\(\int \frac {(c+\frac {d}{x})^3}{(a+\frac {b}{x})^{5/2}} \, dx\) [39]

Optimal result

Integrand size = 21, antiderivative size = 125 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {2 (b c-a d)^3}{3 a^2 b^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {2 (b c-a d)^2 (2 b c+a d)}{a^3 b^2 \sqrt {a+\frac {b}{x}}}+\frac {c^3 \sqrt {a+\frac {b}{x}} x}{a^3}-\frac {c^2 (5 b c-6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.07 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} x \left (15 b^4 c^3+4 a^4 d^3 x+3 a^2 b^2 c^2 x (-8 d+c x)+6 a^3 b d^2 (d+c x)+2 a b^3 c^2 (-9 d+10 c x)\right )}{3 a^3 b^2 (b+a x)^2}+\frac {c^2 (-5 b c+6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \] Input:

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

Output:

(Sqrt[a + b/x]*x*(15*b^4*c^3 + 4*a^4*d^3*x + 3*a^2*b^2*c^2*x*(-8*d + c*x) 
+ 6*a^3*b*d^2*(d + c*x) + 2*a*b^3*c^2*(-9*d + 10*c*x)))/(3*a^3*b^2*(b + a* 
x)^2) + (c^2*(-5*b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(7/2)
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.32, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {899, 109, 27, 162, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 162

\(\displaystyle \frac {\frac {c^2 (5 b c-6 a d) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{a^2}+\frac {2 \left (\frac {3 b (b c-a d) \left (-2 a^2 d^2-a b c d+5 b^2 c^2\right )}{x}+2 a (b c-a d) \left (-2 a^2 d^2-5 a b c d+10 b^2 c^2\right )\right )}{3 a^2 b^2 \left (a+\frac {b}{x}\right )^{3/2}}}{2 a}+\frac {c x \left (c+\frac {d}{x}\right )^2}{a \left (a+\frac {b}{x}\right )^{3/2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {2 c^2 (5 b c-6 a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{a^2 b}+\frac {2 \left (\frac {3 b (b c-a d) \left (-2 a^2 d^2-a b c d+5 b^2 c^2\right )}{x}+2 a (b c-a d) \left (-2 a^2 d^2-5 a b c d+10 b^2 c^2\right )\right )}{3 a^2 b^2 \left (a+\frac {b}{x}\right )^{3/2}}}{2 a}+\frac {c x \left (c+\frac {d}{x}\right )^2}{a \left (a+\frac {b}{x}\right )^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {2 \left (\frac {3 b (b c-a d) \left (-2 a^2 d^2-a b c d+5 b^2 c^2\right )}{x}+2 a (b c-a d) \left (-2 a^2 d^2-5 a b c d+10 b^2 c^2\right )\right )}{3 a^2 b^2 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2 c^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (5 b c-6 a d)}{a^{5/2}}}{2 a}+\frac {c x \left (c+\frac {d}{x}\right )^2}{a \left (a+\frac {b}{x}\right )^{3/2}}\)

Input:

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

Output:

(c*(c + d/x)^2*x)/(a*(a + b/x)^(3/2)) + ((2*(2*a*(b*c - a*d)*(10*b^2*c^2 - 
 5*a*b*c*d - 2*a^2*d^2) + (3*b*(b*c - a*d)*(5*b^2*c^2 - a*b*c*d - 2*a^2*d^ 
2))/x))/(3*a^2*b^2*(a + b/x)^(3/2)) - (2*c^2*(5*b*c - 6*a*d)*ArcTanh[Sqrt[ 
a + b/x]/Sqrt[a]])/a^(5/2))/(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 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 109

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

rule 162

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) 
 - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e 
*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + 
e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b 
*c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim 
p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d 
*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( 
b^2*(b*c - a*d)^2*(m + 1)*(m + 2)))   Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] 
, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + 
 n + 3, 0] &&  !LtQ[n, -2]))
 

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. \(301\) vs. \(2(111)=222\).

