\(\int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^3} \, dx\) [167]

Optimal result

Integrand size = 21, antiderivative size = 88 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^3} \, dx=-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{2 c x^2}+\frac {b d \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {b+a c}}\right )}{2 c^{3/2} \sqrt {b+a c}} \] Output:

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^3} \, dx=-\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{2 c x^2}-\frac {b d \arctan \left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {-b-a c}}\right )}{2 c^{3/2} \sqrt {-b-a c}} \] Input:

Integrate[Sqrt[a + b/(c + d*x^2)]/x^3,x]
 

Output:

-1/2*((c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(c*x^2) - (b*d*Ar 
cTan[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[-b - a*c]])/(2*c 
^(3/2)*Sqrt[-b - a*c])
 

Rubi [A] (warning: unable to verify)

Time = 0.55 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.24, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2057, 2053, 2052, 252, 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.

\(\displaystyle \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^3} \, dx\)

\(\Big \downarrow \) 2057

\(\displaystyle \int \frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{x^3}dx\)

\(\Big \downarrow \) 2053

\(\displaystyle \frac {1}{2} \int \frac {\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{x^4}dx^2\)

\(\Big \downarrow \) 2052

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 221

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

Input:

Int[Sqrt[a + b/(c + d*x^2)]/x^3,x]
 

Output:

-(b*d*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]/(2*c*(b + a*c - c*x^4)) - Arc 
Tanh[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[b + a*c]]/(2*c^( 
3/2)*Sqrt[b + a*c])))
 

Defintions of rubi rules used

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 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(188\) vs. \(2(72)=144\).

Time = 0.14 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.15

method	result	size
risch	\(-\frac {\left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{2 c \,x^{2}}+\frac {d b \ln \left (\frac {2 a \,c^{2}+2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,c^{2}+b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{4 c \sqrt {a \,c^{2}+b c}\, \left (a d \,x^{2}+a c +b \right )}\)	\(189\)
default	\(-\frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-2 a \,d^{2} \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, x^{4} \sqrt {a \,c^{2}+b c}-\ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) a b \,c^{2} d \,x^{2}-4 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, a c d \,x^{2} \sqrt {a \,c^{2}+b c}-\ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) b^{2} c d \,x^{2}-2 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, b d \,x^{2} \sqrt {a \,c^{2}+b c}+2 \left (a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \sqrt {a \,c^{2}+b c}\right )}{4 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, c^{2} \left (a c +b \right ) x^{2} \sqrt {a \,c^{2}+b c}}\)	\(454\)

Input:

int((a+b/(d*x^2+c))^(1/2)/x^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2/c*(d*x^2+c)/x^2*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+1/4/c*d*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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (72) = 144\).

Time = 0.12 (sec) , antiderivative size = 433, normalized size of antiderivative = 4.92 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^3} \, dx=\left [\frac {\sqrt {a c^{2} + b c} b d x^{2} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} + 4 \, {\left ({\left (2 \, a c + b\right )} d^{2} x^{4} + 2 \, a c^{3} + {\left (4 \, a c^{2} + 3 \, b c\right )} d x^{2} + 2 \, b c^{2}\right )} \sqrt {a c^{2} + b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{x^{4}}\right ) - 4 \, {\left (a c^{3} + {\left (a c^{2} + b c\right )} d x^{2} + b c^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, {\left (a c^{3} + b c^{2}\right )} x^{2}}, -\frac {\sqrt {-a c^{2} - b c} b d x^{2} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {-a c^{2} - b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} c^{3} + 2 \, a b c^{2} + {\left (a^{2} c^{2} + a b c\right )} d x^{2} + b^{2} c\right )}}\right ) + 2 \, {\left (a c^{3} + {\left (a c^{2} + b c\right )} d x^{2} + b c^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{4 \, {\left (a c^{3} + b c^{2}\right )} x^{2}}\right ] \] Input:

integrate((a+b/(d*x^2+c))^(1/2)/x^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^3} \, dx=\int \frac {\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}{x^{3}}\, dx \] Input:

integrate((a+b/(d*x**2+c))**(1/2)/x**3,x)
 

Output:

Integral(sqrt((a*c + a*d*x**2 + b)/(c + d*x**2))/x**3, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (72) = 144\).

Time = 0.13 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^3} \, dx=-\frac {b d \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a c^{2} + b c - \frac {{\left (a d x^{2} + a c + b\right )} c^{2}}{d x^{2} + c}\right )}} - \frac {b d \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{4 \, \sqrt {{\left (a c + b\right )} c} c} \] Input:

integrate((a+b/(d*x^2+c))^(1/2)/x^3,x, algorithm="maxima")
 

Output:

-1/2*b*d*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/(a*c^2 + b*c - (a*d*x^2 + a 
*c + b)*c^2/(d*x^2 + c)) - 1/4*b*d*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)))/(sqrt((a*c + b)*c)*c)
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (72) = 144\).

Time = 0.18 (sec) , antiderivative size = 281, normalized size of antiderivative = 3.19 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^3} \, dx=-\frac {1}{2} \, {\left (\frac {b d \arctan \left (-\frac {\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}}{\sqrt {-a c^{2} - b c}}\right )}{\sqrt {-a c^{2} - b c} c} + \frac {2 \, a^{\frac {3}{2}} c^{2} {\left | d \right |} + 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a c d + 2 \, \sqrt {a} b c {\left | d \right |} + {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} b d}{{\left (a c^{2} - {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} + b c\right )} c}\right )} \mathrm {sgn}\left (d x^{2} + c\right ) \] Input:

integrate((a+b/(d*x^2+c))^(1/2)/x^3,x, algorithm="giac")
 

Output:

-1/2*(b*d*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))/(sqrt(-a*c^2 - b*c)*c) + (2*a^(3/2)* 
c^2*abs(d) + 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))*a*c*d + 2*sqrt(a)*b*c*abs(d) + (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*d)/((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)*c))* 
sgn(d*x^2 + c)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^3} \, dx=\int \frac {\sqrt {a+\frac {b}{d\,x^2+c}}}{x^3} \,d x \] Input:

int((a + b/(c + d*x^2))^(1/2)/x^3,x)
 

Output:

int((a + b/(c + d*x^2))^(1/2)/x^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^3} \, dx=\frac {-\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a \,c^{2}-\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, b c +\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c +\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c +\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) b d \,x^{2}-\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) b d \,x^{2}}{2 c^{2} x^{2} \left (a c +b \right )} \] Input:

int((a+b/(d*x^2+c))^(1/2)/x^3,x)
 

Output:

( - sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*a*c**2 - sqrt(c + d*x**2)*sq 
rt(a*c + a*d*x**2 + b)*b*c + 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*d*x**2 - sqrt(c)*sqrt(a*c + b)*log(x)*b*d*x**2)/(2*c**2*x**2*(a 
*c + b))