Integrand size = 22, antiderivative size = 123 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^2} \, dx=\frac {c (b c-6 a d) \sqrt {c+\frac {d}{x^2}}}{16 d x}+\frac {(b c-6 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{24 d x}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 d x}+\frac {c^2 (b c-6 a d) \text {arctanh}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )}{16 d^{3/2}} \]
1/24*(-6*a*d+b*c)*(c+d/x^2)^(3/2)/d/x-1/6*b*(c+d/x^2)^(5/2)/d/x+1/16*c^2*( -6*a*d+b*c)*arctanh(d^(1/2)/x/(c+d/x^2)^(1/2))/d^(3/2)+1/16*c*(-6*a*d+b*c) *(c+d/x^2)^(1/2)/d/x
Time = 0.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^2} \, dx=\frac {\sqrt {c+\frac {d}{x^2}} \left (-\sqrt {d} \left (6 a d x^2 \left (2 d+5 c x^2\right )+b \left (8 d^2+14 c d x^2+3 c^2 x^4\right )\right )+\frac {3 c^2 (b c-6 a d) x^6 \text {arctanh}\left (\frac {\sqrt {d+c x^2}}{\sqrt {d}}\right )}{\sqrt {d+c x^2}}\right )}{48 d^{3/2} x^5} \]
(Sqrt[c + d/x^2]*(-(Sqrt[d]*(6*a*d*x^2*(2*d + 5*c*x^2) + b*(8*d^2 + 14*c*d *x^2 + 3*c^2*x^4))) + (3*c^2*(b*c - 6*a*d)*x^6*ArcTanh[Sqrt[d + c*x^2]/Sqr t[d]])/Sqrt[d + c*x^2]))/(48*d^(3/2)*x^5)
Time = 0.23 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {959, 858, 211, 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.
| \(\displaystyle \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^2} \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle -\frac {(b c-6 a d) \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2}}{x^2}dx}{6 d}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 d x}\) |
| \(\Big \downarrow \) 858 |
| \(\displaystyle \frac {(b c-6 a d) \int \left (c+\frac {d}{x^2}\right )^{3/2}d\frac {1}{x}}{6 d}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 d x}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {(b c-6 a d) \left (\frac {3}{4} c \int \sqrt {c+\frac {d}{x^2}}d\frac {1}{x}+\frac {\left (c+\frac {d}{x^2}\right )^{3/2}}{4 x}\right )}{6 d}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 d x}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {(b c-6 a d) \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}+\frac {\sqrt {c+\frac {d}{x^2}}}{2 x}\right )+\frac {\left (c+\frac {d}{x^2}\right )^{3/2}}{4 x}\right )}{6 d}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 d x}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {(b c-6 a d) \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{1-\frac {d}{x^2}}d\frac {1}{\sqrt {c+\frac {d}{x^2}} x}+\frac {\sqrt {c+\frac {d}{x^2}}}{2 x}\right )+\frac {\left (c+\frac {d}{x^2}\right )^{3/2}}{4 x}\right )}{6 d}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 d x}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(b c-6 a d) \left (\frac {3}{4} c \left (\frac {c \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{2 \sqrt {d}}+\frac {\sqrt {c+\frac {d}{x^2}}}{2 x}\right )+\frac {\left (c+\frac {d}{x^2}\right )^{3/2}}{4 x}\right )}{6 d}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 d x}\) |
-1/6*(b*(c + d/x^2)^(5/2))/(d*x) + ((b*c - 6*a*d)*((c + d/x^2)^(3/2)/(4*x) + (3*c*(Sqrt[c + d/x^2]/(2*x) + (c*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/ (2*Sqrt[d])))/4))/(6*d)
3.10.60.3.1 Defintions of rubi rules used
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 = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(-\frac {\left (30 a c d \,x^{4}+3 b \,c^{2} x^{4}+12 a \,d^{2} x^{2}+14 b c d \,x^{2}+8 b \,d^{2}\right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{48 x^{5} d}-\frac {c^{2} \left (6 a d -b c \right ) \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}{16 d^{\frac {3}{2}} \sqrt {c \,x^{2}+d}}\) | \(127\) |
