\(\int \frac {(c+\frac {d}{x})^2}{(a+\frac {b}{x})^{5/2}} \, dx\) [40]

Optimal result

Integrand size = 21, antiderivative size = 111 \[ \int \frac {\left (c+\frac {d}{x}\right )^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {2 (b c-a d)^2}{3 a^2 b \left (a+\frac {b}{x}\right )^{3/2}}+\frac {4 c (b c-a d)}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {c^2 \sqrt {a+\frac {b}{x}} x}{a^3}-\frac {c (5 b c-4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.01 \[ \int \frac {\left (c+\frac {d}{x}\right )^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} x \left (15 b^3 c^2+2 a^3 d^2 x+a^2 b c x (-16 d+3 c x)+4 a b^2 c (-3 d+5 c x)\right )}{3 a^3 b (b+a x)^2}+\frac {c (-5 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \] Input:

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

Output:

(Sqrt[a + b/x]*x*(15*b^3*c^2 + 2*a^3*d^2*x + a^2*b*c*x*(-16*d + 3*c*x) + 4 
*a*b^2*c*(-3*d + 5*c*x)))/(3*a^3*b*(b + a*x)^2) + (c*(-5*b*c + 4*a*d)*ArcT 
anh[Sqrt[a + b/x]/Sqrt[a]])/a^(7/2)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {899, 100, 27, 87, 61, 73, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, 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 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 100

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

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. \(286\) vs. \(2(97)=194\).

Time = 0.36 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.59

method	result	size
risch	\(\frac {c^{2} \left (a x +b \right )}{a^{3} \sqrt {\frac {a x +b}{x}}}+\frac {\left (\frac {2 \left (2 a^{2} d^{2}-8 a b c d +6 b^{2} c^{2}\right ) \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{a b \left (x +\frac {b}{a}\right )}-\frac {5 b \,c^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{\sqrt {a}}+4 \sqrt {a}\, c d \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )-\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \left (\frac {2 \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 b \left (x +\frac {b}{a}\right )^{2}}+\frac {4 a \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 b^{2} \left (x +\frac {b}{a}\right )}\right )}{a^{2}}\right ) \sqrt {x \left (a x +b \right )}}{2 a^{3} x \sqrt {\frac {a x +b}{x}}}\)	\(287\)
default	\(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (-24 \sqrt {x \left (a x +b \right )}\, a^{\frac {9}{2}} c d \,x^{3}+30 \sqrt {x \left (a x +b \right )}\, a^{\frac {7}{2}} b \,c^{2} x^{3}+12 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b c d \,x^{3}-15 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{2} c^{2} x^{3}+24 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {7}{2}} c d x -24 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {5}{2}} b \,c^{2} x -72 \sqrt {x \left (a x +b \right )}\, a^{\frac {7}{2}} b c d \,x^{2}+90 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}} b^{2} c^{2} x^{2}+36 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{2} c d \,x^{2}-45 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{3} c^{2} x^{2}+4 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {7}{2}} d^{2}+16 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {5}{2}} b c d -20 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{2} c^{2}-72 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}} b^{2} c d x +90 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b^{3} c^{2} x +36 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{3} c d x -45 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{4} c^{2} x -24 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b^{3} c d +30 \sqrt {x \left (a x +b \right )}\, \sqrt {a}\, b^{4} c^{2}+12 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{4} c d -15 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{5} c^{2}\right )}{6 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b \left (a x +b \right )^{3}}\)	\(588\)

Input:

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

Output:

c^2/a^3*(a*x+b)/((a*x+b)/x)^(1/2)+1/2/a^3*(2*(2*a^2*d^2-8*a*b*c*d+6*b^2*c^ 
2)/a/b/(x+b/a)*(a*(x+b/a)^2-b*(x+b/a))^(1/2)-5*b*c^2*ln((1/2*b+a*x)/a^(1/2 
)+(a*x^2+b*x)^(1/2))/a^(1/2)+4*a^(1/2)*c*d*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b 
*x)^(1/2))-2*(a^2*d^2-2*a*b*c*d+b^2*c^2)*b/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)))/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. 207 vs. \(2 (97) = 194\).