Time = 0.37 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.42

method	result	size
risch	\(\frac {c^{3} \left (a x +b \right )}{a^{3} \sqrt {\frac {a x +b}{x}}}+\frac {\left (\frac {\left (2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) \left (\frac {2 \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 b \left (x +\frac {b}{a}\right )^{2}}+\frac {4 a \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 b^{2} \left (x +\frac {b}{a}\right )}\right )}{a^{2}}-\frac {5 c^{3} b \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{\sqrt {a}}+6 \sqrt {a}\, c^{2} d \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )+\frac {12 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{a b \left (x +\frac {b}{a}\right )}\right ) \sqrt {x \left (a x +b \right )}}{2 a^{3} x \sqrt {\frac {a x +b}{x}}}\)	\(302\)
default	\(\text {Expression too large to display}\)	\(1150\)

Input:

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

Output:

c^3/a^3*(a*x+b)/((a*x+b)/x)^(1/2)+1/2/a^3*((2*a^3*d^3-6*a^2*b*c*d^2+6*a*b^ 
2*c^2*d-2*b^3*c^3)/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))-5*c^3*b*ln((1/2*b+a*x)/a^(1/2 
)+(a*x^2+b*x)^(1/2))/a^(1/2)+6*a^(1/2)*c^2*d*ln((1/2*b+a*x)/a^(1/2)+(a*x^2 
+b*x)^(1/2))+12*c*(a^2*d^2-2*a*b*c*d+b^2*c^2)/a/b/(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. 245 vs. \(2 (111) = 222\).

Time = 0.16 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.90 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{5} c^{3} - 6 \, a b^{4} c^{2} d + {\left (5 \, a^{2} b^{3} c^{3} - 6 \, a^{3} b^{2} c^{2} d\right )} x^{2} + 2 \, {\left (5 \, a b^{4} c^{3} - 6 \, a^{2} b^{3} c^{2} d\right )} x\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (3 \, a^{3} b^{2} c^{3} x^{3} + 2 \, {\left (10 \, a^{2} b^{3} c^{3} - 12 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} + 2 \, a^{5} d^{3}\right )} x^{2} + 3 \, {\left (5 \, a b^{4} c^{3} - 6 \, a^{2} b^{3} c^{2} d + 2 \, a^{4} b d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{6 \, {\left (a^{6} b^{2} x^{2} + 2 \, a^{5} b^{3} x + a^{4} b^{4}\right )}}, \frac {3 \, {\left (5 \, b^{5} c^{3} - 6 \, a b^{4} c^{2} d + {\left (5 \, a^{2} b^{3} c^{3} - 6 \, a^{3} b^{2} c^{2} d\right )} x^{2} + 2 \, {\left (5 \, a b^{4} c^{3} - 6 \, a^{2} b^{3} c^{2} d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + {\left (3 \, a^{3} b^{2} c^{3} x^{3} + 2 \, {\left (10 \, a^{2} b^{3} c^{3} - 12 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} + 2 \, a^{5} d^{3}\right )} x^{2} + 3 \, {\left (5 \, a b^{4} c^{3} - 6 \, a^{2} b^{3} c^{2} d + 2 \, a^{4} b d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{6} b^{2} x^{2} + 2 \, a^{5} b^{3} x + a^{4} b^{4}\right )}}\right ] \] Input:

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

Output:

[-1/6*(3*(5*b^5*c^3 - 6*a*b^4*c^2*d + (5*a^2*b^3*c^3 - 6*a^3*b^2*c^2*d)*x^ 
2 + 2*(5*a*b^4*c^3 - 6*a^2*b^3*c^2*d)*x)*sqrt(a)*log(2*a*x + 2*sqrt(a)*x*s 
qrt((a*x + b)/x) + b) - 2*(3*a^3*b^2*c^3*x^3 + 2*(10*a^2*b^3*c^3 - 12*a^3* 
b^2*c^2*d + 3*a^4*b*c*d^2 + 2*a^5*d^3)*x^2 + 3*(5*a*b^4*c^3 - 6*a^2*b^3*c^ 
2*d + 2*a^4*b*d^3)*x)*sqrt((a*x + b)/x))/(a^6*b^2*x^2 + 2*a^5*b^3*x + a^4* 
b^4), 1/3*(3*(5*b^5*c^3 - 6*a*b^4*c^2*d + (5*a^2*b^3*c^3 - 6*a^3*b^2*c^2*d 
)*x^2 + 2*(5*a*b^4*c^3 - 6*a^2*b^3*c^2*d)*x)*sqrt(-a)*arctan(sqrt(-a)*x*sq 
rt((a*x + b)/x)/(a*x + b)) + (3*a^3*b^2*c^3*x^3 + 2*(10*a^2*b^3*c^3 - 12*a 
^3*b^2*c^2*d + 3*a^4*b*c*d^2 + 2*a^5*d^3)*x^2 + 3*(5*a*b^4*c^3 - 6*a^2*b^3 
*c^2*d + 2*a^4*b*d^3)*x)*sqrt((a*x + b)/x))/(a^6*b^2*x^2 + 2*a^5*b^3*x + a 
^4*b^4)]
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (111) = 222\).

