\(\int \frac {1}{(a+\frac {b}{x})^{3/2} (c+\frac {d}{x})^2} \, dx\) [37]

Optimal result

Integrand size = 21, antiderivative size = 224 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2} \, dx=\frac {b \left (3 b^2 c^2-2 a b c d+2 a^2 d^2\right )}{a^2 c^2 (b c-a d)^2 \sqrt {a+\frac {b}{x}}}+\frac {d (b c-2 a d)}{a c^2 (b c-a d) \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}+\frac {d^{5/2} (7 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 (b c-a d)^{5/2}}-\frac {(3 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2} c^3} \] Output:

b*(2*a^2*d^2-2*a*b*c*d+3*b^2*c^2)/a^2/c^2/(-a*d+b*c)^2/(a+b/x)^(1/2)+d*(-2 
*a*d+b*c)/a/c^2/(-a*d+b*c)/(a+b/x)^(1/2)/(c+d/x)+x/a/c/(a+b/x)^(1/2)/(c+d/ 
x)+d^(5/2)*(-4*a*d+7*b*c)*arctan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+b*c)^(1/2))/c 
^3/(-a*d+b*c)^(5/2)-(4*a*d+3*b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(5/2)/c 
^3
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x \left (3 b^3 c^2 (d+c x)+a^3 d^2 x (2 d+c x)+a^2 b d \left (2 d^2-c d x-2 c^2 x^2\right )+a b^2 c \left (-2 d^2-c d x+c^2 x^2\right )\right )}{a^2 (b c-a d)^2 (b+a x) (d+c x)}+\frac {d^{5/2} (7 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{5/2}}-\frac {(3 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}}{c^3} \] Input:

Integrate[1/((a + b/x)^(3/2)*(c + d/x)^2),x]
 

Output:

((c*Sqrt[a + b/x]*x*(3*b^3*c^2*(d + c*x) + a^3*d^2*x*(2*d + c*x) + a^2*b*d 
*(2*d^2 - c*d*x - 2*c^2*x^2) + a*b^2*c*(-2*d^2 - c*d*x + c^2*x^2)))/(a^2*( 
b*c - a*d)^2*(b + a*x)*(d + c*x)) + (d^(5/2)*(7*b*c - 4*a*d)*ArcTan[(Sqrt[ 
d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(5/2) - ((3*b*c + 4*a*d)*A 
rcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(5/2))/c^3
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.23, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {899, 114, 27, 168, 25, 169, 27, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {\frac {a^2 d^3 (7 b c-4 a d) \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}+\frac {(b c-a d)^2 (4 a d+3 b c) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}}{a (b c-a d)}+\frac {2 b \left (2 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{c (b c-a d)}+\frac {2 d (b c-2 a d)}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {\frac {2 a^2 d^3 (7 b c-4 a d) \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}+\frac {2 (b c-a d)^2 (4 a d+3 b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}}{a (b c-a d)}+\frac {2 b \left (2 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{c (b c-a d)}+\frac {2 d (b c-2 a d)}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\frac {2 (b c-a d)^2 (4 a d+3 b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}+\frac {2 a^2 d^{5/2} (7 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}}{a (b c-a d)}+\frac {2 b \left (2 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{c (b c-a d)}+\frac {2 d (b c-2 a d)}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {\frac {2 a^2 d^{5/2} (7 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (b c-a d)^2 (4 a d+3 b c)}{\sqrt {a} c}}{a (b c-a d)}+\frac {2 b \left (2 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{c (b c-a d)}+\frac {2 d (b c-2 a d)}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}\)

Input:

Int[1/((a + b/x)^(3/2)*(c + d/x)^2),x]
 

Output:

x/(a*c*Sqrt[a + b/x]*(c + d/x)) + ((2*d*(b*c - 2*a*d))/(c*(b*c - a*d)*Sqrt 
[a + b/x]*(c + d/x)) + ((2*b*(3*b^2*c^2 - 2*a*b*c*d + 2*a^2*d^2))/(a*(b*c 
- a*d)*Sqrt[a + b/x]) + ((2*a^2*d^(5/2)*(7*b*c - 4*a*d)*ArcTan[(Sqrt[d]*Sq 
rt[a + b/x])/Sqrt[b*c - a*d]])/(c*Sqrt[b*c - a*d]) - (2*(b*c - a*d)^2*(3*b 
*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(Sqrt[a]*c))/(a*(b*c - a*d)))/ 
(c*(b*c - a*d)))/(2*a*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 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

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 169

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] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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. \(532\) vs. \(2(202)=404\).

