\(\int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^5} \, dx\) [168]

Optimal result

Integrand size = 21, antiderivative size = 150 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^5} \, dx=\frac {(5 b+4 a c) d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{8 c^2 (b+a c) x^2}-\frac {\left (c+d x^2\right )^2 \sqrt {a+\frac {b}{c+d x^2}}}{4 c^2 x^4}-\frac {b (3 b+4 a c) d^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {b+a c}}\right )}{8 c^{5/2} (b+a c)^{3/2}} \] Output:

1/8*(4*a*c+5*b)*d*(d*x^2+c)*(a+b/(d*x^2+c))^(1/2)/c^2/(a*c+b)/x^2-1/4*(d*x 
^2+c)^2*(a+b/(d*x^2+c))^(1/2)/c^2/x^4-1/8*b*(4*a*c+3*b)*d^2*arctanh(c^(1/2 
)*(a+b/(d*x^2+c))^(1/2)/(a*c+b)^(1/2))/c^(5/2)/(a*c+b)^(3/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^5} \, dx=-\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (b \left (2 c-3 d x^2\right )+2 a c \left (c-d x^2\right )\right )}{8 c^2 (b+a c) x^4}-\frac {b (3 b+4 a c) d^2 \arctan \left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {-b-a c}}\right )}{8 c^{5/2} (-b-a c)^{3/2}} \] Input:

Integrate[Sqrt[a + b/(c + d*x^2)]/x^5,x]
 

Output:

-1/8*((c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(b*(2*c - 3*d*x^2) 
 + 2*a*c*(c - d*x^2)))/(c^2*(b + a*c)*x^4) - (b*(3*b + 4*a*c)*d^2*ArcTan[( 
Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[-b - a*c]])/(8*c^(5/2) 
*(-b - a*c)^(3/2))
 

Rubi [A] (warning: unable to verify)

Time = 0.70 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.23, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2057, 2053, 2052, 25, 27, 360, 25, 298, 221}

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.

\(\displaystyle \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^5} \, dx\)

\(\Big \downarrow \) 2057

\(\displaystyle \int \frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{x^5}dx\)

\(\Big \downarrow \) 2053

\(\displaystyle \frac {1}{2} \int \frac {\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{x^6}dx^2\)

\(\Big \downarrow \) 2052

\(\displaystyle -b d \int -\frac {d x^4 \left (a-x^4\right )}{\left (-c x^4+b+a c\right )^3}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\)

\(\Big \downarrow \) 25

\(\displaystyle b d \int \frac {d x^4 \left (a-x^4\right )}{\left (-c x^4+b+a c\right )^3}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\)

\(\Big \downarrow \) 27

\(\displaystyle b d^2 \int \frac {x^4 \left (a-x^4\right )}{\left (-c x^4+b+a c\right )^3}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\)

\(\Big \downarrow \) 360

\(\displaystyle b d^2 \left (-\frac {\int -\frac {4 c x^4+b}{\left (-c x^4+b+a c\right )^2}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{4 c^2}-\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^2 \left (a c+b-c x^4\right )^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle b d^2 \left (\frac {\int \frac {4 c x^4+b}{\left (-c x^4+b+a c\right )^2}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{4 c^2}-\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^2 \left (a c+b-c x^4\right )^2}\right )\)

\(\Big \downarrow \) 298

\(\displaystyle b d^2 \left (\frac {\frac {(4 a c+5 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 (a c+b) \left (a c+b-c x^4\right )}-\frac {(4 a c+3 b) \int \frac {1}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{2 (a c+b)}}{4 c^2}-\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^2 \left (a c+b-c x^4\right )^2}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle b d^2 \left (\frac {\frac {(4 a c+5 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 (a c+b) \left (a c+b-c x^4\right )}-\frac {(4 a c+3 b) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{2 \sqrt {c} (a c+b)^{3/2}}}{4 c^2}-\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^2 \left (a c+b-c x^4\right )^2}\right )\)

Input:

Int[Sqrt[a + b/(c + d*x^2)]/x^5,x]
 

Output:

b*d^2*(-1/4*(b*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(c^2*(b + a*c - c*x^ 
4)^2) + (((5*b + 4*a*c)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(2*(b + a*c 
)*(b + a*c - c*x^4)) - ((3*b + 4*a*c)*ArcTanh[(Sqrt[c]*Sqrt[(b + a*c + a*d 
*x^2)/(c + d*x^2)])/Sqrt[b + a*c]])/(2*Sqrt[c]*(b + a*c)^(3/2)))/(4*c^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 360

