\(\int \frac {1}{(a+\frac {b}{x})^{3/2} (c+\frac {d}{x})^3} \, dx\) [38]

Optimal result

Integrand size = 21, antiderivative size = 320 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3} \, dx=\frac {3 b (2 b c-a d) \left (2 b^2 c^2-a b c d+4 a^2 d^2\right )}{4 a^2 c^3 (b c-a d)^3 \sqrt {a+\frac {b}{x}}}+\frac {d (2 b c-3 a d)}{2 a c^2 (b c-a d) \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}+\frac {d \left (4 b^2 c^2-21 a b c d+12 a^2 d^2\right )}{4 a c^3 (b c-a d)^2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}+\frac {3 d^{5/2} \left (21 b^2 c^2-24 a b c d+8 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 (b c-a d)^{7/2}}-\frac {3 (b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2} c^4} \] Output:

3/4*b*(-a*d+2*b*c)*(4*a^2*d^2-a*b*c*d+2*b^2*c^2)/a^2/c^3/(-a*d+b*c)^3/(a+b 
/x)^(1/2)+1/2*d*(-3*a*d+2*b*c)/a/c^2/(-a*d+b*c)/(a+b/x)^(1/2)/(c+d/x)^2+1/ 
4*d*(12*a^2*d^2-21*a*b*c*d+4*b^2*c^2)/a/c^3/(-a*d+b*c)^2/(a+b/x)^(1/2)/(c+ 
d/x)+x/a/c/(a+b/x)^(1/2)/(c+d/x)^2+3/4*d^(5/2)*(8*a^2*d^2-24*a*b*c*d+21*b^ 
2*c^2)*arctan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+b*c)^(1/2))/c^4/(-a*d+b*c)^(7/2) 
-3*(2*a*d+b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(5/2)/c^4
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 2.10 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x \left (-12 b^4 c^3 (d+c x)^2-4 a b^3 c^2 (-3 d+c x) (d+c x)^2+2 a^4 d^3 x \left (6 d^2+9 c d x+2 c^2 x^2\right )+a^3 b d^2 \left (12 d^3-9 c d^2 x-37 c^2 d x^2-12 c^3 x^3\right )+a^2 b^2 c d \left (-27 d^3-29 c d^2 x+12 c^2 d x^2+12 c^3 x^3\right )\right )}{a^2 (-b c+a d)^3 (b+a x) (d+c x)^2}+\frac {3 d^{5/2} \left (21 b^2 c^2-24 a b c d+8 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{7/2}}-\frac {12 (b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}}{4 c^4} \] Input:

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

Output:

((c*Sqrt[a + b/x]*x*(-12*b^4*c^3*(d + c*x)^2 - 4*a*b^3*c^2*(-3*d + c*x)*(d 
 + c*x)^2 + 2*a^4*d^3*x*(6*d^2 + 9*c*d*x + 2*c^2*x^2) + a^3*b*d^2*(12*d^3 
- 9*c*d^2*x - 37*c^2*d*x^2 - 12*c^3*x^3) + a^2*b^2*c*d*(-27*d^3 - 29*c*d^2 
*x + 12*c^2*d*x^2 + 12*c^3*x^3)))/(a^2*(-(b*c) + a*d)^3*(b + a*x)*(d + c*x 
)^2) + (3*d^(5/2)*(21*b^2*c^2 - 24*a*b*c*d + 8*a^2*d^2)*ArcTan[(Sqrt[d]*Sq 
rt[a + b/x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(7/2) - (12*(b*c + 2*a*d)*ArcTa 
nh[Sqrt[a + b/x]/Sqrt[a]])/a^(5/2))/(4*c^4)
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.18, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {899, 114, 27, 168, 25, 168, 27, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {d (2 b c-3 a d)}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2 (b c-a d)}-\frac {\int -\frac {\left (\frac {5 b d (2 b c-3 a d)}{x}+6 (b c-a d) (b c+2 a d)\right ) x}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2}d\frac {1}{x}}{2 c (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\left (\frac {5 b d (2 b c-3 a d)}{x}+6 (b c-a d) (b c+2 a d)\right ) x}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2}d\frac {1}{x}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\frac {d \left (12 a^2 d^2-21 a b c d+4 b^2 c^2\right )}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) (b c-a d)}-\frac {\int -\frac {3 \left (4 (b c+2 a d) (b c-a d)^2+\frac {b d \left (4 b^2 c^2-21 a b d c+12 a^2 d^2\right )}{x}\right ) x}{2 \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \int \frac {\left (4 (b c+2 a d) (b c-a d)^2+\frac {b d \left (4 b^2 c^2-21 a b d c+12 a^2 d^2\right )}{x}\right ) x}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-21 a b c d+4 b^2 c^2\right )}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 \int \frac {\left (4 (b c+2 a d) (b c-a d)^3+\frac {b d (2 b c-a d) \left (2 b^2 c^2-a b d c+4 a^2 d^2\right )}{x}\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a (b c-a d)}+\frac {2 b (2 b c-a d) \left (4 a^2 d^2-a b c d+2 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}\right )}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-21 a b c d+4 b^2 c^2\right )}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {\left (4 (b c+2 a d) (b c-a d)^3+\frac {b d (2 b c-a d) \left (2 b^2 c^2-a b d c+4 a^2 d^2\right )}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a (b c-a d)}+\frac {2 b (2 b c-a d) \left (4 a^2 d^2-a b c d+2 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}\right )}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-21 a b c d+4 b^2 c^2\right )}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {a^2 d^3 \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right ) \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}+\frac {4 (b c-a d)^3 (2 a d+b c) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}}{a (b c-a d)}+\frac {2 b (2 b c-a d) \left (4 a^2 d^2-a b c d+2 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}\right )}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-21 a b c d+4 b^2 c^2\right )}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {2 a^2 d^3 \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right ) \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}+\frac {8 (b c-a d)^3 (2 a d+b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}}{a (b c-a d)}+\frac {2 b (2 b c-a d) \left (4 a^2 d^2-a b c d+2 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}\right )}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-21 a b c d+4 b^2 c^2\right )}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {8 (2 a d+b c) (b c-a d)^3 \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}+\frac {2 a^2 d^{5/2} \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}}{a (b c-a d)}+\frac {2 b (2 b c-a d) \left (4 a^2 d^2-a b c d+2 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}\right )}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-21 a b c d+4 b^2 c^2\right )}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {2 a^2 d^{5/2} \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}-\frac {8 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (b c-a d)^3 (2 a d+b c)}{\sqrt {a} c}}{a (b c-a d)}+\frac {2 b (2 b c-a d) \left (4 a^2 d^2-a b c d+2 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}\right )}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-21 a b c d+4 b^2 c^2\right )}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}\)

Input:

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

Output:

x/(a*c*Sqrt[a + b/x]*(c + d/x)^2) + ((d*(2*b*c - 3*a*d))/(c*(b*c - a*d)*Sq 
rt[a + b/x]*(c + d/x)^2) + ((d*(4*b^2*c^2 - 21*a*b*c*d + 12*a^2*d^2))/(c*( 
b*c - a*d)*Sqrt[a + b/x]*(c + d/x)) + (3*((2*b*(2*b*c - a*d)*(2*b^2*c^2 - 
a*b*c*d + 4*a^2*d^2))/(a*(b*c - a*d)*Sqrt[a + b/x]) + ((2*a^2*d^(5/2)*(21* 
b^2*c^2 - 24*a*b*c*d + 8*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c 
- a*d]])/(c*Sqrt[b*c - a*d]) - (8*(b*c - a*d)^3*(b*c + 2*a*d)*ArcTanh[Sqrt 
[a + b/x]/Sqrt[a]])/(Sqrt[a]*c))/(a*(b*c - a*d))))/(2*c*(b*c - a*d)))/(2*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. \(982\) vs. \(2(288)=576\).

Time = 0.69 (sec) , antiderivative size = 983, normalized size of antiderivative = 3.07

method	result	size
risch	\(\frac {a x +b}{a^{2} c^{3} \sqrt {\frac {a x +b}{x}}}+\frac {\left (-\frac {3 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) d}{a^{\frac {3}{2}} c^{4}}-\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^{3}}-\frac {2 b^{4} \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{a^{3} \left (a d -b c \right )^{3} \left (x +\frac {b}{a}\right )}-\frac {d^{4} \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}}}}{2 c^{5} \left (a d -b c \right )^{2} \left (x +\frac {d}{c}\right )^{2}}+\frac {5 a \,d^{4} \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}}}}{2 c^{4} \left (a d -b c \right )^{3} \left (x +\frac {d}{c}\right )}-\frac {17 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}}}\, b}{4 c^{3} \left (a d -b c \right )^{3} \left (x +\frac {d}{c}\right )}-\frac {7 a^{2} d^{5} \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 c^{5} \left (a d -b c \right )^{3} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}+\frac {19 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 ) b}{2 c^{4} \left (a d -b c \right )^{3} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}-\frac {63 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}}{8 c^{3} \left (a d -b c \right )^{3} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}+\frac {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 )}{2 c^{5} \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}}}\)	\(983\)
default	\(\text {Expression too large to display}\)	\(5158\)

