\(\int \frac {1}{(a+\frac {b}{x})^{5/2} (c+\frac {d}{x})^2} \, dx\) [44]

Optimal result

Integrand size = 21, antiderivative size = 287 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2} \, dx=\frac {b \left (5 b^2 c^2-6 a b c d+6 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {b (b c-2 a d) \left (5 b^2 c^2-a b c d+a^2 d^2\right )}{a^3 c^2 (b c-a d)^3 \sqrt {a+\frac {b}{x}}}+\frac {d (b c-2 a d)}{a c^2 (b c-a d) \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}-\frac {d^{7/2} (9 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 (b c-a d)^{7/2}}-\frac {(5 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2} c^3} \] Output:

1/3*b*(6*a^2*d^2-6*a*b*c*d+5*b^2*c^2)/a^2/c^2/(-a*d+b*c)^2/(a+b/x)^(3/2)+b 
*(-2*a*d+b*c)*(a^2*d^2-a*b*c*d+5*b^2*c^2)/a^3/c^2/(-a*d+b*c)^3/(a+b/x)^(1/ 
2)+d*(-2*a*d+b*c)/a/c^2/(-a*d+b*c)/(a+b/x)^(3/2)/(c+d/x)+x/a/c/(a+b/x)^(3/ 
2)/(c+d/x)-d^(7/2)*(-4*a*d+9*b*c)*arctan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+b*c)^ 
(1/2))/c^3/(-a*d+b*c)^(7/2)-(4*a*d+5*b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/a 
^(7/2)/c^3
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.71 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x \left (-15 b^5 c^3 (d+c x)+3 a^5 d^3 x^2 (2 d+c x)+a b^4 c^2 \left (33 d^2+13 c d x-20 c^2 x^2\right )-3 a^4 b d^2 x \left (-4 d^2+c d x+3 c^2 x^2\right )+a^2 b^3 c \left (-9 d^3+35 c d^2 x+41 c^2 d x^2-3 c^3 x^3\right )+3 a^3 b^2 d \left (2 d^3-5 c d^2 x-3 c^2 d x^2+3 c^3 x^3\right )\right )}{a^3 (-b c+a d)^3 (b+a x)^2 (d+c x)}+\frac {3 d^{7/2} (-9 b c+4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{7/2}}-\frac {3 (5 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}}}{3 c^3} \] Input:

Integrate[1/((a + b/x)^(5/2)*(c + d/x)^2),x]
 

Output:

((c*Sqrt[a + b/x]*x*(-15*b^5*c^3*(d + c*x) + 3*a^5*d^3*x^2*(2*d + c*x) + a 
*b^4*c^2*(33*d^2 + 13*c*d*x - 20*c^2*x^2) - 3*a^4*b*d^2*x*(-4*d^2 + c*d*x 
+ 3*c^2*x^2) + a^2*b^3*c*(-9*d^3 + 35*c*d^2*x + 41*c^2*d*x^2 - 3*c^3*x^3) 
+ 3*a^3*b^2*d*(2*d^3 - 5*c*d^2*x - 3*c^2*d*x^2 + 3*c^3*x^3)))/(a^3*(-(b*c) 
 + a*d)^3*(b + a*x)^2*(d + c*x)) + (3*d^(7/2)*(-9*b*c + 4*a*d)*ArcTan[(Sqr 
t[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(7/2) - (3*(5*b*c + 4*a* 
d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(7/2))/(3*c^3)
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.22, 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, 169, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {\frac {2 \int \frac {3 \left ((5 b c+4 a d) (b c-a d)^2+\frac {b d \left (5 b^2 c^2-6 a b d c+6 a^2 d^2\right )}{x}\right ) x}{2 \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{3 a (b c-a d)}+\frac {2 b \left (6 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{c (b c-a d)}+\frac {2 d (b c-2 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {\left ((5 b c+4 a d) (b c-a d)^2+\frac {b d \left (5 b^2 c^2-6 a b d c+6 a^2 d^2\right )}{x}\right ) x}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a (b c-a d)}+\frac {2 b \left (6 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{c (b c-a d)}+\frac {2 d (b c-2 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {\frac {\frac {2 \int \frac {\left ((5 b c+4 a d) (b c-a d)^3+\frac {b d (b c-2 a d) \left (5 b^2 c^2-a b d c+a^2 d^2\right )}{x}\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a (b c-a d)}+\frac {2 b (b c-2 a d) \left (a^2 d^2-a b c d+5 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (6 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{c (b c-a d)}+\frac {2 d (b c-2 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\left ((5 b c+4 a d) (b c-a d)^3+\frac {b d (b c-2 a d) \left (5 b^2 c^2-a b d c+a^2 d^2\right )}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a (b c-a d)}+\frac {2 b (b c-2 a d) \left (a^2 d^2-a b c d+5 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (6 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{c (b c-a d)}+\frac {2 d (b c-2 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {\frac {\frac {(b c-a d)^3 (4 a d+5 b c) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}-\frac {a^3 d^4 (9 b c-4 a d) \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}}{a (b c-a d)}+\frac {2 b (b c-2 a d) \left (a^2 d^2-a b c d+5 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (6 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{c (b c-a d)}+\frac {2 d (b c-2 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {\frac {\frac {2 (b c-a d)^3 (4 a d+5 b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {2 a^3 d^4 (9 b c-4 a d) \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}}{a (b c-a d)}+\frac {2 b (b c-2 a d) \left (a^2 d^2-a b c d+5 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (6 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{c (b c-a d)}+\frac {2 d (b c-2 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\frac {\frac {2 (b c-a d)^3 (4 a d+5 b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {2 a^3 d^{7/2} (9 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}}{a (b c-a d)}+\frac {2 b (b c-2 a d) \left (a^2 d^2-a b c d+5 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (6 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{c (b c-a d)}+\frac {2 d (b c-2 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {2 b \left (6 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}+\frac {\frac {-\frac {2 a^3 d^{7/2} (9 b c-4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (b c-a d)^3 (4 a d+5 b c)}{\sqrt {a} c}}{a (b c-a d)}+\frac {2 b (b c-2 a d) \left (a^2 d^2-a b c d+5 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}}{c (b c-a d)}+\frac {2 d (b c-2 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}\)

