\(\int \frac {(a+\frac {b}{x})^{3/2}}{(c+\frac {d}{x})^2} \, dx\) [16]

Optimal result

Integrand size = 21, antiderivative size = 156 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^2} \, dx=-\frac {(b c-2 a d) \sqrt {a+\frac {b}{x}}}{c^2 \left (c+\frac {d}{x}\right )}+\frac {a \sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )}-\frac {(b c-4 a d) \sqrt {b c-a d} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 \sqrt {d}}+\frac {\sqrt {a} (3 b c-4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^2} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x (-b c+2 a d+a c x)}{d+c x}-\frac {\left (b^2 c^2-5 a b c d+4 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{\sqrt {d} \sqrt {b c-a d}}-\sqrt {a} (-3 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.21, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {899, 109, 27, 168, 25, 174, 73, 218, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 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 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || 
IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 218

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

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 899

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(500\) vs. \(2(136)=272\).

Time = 0.49 (sec) , antiderivative size = 501, normalized size of antiderivative = 3.21

method	result	size
risch	\(\frac {x a \sqrt {\frac {a x +b}{x}}}{c^{2}}-\frac {\left (\frac {\sqrt {a}\, \left (4 a d -3 b c \right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{c}+\frac {2 \left (3 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{c^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}+\frac {2 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {c^{2} \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{\left (a d -b c \right ) d \left (x +\frac {d}{c}\right )}-\frac {\left (2 a d -b c \right ) c \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{2 \left (a d -b c \right ) d \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right )}{c^{3}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 c^{2} \left (a x +b \right )}\)	\(501\)
default	\(-\frac {\left (4 a^{\frac {7}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c -2 a d x +b c x -b d}{c x +d}\right ) c \,d^{3} x +2 a^{\frac {5}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c^{4} x^{2}+4 a^{\frac {7}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c -2 a d x +b c x -b d}{c x +d}\right ) d^{4}-2 a^{\frac {5}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c^{3} d x -5 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c -2 a d x +b c x -b d}{c x +d}\right ) b \,c^{2} d^{2} x -4 a^{\frac {5}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c^{2} d^{2}-5 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c -2 a d x +b c x -b d}{c x +d}\right ) b c \,d^{3}-2 c^{4} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}+2 a^{\frac {3}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, b \,c^{4} x +a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c -2 a d x +b c x -b d}{c x +d}\right ) b^{2} c^{3} d x +4 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c^{2} d^{2} x -3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b \,c^{3} d x +2 a^{\frac {3}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, b \,c^{3} d +a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c -2 a d x +b c x -b d}{c x +d}\right ) b^{2} c^{2} d^{2}+4 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c \,d^{3}-3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b \,c^{2} d^{2}\right ) x \sqrt {\frac {a x +b}{x}}}{2 c^{4} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{\frac {3}{2}} \left (c x +d \right ) d \sqrt {x \left (a x +b \right )}}\)	\(834\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 781, normalized size of antiderivative = 5.01 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/2*((3*b*c*d - 4*a*d^2 + (3*b*c^2 - 4*a*c*d)*x)*sqrt(a)*log(2*a*x - 2*s 
qrt(a)*x*sqrt((a*x + b)/x) + b) + (b*c*d - 4*a*d^2 + (b*c^2 - 4*a*c*d)*x)* 
sqrt(-(b*c - a*d)/d)*log((2*d*x*sqrt(-(b*c - a*d)/d)*sqrt((a*x + b)/x) + b 
*d - (b*c - 2*a*d)*x)/(c*x + d)) - 2*(a*c^2*x^2 - (b*c^2 - 2*a*c*d)*x)*sqr 
t((a*x + b)/x))/(c^4*x + c^3*d), 1/2*(2*(b*c*d - 4*a*d^2 + (b*c^2 - 4*a*c* 
d)*x)*sqrt((b*c - a*d)/d)*arctan(-d*sqrt((b*c - a*d)/d)*sqrt((a*x + b)/x)/ 
(b*c - a*d)) - (3*b*c*d - 4*a*d^2 + (3*b*c^2 - 4*a*c*d)*x)*sqrt(a)*log(2*a 
*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(a*c^2*x^2 - (b*c^2 - 2*a*c*d) 
*x)*sqrt((a*x + b)/x))/(c^4*x + c^3*d), -1/2*(2*(3*b*c*d - 4*a*d^2 + (3*b* 
c^2 - 4*a*c*d)*x)*sqrt(-a)*arctan(sqrt(-a)*x*sqrt((a*x + b)/x)/(a*x + b)) 
+ (b*c*d - 4*a*d^2 + (b*c^2 - 4*a*c*d)*x)*sqrt(-(b*c - a*d)/d)*log((2*d*x* 
sqrt(-(b*c - a*d)/d)*sqrt((a*x + b)/x) + b*d - (b*c - 2*a*d)*x)/(c*x + d)) 
 - 2*(a*c^2*x^2 - (b*c^2 - 2*a*c*d)*x)*sqrt((a*x + b)/x))/(c^4*x + c^3*d), 
 -((3*b*c*d - 4*a*d^2 + (3*b*c^2 - 4*a*c*d)*x)*sqrt(-a)*arctan(sqrt(-a)*x* 
sqrt((a*x + b)/x)/(a*x + b)) - (b*c*d - 4*a*d^2 + (b*c^2 - 4*a*c*d)*x)*sqr 
t((b*c - a*d)/d)*arctan(-d*sqrt((b*c - a*d)/d)*sqrt((a*x + b)/x)/(b*c - a* 
d)) - (a*c^2*x^2 - (b*c^2 - 2*a*c*d)*x)*sqrt((a*x + b)/x))/(c^4*x + c^3*d) 
]
 

