\(\int \frac {\sqrt {a+\frac {b}{c+d x}}}{x^2} \, dx\) [17]

Optimal result

Integrand size = 19, antiderivative size = 77 \[ \int \frac {\sqrt {a+\frac {b}{c+d x}}}{x^2} \, dx=-\frac {(c+d x) \sqrt {a+\frac {b}{c+d x}}}{c x}+\frac {b d \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{c+d x}}}{\sqrt {b+a c}}\right )}{c^{3/2} \sqrt {b+a c}} \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {a+\frac {b}{c+d x}}}{x^2} \, dx=-\frac {(c+d x) \sqrt {\frac {b+a c+a d x}{c+d x}}}{c x}-\frac {b d \arctan \left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x}{c+d x}}}{\sqrt {-b-a c}}\right )}{c^{3/2} \sqrt {-b-a c}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {896, 941, 946, 51, 73, 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.

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

\(\Big \downarrow \) 896

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

\(\Big \downarrow \) 941

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

\(\Big \downarrow \) 946

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

\(\displaystyle -d \left (\frac {\sqrt {a+\frac {b}{c+d x}}}{c \left (1-\frac {c}{c+d x}\right )}-\frac {b \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{c+d x}}}{\sqrt {a c+b}}\right )}{c^{3/2} \sqrt {a c+b}}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, 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 896

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff 
icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1)   Subst[Int[Si 
mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; 
 FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
 

rule 941

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Sym 
bol] :> Int[(a + b*x^n)^p*((d + c*x^n)^q/x^(n*q)), x] /; FreeQ[{a, b, c, d, 
 n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] ||  !IntegerQ[p])
 

rule 946

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^n], 
x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m - n 
+ 1, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(765\) vs. \(2(65)=130\).

Time = 0.12 (sec) , antiderivative size = 766, normalized size of antiderivative = 9.95

method	result	size
default	\(\frac {\sqrt {\frac {a d x +a c +b}{d x +c}}\, \left (d x +c \right ) \left (2 \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b d x +b c}\, \sqrt {a \,d^{2}}\, \sqrt {\left (a c +b \right ) c}\, a \,d^{2} x^{2}-\ln \left (\frac {2 a \,d^{2} x +2 a c d +2 \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b d x +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) \sqrt {\left (a c +b \right ) c}\, a b c \,d^{2} x +\sqrt {a \,d^{2}}\, \ln \left (\frac {2 a d x c +2 a \,c^{2}+b d x +2 \sqrt {\left (a c +b \right ) c}\, \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b d x +b c}+2 b c}{x}\right ) a b \,c^{2} d x +\sqrt {\left (a c +b \right ) c}\, \ln \left (\frac {2 a \,d^{2} x +2 a c d +2 \sqrt {\left (a d x +a c +b \right ) \left (d x +c \right )}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a b c \,d^{2} x +2 \sqrt {\left (a d x +a c +b \right ) \left (d x +c \right )}\, \sqrt {a \,d^{2}}\, \sqrt {\left (a c +b \right ) c}\, a c d x +2 \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b d x +b c}\, \sqrt {a \,d^{2}}\, \sqrt {\left (a c +b \right ) c}\, a c d x -\ln \left (\frac {2 a \,d^{2} x +2 a c d +2 \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b d x +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) \sqrt {\left (a c +b \right ) c}\, b^{2} d^{2} x +\sqrt {a \,d^{2}}\, \ln \left (\frac {2 a d x c +2 a \,c^{2}+b d x +2 \sqrt {\left (a c +b \right ) c}\, \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b d x +b c}+2 b c}{x}\right ) b^{2} c d x +\sqrt {\left (a c +b \right ) c}\, \ln \left (\frac {2 a \,d^{2} x +2 a c d +2 \sqrt {\left (a d x +a c +b \right ) \left (d x +c \right )}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) b^{2} d^{2} x +2 \sqrt {\left (a d x +a c +b \right ) \left (d x +c \right )}\, \sqrt {a \,d^{2}}\, \sqrt {\left (a c +b \right ) c}\, b d x -2 \left (a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b d x +b c \right )^{\frac {3}{2}} \sqrt {a \,d^{2}}\, \sqrt {\left (a c +b \right ) c}\right )}{2 \sqrt {\left (a d x +a c +b \right ) \left (d x +c \right )}\, c^{2} \left (a c +b \right ) x \sqrt {a \,d^{2}}\, \sqrt {\left (a c +b \right ) c}}\)	\(766\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (65) = 130\).