Time = 0.11 (sec) , antiderivative size = 412, normalized size of antiderivative = 3.71 \[ \int \frac {\left (c+\frac {d}{x}\right )^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{4} c^{2} - 4 \, a b^{3} c d + {\left (5 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d\right )} x^{2} + 2 \, {\left (5 \, a b^{3} c^{2} - 4 \, a^{2} b^{2} c d\right )} x\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (3 \, a^{3} b c^{2} x^{3} + 2 \, {\left (10 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2} + 3 \, {\left (5 \, a b^{3} c^{2} - 4 \, a^{2} b^{2} c d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{6 \, {\left (a^{6} b x^{2} + 2 \, a^{5} b^{2} x + a^{4} b^{3}\right )}}, \frac {3 \, {\left (5 \, b^{4} c^{2} - 4 \, a b^{3} c d + {\left (5 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d\right )} x^{2} + 2 \, {\left (5 \, a b^{3} c^{2} - 4 \, a^{2} b^{2} c d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + {\left (3 \, a^{3} b c^{2} x^{3} + 2 \, {\left (10 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2} + 3 \, {\left (5 \, a b^{3} c^{2} - 4 \, a^{2} b^{2} c d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{6} b x^{2} + 2 \, a^{5} b^{2} x + a^{4} b^{3}\right )}}\right ] \] Input:

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

Output:

[-1/6*(3*(5*b^4*c^2 - 4*a*b^3*c*d + (5*a^2*b^2*c^2 - 4*a^3*b*c*d)*x^2 + 2* 
(5*a*b^3*c^2 - 4*a^2*b^2*c*d)*x)*sqrt(a)*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x 
 + b)/x) + b) - 2*(3*a^3*b*c^2*x^3 + 2*(10*a^2*b^2*c^2 - 8*a^3*b*c*d + a^4 
*d^2)*x^2 + 3*(5*a*b^3*c^2 - 4*a^2*b^2*c*d)*x)*sqrt((a*x + b)/x))/(a^6*b*x 
^2 + 2*a^5*b^2*x + a^4*b^3), 1/3*(3*(5*b^4*c^2 - 4*a*b^3*c*d + (5*a^2*b^2* 
c^2 - 4*a^3*b*c*d)*x^2 + 2*(5*a*b^3*c^2 - 4*a^2*b^2*c*d)*x)*sqrt(-a)*arcta 
n(sqrt(-a)*x*sqrt((a*x + b)/x)/(a*x + b)) + (3*a^3*b*c^2*x^3 + 2*(10*a^2*b 
^2*c^2 - 8*a^3*b*c*d + a^4*d^2)*x^2 + 3*(5*a*b^3*c^2 - 4*a^2*b^2*c*d)*x)*s 
qrt((a*x + b)/x))/(a^6*b*x^2 + 2*a^5*b^2*x + a^4*b^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.71 \[ \int \frac {\left (c+\frac {d}{x}\right )^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {1}{6} \, c^{2} {\left (\frac {2 \, {\left (15 \, {\left (a + \frac {b}{x}\right )}^{2} b - 10 \, {\left (a + \frac {b}{x}\right )} a b - 2 \, a^{2} b\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3} - {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{4}} + \frac {15 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )} - \frac {2}{3} \, c d {\left (\frac {3 \, \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (4 \, a + \frac {3 \, b}{x}\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2}}\right )} + \frac {2 \, d^{2}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b} \] Input:

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

Output:

1/6*c^2*(2*(15*(a + b/x)^2*b - 10*(a + b/x)*a*b - 2*a^2*b)/((a + b/x)^(5/2 
)*a^3 - (a + b/x)^(3/2)*a^4) + 15*b*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a 
+ b/x) + sqrt(a)))/a^(7/2)) - 2/3*c*d*(3*log((sqrt(a + b/x) - sqrt(a))/(sq 
rt(a + b/x) + sqrt(a)))/a^(5/2) + 2*(4*a + 3*b/x)/((a + b/x)^(3/2)*a^2)) + 
 2/3*d^2/((a + b/x)^(3/2)*b)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (97) = 194\).

Time = 0.16 (sec) , antiderivative size = 363, normalized size of antiderivative = 3.27 \[ \int \frac {\left (c+\frac {d}{x}\right )^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\sqrt {a x^{2} + b x} c^{2}}{a^{3} \mathrm {sgn}\left (x\right )} + \frac {{\left (5 \, b c^{2} - 4 \, a c d\right )} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{2 \, a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} - \frac {{\left (15 \, b^{2} c^{2} \log \left ({\left | b \right |}\right ) - 12 \, a b c d \log \left ({\left | b \right |}\right ) + 28 \, b^{2} c^{2} - 32 \, a b c d + 4 \, a^{2} d^{2}\right )} \mathrm {sgn}\left (x\right )}{6 \, a^{\frac {7}{2}} b} + \frac {2 \, {\left (9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{2} c^{2} - 12 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{2} b c d + 3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{3} d^{2} + 15 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{3} c^{2} - 18 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{\frac {3}{2}} b^{2} c d + 3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{\frac {5}{2}} b d^{2} + 7 \, b^{4} c^{2} - 8 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )}^{3} a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \] Input:

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

Output:

sqrt(a*x^2 + b*x)*c^2/(a^3*sgn(x)) + 1/2*(5*b*c^2 - 4*a*c*d)*log(abs(2*(sq 
rt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b))/(a^(7/2)*sgn(x)) - 1/6*(15*b^2* 
c^2*log(abs(b)) - 12*a*b*c*d*log(abs(b)) + 28*b^2*c^2 - 32*a*b*c*d + 4*a^2 
*d^2)*sgn(x)/(a^(7/2)*b) + 2/3*(9*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^2* 
c^2 - 12*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^2*b*c*d + 3*(sqrt(a)*x - sqrt 
(a*x^2 + b*x))^2*a^3*d^2 + 15*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^3* 
c^2 - 18*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^(3/2)*b^2*c*d + 3*(sqrt(a)*x - 
sqrt(a*x^2 + b*x))*a^(5/2)*b*d^2 + 7*b^4*c^2 - 8*a*b^3*c*d + a^2*b^2*d^2)/ 
(((sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b)^3*a^(7/2)*sgn(x))
 

Mupad [B] (verification not implemented)

Time = 1.53 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.30 \[ \int \frac {\left (c+\frac {d}{x}\right )^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\frac {2\,\left (a+\frac {b}{x}\right )\,\left (a^2\,d^2+4\,a\,b\,c\,d-5\,b^2\,c^2\right )}{3\,a^2}-\frac {2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{3\,a}+\frac {b\,{\left (a+\frac {b}{x}\right )}^2\,\left (5\,b\,c^2-4\,a\,c\,d\right )}{a^3}}{b\,{\left (a+\frac {b}{x}\right )}^{5/2}-a\,b\,{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {c\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\,\left (4\,a\,d-5\,b\,c\right )}{a^{7/2}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.74 \[ \int \frac {\left (c+\frac {d}{x}\right )^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {24 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a^{2} b c d x -30 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a \,b^{2} c^{2} x +24 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a \,b^{2} c d -30 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b^{3} c^{2}+4 \sqrt {a}\, \sqrt {a x +b}\, a^{3} d^{2} x +4 \sqrt {a}\, \sqrt {a x +b}\, a^{2} b \,d^{2}-5 \sqrt {a}\, \sqrt {a x +b}\, a \,b^{2} c^{2} x -5 \sqrt {a}\, \sqrt {a x +b}\, b^{3} c^{2}+4 \sqrt {x}\, a^{4} d^{2} x +6 \sqrt {x}\, a^{3} b \,c^{2} x^{2}-32 \sqrt {x}\, a^{3} b c d x +40 \sqrt {x}\, a^{2} b^{2} c^{2} x -24 \sqrt {x}\, a^{2} b^{2} c d +30 \sqrt {x}\, a \,b^{3} c^{2}}{6 \sqrt {a x +b}\, a^{4} b \left (a x +b \right )} \] Input:

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

Output:

(24*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*a 
**2*b*c*d*x - 30*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a 
))/sqrt(b))*a*b**2*c**2*x + 24*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + 
sqrt(x)*sqrt(a))/sqrt(b))*a*b**2*c*d - 30*sqrt(a)*sqrt(a*x + b)*log((sqrt( 
a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*b**3*c**2 + 4*sqrt(a)*sqrt(a*x + b)*a 
**3*d**2*x + 4*sqrt(a)*sqrt(a*x + b)*a**2*b*d**2 - 5*sqrt(a)*sqrt(a*x + b) 
*a*b**2*c**2*x - 5*sqrt(a)*sqrt(a*x + b)*b**3*c**2 + 4*sqrt(x)*a**4*d**2*x 
 + 6*sqrt(x)*a**3*b*c**2*x**2 - 32*sqrt(x)*a**3*b*c*d*x + 40*sqrt(x)*a**2* 
b**2*c**2*x - 24*sqrt(x)*a**2*b**2*c*d + 30*sqrt(x)*a*b**3*c**2)/(6*sqrt(a 
*x + b)*a**4*b*(a*x + b))