Input:

Int[1/((a + b/x)^(5/2)*(c + d/x)^2),x]
 

Output:

x/(a*c*(a + b/x)^(3/2)*(c + d/x)) + ((2*d*(b*c - 2*a*d))/(c*(b*c - a*d)*(a 
 + b/x)^(3/2)*(c + d/x)) + ((2*b*(5*b^2*c^2 - 6*a*b*c*d + 6*a^2*d^2))/(3*a 
*(b*c - a*d)*(a + b/x)^(3/2)) + ((2*b*(b*c - 2*a*d)*(5*b^2*c^2 - a*b*c*d + 
 a^2*d^2))/(a*(b*c - a*d)*Sqrt[a + b/x]) + ((-2*a^3*d^(7/2)*(9*b*c - 4*a*d 
)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(c*Sqrt[b*c - a*d]) - ( 
2*(b*c - a*d)^3*(5*b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(Sqrt[a]*c 
))/(a*(b*c - a*d)))/(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. \(664\) vs. \(2(261)=522\).

Time = 0.69 (sec) , antiderivative size = 665, normalized size of antiderivative = 2.32

method	result	size
risch	\(\frac {a x +b}{a^{3} c^{2} \sqrt {\frac {a x +b}{x}}}-\frac {\left (\frac {\left (4 a d +5 b c \right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{c \sqrt {a}}+\frac {2 c^{2} b^{5} \left (\frac {2 \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 b \left (x +\frac {b}{a}\right )^{2}}+\frac {4 a \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 b^{2} \left (x +\frac {b}{a}\right )}\right )}{\left (a d -b c \right )^{2} a^{2}}+\frac {2 a^{3} d^{5} \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} \left (a d -b c \right )^{2}}-\frac {4 c^{2} b^{3} \left (5 a d -3 b c \right ) \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{\left (a d -b c \right )^{3} a \left (x +\frac {b}{a}\right )}+\frac {2 a^{3} d^{4} \left (3 a d -5 b c \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} \left (a d -b c \right )^{3} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right ) \sqrt {x \left (a x +b \right )}}{2 c^{2} a^{3} x \sqrt {\frac {a x +b}{x}}}\)	\(665\)
default	\(\text {Expression too large to display}\)	\(4644\)

Input:

int(1/(a+b/x)^(5/2)/(c+1/x*d)^2,x,method=_RETURNVERBOSE)
 

Output:

1/a^3/c^2*(a*x+b)/((a*x+b)/x)^(1/2)-1/2/c^2/a^3*((4*a*d+5*b*c)/c*ln((1/2*b 
+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))/a^(1/2)+2*c^2*b^5/(a*d-b*c)^2/a^2*(2/3/b/ 
(x+b/a)^2*(a*(x+b/a)^2-b*(x+b/a))^(1/2)+4/3*a/b^2/(x+b/a)*(a*(x+b/a)^2-b*( 
x+b/a))^(1/2))+2/c^3*a^3*d^5/(a*d-b*c)^2*(-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)))-4*c^2*b^3*(5*a*d-3*b*c)/(a*d-b*c)^3/ 
a/(x+b/a)*(a*(x+b/a)^2-b*(x+b/a))^(1/2)+2/c^2*a^3*d^4*(3*a*d-5*b*c)/(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)))/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. 947 vs. \(2 (261) = 522\).

Time = 1.44 (sec) , antiderivative size = 3856, normalized size of antiderivative = 13.44 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b/x)^(5/2)/(c+d/x)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2} \, dx=\int \frac {x^{2}}{\left (a + \frac {b}{x}\right )^{\frac {5}{2}} \left (c x + d\right )^{2}}\, dx \] Input:

integrate(1/(a+b/x)**(5/2)/(c+d/x)**2,x)
 

Output:

Integral(x**2/((a + b/x)**(5/2)*(c*x + d)**2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} {\left (c + \frac {d}{x}\right )}^{2}} \,d x } \] Input:

integrate(1/(a+b/x)^(5/2)/(c+d/x)^2,x, algorithm="maxima")
 

Output:

integrate(1/((a + b/x)^(5/2)*(c + d/x)^2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 924 vs. \(2 (261) = 522\).

Time = 0.23 (sec) , antiderivative size = 924, normalized size of antiderivative = 3.22 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(1/(a+b/x)^(5/2)/(c+d/x)^2,x, algorithm="giac")
 

Output:

1/6*(54*a^(7/2)*b*c*d^4*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 24*a^(9/2) 
*d^5*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 15*sqrt(b*c*d - a*d^2)*b^4*c^ 
4*log(abs(b)) + 33*sqrt(b*c*d - a*d^2)*a*b^3*c^3*d*log(abs(b)) - 9*sqrt(b* 
c*d - a*d^2)*a^2*b^2*c^2*d^2*log(abs(b)) - 21*sqrt(b*c*d - a*d^2)*a^3*b*c* 
d^3*log(abs(b)) + 12*sqrt(b*c*d - a*d^2)*a^4*d^4*log(abs(b)) - 28*sqrt(b*c 
*d - a*d^2)*b^4*c^4 + 52*sqrt(b*c*d - a*d^2)*a*b^3*c^3*d + 6*sqrt(b*c*d - 
a*d^2)*a^4*d^4)*sgn(x)/(sqrt(b*c*d - a*d^2)*a^(7/2)*b^3*c^6 - 3*sqrt(b*c*d 
 - a*d^2)*a^(9/2)*b^2*c^5*d + 3*sqrt(b*c*d - a*d^2)*a^(11/2)*b*c^4*d^2 - s 
qrt(b*c*d - a*d^2)*a^(13/2)*c^3*d^3) + (9*b*c*d^4 - 4*a*d^5)*arctan(-((sqr 
t(a)*x - sqrt(a*x^2 + b*x))*c + sqrt(a)*d)/sqrt(b*c*d - a*d^2))/((b^3*c^6* 
sgn(x) - 3*a*b^2*c^5*d*sgn(x) + 3*a^2*b*c^4*d^2*sgn(x) - a^3*c^3*d^3*sgn(x 
))*sqrt(b*c*d - a*d^2)) + ((sqrt(a)*x - sqrt(a*x^2 + b*x))*b*c*d^4 - 2*(sq 
rt(a)*x - sqrt(a*x^2 + b*x))*a*d^5 - sqrt(a)*b*d^5)/((b^3*c^6*sgn(x) - 3*a 
*b^2*c^5*d*sgn(x) + 3*a^2*b*c^4*d^2*sgn(x) - a^3*c^3*d^3*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)) + 2/3*(9*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^5*c - 15*(sqrt(a)*x 
 - sqrt(a*x^2 + b*x))^2*a^2*b^4*d + 15*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqr 
t(a)*b^6*c - 27*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^(3/2)*b^5*d + 7*b^7*c - 
13*a*b^6*d)/((a^(7/2)*b^3*c^3*sgn(x) - 3*a^(9/2)*b^2*c^2*d*sgn(x) + 3*a^(1 
1/2)*b*c*d^2*sgn(x) - a^(13/2)*d^3*sgn(x))*((sqrt(a)*x - sqrt(a*x^2 + b...
 