Sympy [F]

\[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^2} \, dx=\int \frac {x^{2} \left (a + \frac {b}{x}\right )^{\frac {3}{2}}}{\left (c x + d\right )^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^2} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}}}{{\left (c + \frac {d}{x}\right )}^{2}} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 514 vs. \(2 (136) = 272\).

Time = 0.17 (sec) , antiderivative size = 514, normalized size of antiderivative = 3.29 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^2} \, dx=\frac {\sqrt {a x^{2} + b x} a \mathrm {sgn}\left (x\right )}{c^{2}} + \frac {{\left (b^{2} c^{2} \mathrm {sgn}\left (x\right ) - 5 \, a b c d \mathrm {sgn}\left (x\right ) + 4 \, a^{2} d^{2} \mathrm {sgn}\left (x\right )\right )} \arctan \left (-\frac {{\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} c + \sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right )}{\sqrt {b c d - a d^{2}} c^{3}} - \frac {{\left (3 \, a b c \mathrm {sgn}\left (x\right ) - 4 \, a^{2} d \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} - b \right |}\right )}{2 \, \sqrt {a} c^{3}} + \frac {{\left (2 \, \sqrt {a} b^{2} c^{2} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - 10 \, a^{\frac {3}{2}} b c d \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) + 8 \, a^{\frac {5}{2}} d^{2} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) + 3 \, \sqrt {b c d - a d^{2}} a b c \log \left ({\left | b \right |}\right ) - 4 \, \sqrt {b c d - a d^{2}} a^{2} d \log \left ({\left | b \right |}\right ) + 2 \, \sqrt {b c d - a d^{2}} a b c - 2 \, \sqrt {b c d - a d^{2}} a^{2} d\right )} \mathrm {sgn}\left (x\right )}{2 \, \sqrt {b c d - a d^{2}} \sqrt {a} c^{3}} + \frac {{\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} b^{2} c^{2} \mathrm {sgn}\left (x\right ) - 3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a b c d \mathrm {sgn}\left (x\right ) + 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{2} d^{2} \mathrm {sgn}\left (x\right ) - \sqrt {a} b^{2} c d \mathrm {sgn}\left (x\right ) + a^{\frac {3}{2}} b d^{2} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} c + 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} d + b d\right )} c^{3}} \] Input:

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

Output:

sqrt(a*x^2 + b*x)*a*sgn(x)/c^2 + (b^2*c^2*sgn(x) - 5*a*b*c*d*sgn(x) + 4*a^ 
2*d^2*sgn(x))*arctan(-((sqrt(a)*x - sqrt(a*x^2 + b*x))*c + sqrt(a)*d)/sqrt 
(b*c*d - a*d^2))/(sqrt(b*c*d - a*d^2)*c^3) - 1/2*(3*a*b*c*sgn(x) - 4*a^2*d 
*sgn(x))*log(abs(-2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) - b))/(sqrt(a) 
*c^3) + 1/2*(2*sqrt(a)*b^2*c^2*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 10* 
a^(3/2)*b*c*d*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 8*a^(5/2)*d^2*arctan 
(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 3*sqrt(b*c*d - a*d^2)*a*b*c*log(abs(b)) 
- 4*sqrt(b*c*d - a*d^2)*a^2*d*log(abs(b)) + 2*sqrt(b*c*d - a*d^2)*a*b*c - 
2*sqrt(b*c*d - a*d^2)*a^2*d)*sgn(x)/(sqrt(b*c*d - a*d^2)*sqrt(a)*c^3) + (( 
sqrt(a)*x - sqrt(a*x^2 + b*x))*b^2*c^2*sgn(x) - 3*(sqrt(a)*x - sqrt(a*x^2 
+ b*x))*a*b*c*d*sgn(x) + 2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^2*d^2*sgn(x) 
- sqrt(a)*b^2*c*d*sgn(x) + a^(3/2)*b*d^2*sgn(x))/(((sqrt(a)*x - sqrt(a*x^2 
 + b*x))^2*c + 2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*d + b*d)*c^3)
 

Mupad [B] (verification not implemented)

