\(\int \frac {1}{\sqrt {a+\frac {b}{x}} (c+\frac {d}{x})^3} \, dx\) [31]

Optimal result

Integrand size = 21, antiderivative size = 250 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3} \, dx=\frac {d (2 b c-3 a d) \sqrt {a+\frac {b}{x}}}{2 a c^2 (b c-a d) \left (c+\frac {d}{x}\right )^2}+\frac {d (b c-4 a d) (4 b c-3 a d) \sqrt {a+\frac {b}{x}}}{4 a c^3 (b c-a d)^2 \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{a c \left (c+\frac {d}{x}\right )^2}-\frac {d^{3/2} \left (35 b^2 c^2-56 a b c d+24 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 (b c-a d)^{5/2}}-\frac {(b c+6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2} c^4} \] Output:

1/2*d*(-3*a*d+2*b*c)*(a+b/x)^(1/2)/a/c^2/(-a*d+b*c)/(c+d/x)^2+1/4*d*(-4*a* 
d+b*c)*(-3*a*d+4*b*c)*(a+b/x)^(1/2)/a/c^3/(-a*d+b*c)^2/(c+d/x)+(a+b/x)^(1/ 
2)*x/a/c/(c+d/x)^2-1/4*d^(3/2)*(24*a^2*d^2-56*a*b*c*d+35*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)^(5/2)-(6*a*d+b*c)*ar 
ctanh((a+b/x)^(1/2)/a^(1/2))/a^(3/2)/c^4
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.86 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x \left (4 b^2 c^2 (d+c x)^2+2 a^2 d^2 \left (6 d^2+9 c d x+2 c^2 x^2\right )-a b c d \left (19 d^2+29 c d x+8 c^2 x^2\right )\right )}{a (b c-a d)^2 (d+c x)^2}-\frac {d^{3/2} \left (35 b^2 c^2-56 a b c d+24 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{5/2}}-\frac {4 (b c+6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}}{4 c^4} \] Input:

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

Output:

((c*Sqrt[a + b/x]*x*(4*b^2*c^2*(d + c*x)^2 + 2*a^2*d^2*(6*d^2 + 9*c*d*x + 
2*c^2*x^2) - a*b*c*d*(19*d^2 + 29*c*d*x + 8*c^2*x^2)))/(a*(b*c - a*d)^2*(d 
 + c*x)^2) - (d^(3/2)*(35*b^2*c^2 - 56*a*b*c*d + 24*a^2*d^2)*ArcTan[(Sqrt[ 
d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(5/2) - (4*(b*c + 6*a*d)*A 
rcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2))/(4*c^4)
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.18, 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, 168, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\frac {d \sqrt {a+\frac {b}{x}} (b c-4 a d) (4 b c-3 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}-\frac {\int -\frac {\left (4 (b c+6 a d) (b c-a d)^2+\frac {b d (b c-4 a d) (4 b c-3 a d)}{x}\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c (b c-a d)}}{2 c (b c-a d)}+\frac {d \sqrt {a+\frac {b}{x}} (2 b c-3 a d)}{c \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (4 (b c+6 a d) (b c-a d)^2+\frac {b d (b c-4 a d) (4 b c-3 a d)}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{2 c (b c-a d)}+\frac {d \sqrt {a+\frac {b}{x}} (b c-4 a d) (4 b c-3 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d \sqrt {a+\frac {b}{x}} (2 b c-3 a d)}{c \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {\frac {4 (b c-a d)^2 (6 a d+b c) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}-\frac {a d^2 \left (24 a^2 d^2-56 a b c d+35 b^2 c^2\right ) \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}}{2 c (b c-a d)}+\frac {d \sqrt {a+\frac {b}{x}} (b c-4 a d) (4 b c-3 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d \sqrt {a+\frac {b}{x}} (2 b c-3 a d)}{c \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {\frac {8 (b c-a d)^2 (6 a d+b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {2 a d^2 \left (24 a^2 d^2-56 a b c d+35 b^2 c^2\right ) \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}}{2 c (b c-a d)}+\frac {d \sqrt {a+\frac {b}{x}} (b c-4 a d) (4 b c-3 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d \sqrt {a+\frac {b}{x}} (2 b c-3 a d)}{c \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\frac {8 (b c-a d)^2 (6 a d+b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {2 a d^{3/2} \left (24 a^2 d^2-56 a b c d+35 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}}{2 c (b c-a d)}+\frac {d \sqrt {a+\frac {b}{x}} (b c-4 a d) (4 b c-3 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d \sqrt {a+\frac {b}{x}} (2 b c-3 a d)}{c \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {-\frac {2 a d^{3/2} \left (24 a^2 d^2-56 a b c d+35 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}-\frac {8 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (6 a d+b c) (b c-a d)^2}{\sqrt {a} c}}{2 c (b c-a d)}+\frac {d \sqrt {a+\frac {b}{x}} (b c-4 a d) (4 b c-3 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d \sqrt {a+\frac {b}{x}} (2 b c-3 a d)}{c \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x \sqrt {a+\frac {b}{x}}}{a c \left (c+\frac {d}{x}\right )^2}\)