Time = 0.09 (sec) , antiderivative size = 276, normalized size of antiderivative = 3.58 \[ \int \frac {\sqrt {a+\frac {b}{c+d x}}}{x^2} \, dx=\left [\frac {\sqrt {a c^{2} + b c} b d x \log \left (-\frac {2 \, a c^{2} + {\left (2 \, a c + b\right )} d x + 2 \, b c + 2 \, \sqrt {a c^{2} + b c} {\left (d x + c\right )} \sqrt {\frac {a d x + a c + b}{d x + c}}}{x}\right ) - 2 \, {\left (a c^{3} + b c^{2} + {\left (a c^{2} + b c\right )} d x\right )} \sqrt {\frac {a d x + a c + b}{d x + c}}}{2 \, {\left (a c^{3} + b c^{2}\right )} x}, -\frac {\sqrt {-a c^{2} - b c} b d x \arctan \left (\frac {\sqrt {-a c^{2} - b c} {\left (d x + c\right )} \sqrt {\frac {a d x + a c + b}{d x + c}}}{a c d x + a c^{2} + b c}\right ) + {\left (a c^{3} + b c^{2} + {\left (a c^{2} + b c\right )} d x\right )} \sqrt {\frac {a d x + a c + b}{d x + c}}}{{\left (a c^{3} + b c^{2}\right )} x}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(sqrt(a + b/(d*x + c))/x^2, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (65) = 130\).

Time = 0.15 (sec) , antiderivative size = 279, normalized size of antiderivative = 3.62 \[ \int \frac {\sqrt {a+\frac {b}{c+d x}}}{x^2} \, dx=-\frac {b d \arctan \left (-\frac {\sqrt {a d^{2}} x - \sqrt {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b d x + b c}}{\sqrt {-a c^{2} - b c}}\right ) \mathrm {sgn}\left (d x + c\right )}{\sqrt {-a c^{2} - b c} c} - \frac {2 \, a^{\frac {3}{2}} c^{2} {\left | d \right |} \mathrm {sgn}\left (d x + c\right ) + 2 \, {\left (\sqrt {a d^{2}} x - \sqrt {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b d x + b c}\right )} a c d \mathrm {sgn}\left (d x + c\right ) + 2 \, \sqrt {a} b c {\left | d \right |} \mathrm {sgn}\left (d x + c\right ) + {\left (\sqrt {a d^{2}} x - \sqrt {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b d x + b c}\right )} b d \mathrm {sgn}\left (d x + c\right )}{{\left (a c^{2} - {\left (\sqrt {a d^{2}} x - \sqrt {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b d x + b c}\right )}^{2} + b c\right )} c} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.82 \[ \int \frac {\sqrt {a+\frac {b}{c+d x}}}{x^2} \, dx=\frac {-2 \sqrt {d x +c}\, \sqrt {a d x +a c +b}\, a \,c^{2}-2 \sqrt {d x +c}\, \sqrt {a d x +a c +b}\, b c -\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a d x +a c +b}-\sqrt {2 \sqrt {c}\, \sqrt {a}\, \sqrt {a c +b}+2 a c +b}+\sqrt {a}\, \sqrt {d x +c}\right ) b d x -\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a d x +a c +b}+\sqrt {2 \sqrt {c}\, \sqrt {a}\, \sqrt {a c +b}+2 a c +b}+\sqrt {a}\, \sqrt {d x +c}\right ) b d x +\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {d x +c}\, \sqrt {a d x +a c +b}+2 \sqrt {c}\, \sqrt {a}\, \sqrt {a c +b}+2 a d x \right ) b d x}{2 c^{2} x \left (a c +b \right )} \] Input:

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

Output:

( - 2*sqrt(c + d*x)*sqrt(a*c + a*d*x + b)*a*c**2 - 2*sqrt(c + d*x)*sqrt(a* 
c + a*d*x + b)*b*c - sqrt(c)*sqrt(a*c + b)*log(sqrt(a*c + a*d*x + b) - sqr 
t(2*sqrt(c)*sqrt(a)*sqrt(a*c + b) + 2*a*c + b) + sqrt(a)*sqrt(c + d*x))*b* 
d*x - sqrt(c)*sqrt(a*c + b)*log(sqrt(a*c + a*d*x + b) + sqrt(2*sqrt(c)*sqr 
t(a)*sqrt(a*c + b) + 2*a*c + b) + sqrt(a)*sqrt(c + d*x))*b*d*x + sqrt(c)*s 
qrt(a*c + b)*log(2*sqrt(a)*sqrt(c + d*x)*sqrt(a*c + a*d*x + b) + 2*sqrt(c) 
*sqrt(a)*sqrt(a*c + b) + 2*a*d*x)*b*d*x)/(2*c**2*x*(a*c + b))