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : 
> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p 
+ 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1))   Int[(a + b*x^2)^(p + 1)*Expan 
dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 
- 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; 
FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & 
& (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
 

rule 2052

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S 
ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d)   Subst[Int[x^(q* 
(p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* 
x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] 
&& IntegerQ[m]
 

rule 2053

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) 
))^(p_), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( 
(a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, 
 x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.59

method	result	size
risch	\(-\frac {\left (d \,x^{2}+c \right ) \left (-2 a d \,x^{2} c -3 b d \,x^{2}+2 a \,c^{2}+2 b c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{8 c^{2} x^{4} \left (a c +b \right )}-\frac {d^{2} b \left (4 a c +3 b \right ) \ln \left (\frac {2 a \,c^{2}+2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,c^{2}+b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{16 c^{2} \left (a c +b \right ) \sqrt {a \,c^{2}+b c}\, \left (a d \,x^{2}+a c +b \right )}\)	\(239\)
default	\(-\frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (12 a^{2} d^{3} \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, x^{6} c \left (a \,c^{2}+b c \right )^{\frac {3}{2}}+4 \ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) a^{3} b \,c^{5} d^{2} x^{4}+10 a \,d^{3} \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, x^{6} b \left (a \,c^{2}+b c \right )^{\frac {3}{2}}+11 \ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) a^{2} b^{2} c^{4} d^{2} x^{4}+20 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, a^{2} c^{2} d^{2} x^{4} \left (a \,c^{2}+b c \right )^{\frac {3}{2}}+10 \ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) a \,b^{3} c^{3} d^{2} x^{4}+28 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, a c \,d^{2} b \,x^{4} \left (a \,c^{2}+b c \right )^{\frac {3}{2}}+3 \ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) b^{4} c^{2} d^{2} x^{4}+10 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, b^{2} d^{2} x^{4} \left (a \,c^{2}+b c \right )^{\frac {3}{2}}-12 \left (a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} a c d \,x^{2} \left (a \,c^{2}+b c \right )^{\frac {3}{2}}-10 \left (a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} b d \,x^{2} \left (a \,c^{2}+b c \right )^{\frac {3}{2}}+4 \left (a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \left (a \,c^{2}+b c \right )^{\frac {3}{2}} a \,c^{2}+4 \left (a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \left (a \,c^{2}+b c \right )^{\frac {3}{2}} b c \right )}{16 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, c^{3} \left (a c +b \right )^{2} x^{4} \left (a \,c^{2}+b c \right )^{\frac {3}{2}}}\)	\(923\)

Input:

int((a+b/(d*x^2+c))^(1/2)/x^5,x,method=_RETURNVERBOSE)
 

Output:

-1/8*(d*x^2+c)*(-2*a*c*d*x^2-3*b*d*x^2+2*a*c^2+2*b*c)/c^2/x^4/(a*c+b)*((a* 
d*x^2+a*c+b)/(d*x^2+c))^(1/2)-1/16*d^2*b*(4*a*c+3*b)/c^2/(a*c+b)/(a*c^2+b* 
c)^(1/2)*ln((2*a*c^2+2*b*c+(2*a*c*d+b*d)*x^2+2*(a*c^2+b*c)^(1/2)*(a*c^2+b* 
c+(2*a*c*d+b*d)*x^2+a*d^2*x^4)^(1/2))/x^2)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/ 
2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)/(a*d*x^2+a*c+b)
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 577, normalized size of antiderivative = 3.85 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^5} \, dx=\left [\frac {{\left (4 \, a b c + 3 \, b^{2}\right )} \sqrt {a c^{2} + b c} d^{2} x^{4} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} - 4 \, {\left ({\left (2 \, a c + b\right )} d^{2} x^{4} + 2 \, a c^{3} + {\left (4 \, a c^{2} + 3 \, b c\right )} d x^{2} + 2 \, b c^{2}\right )} \sqrt {a c^{2} + b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{x^{4}}\right ) - 4 \, {\left (2 \, a^{2} c^{5} - {\left (2 \, a^{2} c^{3} + 5 \, a b c^{2} + 3 \, b^{2} c\right )} d^{2} x^{4} + 4 \, a b c^{4} + 2 \, b^{2} c^{3} - {\left (a b c^{3} + b^{2} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{32 \, {\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} x^{4}}, \frac {{\left (4 \, a b c + 3 \, b^{2}\right )} \sqrt {-a c^{2} - b c} d^{2} x^{4} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {-a c^{2} - b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} c^{3} + 2 \, a b c^{2} + {\left (a^{2} c^{2} + a b c\right )} d x^{2} + b^{2} c\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{5} - {\left (2 \, a^{2} c^{3} + 5 \, a b c^{2} + 3 \, b^{2} c\right )} d^{2} x^{4} + 4 \, a b c^{4} + 2 \, b^{2} c^{3} - {\left (a b c^{3} + b^{2} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{16 \, {\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} x^{4}}\right ] \] Input:

integrate((a+b/(d*x^2+c))^(1/2)/x^5,x, algorithm="fricas")
 

Output:

[1/32*((4*a*b*c + 3*b^2)*sqrt(a*c^2 + b*c)*d^2*x^4*log(((8*a^2*c^2 + 8*a*b 
*c + b^2)*d^2*x^4 + 8*a^2*c^4 + 16*a*b*c^3 + 8*b^2*c^2 + 8*(2*a^2*c^3 + 3* 
a*b*c^2 + b^2*c)*d*x^2 - 4*((2*a*c + b)*d^2*x^4 + 2*a*c^3 + (4*a*c^2 + 3*b 
*c)*d*x^2 + 2*b*c^2)*sqrt(a*c^2 + b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c 
)))/x^4) - 4*(2*a^2*c^5 - (2*a^2*c^3 + 5*a*b*c^2 + 3*b^2*c)*d^2*x^4 + 4*a* 
b*c^4 + 2*b^2*c^3 - (a*b*c^3 + b^2*c^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d 
*x^2 + c)))/((a^2*c^5 + 2*a*b*c^4 + b^2*c^3)*x^4), 1/16*((4*a*b*c + 3*b^2) 
*sqrt(-a*c^2 - b*c)*d^2*x^4*arctan(1/2*((2*a*c + b)*d*x^2 + 2*a*c^2 + 2*b* 
c)*sqrt(-a*c^2 - b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/(a^2*c^3 + 2*a 
*b*c^2 + (a^2*c^2 + a*b*c)*d*x^2 + b^2*c)) - 2*(2*a^2*c^5 - (2*a^2*c^3 + 5 
*a*b*c^2 + 3*b^2*c)*d^2*x^4 + 4*a*b*c^4 + 2*b^2*c^3 - (a*b*c^3 + b^2*c^2)* 
d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^2*c^5 + 2*a*b*c^4 + b^2* 
c^3)*x^4)]
 

Sympy [F]

\[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^5} \, dx=\int \frac {\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}{x^{5}}\, dx \] Input:

integrate((a+b/(d*x**2+c))**(1/2)/x**5,x)
 

Output:

Integral(sqrt((a*c + a*d*x**2 + b)/(c + d*x**2))/x**5, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (130) = 260\).

Time = 0.15 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.15 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^5} \, dx=\frac {{\left (4 \, a b c + 3 \, b^{2}\right )} d^{2} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{16 \, {\left (a c^{3} + b c^{2}\right )} \sqrt {{\left (a c + b\right )} c}} - \frac {{\left (4 \, a b c^{2} + 5 \, b^{2} c\right )} d^{2} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} - {\left (4 \, a^{2} b c^{2} + 7 \, a b^{2} c + 3 \, b^{3}\right )} d^{2} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, {\left (a^{3} c^{5} + 3 \, a^{2} b c^{4} + 3 \, a b^{2} c^{3} + b^{3} c^{2} + \frac {{\left (a c^{5} + b c^{4}\right )} {\left (a d x^{2} + a c + b\right )}^{2}}{{\left (d x^{2} + c\right )}^{2}} - \frac {2 \, {\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} {\left (a d x^{2} + a c + b\right )}}{d x^{2} + c}\right )}} \] Input:

integrate((a+b/(d*x^2+c))^(1/2)/x^5,x, algorithm="maxima")
 

Output:

1/16*(4*a*b*c + 3*b^2)*d^2*log((c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) - 
sqrt((a*c + b)*c))/(c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) + sqrt((a*c + 
b)*c)))/((a*c^3 + b*c^2)*sqrt((a*c + b)*c)) - 1/8*((4*a*b*c^2 + 5*b^2*c)*d 
^2*((a*d*x^2 + a*c + b)/(d*x^2 + c))^(3/2) - (4*a^2*b*c^2 + 7*a*b^2*c + 3* 
b^3)*d^2*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^3*c^5 + 3*a^2*b*c^4 + 3 
*a*b^2*c^3 + b^3*c^2 + (a*c^5 + b*c^4)*(a*d*x^2 + a*c + b)^2/(d*x^2 + c)^2 
 - 2*(a^2*c^5 + 2*a*b*c^4 + b^2*c^3)*(a*d*x^2 + a*c + b)/(d*x^2 + c))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 713 vs. \(2 (130) = 260\).