Input:

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

Output:

(Sqrt[a + b/x]*x)/(a*c*(c + d/x)^2) + ((d*(2*b*c - 3*a*d)*Sqrt[a + b/x])/( 
c*(b*c - a*d)*(c + d/x)^2) + ((d*(b*c - 4*a*d)*(4*b*c - 3*a*d)*Sqrt[a + b/ 
x])/(c*(b*c - a*d)*(c + d/x)) + ((-2*a*d^(3/2)*(35*b^2*c^2 - 56*a*b*c*d + 
24*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)^2*(b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(S 
qrt[a]*c))/(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 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. \(952\) vs. \(2(222)=444\).

Time = 0.65 (sec) , antiderivative size = 953, normalized size of antiderivative = 3.81

method	result	size
risch	\(\frac {a x +b}{a \,c^{3} \sqrt {\frac {a x +b}{x}}}-\frac {\left (\frac {\left (6 a d +b c \right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{c \sqrt {a}}+\frac {12 a \,d^{2} \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 {8 a \,d^{3} \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}}-\frac {2 a \,d^{4} \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}}}}{2 \left (a d -b c \right ) d \left (x +\frac {d}{c}\right )^{2}}+\frac {3 \left (2 a d -b c \right ) c \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 )}{4 \left (a d -b c \right ) d}+\frac {a \,c^{2} \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^{4}}\right ) \sqrt {x \left (a x +b \right )}}{2 c^{3} a x \sqrt {\frac {a x +b}{x}}}\)	\(953\)
default	\(\text {Expression too large to display}\)	\(2269\)

Input:

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

Output:

1/a/c^3*(a*x+b)/((a*x+b)/x)^(1/2)-1/2/c^3/a*((6*a*d+b*c)/c*ln((1/2*b+a*x)/ 
a^(1/2)+(a*x^2+b*x)^(1/2))/a^(1/2)+12*a*d^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))+8*a* 
d^3/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)))-2*a*d^4/c^4*(-1/2/(a*d-b*c)/d*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)+3/4*(2*a*d-b*c)*c/(a*d-b*c)/d*(-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)))+1/2*a/(a 
*d-b*c)/d*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. 553 vs. \(2 (222) = 444\).