Time = 0.12 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.82 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {1}{6} \, c^{3} {\left (\frac {2 \, {\left (15 \, {\left (a + \frac {b}{x}\right )}^{2} b - 10 \, {\left (a + \frac {b}{x}\right )} a b - 2 \, a^{2} b\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3} - {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{4}} + \frac {15 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )} - c^{2} d {\left (\frac {3 \, \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (4 \, a + \frac {3 \, b}{x}\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2}}\right )} + \frac {2}{3} \, d^{3} {\left (\frac {3}{\sqrt {a + \frac {b}{x}} b^{2}} - \frac {a}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {2 \, c d^{2}}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b} \] Input:

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

Output:

1/6*c^3*(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)) - c^2*d*(3*log((sqrt(a + b/x) - sqrt(a))/(sqrt 
(a + b/x) + sqrt(a)))/a^(5/2) + 2*(4*a + 3*b/x)/((a + b/x)^(3/2)*a^2)) + 2 
/3*d^3*(3/(sqrt(a + b/x)*b^2) - a/((a + b/x)^(3/2)*b^2)) + 2*c*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. 429 vs. \(2 (111) = 222\).

Time = 0.16 (sec) , antiderivative size = 429, normalized size of antiderivative = 3.43 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\sqrt {a x^{2} + b x} c^{3}}{a^{3} \mathrm {sgn}\left (x\right )} + \frac {{\left (5 \, b c^{3} - 6 \, a c^{2} d\right )} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{2 \, a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} - \frac {{\left (15 \, b^{3} c^{3} \log \left ({\left | b \right |}\right ) - 18 \, a b^{2} c^{2} d \log \left ({\left | b \right |}\right ) + 28 \, b^{3} c^{3} - 48 \, a b^{2} c^{2} d + 12 \, a^{2} b c d^{2} + 8 \, a^{3} d^{3}\right )} \mathrm {sgn}\left (x\right )}{6 \, a^{\frac {7}{2}} b^{2}} + \frac {2 \, {\left (9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{2} c^{3} - 18 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{2} b c^{2} d + 9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{3} c d^{2} + 15 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{3} c^{3} - 27 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{\frac {3}{2}} b^{2} c^{2} d + 9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{\frac {5}{2}} b c d^{2} + 3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{\frac {7}{2}} d^{3} + 7 \, b^{4} c^{3} - 12 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} + 2 \, a^{3} b d^{3}\right )}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )}^{3} a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \] Input:

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

Output:

sqrt(a*x^2 + b*x)*c^3/(a^3*sgn(x)) + 1/2*(5*b*c^3 - 6*a*c^2*d)*log(abs(2*( 
sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b))/(a^(7/2)*sgn(x)) - 1/6*(15*b^ 
3*c^3*log(abs(b)) - 18*a*b^2*c^2*d*log(abs(b)) + 28*b^3*c^3 - 48*a*b^2*c^2 
*d + 12*a^2*b*c*d^2 + 8*a^3*d^3)*sgn(x)/(a^(7/2)*b^2) + 2/3*(9*(sqrt(a)*x 
- sqrt(a*x^2 + b*x))^2*a*b^2*c^3 - 18*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^ 
2*b*c^2*d + 9*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^3*c*d^2 + 15*(sqrt(a)*x 
- sqrt(a*x^2 + b*x))*sqrt(a)*b^3*c^3 - 27*(sqrt(a)*x - sqrt(a*x^2 + b*x))* 
a^(3/2)*b^2*c^2*d + 9*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^(5/2)*b*c*d^2 + 3* 
(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^(7/2)*d^3 + 7*b^4*c^3 - 12*a*b^3*c^2*d + 
 3*a^2*b^2*c*d^2 + 2*a^3*b*d^3)/(((sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) 
+ b)^3*a^(7/2)*sgn(x))
 

