\(\int \frac {1}{(a+\frac {b}{x})^{5/2} (c+\frac {d}{x})^3} \, dx\) [45]

Optimal result

Integrand size = 21, antiderivative size = 409 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^3} \, dx=\frac {b \left (20 b^3 c^3-36 a b^2 c^2 d+87 a^2 b c d^2-36 a^3 d^3\right )}{12 a^2 c^3 (b c-a d)^3 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {b \left (20 b^4 c^4-56 a b^3 c^3 d+24 a^2 b^2 c^2 d^2-35 a^3 b c d^3+12 a^4 d^4\right )}{4 a^3 c^3 (b c-a d)^4 \sqrt {a+\frac {b}{x}}}+\frac {d (2 b c-3 a d)}{2 a c^2 (b c-a d) \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2}+\frac {d \left (4 b^2 c^2-23 a b c d+12 a^2 d^2\right )}{4 a c^3 (b c-a d)^2 \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2}-\frac {d^{7/2} \left (99 b^2 c^2-88 a b c d+24 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 (b c-a d)^{9/2}}-\frac {(5 b c+6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2} c^4} \] Output:

1/12*b*(-36*a^3*d^3+87*a^2*b*c*d^2-36*a*b^2*c^2*d+20*b^3*c^3)/a^2/c^3/(-a* 
d+b*c)^3/(a+b/x)^(3/2)+1/4*b*(12*a^4*d^4-35*a^3*b*c*d^3+24*a^2*b^2*c^2*d^2 
-56*a*b^3*c^3*d+20*b^4*c^4)/a^3/c^3/(-a*d+b*c)^4/(a+b/x)^(1/2)+1/2*d*(-3*a 
*d+2*b*c)/a/c^2/(-a*d+b*c)/(a+b/x)^(3/2)/(c+d/x)^2+1/4*d*(12*a^2*d^2-23*a* 
b*c*d+4*b^2*c^2)/a/c^3/(-a*d+b*c)^2/(a+b/x)^(3/2)/(c+d/x)+x/a/c/(a+b/x)^(3 
/2)/(c+d/x)^2-1/4*d^(7/2)*(24*a^2*d^2-88*a*b*c*d+99*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)^(9/2)-(6*a*d+5*b*c)*arcta 
nh((a+b/x)^(1/2)/a^(1/2))/a^(7/2)/c^4
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 2.53 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^3} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x \left (60 b^6 c^4 (d+c x)^2+8 a b^5 c^3 (d+c x)^2 (-21 d+10 c x)+6 a^6 d^4 x^2 \left (6 d^2+9 c d x+2 c^2 x^2\right )+4 a^2 b^4 c^2 (d+c x)^2 \left (18 d^2-56 c d x+3 c^2 x^2\right )+3 a^5 b d^3 x \left (24 d^3+c d^2 x-45 c^2 d x^2-16 c^3 x^3\right )+6 a^4 b^2 d^2 \left (6 d^4-26 c d^3 x-39 c^2 d^2 x^2+8 c^3 d x^3+12 c^4 x^4\right )-3 a^3 b^3 c d \left (35 d^4+5 c d^3 x-64 c^2 d^2 x^2-16 c^3 d x^3+16 c^4 x^4\right )\right )}{a^3 (b c-a d)^4 (b+a x)^2 (d+c x)^2}-\frac {3 d^{7/2} \left (99 b^2 c^2-88 a b c d+24 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{9/2}}-\frac {12 (5 b c+6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}}}{12 c^4} \] Input:

Integrate[1/((a + b/x)^(5/2)*(c + d/x)^3),x]
 

Output:

((c*Sqrt[a + b/x]*x*(60*b^6*c^4*(d + c*x)^2 + 8*a*b^5*c^3*(d + c*x)^2*(-21 
*d + 10*c*x) + 6*a^6*d^4*x^2*(6*d^2 + 9*c*d*x + 2*c^2*x^2) + 4*a^2*b^4*c^2 
*(d + c*x)^2*(18*d^2 - 56*c*d*x + 3*c^2*x^2) + 3*a^5*b*d^3*x*(24*d^3 + c*d 
^2*x - 45*c^2*d*x^2 - 16*c^3*x^3) + 6*a^4*b^2*d^2*(6*d^4 - 26*c*d^3*x - 39 
*c^2*d^2*x^2 + 8*c^3*d*x^3 + 12*c^4*x^4) - 3*a^3*b^3*c*d*(35*d^4 + 5*c*d^3 
*x - 64*c^2*d^2*x^2 - 16*c^3*d*x^3 + 16*c^4*x^4)))/(a^3*(b*c - a*d)^4*(b + 
 a*x)^2*(d + c*x)^2) - (3*d^(7/2)*(99*b^2*c^2 - 88*a*b*c*d + 24*a^2*d^2)*A 
rcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(9/2) - (12*(5 
*b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(7/2))/(12*c^4)
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.17, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {899, 114, 27, 168, 25, 168, 27, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\frac {d \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}-\frac {\int -\frac {\left (4 (5 b c+6 a d) (b c-a d)^2+\frac {5 b d \left (4 b^2 c^2-23 a b d c+12 a^2 d^2\right )}{x}\right ) x}{2 \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (4 (5 b c+6 a d) (b c-a d)^2+\frac {5 b d \left (4 b^2 c^2-23 a b d c+12 a^2 d^2\right )}{x}\right ) x}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {\frac {\frac {2 \int \frac {3 \left (4 (5 b c+6 a d) (b c-a d)^3+\frac {b d \left (20 b^3 c^3-36 a b^2 d c^2+87 a^2 b d^2 c-36 a^3 d^3\right )}{x}\right ) x}{2 \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{3 a (b c-a d)}+\frac {2 b \left (-36 a^3 d^3+87 a^2 b c d^2-36 a b^2 c^2 d+20 b^3 c^3\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\left (4 (5 b c+6 a d) (b c-a d)^3+\frac {b d \left (20 b^3 c^3-36 a b^2 d c^2+87 a^2 b d^2 c-36 a^3 d^3\right )}{x}\right ) x}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a (b c-a d)}+\frac {2 b \left (-36 a^3 d^3+87 a^2 b c d^2-36 a b^2 c^2 d+20 b^3 c^3\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {\frac {\frac {\frac {2 \int \frac {\left (4 (5 b c+6 a d) (b c-a d)^4+\frac {b d \left (20 b^4 c^4-56 a b^3 d c^3+24 a^2 b^2 d^2 c^2-35 a^3 b d^3 c+12 a^4 d^4\right )}{x}\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a (b c-a d)}+\frac {2 b \left (12 a^4 d^4-35 a^3 b c d^3+24 a^2 b^2 c^2 d^2-56 a b^3 c^3 d+20 b^4 c^4\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-36 a^3 d^3+87 a^2 b c d^2-36 a b^2 c^2 d+20 b^3 c^3\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {\left (4 (5 b c+6 a d) (b c-a d)^4+\frac {b d \left (20 b^4 c^4-56 a b^3 d c^3+24 a^2 b^2 d^2 c^2-35 a^3 b d^3 c+12 a^4 d^4\right )}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a (b c-a d)}+\frac {2 b \left (12 a^4 d^4-35 a^3 b c d^3+24 a^2 b^2 c^2 d^2-56 a b^3 c^3 d+20 b^4 c^4\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-36 a^3 d^3+87 a^2 b c d^2-36 a b^2 c^2 d+20 b^3 c^3\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {4 (b c-a d)^4 (6 a d+5 b c) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}-\frac {a^3 d^4 \left (24 a^2 d^2-88 a b c d+99 b^2 c^2\right ) \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}}{a (b c-a d)}+\frac {2 b \left (12 a^4 d^4-35 a^3 b c d^3+24 a^2 b^2 c^2 d^2-56 a b^3 c^3 d+20 b^4 c^4\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-36 a^3 d^3+87 a^2 b c d^2-36 a b^2 c^2 d+20 b^3 c^3\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {8 (b c-a d)^4 (6 a d+5 b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {2 a^3 d^4 \left (24 a^2 d^2-88 a b c d+99 b^2 c^2\right ) \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}}{a (b c-a d)}+\frac {2 b \left (12 a^4 d^4-35 a^3 b c d^3+24 a^2 b^2 c^2 d^2-56 a b^3 c^3 d+20 b^4 c^4\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-36 a^3 d^3+87 a^2 b c d^2-36 a b^2 c^2 d+20 b^3 c^3\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {8 (b c-a d)^4 (6 a d+5 b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {2 a^3 d^{7/2} \left (24 a^2 d^2-88 a b c d+99 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}}{a (b c-a d)}+\frac {2 b \left (12 a^4 d^4-35 a^3 b c d^3+24 a^2 b^2 c^2 d^2-56 a b^3 c^3 d+20 b^4 c^4\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-36 a^3 d^3+87 a^2 b c d^2-36 a b^2 c^2 d+20 b^3 c^3\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 c (b c-a d)}+\frac {d \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {d \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}+\frac {\frac {2 b \left (-36 a^3 d^3+87 a^2 b c d^2-36 a b^2 c^2 d+20 b^3 c^3\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}+\frac {\frac {-\frac {2 a^3 d^{7/2} \left (24 a^2 d^2-88 a b c d+99 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}-\frac {8 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (6 a d+5 b c) (b c-a d)^4}{\sqrt {a} c}}{a (b c-a d)}+\frac {2 b \left (12 a^4 d^4-35 a^3 b c d^3+24 a^2 b^2 c^2 d^2-56 a b^3 c^3 d+20 b^4 c^4\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}}{2 c (b c-a d)}}{2 c (b c-a d)}+\frac {d (2 b c-3 a d)}{c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2}\)