Time = 0.59 (sec) , antiderivative size = 2277, normalized size of antiderivative = 9.11 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/8*(4*(b^3*c^3*d^2 + 4*a*b^2*c^2*d^3 - 11*a^2*b*c*d^4 + 6*a^3*d^5 + (b^3 
*c^5 + 4*a*b^2*c^4*d - 11*a^2*b*c^3*d^2 + 6*a^3*c^2*d^3)*x^2 + 2*(b^3*c^4* 
d + 4*a*b^2*c^3*d^2 - 11*a^2*b*c^2*d^3 + 6*a^3*c*d^4)*x)*sqrt(a)*log(2*a*x 
 - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (35*a^2*b^2*c^2*d^3 - 56*a^3*b*c*d 
^4 + 24*a^4*d^5 + (35*a^2*b^2*c^4*d - 56*a^3*b*c^3*d^2 + 24*a^4*c^2*d^3)*x 
^2 + 2*(35*a^2*b^2*c^3*d^2 - 56*a^3*b*c^2*d^3 + 24*a^4*c*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*(4*(a*b^2*c^5 - 2*a^2*b*c^4*d + a^3 
*c^3*d^2)*x^3 + (8*a*b^2*c^4*d - 29*a^2*b*c^3*d^2 + 18*a^3*c^2*d^3)*x^2 + 
(4*a*b^2*c^3*d^2 - 19*a^2*b*c^2*d^3 + 12*a^3*c*d^4)*x)*sqrt((a*x + b)/x))/ 
(a^2*b^2*c^6*d^2 - 2*a^3*b*c^5*d^3 + a^4*c^4*d^4 + (a^2*b^2*c^8 - 2*a^3*b* 
c^7*d + a^4*c^6*d^2)*x^2 + 2*(a^2*b^2*c^7*d - 2*a^3*b*c^6*d^2 + a^4*c^5*d^ 
3)*x), 1/8*(8*(b^3*c^3*d^2 + 4*a*b^2*c^2*d^3 - 11*a^2*b*c*d^4 + 6*a^3*d^5 
+ (b^3*c^5 + 4*a*b^2*c^4*d - 11*a^2*b*c^3*d^2 + 6*a^3*c^2*d^3)*x^2 + 2*(b^ 
3*c^4*d + 4*a*b^2*c^3*d^2 - 11*a^2*b*c^2*d^3 + 6*a^3*c*d^4)*x)*sqrt(-a)*ar 
ctan(sqrt(-a)*x*sqrt((a*x + b)/x)/(a*x + b)) + (35*a^2*b^2*c^2*d^3 - 56*a^ 
3*b*c*d^4 + 24*a^4*d^5 + (35*a^2*b^2*c^4*d - 56*a^3*b*c^3*d^2 + 24*a^4*c^2 
*d^3)*x^2 + 2*(35*a^2*b^2*c^3*d^2 - 56*a^3*b*c^2*d^3 + 24*a^4*c*d^4)*x)*sq 
rt(-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*(4*(a*b^2*c^5 - 2*a^2*b*c...
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 890 vs. \(2 (222) = 444\).

Time = 0.19 (sec) , antiderivative size = 890, normalized size of antiderivative = 3.56 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*(35*a^(3/2)*b^2*c^2*d^2*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 56*a^( 
5/2)*b*c*d^3*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 24*a^(7/2)*d^4*arctan 
(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 2*sqrt(b*c*d - a*d^2)*b^3*c^3*log(abs(b) 
) - 8*sqrt(b*c*d - a*d^2)*a*b^2*c^2*d*log(abs(b)) + 22*sqrt(b*c*d - a*d^2) 
*a^2*b*c*d^2*log(abs(b)) - 12*sqrt(b*c*d - a*d^2)*a^3*d^3*log(abs(b)) + 13 
*sqrt(b*c*d - a*d^2)*a^2*b*c*d^2 - 10*sqrt(b*c*d - a*d^2)*a^3*d^3)*sgn(x)/ 
(sqrt(b*c*d - a*d^2)*a^(3/2)*b^2*c^6 - 2*sqrt(b*c*d - a*d^2)*a^(5/2)*b*c^5 
*d + sqrt(b*c*d - a*d^2)*a^(7/2)*c^4*d^2) + 1/4*(35*b^2*c^2*d^2 - 56*a*b*c 
*d^3 + 24*a^2*d^4)*arctan(-((sqrt(a)*x - sqrt(a*x^2 + b*x))*c + sqrt(a)*d) 
/sqrt(b*c*d - a*d^2))/((b^2*c^6*sgn(x) - 2*a*b*c^5*d*sgn(x) + a^2*c^4*d^2* 
sgn(x))*sqrt(b*c*d - a*d^2)) + 1/4*(13*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*b 
^2*c^3*d^2 - 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a*b*c^2*d^3 + 24*(sqrt(a 
)*x - sqrt(a*x^2 + b*x))^3*a^2*c*d^4 + 7*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2 
*sqrt(a)*b^2*c^2*d^3 - 56*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^(3/2)*b*c*d^ 
4 + 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^(5/2)*d^5 + 11*(sqrt(a)*x - sqr 
t(a*x^2 + b*x))*b^3*c^2*d^3 - 60*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a*b^2*c*d 
^4 + 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^2*b*d^5 - 13*sqrt(a)*b^3*c*d^4 + 
 10*a^(3/2)*b^2*d^5)/((b^2*c^6*sgn(x) - 2*a*b*c^5*d*sgn(x) + a^2*c^4*d^2*s 
gn(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)^2) + sqrt(a*x^2 + b*x)/(a*c^3*sgn(x)) + 1/2*(b*c...
 