Time = 0.65 (sec) , antiderivative size = 533, normalized size of antiderivative = 2.38

method	result	size
risch	\(\frac {a x +b}{a^{2} c^{2} \sqrt {\frac {a x +b}{x}}}+\frac {\left (-\frac {2 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) d}{a^{\frac {3}{2}} c^{3}}-\frac {3 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) b}{2 a^{\frac {5}{2}} c^{2}}+\frac {2 b^{3} \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{a^{3} \left (a d -b c \right )^{2} \left (x +\frac {b}{a}\right )}+\frac {d^{3} \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{c^{3} \left (a d -b c \right )^{2} \left (x +\frac {d}{c}\right )}-\frac {2 a \,d^{4} \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{c^{4} \left (a d -b c \right )^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}+\frac {7 d^{3} \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right ) b}{2 c^{3} \left (a d -b c \right )^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right ) \sqrt {x \left (a x +b \right )}}{x \sqrt {\frac {a x +b}{x}}}\)	\(533\)
default	\(\text {Expression too large to display}\)	\(3119\)

Input:

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

Output:

1/a^2/c^2*(a*x+b)/((a*x+b)/x)^(1/2)+(-2/a^(3/2)/c^3*ln((1/2*b+a*x)/a^(1/2) 
+(a*x^2+b*x)^(1/2))*d-3/2/a^(5/2)/c^2*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^( 
1/2))*b+2/a^3*b^3/(a*d-b*c)^2/(x+b/a)*(a*(x+b/a)^2-b*(x+b/a))^(1/2)+1/c^3* 
d^3/(a*d-b*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)-2*a/c^4*d^4/(a*d-b*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))+7/2/c^3*d^3/(a 
*d-b*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))*b)/x/((a*x+b)/x)^(1/2)*(x*(a*x+b))^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 554 vs. \(2 (202) = 404\).

Time = 0.71 (sec) , antiderivative size = 2291, normalized size of antiderivative = 10.23 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*((3*b^4*c^3*d - 2*a*b^3*c^2*d^2 - 5*a^2*b^2*c*d^3 + 4*a^3*b*d^4 + (3* 
a*b^3*c^4 - 2*a^2*b^2*c^3*d - 5*a^3*b*c^2*d^2 + 4*a^4*c*d^3)*x^2 + (3*b^4* 
c^4 + a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 - a^3*b*c*d^3 + 4*a^4*d^4)*x)*sqrt(a 
)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - (7*a^3*b^2*c*d^3 - 4*a^ 
4*b*d^4 + (7*a^4*b*c^2*d^2 - 4*a^5*c*d^3)*x^2 + (7*a^3*b^2*c^2*d^2 + 3*a^4 
*b*c*d^3 - 4*a^5*d^4)*x)*sqrt(-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sqrt(- 
d/(b*c - a*d))*sqrt((a*x + b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x + d)) + 2*( 
(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*x^3 + (3*a*b^3*c^4 - a^2*b^2*c 
^3*d - a^3*b*c^2*d^2 + 2*a^4*c*d^3)*x^2 + (3*a*b^3*c^3*d - 2*a^2*b^2*c^2*d 
^2 + 2*a^3*b*c*d^3)*x)*sqrt((a*x + b)/x))/(a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d 
^2 + a^5*b*c^3*d^3 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x^2 + (a^ 
3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x), 1/2*(2*(7*a^3 
*b^2*c*d^3 - 4*a^4*b*d^4 + (7*a^4*b*c^2*d^2 - 4*a^5*c*d^3)*x^2 + (7*a^3*b^ 
2*c^2*d^2 + 3*a^4*b*c*d^3 - 4*a^5*d^4)*x)*sqrt(d/(b*c - a*d))*arctan(sqrt( 
d/(b*c - a*d))*sqrt((a*x + b)/x)) + (3*b^4*c^3*d - 2*a*b^3*c^2*d^2 - 5*a^2 
*b^2*c*d^3 + 4*a^3*b*d^4 + (3*a*b^3*c^4 - 2*a^2*b^2*c^3*d - 5*a^3*b*c^2*d^ 
2 + 4*a^4*c*d^3)*x^2 + (3*b^4*c^4 + a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 - a^3* 
b*c*d^3 + 4*a^4*d^4)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) 
+ b) + 2*((a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*x^3 + (3*a*b^3*c^4 - 
 a^2*b^2*c^3*d - a^3*b*c^2*d^2 + 2*a^4*c*d^3)*x^2 + (3*a*b^3*c^3*d - 2*...
 