Input:

Int[1/((a + b/x)^(5/2)*(c + d/x)^3),x]
 

Output:

x/(a*c*(a + b/x)^(3/2)*(c + d/x)^2) + ((d*(2*b*c - 3*a*d))/(c*(b*c - a*d)* 
(a + b/x)^(3/2)*(c + d/x)^2) + ((d*(4*b^2*c^2 - 23*a*b*c*d + 12*a^2*d^2))/ 
(c*(b*c - a*d)*(a + b/x)^(3/2)*(c + d/x)) + ((2*b*(20*b^3*c^3 - 36*a*b^2*c 
^2*d + 87*a^2*b*c*d^2 - 36*a^3*d^3))/(3*a*(b*c - a*d)*(a + b/x)^(3/2)) + ( 
(2*b*(20*b^4*c^4 - 56*a*b^3*c^3*d + 24*a^2*b^2*c^2*d^2 - 35*a^3*b*c*d^3 + 
12*a^4*d^4))/(a*(b*c - a*d)*Sqrt[a + b/x]) + ((-2*a^3*d^(7/2)*(99*b^2*c^2 
- 88*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)^4*(5*b*c + 6*a*d)*ArcTanh[Sqrt[a + 
b/x]/Sqrt[a]])/(Sqrt[a]*c))/(a*(b*c - a*d)))/(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. \(1137\) vs. \(2(373)=746\).

Time = 0.78 (sec) , antiderivative size = 1138, normalized size of antiderivative = 2.78

method	result	size
risch	\(\text {Expression too large to display}\)	\(1138\)
default	\(\text {Expression too large to display}\)	\(7300\)

Input:

int(1/(a+b/x)^(5/2)/(c+1/x*d)^3,x,method=_RETURNVERBOSE)
 

Output:

1/a^3/c^3*(a*x+b)/((a*x+b)/x)^(1/2)+(-3/a^(5/2)/c^4*ln((1/2*b+a*x)/a^(1/2) 
+(a*x^2+b*x)^(1/2))*d-5/2/a^(7/2)/c^3*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^( 
1/2))*b+2/3/a^5*b^5/(a*d-b*c)^3/(x+b/a)^2*(a*(x+b/a)^2-b*(x+b/a))^(1/2)+4/ 
3/a^4*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^5/(a 
*d-b*c)^3/(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^5/(a*d-b*c)^4/(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)-21/4/c^3*d^4/(a*d-b*c)^4/(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^6 
/(a*d-b*c)^4/((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))+23/2*a/c^4*d^5/(a*d-b*c)^4/((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-99/8/c^3*d^4/(a*d-b*c)^4/((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 
^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))-12/a^3*b^4/(a*d-b*c)^4/(x+b/a)*(a*(x+ 
b/a)^2-b*(x+b/a))^(1/2)*d+6/a^4*c*b^5/(a*d-b*c)^4/(x+b/a)*(a*(x+b/a)^2-...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1518 vs. \(2 (373) = 746\).