Mupad [B] (verification not implemented)

Time = 1.39 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.55 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\frac {2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{3\,a}+\frac {{\left (a+\frac {b}{x}\right )}^2\,\left (2\,a^3\,d^3-6\,a\,b^2\,c^2\,d+5\,b^3\,c^3\right )}{a^3}-\frac {2\,\left (a+\frac {b}{x}\right )\,\left (4\,a^3\,d^3-3\,a^2\,b\,c\,d^2-6\,a\,b^2\,c^2\,d+5\,b^3\,c^3\right )}{3\,a^2}}{b^2\,{\left (a+\frac {b}{x}\right )}^{5/2}-a\,b^2\,{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {c^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\,\left (6\,a\,d-5\,b\,c\right )}{a^{7/2}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.05 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {36 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a^{2} b^{2} c^{2} d x -30 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a \,b^{3} c^{3} x +36 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a \,b^{3} c^{2} d -30 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b^{4} c^{3}-8 \sqrt {a}\, \sqrt {a x +b}\, a^{4} d^{3} x +12 \sqrt {a}\, \sqrt {a x +b}\, a^{3} b c \,d^{2} x -8 \sqrt {a}\, \sqrt {a x +b}\, a^{3} b \,d^{3}+12 \sqrt {a}\, \sqrt {a x +b}\, a^{2} b^{2} c \,d^{2}-5 \sqrt {a}\, \sqrt {a x +b}\, a \,b^{3} c^{3} x -5 \sqrt {a}\, \sqrt {a x +b}\, b^{4} c^{3}+8 \sqrt {x}\, a^{5} d^{3} x +12 \sqrt {x}\, a^{4} b c \,d^{2} x +12 \sqrt {x}\, a^{4} b \,d^{3}+6 \sqrt {x}\, a^{3} b^{2} c^{3} x^{2}-48 \sqrt {x}\, a^{3} b^{2} c^{2} d x +40 \sqrt {x}\, a^{2} b^{3} c^{3} x -36 \sqrt {x}\, a^{2} b^{3} c^{2} d +30 \sqrt {x}\, a \,b^{4} c^{3}}{6 \sqrt {a x +b}\, a^{4} b^{2} \left (a x +b \right )} \] Input:

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

Output:

(36*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*a 
**2*b**2*c**2*d*x - 30*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + sqrt(x)* 
sqrt(a))/sqrt(b))*a*b**3*c**3*x + 36*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + 
 b) + sqrt(x)*sqrt(a))/sqrt(b))*a*b**3*c**2*d - 30*sqrt(a)*sqrt(a*x + b)*l 
og((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*b**4*c**3 - 8*sqrt(a)*sqrt(a 
*x + b)*a**4*d**3*x + 12*sqrt(a)*sqrt(a*x + b)*a**3*b*c*d**2*x - 8*sqrt(a) 
*sqrt(a*x + b)*a**3*b*d**3 + 12*sqrt(a)*sqrt(a*x + b)*a**2*b**2*c*d**2 - 5 
*sqrt(a)*sqrt(a*x + b)*a*b**3*c**3*x - 5*sqrt(a)*sqrt(a*x + b)*b**4*c**3 + 
 8*sqrt(x)*a**5*d**3*x + 12*sqrt(x)*a**4*b*c*d**2*x + 12*sqrt(x)*a**4*b*d* 
*3 + 6*sqrt(x)*a**3*b**2*c**3*x**2 - 48*sqrt(x)*a**3*b**2*c**2*d*x + 40*sq 
rt(x)*a**2*b**3*c**3*x - 36*sqrt(x)*a**2*b**3*c**2*d + 30*sqrt(x)*a*b**4*c 
**3)/(6*sqrt(a*x + b)*a**4*b**2*(a*x + b))