Sympy [F]

\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2} \, dx=\int \frac {x^{2}}{\left (a + \frac {b}{x}\right )^{\frac {3}{2}} \left (c x + d\right )^{2}}\, dx \] Input:

integrate(1/(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]

\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} {\left (c + \frac {d}{x}\right )}^{2}} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 5.64 (sec) , antiderivative size = 4274, normalized size of antiderivative = 19.08 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((2*b^3)/(a^2*d - a*b*c) + (b*(a + b/x)^2*(2*a^2*d^3 + 3*b^2*c^2*d - 2*a*b 
*c*d^2))/(c^2*(a^2*d - a*b*c)^2) - (b*(a + b/x)*(2*a*d - b*c)*(a^2*d^2 + 3 
*b^2*c^2 - a*b*c*d))/(c^2*(a^2*d - a*b*c)^2))/(d*(a + b/x)^(5/2) + (a + b/ 
x)^(1/2)*(a^2*d - a*b*c) - (a + b/x)^(3/2)*(2*a*d - b*c)) + (atan((a^13*b^ 
11*c^11*d^3*(a + b/x)^(1/2)*35i - a^12*b^12*c^12*d^2*(a + b/x)^(1/2)*441i 
- a^10*b^14*c^14*(a + b/x)^(1/2)*27i + a^14*b^10*c^10*d^4*(a + b/x)^(1/2)* 
1694i - a^15*b^9*c^9*d^5*(a + b/x)^(1/2)*3073i + a^16*b^8*c^8*d^6*(a + b/x 
)^(1/2)*1316i + a^17*b^7*c^7*d^7*(a + b/x)^(1/2)*2561i - a^18*b^6*c^6*d^8* 
(a + b/x)^(1/2)*4375i + a^19*b^5*c^5*d^9*(a + b/x)^(1/2)*2996i - a^20*b^4* 
c^4*d^10*(a + b/x)^(1/2)*1015i + a^21*b^3*c^3*d^11*(a + b/x)^(1/2)*140i + 
a^11*b^13*c^13*d*(a + b/x)^(1/2)*189i)/(a^5*(a^5)^(1/2)*(a^5*(a^5*(2561*b^ 
7*c^7*d^7 - 4375*a*b^6*c^6*d^8 + 2996*a^2*b^5*c^5*d^9 - 1015*a^3*b^4*c^4*d 
^10 + 140*a^4*b^3*c^3*d^11) - 441*b^12*c^12*d^2 + 35*a*b^11*c^11*d^3 + 169 
4*a^2*b^10*c^10*d^4 - 3073*a^3*b^9*c^9*d^5 + 1316*a^4*b^8*c^8*d^6) - 27*a^ 
3*b^14*c^14 + 189*a^4*b^13*c^13*d)))*(4*a*d + 3*b*c)*1i)/(c^3*(a^5)^(1/2)) 
 - (atan((((d^5*(a*d - b*c)^5)^(1/2)*(4*a*d - 7*b*c)*((a + b/x)^(1/2)*(18* 
a^6*b^14*c^18*d^3 - 132*a^7*b^13*c^17*d^4 + 362*a^8*b^12*c^16*d^5 - 320*a^ 
9*b^11*c^15*d^6 - 442*a^10*b^10*c^14*d^7 + 1004*a^11*b^9*c^13*d^8 + 578*a^ 
12*b^8*c^12*d^9 - 3976*a^13*b^7*c^11*d^10 + 5960*a^14*b^6*c^10*d^11 - 4768 
*a^15*b^5*c^9*d^12 + 2228*a^16*b^4*c^8*d^13 - 576*a^17*b^3*c^7*d^14 + 6...
 

Reduce [B] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 2294, normalized size of antiderivative = 10.24 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(16*sqrt(d)*sqrt(a*x + b)*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**5*c*d**4*x + 16*sqrt(d)*sqrt(a*x + b)*sqrt(a*d - b*c)*log(sqrt(c)*sq 
rt(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**5*d**5 - 32*sqrt(d)*sqrt(a*x + b)*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**4*b*c**2*d**3*x - 32*sqrt(d)*sqrt(a* 
x + b)*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**4*b*c*d**4 + 
7*sqrt(d)*sqrt(a*x + b)*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**3*b**2*c**3*d**2*x + 7*sqrt(d)*sqrt(a*x + b)*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**3*b**2*c**2*d**3 + 16*sqrt(d)*sqrt(a*x + b)*s 
qrt(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**5*c*d**4*x + 16*sqrt( 
d)*sqrt(a*x + b)*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**5*d 
**5 - 32*sqrt(d)*sqrt(a*x + b)*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...