Time = 1.61 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.87 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^2} \, dx=\frac {\mathrm {atanh}\left (\frac {8\,a^2\,b^5\,d^2\,\sqrt {a+\frac {b}{x}}\,\sqrt {a\,d^2-b\,c\,d}}{8\,a^3\,b^5\,d^3-10\,a^2\,b^6\,c\,d^2+2\,a\,b^7\,c^2\,d}-\frac {2\,a\,b^6\,d\,\sqrt {a+\frac {b}{x}}\,\sqrt {a\,d^2-b\,c\,d}}{2\,a\,b^7\,c\,d-10\,a^2\,b^6\,d^2+\frac {8\,a^3\,b^5\,d^3}{c}}\right )\,\sqrt {d\,\left (a\,d-b\,c\right )}\,\left (4\,a\,d-b\,c\right )}{c^3\,d}-\frac {\sqrt {a}\,\mathrm {atanh}\left (\frac {6\,\sqrt {a}\,b^7\,d\,\sqrt {a+\frac {b}{x}}}{6\,a\,b^7\,d-\frac {14\,a^2\,b^6\,d^2}{c}+\frac {8\,a^3\,b^5\,d^3}{c^2}}-\frac {14\,a^{3/2}\,b^6\,d^2\,\sqrt {a+\frac {b}{x}}}{6\,a\,b^7\,c\,d-14\,a^2\,b^6\,d^2+\frac {8\,a^3\,b^5\,d^3}{c}}+\frac {8\,a^{5/2}\,b^5\,d^3\,\sqrt {a+\frac {b}{x}}}{8\,a^3\,b^5\,d^3-14\,a^2\,b^6\,c\,d^2+6\,a\,b^7\,c^2\,d}\right )\,\left (4\,a\,d-3\,b\,c\right )}{c^3}-\frac {\frac {2\,\left (a\,b^2\,c-a^2\,b\,d\right )\,\sqrt {a+\frac {b}{x}}}{c^2}+\frac {b\,{\left (a+\frac {b}{x}\right )}^{3/2}\,\left (2\,a\,d-b\,c\right )}{c^2}}{\left (a+\frac {b}{x}\right )\,\left (2\,a\,d-b\,c\right )-d\,{\left (a+\frac {b}{x}\right )}^2-a^2\,d+a\,b\,c} \] Input:

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

Output:

(atanh((8*a^2*b^5*d^2*(a + b/x)^(1/2)*(a*d^2 - b*c*d)^(1/2))/(8*a^3*b^5*d^ 
3 - 10*a^2*b^6*c*d^2 + 2*a*b^7*c^2*d) - (2*a*b^6*d*(a + b/x)^(1/2)*(a*d^2 
- b*c*d)^(1/2))/(2*a*b^7*c*d - 10*a^2*b^6*d^2 + (8*a^3*b^5*d^3)/c))*(d*(a* 
d - b*c))^(1/2)*(4*a*d - b*c))/(c^3*d) - (a^(1/2)*atanh((6*a^(1/2)*b^7*d*( 
a + b/x)^(1/2))/(6*a*b^7*d - (14*a^2*b^6*d^2)/c + (8*a^3*b^5*d^3)/c^2) - ( 
14*a^(3/2)*b^6*d^2*(a + b/x)^(1/2))/(6*a*b^7*c*d - 14*a^2*b^6*d^2 + (8*a^3 
*b^5*d^3)/c) + (8*a^(5/2)*b^5*d^3*(a + b/x)^(1/2))/(8*a^3*b^5*d^3 - 14*a^2 
*b^6*c*d^2 + 6*a*b^7*c^2*d))*(4*a*d - 3*b*c))/c^3 - ((2*(a*b^2*c - a^2*b*d 
)*(a + b/x)^(1/2))/c^2 + (b*(a + b/x)^(3/2)*(2*a*d - b*c))/c^2)/((a + b/x) 
*(2*a*d - b*c) - d*(a + b/x)^2 - a^2*d + a*b*c)
 

Reduce [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 873, normalized size of antiderivative = 5.60 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt 
(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a*c*d*x + 4* 
sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt(a) 
*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a*d**2 - sqrt(d 
)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt(a)*sqrt( 
a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*b*c**2*x - sqrt(d)*sq 
rt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d 
- b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*b*c*d + 4*sqrt(d)*sqrt(a* 
d - b*c)*log(sqrt(c)*sqrt(a*x + b) + sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c 
) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a*c*d*x + 4*sqrt(d)*sqrt(a*d - 
 b*c)*log(sqrt(c)*sqrt(a*x + b) + sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 
 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a*d**2 - sqrt(d)*sqrt(a*d - b*c)* 
log(sqrt(c)*sqrt(a*x + b) + sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d 
 + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*b*c**2*x - sqrt(d)*sqrt(a*d - b*c)*log( 
sqrt(c)*sqrt(a*x + b) + sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b 
*c) + sqrt(x)*sqrt(c)*sqrt(a))*b*c*d - 4*sqrt(d)*sqrt(a*d - b*c)*log(2*sqr 
t(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(x)*sqrt(a)*sqrt(a*x + b)*c + 2*a*c*x 
 + 2*a*d)*a*c*d*x - 4*sqrt(d)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a 
*d - b*c) + 2*sqrt(x)*sqrt(a)*sqrt(a*x + b)*c + 2*a*c*x + 2*a*d)*a*d**2 + 
sqrt(d)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(...