Input:

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

Output:

1/a^2/c^3*(a*x+b)/((a*x+b)/x)^(1/2)+(-3/a^(3/2)/c^4*ln((1/2*b+a*x)/a^(1/2) 
+(a*x^2+b*x)^(1/2))*d-3/2/a^(5/2)/c^3*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^( 
1/2))*b-2/a^3*b^4/(a*d-b*c)^3/(x+b/a)*(a*(x+b/a)^2-b*(x+b/a))^(1/2)-1/2/c^ 
5*d^4/(a*d-b*c)^2/(x+1/c*d)^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)+5/2*a/c^4*d^4/(a*d-b*c)^3/(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)-17/4/c^3*d^3/(a*d-b*c)^3/(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)*b-7/2*a^2 
/c^5*d^5/(a*d-b*c)^3/((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))+19/2*a/c^4*d^4/(a*d-b*c)^3/((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-63/8/c^3*d^3/(a*d-b*c)^3/((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^2+1/2 
*a/c^5*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)))/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. 999 vs. \(2 (288) = 576\).

Time = 1.59 (sec) , antiderivative size = 4063, normalized size of antiderivative = 12.70 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1089 vs. \(2 (288) = 576\).

Time = 0.25 (sec) , antiderivative size = 1089, normalized size of antiderivative = 3.40 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

2*sqrt(a)*b^5/((a^3*b^3*c^3*sgn(x) - 3*a^4*b^2*c^2*d*sgn(x) + 3*a^5*b*c*d^ 
2*sgn(x) - a^6*d^3*sgn(x))*((sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b)) 
- 1/4*(63*a^(5/2)*b^2*c^2*d^3*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 72*a 
^(7/2)*b*c*d^4*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 24*a^(9/2)*d^5*arct 
an(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 6*sqrt(b*c*d - a*d^2)*b^4*c^4*log(abs( 
b)) - 6*sqrt(b*c*d - a*d^2)*a*b^3*c^3*d*log(abs(b)) - 18*sqrt(b*c*d - a*d^ 
2)*a^2*b^2*c^2*d^2*log(abs(b)) + 30*sqrt(b*c*d - a*d^2)*a^3*b*c*d^3*log(ab 
s(b)) - 12*sqrt(b*c*d - a*d^2)*a^4*d^4*log(abs(b)) + 8*sqrt(b*c*d - a*d^2) 
*b^4*c^4 + 17*sqrt(b*c*d - a*d^2)*a^3*b*c*d^3 - 10*sqrt(b*c*d - a*d^2)*a^4 
*d^4)*sgn(x)/(sqrt(b*c*d - a*d^2)*a^(5/2)*b^3*c^7 - 3*sqrt(b*c*d - a*d^2)* 
a^(7/2)*b^2*c^6*d + 3*sqrt(b*c*d - a*d^2)*a^(9/2)*b*c^5*d^2 - sqrt(b*c*d - 
 a*d^2)*a^(11/2)*c^4*d^3) - 3/4*(21*b^2*c^2*d^3 - 24*a*b*c*d^4 + 8*a^2*d^5 
)*arctan(-((sqrt(a)*x - sqrt(a*x^2 + b*x))*c + sqrt(a)*d)/sqrt(b*c*d - a*d 
^2))/((b^3*c^7*sgn(x) - 3*a*b^2*c^6*d*sgn(x) + 3*a^2*b*c^5*d^2*sgn(x) - a^ 
3*c^4*d^3*sgn(x))*sqrt(b*c*d - a*d^2)) - 1/4*(17*(sqrt(a)*x - sqrt(a*x^2 + 
 b*x))^3*b^2*c^3*d^3 - 48*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a*b*c^2*d^4 + 
24*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^2*c*d^5 + 11*(sqrt(a)*x - sqrt(a*x^ 
2 + b*x))^2*sqrt(a)*b^2*c^2*d^4 - 72*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^( 
3/2)*b*c*d^5 + 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^(5/2)*d^6 + 15*(sqrt 
(a)*x - sqrt(a*x^2 + b*x))*b^3*c^2*d^4 - 76*(sqrt(a)*x - sqrt(a*x^2 + b...
 