Time = 0.20 (sec) , antiderivative size = 713, normalized size of antiderivative = 4.75 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^5} \, dx=\frac {1}{8} \, {\left (\frac {{\left (4 \, a b c d^{2} + 3 \, b^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}}{\sqrt {-a c^{2} - b c}}\right )}{{\left (a c^{3} + b c^{2}\right )} \sqrt {-a c^{2} - b c}} + \frac {8 \, a^{\frac {7}{2}} c^{5} d {\left | d \right |} + 16 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{3} c^{4} d^{2} + 8 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac {5}{2}} c^{3} d {\left | d \right |} + 24 \, a^{\frac {5}{2}} b c^{4} d {\left | d \right |} + 36 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{2} b c^{3} d^{2} + 8 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac {3}{2}} b c^{2} d {\left | d \right |} + 24 \, a^{\frac {3}{2}} b^{2} c^{3} d {\left | d \right |} - 4 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} a b c d^{2} + 25 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a b^{2} c^{2} d^{2} + 8 \, \sqrt {a} b^{3} c^{2} d {\left | d \right |} - 3 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} b^{2} d^{2} + 5 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} b^{3} c d^{2}}{{\left (a c^{3} + b c^{2}\right )} {\left (a c^{2} - {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} + b c\right )}^{2}}\right )} \mathrm {sgn}\left (d x^{2} + c\right ) \] Input:

integrate((a+b/(d*x^2+c))^(1/2)/x^5,x, algorithm="giac")
 

Output:

1/8*((4*a*b*c*d^2 + 3*b^2*d^2)*arctan(-(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 
 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))/sqrt(-a*c^2 - b*c))/((a*c^3 + b*c^2 
)*sqrt(-a*c^2 - b*c)) + (8*a^(7/2)*c^5*d*abs(d) + 16*(sqrt(a*d^2)*x^2 - sq 
rt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*a^3*c^4*d^2 + 8*(sqrt 
(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^2*a^( 
5/2)*c^3*d*abs(d) + 24*a^(5/2)*b*c^4*d*abs(d) + 36*(sqrt(a*d^2)*x^2 - sqrt 
(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*a^2*b*c^3*d^2 + 8*(sqrt 
(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^2*a^( 
3/2)*b*c^2*d*abs(d) + 24*a^(3/2)*b^2*c^3*d*abs(d) - 4*(sqrt(a*d^2)*x^2 - s 
qrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^3*a*b*c*d^2 + 25*(sq 
rt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*a*b 
^2*c^2*d^2 + 8*sqrt(a)*b^3*c^2*d*abs(d) - 3*(sqrt(a*d^2)*x^2 - sqrt(a*d^2* 
x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^3*b^2*d^2 + 5*(sqrt(a*d^2)*x^2 
 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*b^3*c*d^2)/((a*c 
^3 + b*c^2)*(a*c^2 - (sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d 
*x^2 + a*c^2 + b*c))^2 + b*c)^2))*sgn(d*x^2 + c)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^5} \, dx=\int \frac {\sqrt {a+\frac {b}{d\,x^2+c}}}{x^5} \,d x \] Input:

int((a + b/(c + d*x^2))^(1/2)/x^5,x)
 

Output:

int((a + b/(c + d*x^2))^(1/2)/x^5, x)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.54 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^5} \, dx=\frac {-2 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a^{2} c^{4}+2 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a^{2} c^{3} d \,x^{2}-4 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a b \,c^{3}+5 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a b \,c^{2} d \,x^{2}-2 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, b^{2} c^{2}+3 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, b^{2} c d \,x^{2}+4 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) a b c \,d^{2} x^{4}+3 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) b^{2} d^{2} x^{4}-4 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) a b c \,d^{2} x^{4}-3 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) b^{2} d^{2} x^{4}}{8 c^{3} x^{4} \left (a^{2} c^{2}+2 a b c +b^{2}\right )} \] Input:

int((a+b/(d*x^2+c))^(1/2)/x^5,x)
 

Output:

( - 2*sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*a**2*c**4 + 2*sqrt(c + d*x 
**2)*sqrt(a*c + a*d*x**2 + b)*a**2*c**3*d*x**2 - 4*sqrt(c + d*x**2)*sqrt(a 
*c + a*d*x**2 + b)*a*b*c**3 + 5*sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)* 
a*b*c**2*d*x**2 - 2*sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*b**2*c**2 + 
3*sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*b**2*c*d*x**2 + 4*sqrt(c)*sqrt 
(a*c + b)*log(sqrt(a*c + b)*sqrt(a*c + a*d*x**2 + b)*c - sqrt(c)*sqrt(c + 
d*x**2)*a*c - sqrt(c)*sqrt(c + d*x**2)*b)*a*b*c*d**2*x**4 + 3*sqrt(c)*sqrt 
(a*c + b)*log(sqrt(a*c + b)*sqrt(a*c + a*d*x**2 + b)*c - sqrt(c)*sqrt(c + 
d*x**2)*a*c - sqrt(c)*sqrt(c + d*x**2)*b)*b**2*d**2*x**4 - 4*sqrt(c)*sqrt( 
a*c + b)*log(x)*a*b*c*d**2*x**4 - 3*sqrt(c)*sqrt(a*c + b)*log(x)*b**2*d**2 
*x**4)/(8*c**3*x**4*(a**2*c**2 + 2*a*b*c + b**2))