| default | \(-\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (18 d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) a \,c^{2} x^{6}-3 d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) b \,c^{3} x^{6}-6 \left (c \,x^{2}+d \right )^{\frac {3}{2}} a \,c^{2} d \,x^{6}+\left (c \,x^{2}+d \right )^{\frac {3}{2}} b \,c^{3} x^{6}+6 \left (c \,x^{2}+d \right )^{\frac {5}{2}} a c d \,x^{4}-\left (c \,x^{2}+d \right )^{\frac {5}{2}} b \,c^{2} x^{4}-18 \sqrt {c \,x^{2}+d}\, a \,c^{2} d^{2} x^{6}+3 \sqrt {c \,x^{2}+d}\, b \,c^{3} d \,x^{6}+12 \left (c \,x^{2}+d \right )^{\frac {5}{2}} a \,d^{2} x^{2}-2 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b c d \,x^{2}+8 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b \,d^{2}\right )}{48 x^{3} \left (c \,x^{2}+d \right )^{\frac {3}{2}} d^{3}}\) | \(259\) |
-1/48*(30*a*c*d*x^4+3*b*c^2*x^4+12*a*d^2*x^2+14*b*c*d*x^2+8*b*d^2)/x^5/d*( (c*x^2+d)/x^2)^(1/2)-1/16*c^2*(6*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 = 0.30 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.00 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^2} \, dx=\left [-\frac {3 \, {\left (b c^{3} - 6 \, a c^{2} d\right )} \sqrt {d} x^{5} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (3 \, {\left (b c^{2} d + 10 \, a c d^{2}\right )} x^{4} + 8 \, b d^{3} + 2 \, {\left (7 \, b c d^{2} + 6 \, a d^{3}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{96 \, d^{2} x^{5}}, -\frac {3 \, {\left (b c^{3} - 6 \, a c^{2} d\right )} \sqrt {-d} x^{5} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + {\left (3 \, {\left (b c^{2} d + 10 \, a c d^{2}\right )} x^{4} + 8 \, b d^{3} + 2 \, {\left (7 \, b c d^{2} + 6 \, a d^{3}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{48 \, d^{2} x^{5}}\right ] \]
[-1/96*(3*(b*c^3 - 6*a*c^2*d)*sqrt(d)*x^5*log(-(c*x^2 - 2*sqrt(d)*x*sqrt(( c*x^2 + d)/x^2) + 2*d)/x^2) + 2*(3*(b*c^2*d + 10*a*c*d^2)*x^4 + 8*b*d^3 + 2*(7*b*c*d^2 + 6*a*d^3)*x^2)*sqrt((c*x^2 + d)/x^2))/(d^2*x^5), -1/48*(3*(b *c^3 - 6*a*c^2*d)*sqrt(-d)*x^5*arctan(sqrt(-d)*x*sqrt((c*x^2 + d)/x^2)/(c* x^2 + d)) + (3*(b*c^2*d + 10*a*c*d^2)*x^4 + 8*b*d^3 + 2*(7*b*c*d^2 + 6*a*d ^3)*x^2)*sqrt((c*x^2 + d)/x^2))/(d^2*x^5)]
Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (102) = 204\).
Time = 10.45 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.06 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^2} \, dx=- \frac {a c^{\frac {3}{2}} \sqrt {1 + \frac {d}{c x^{2}}}}{2 x} - \frac {a c^{\frac {3}{2}}}{8 x \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 a \sqrt {c} d}{8 x^{3} \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 a c^{2} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{8 \sqrt {d}} - \frac {a d^{2}}{4 \sqrt {c} x^{5} \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {b c^{\frac {5}{2}}}{16 d x \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {17 b c^{\frac {3}{2}}}{48 x^{3} \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {11 b \sqrt {c} d}{24 x^{5} \sqrt {1 + \frac {d}{c x^{2}}}} + \frac {b c^{3} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{16 d^{\frac {3}{2}}} - \frac {b d^{2}}{6 \sqrt {c} x^{7} \sqrt {1 + \frac {d}{c x^{2}}}} \]
-a*c**(3/2)*sqrt(1 + d/(c*x**2))/(2*x) - a*c**(3/2)/(8*x*sqrt(1 + d/(c*x** 2))) - 3*a*sqrt(c)*d/(8*x**3*sqrt(1 + d/(c*x**2))) - 3*a*c**2*asinh(sqrt(d )/(sqrt(c)*x))/(8*sqrt(d)) - a*d**2/(4*sqrt(c)*x**5*sqrt(1 + d/(c*x**2))) - b*c**(5/2)/(16*d*x*sqrt(1 + d/(c*x**2))) - 17*b*c**(3/2)/(48*x**3*sqrt(1 + d/(c*x**2))) - 11*b*sqrt(c)*d/(24*x**5*sqrt(1 + d/(c*x**2))) + b*c**3*a sinh(sqrt(d)/(sqrt(c)*x))/(16*d**(3/2)) - b*d**2/(6*sqrt(c)*x**7*sqrt(1 + d/(c*x**2)))
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (103) = 206\).