Mupad [B] (verification not implemented)

Time = 9.37 (sec) , antiderivative size = 8936, normalized size of antiderivative = 27.92 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((2*b^4)/(a^2*d - a*b*c) + (b*(a + b/x)*(12*a^4*d^4 + 12*b^4*c^4 + 24*a^2* 
b^2*c^2*d^2 - 40*a*b^3*c^3*d - 33*a^3*b*c*d^3))/(4*a*c^3*(a^2*d - a*b*c)*( 
a*d - b*c)) + (3*b*(a + b/x)^3*(4*a^3*d^5 - 4*b^3*c^3*d^2 + 4*a*b^2*c^2*d^ 
3 - 9*a^2*b*c*d^4))/(4*a*c^3*(a^2*d - a*b*c)*(a*d - b*c)^2) - (b*(a + b/x) 
^2*(24*a^4*d^5 + 24*b^4*c^4*d - 56*a*b^3*c^3*d^2 + 65*a^2*b^2*c^2*d^3 - 72 
*a^3*b*c*d^4))/(4*a*c^3*(a^2*d - a*b*c)*(a*d - b*c)^2))/((a + b/x)^(3/2)*( 
3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d) - (a + b/x)^(5/2)*(3*a*d^2 - 2*b*c*d) + d 
^2*(a + b/x)^(7/2) - (a + b/x)^(1/2)*(a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d)) 
+ (atan(((((a + b/x)^(1/2)*(18432*a^6*b^19*c^26*d^3 - 202752*a^7*b^18*c^25 
*d^4 + 903168*a^8*b^17*c^24*d^5 - 1751040*a^9*b^16*c^23*d^6 - 137088*a^10* 
b^15*c^22*d^7 + 6007680*a^11*b^14*c^21*d^8 + 1276416*a^12*b^13*c^20*d^9 - 
65382912*a^13*b^12*c^19*d^10 + 216610560*a^14*b^11*c^18*d^11 - 407418624*a 
^15*b^10*c^17*d^12 + 521961984*a^16*b^9*c^16*d^13 - 482904576*a^17*b^8*c^1 
5*d^14 + 328809600*a^18*b^7*c^14*d^15 - 164257920*a^19*b^6*c^13*d^16 + 588 
16512*a^20*b^5*c^12*d^17 - 14340096*a^21*b^4*c^11*d^18 + 2138112*a^22*b^3* 
c^10*d^19 - 147456*a^23*b^2*c^9*d^20) - (3*(d^5*(a*d - b*c)^7)^(1/2)*(8*a^ 
2*d^2 + 21*b^2*c^2 - 24*a*b*c*d)*(12288*a^8*b^19*c^30*d^2 - 172032*a^9*b^1 
8*c^29*d^3 + 1081344*a^10*b^17*c^28*d^4 - 3996672*a^11*b^16*c^27*d^5 + 944 
9472*a^12*b^15*c^26*d^6 - 14112768*a^13*b^14*c^25*d^7 + 10407936*a^14*b^13 
*c^24*d^8 + 6454272*a^15*b^12*c^23*d^9 - 30007296*a^16*b^11*c^22*d^10 +...
 

Reduce [B] (verification not implemented)

Time = 2.73 (sec) , antiderivative size = 4543, normalized size of antiderivative = 14.20 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(192*sqrt(d)*sqrt(a*x + b)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqr 
t(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt( 
a))*a**6*c**2*d**5*x**2 + 384*sqrt(d)*sqrt(a*x + b)*sqrt(a*d - b*c)*log(sq 
rt(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**6*c*d**6*x + 192*sqrt(d)*sqrt(a*x + b)*sqr 
t(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**6*d**7 - 648*sqrt(d)*sq 
rt(a*x + b)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqr 
t(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a**5*b*c**3 
*d**4*x**2 - 1296*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*b*c**2*d**5*x - 648*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*b*c*d**6 + 720*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**2*c**4 
*d**3*x**2 + 1440*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**2*c**3*d**4*x + 720*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 - ...