) + \sqrt {c x^{2} + d} b c^{3} d \mathrm {sgn}\left (x\right ) - 4 \, \sqrt {c x^{2} + d} a c^{2} d^{2} \mathrm {sgn}\left (x\right )}{c^{2} d x^{4}}}{8 \, c} \] Input:
integrate((a+b/x^2)*(c+d/x^2)^(1/2)/x^2,x, algorithm="giac")
Output:
-1/8*((b*c^3*sgn(x) - 4*a*c^2*d*sgn(x))*arctan(sqrt(c*x^2 + d)/sqrt(-d))/( sqrt(-d)*d) + ((c*x^2 + d)^(3/2)*b*c^3*sgn(x) + 4*(c*x^2 + d)^(3/2)*a*c^2* d*sgn(x) + sqrt(c*x^2 + d)*b*c^3*d*sgn(x) - 4*sqrt(c*x^2 + d)*a*c^2*d^2*sg n(x))/(c^2*d*x^4))/c
Timed out. \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^2} \, dx=\int \frac {\left (a+\frac {b}{x^2}\right )\,\sqrt {c+\frac {d}{x^2}}}{x^2} \,d x \] Input:
int(((a + b/x^2)*(c + d/x^2)^(1/2))/x^2,x)
Output:
int(((a + b/x^2)*(c + d/x^2)^(1/2))/x^2, x)
Time = 0.23 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.03 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^2} \, dx=\frac {-4 \sqrt {c \,x^{2}+d}\, a \,d^{2} x^{2}-\sqrt {c \,x^{2}+d}\, b c d \,x^{2}-2 \sqrt {c \,x^{2}+d}\, b \,d^{2}+4 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x -\sqrt {d}}{\sqrt {d}}\right ) a c d \,x^{4}-\sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x -\sqrt {d}}{\sqrt {d}}\right ) b \,c^{2} x^{4}-4 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x +\sqrt {d}}{\sqrt {d}}\right ) a c d \,x^{4}+\sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x +\sqrt {d}}{\sqrt {d}}\right ) b \,c^{2} x^{4}}{8 d^{2} x^{4}} \] Input:
int((a+b/x^2)*(c+d/x^2)^(1/2)/x^2,x)
Output:
( - 4*sqrt(c*x**2 + d)*a*d**2*x**2 - sqrt(c*x**2 + d)*b*c*d*x**2 - 2*sqrt( c*x**2 + d)*b*d**2 + 4*sqrt(d)*log((sqrt(c*x**2 + d) + sqrt(c)*x - sqrt(d) )/sqrt(d))*a*c*d*x**4 - sqrt(d)*log((sqrt(c*x**2 + d) + sqrt(c)*x - sqrt(d ))/sqrt(d))*b*c**2*x**4 - 4*sqrt(d)*log((sqrt(c*x**2 + d) + sqrt(c)*x + sq rt(d))/sqrt(d))*a*c*d*x**4 + sqrt(d)*log((sqrt(c*x**2 + d) + sqrt(c)*x + s qrt(d))/sqrt(d))*b*c**2*x**4)/(8*d**2*x**4)