Mupad [B] (verification not implemented)

Time = 8.38 (sec) , antiderivative size = 5789, normalized size of antiderivative = 20.17 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2} \, dx=\text {Too large to display} \] Input:

int(1/((a + b/x)^(5/2)*(c + d/x)^2),x)
 

Output:

((2*b^3)/(3*(a^2*d - a*b*c)) + (10*b^3*(a + b/x)*(2*a*d - b*c))/(3*(a^2*d 
- a*b*c)^2) - (b*(a + b/x)^2*(6*a^4*d^4 + 15*b^4*c^4 + 64*a^2*b^2*c^2*d^2 
- 58*a*b^3*c^3*d - 12*a^3*b*c*d^3))/(3*c^2*(a^2*d - a*b*c)^3) + (b*(a + b/ 
x)^3*(2*a*d - b*c)*(a^2*d^3 + 5*b^2*c^2*d - a*b*c*d^2))/(c^2*(a^2*d - a*b* 
c)^3))/(d*(a + b/x)^(7/2) + (a + b/x)^(3/2)*(a^2*d - a*b*c) - (a + b/x)^(5 
/2)*(2*a*d - b*c)) + (atan((a^15*b^19*c^19*(a + b/x)^(1/2)*125i + a^17*b^1 
7*c^17*d^2*(a + b/x)^(1/2)*10440i - a^18*b^16*c^16*d^3*(a + b/x)^(1/2)*377 
76i + a^19*b^15*c^15*d^4*(a + b/x)^(1/2)*87276i - a^20*b^14*c^14*d^5*(a + 
b/x)^(1/2)*126720i + a^21*b^13*c^13*d^6*(a + b/x)^(1/2)*91560i + a^22*b^12 
*c^12*d^7*(a + b/x)^(1/2)*40965i - a^23*b^11*c^11*d^8*(a + b/x)^(1/2)*1845 
63i + a^24*b^10*c^10*d^9*(a + b/x)^(1/2)*212608i - a^25*b^9*c^9*d^10*(a + 
b/x)^(1/2)*107740i - a^26*b^8*c^8*d^11*(a + b/x)^(1/2)*19530i + a^27*b^7*c 
^7*d^12*(a + b/x)^(1/2)*71070i - a^28*b^6*c^6*d^13*(a + b/x)^(1/2)*52836i 
+ a^29*b^5*c^5*d^14*(a + b/x)^(1/2)*20916i - a^30*b^4*c^4*d^15*(a + b/x)^( 
1/2)*4515i + a^31*b^3*c^3*d^16*(a + b/x)^(1/2)*420i - a^16*b^18*c^18*d*(a 
+ b/x)^(1/2)*1700i)/(a^7*(a^7)^(1/2)*(a^7*(a^7*(212608*b^10*c^10*d^9 - 107 
740*a*b^9*c^9*d^10 - 19530*a^2*b^8*c^8*d^11 + 71070*a^3*b^7*c^7*d^12 - 528 
36*a^4*b^6*c^6*d^13 + 20916*a^5*b^5*c^5*d^14 - 4515*a^6*b^4*c^4*d^15 + 420 
*a^7*b^3*c^3*d^16) + 10440*b^17*c^17*d^2 - 37776*a*b^16*c^16*d^3 + 87276*a 
^2*b^15*c^15*d^4 - 126720*a^3*b^14*c^14*d^5 + 91560*a^4*b^13*c^13*d^6 +...
 

Reduce [B] (verification not implemented)

Time = 1.75 (sec) , antiderivative size = 4057, normalized size of antiderivative = 14.14 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2} \, dx =\text {Too large to display} \] Input:

int(1/(a+b/x)^(5/2)/(c+d/x)^2,x)
 

Output:

(96*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**7*c*d**5*x**2 + 96*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) + s 
qrt(x)*sqrt(c)*sqrt(a))*a**7*d**6*x - 192*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**6*b*c**2*d**4*x**2 - 96*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**6*b* 
c*d**5*x + 96*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**6*b*d**6 - 54*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**5*b**2*c**3*d**3*x**2 - 246*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**2*c**2* 
d**4*x - 192*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**2*c*d**5 - 54*sqrt(d)*sqrt(a*x + b)*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*...