Time = 5.00 (sec) , antiderivative size = 6140, normalized size of antiderivative = 15.01 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b/x)^(5/2)/(c+d/x)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b/x)**(5/2)/(c+d/x)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^3} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} {\left (c + \frac {d}{x}\right )}^{3}} \,d x } \] Input:

integrate(1/(a+b/x)^(5/2)/(c+d/x)^3,x, algorithm="maxima")
 

Output:

integrate(1/((a + b/x)^(5/2)*(c + d/x)^3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1336 vs. \(2 (373) = 746\).

Time = 0.32 (sec) , antiderivative size = 1336, normalized size of antiderivative = 3.27 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b/x)^(5/2)/(c+d/x)^3,x, algorithm="giac")
 

Output:

1/12*(297*a^(7/2)*b^2*c^2*d^4*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 264* 
a^(9/2)*b*c*d^5*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 72*a^(11/2)*d^6*ar 
ctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 30*sqrt(b*c*d - a*d^2)*b^5*c^5*log(a 
bs(b)) + 84*sqrt(b*c*d - a*d^2)*a*b^4*c^4*d*log(abs(b)) - 36*sqrt(b*c*d - 
a*d^2)*a^2*b^3*c^3*d^2*log(abs(b)) - 96*sqrt(b*c*d - a*d^2)*a^3*b^2*c^2*d^ 
3*log(abs(b)) + 114*sqrt(b*c*d - a*d^2)*a^4*b*c*d^4*log(abs(b)) - 36*sqrt( 
b*c*d - a*d^2)*a^5*d^5*log(abs(b)) - 56*sqrt(b*c*d - a*d^2)*b^5*c^5 + 128* 
sqrt(b*c*d - a*d^2)*a*b^4*c^4*d + 63*sqrt(b*c*d - a*d^2)*a^4*b*c*d^4 - 30* 
sqrt(b*c*d - a*d^2)*a^5*d^5)*sgn(x)/(sqrt(b*c*d - a*d^2)*a^(7/2)*b^4*c^8 - 
 4*sqrt(b*c*d - a*d^2)*a^(9/2)*b^3*c^7*d + 6*sqrt(b*c*d - a*d^2)*a^(11/2)* 
b^2*c^6*d^2 - 4*sqrt(b*c*d - a*d^2)*a^(13/2)*b*c^5*d^3 + sqrt(b*c*d - a*d^ 
2)*a^(15/2)*c^4*d^4) + 1/4*(99*b^2*c^2*d^4 - 88*a*b*c*d^5 + 24*a^2*d^6)*ar 
ctan(-((sqrt(a)*x - sqrt(a*x^2 + b*x))*c + sqrt(a)*d)/sqrt(b*c*d - a*d^2)) 
/((b^4*c^8*sgn(x) - 4*a*b^3*c^7*d*sgn(x) + 6*a^2*b^2*c^6*d^2*sgn(x) - 4*a^ 
3*b*c^5*d^3*sgn(x) + a^4*c^4*d^4*sgn(x))*sqrt(b*c*d - a*d^2)) + 1/4*(21*(s 
qrt(a)*x - sqrt(a*x^2 + b*x))^3*b^2*c^3*d^4 - 56*(sqrt(a)*x - sqrt(a*x^2 + 
 b*x))^3*a*b*c^2*d^5 + 24*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^2*c*d^6 + 15 
*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*sqrt(a)*b^2*c^2*d^5 - 88*(sqrt(a)*x - s 
qrt(a*x^2 + b*x))^2*a^(3/2)*b*c*d^6 + 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2 
*a^(5/2)*d^7 + 19*(sqrt(a)*x - sqrt(a*x^2 + b*x))*b^3*c^2*d^5 - 92*(sqr...
 