Time = 0.30 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.24 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^2} \, dx=\frac {1}{16} \, {\left (\frac {3 \, c^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{\sqrt {d}} - \frac {2 \, {\left (5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{2} x^{3} - 3 \, \sqrt {c + \frac {d}{x^{2}}} c^{2} d x\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{2} x^{4} - 2 \, {\left (c + \frac {d}{x^{2}}\right )} d x^{2} + d^{2}}\right )} a - \frac {1}{96} \, {\left (\frac {3 \, c^{3} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c^{3} x^{5} + 8 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{3} d x^{3} - 3 \, \sqrt {c + \frac {d}{x^{2}}} c^{3} d^{2} x\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{3} d x^{6} - 3 \, {\left (c + \frac {d}{x^{2}}\right )}^{2} d^{2} x^{4} + 3 \, {\left (c + \frac {d}{x^{2}}\right )} d^{3} x^{2} - d^{4}}\right )} b \]
1/16*(3*c^2*log((sqrt(c + d/x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d) ))/sqrt(d) - 2*(5*(c + d/x^2)^(3/2)*c^2*x^3 - 3*sqrt(c + d/x^2)*c^2*d*x)/( (c + d/x^2)^2*x^4 - 2*(c + d/x^2)*d*x^2 + d^2))*a - 1/96*(3*c^3*log((sqrt( c + d/x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d)))/d^(3/2) + 2*(3*(c + d/x^2)^(5/2)*c^3*x^5 + 8*(c + d/x^2)^(3/2)*c^3*d*x^3 - 3*sqrt(c + d/x^2)* c^3*d^2*x)/((c + d/x^2)^3*d*x^6 - 3*(c + d/x^2)^2*d^2*x^4 + 3*(c + d/x^2)* d^3*x^2 - d^4))*b
Time = 0.30 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^2} \, dx=-\frac {\frac {3 \, {\left (b c^{4} \mathrm {sgn}\left (x\right ) - 6 \, a c^{3} d \mathrm {sgn}\left (x\right )\right )} \arctan \left (\frac {\sqrt {c x^{2} + d}}{\sqrt {-d}}\right )}{\sqrt {-d} d} + \frac {3 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} b c^{4} \mathrm {sgn}\left (x\right ) + 30 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a c^{3} d \mathrm {sgn}\left (x\right ) + 8 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} b c^{4} d \mathrm {sgn}\left (x\right ) - 48 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a c^{3} d^{2} \mathrm {sgn}\left (x\right ) - 3 \, \sqrt {c x^{2} + d} b c^{4} d^{2} \mathrm {sgn}\left (x\right ) + 18 \, \sqrt {c x^{2} + d} a c^{3} d^{3} \mathrm {sgn}\left (x\right )}{c^{3} d x^{6}}}{48 \, c} \]
-1/48*(3*(b*c^4*sgn(x) - 6*a*c^3*d*sgn(x))*arctan(sqrt(c*x^2 + d)/sqrt(-d) )/(sqrt(-d)*d) + (3*(c*x^2 + d)^(5/2)*b*c^4*sgn(x) + 30*(c*x^2 + d)^(5/2)* a*c^3*d*sgn(x) + 8*(c*x^2 + d)^(3/2)*b*c^4*d*sgn(x) - 48*(c*x^2 + d)^(3/2) *a*c^3*d^2*sgn(x) - 3*sqrt(c*x^2 + d)*b*c^4*d^2*sgn(x) + 18*sqrt(c*x^2 + d )*a*c^3*d^3*sgn(x))/(c^3*d*x^6))/c
Timed out. \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^2} \, dx=\int \frac {\left (a+\frac {b}{x^2}\right )\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{x^2} \,d x \]