Mupad [B] (verification not implemented)

Time = 5.21 (sec) , antiderivative size = 2890, normalized size of antiderivative = 11.56 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(log((d^3*(a*d - b*c)^5)^(1/2)*(a + b/x)^(1/2) - a^3*d^4 + b^3*c^3*d - 3*a 
*b^2*c^2*d^2 + 3*a^2*b*c*d^3)*(d^3*(a*d - b*c)^5)^(1/2)*(3*a^2*d^2 + (35*b 
^2*c^2)/8 - 7*a*b*c*d))/(b^5*c^9 - a^5*c^4*d^5 + 5*a^4*b*c^5*d^4 + 10*a^2* 
b^3*c^7*d^2 - 10*a^3*b^2*c^6*d^3 - 5*a*b^4*c^8*d) - ((b*(a + b/x)^(5/2)*(1 
2*a^2*d^4 + 4*b^2*c^2*d^2 - 19*a*b*c*d^3))/(4*a*c^3*(a*d - b*c)^2) - ((a + 
 b/x)^(1/2)*(4*b^4*c^3 - 12*a^3*b*d^3 + 25*a^2*b^2*c*d^2 - 12*a*b^3*c^2*d) 
)/(4*a*c^3*(a*d - b*c)) + (d*(a + b/x)^(3/2)*(8*b^4*c^3 - 24*a^3*b*d^3 + 5 
6*a^2*b^2*c*d^2 - 37*a*b^3*c^2*d))/(4*c^3*(a^2*d - a*b*c)*(a*d - b*c)))/(( 
a + b/x)^2*(3*a*d^2 - 2*b*c*d) - (a + b/x)*(3*a^2*d^2 + b^2*c^2 - 4*a*b*c* 
d) - d^2*(a + b/x)^3 + a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d) - (log((d^3*(a*d 
 - b*c)^5)^(1/2)*(a + b/x)^(1/2) + a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 
 3*a^2*b*c*d^3)*(d^3*(a*d - b*c)^5)^(1/2)*(24*a^2*d^2 + 35*b^2*c^2 - 56*a* 
b*c*d))/(8*(b^5*c^9 - a^5*c^4*d^5 + 5*a^4*b*c^5*d^4 + 10*a^2*b^3*c^7*d^2 - 
 10*a^3*b^2*c^6*d^3 - 5*a*b^4*c^8*d)) - (atan((((((a + b/x)^(1/2)*(1152*a^ 
6*b^2*d^9 + 16*b^8*c^6*d^3 + 128*a*b^7*c^5*d^4 - 4800*a^5*b^3*c*d^8 + 1129 
*a^2*b^6*c^4*d^5 - 5136*a^3*b^5*c^3*d^6 + 7520*a^4*b^4*c^2*d^7))/(8*(a^2*b 
^4*c^10 + a^6*c^6*d^4 - 4*a^3*b^3*c^9*d - 4*a^5*b*c^7*d^3 + 6*a^4*b^2*c^8* 
d^2)) - (((4*a*b^8*c^13*d^2 + 4*a^2*b^7*c^12*d^3 - 45*a^3*b^6*c^11*d^4 + 7 
4*a^4*b^5*c^10*d^5 - 49*a^5*b^4*c^9*d^6 + 12*a^6*b^3*c^8*d^7)/(a^2*b^4*c^1 
3 + a^6*c^9*d^4 - 4*a^3*b^3*c^12*d - 4*a^5*b*c^10*d^3 + 6*a^4*b^2*c^11*...
 

Reduce [B] (verification not implemented)

Time = 1.28 (sec) , antiderivative size = 3752, normalized size of antiderivative = 15.01 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(96*sqrt(d)*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*c**2*d 
**4*x**2 + 192*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**5*c*d**5*x + 96*sqrt(d)*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**5*d**6 - 272*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sq 
rt(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**3*d**3*x**2 - 544*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**4*b*c**2*d**4*x - 272*sqrt(d)*sqrt(a*d - b*c)*log(sqr 
t(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**5 + 252*sqrt(d)*sqrt(a*d - b*c)*lo 
g(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**4*d**2*x**2 + 504*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))*a**3*b**2*c**3*d**3*x + 2 
52*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**3*b**2*c* 
*2*d**4 - 70*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2...