Mupad [B] (verification not implemented)

Time = 8.06 (sec) , antiderivative size = 4284, normalized size of antiderivative = 10.47 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \] Input:

int(1/((a + b/x)^(5/2)*(c + d/x)^3),x)
 

Output:

((2*b^4)/(3*(a^2*d - a*b*c)) + (2*b^4*(a + b/x)*(12*a*d - 5*b*c))/(3*(a^2* 
d - a*b*c)^2) + (b*(a + b/x)^2*(36*a^5*d^5 - 60*b^5*c^5 - 456*a^2*b^3*c^3* 
d^2 + 120*a^3*b^2*c^2*d^3 + 308*a*b^4*c^4*d - 123*a^4*b*c*d^4))/(12*a^2*c^ 
3*(a^2*d - a*b*c)*(a*d - b*c)^2) + (b*(a + b/x)^4*(12*a^4*d^6 + 20*b^4*c^4 
*d^2 - 56*a*b^3*c^3*d^3 + 24*a^2*b^2*c^2*d^4 - 35*a^3*b*c*d^5))/(4*a^2*c^3 
*(a^2*d - a*b*c)*(a*d - b*c)^3) - (b*(a + b/x)^3*(72*a^5*d^6 - 120*b^5*c^5 
*d + 496*a*b^4*c^4*d^2 - 592*a^2*b^3*c^3*d^3 + 303*a^3*b^2*c^2*d^4 - 264*a 
^4*b*c*d^5))/(12*a^2*c^3*(a^2*d - a*b*c)*(a*d - b*c)^3))/((a + b/x)^(5/2)* 
(3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d) - (a + b/x)^(7/2)*(3*a*d^2 - 2*b*c*d) + 
d^2*(a + b/x)^(9/2) - (a + b/x)^(3/2)*(a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d)) 
 + (atan((a^15*b^24*c^24*(a + b/x)^(1/2)*2000i + a^17*b^22*c^22*d^2*(a + b 
/x)^(1/2)*277440i - a^18*b^21*c^21*d^3*(a + b/x)^(1/2)*1325984i + a^19*b^2 
0*c^20*d^4*(a + b/x)^(1/2)*4135824i - a^20*b^19*c^19*d^5*(a + b/x)^(1/2)*8 
371440i + a^21*b^18*c^18*d^6*(a + b/x)^(1/2)*9129120i + a^22*b^17*c^17*d^7 
*(a + b/x)^(1/2)*3058605i - a^23*b^16*c^16*d^8*(a + b/x)^(1/2)*32337558i + 
 a^24*b^15*c^15*d^9*(a + b/x)^(1/2)*63677218i - a^25*b^14*c^14*d^10*(a + b 
/x)^(1/2)*66665280i + a^26*b^13*c^13*d^11*(a + b/x)^(1/2)*24871035i + a^27 
*b^12*c^12*d^12*(a + b/x)^(1/2)*40203170i - a^28*b^11*c^11*d^13*(a + b/x)^ 
(1/2)*85652532i + a^29*b^10*c^10*d^14*(a + b/x)^(1/2)*88170192i - a^30*b^9 
*c^9*d^15*(a + b/x)^(1/2)*60362445i + a^31*b^8*c^8*d^16*(a + b/x)^(1/2)...
 

Reduce [B] (verification not implemented)

Time = 12.95 (sec) , antiderivative size = 7135, normalized size of antiderivative = 17.44 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/(a+b/x)^(5/2)/(c+d/x)^3,x)
 

Output:

(576*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**8*c**2*d**6*x**3 + 1152*sqrt(d)*sqrt(a*x + b)*sqrt(a*d - b*c)*log(s 
qrt(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**8*c*d**7*x**2 + 576*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**8*d**8*x - 2184*sqr 
t(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 
*b*c**3*d**5*x**3 - 3792*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*b*c**2*d**6*x**2 - 1032*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*b*c*d**7*x + 576* 
sqrt(d)*sqrt(a*x + b)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*s 
qrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a 
**7*b*d**8 + 2640*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**2*c**4*d**4*x**3 + 3096*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*...