\(\int \frac {(c+\frac {d}{x})^3}{(a+\frac {b}{x})^{3/2}} \, dx\) [32]

Optimal result

Integrand size = 21, antiderivative size = 104 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {2 (b c-a d)^3}{a^2 b^2 \sqrt {a+\frac {b}{x}}}-\frac {2 d^3 \sqrt {a+\frac {b}{x}}}{b^2}+\frac {c^3 \sqrt {a+\frac {b}{x}} x}{a^2}-\frac {3 c^2 (b c-2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.09 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \left (3 b^3 c^3 x-4 a^3 d^3 x-2 a^2 b d^2 (d-3 c x)+a b^2 c^2 x (-6 d+c x)\right )}{a^2 b^2 (b+a x)}+\frac {3 c^2 (-b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \] Input:

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

Output:

(Sqrt[a + b/x]*(3*b^3*c^3*x - 4*a^3*d^3*x - 2*a^2*b*d^2*(d - 3*c*x) + a*b^ 
2*c^2*x*(-6*d + c*x)))/(a^2*b^2*(b + a*x)) + (3*c^2*(-(b*c) + 2*a*d)*ArcTa 
nh[Sqrt[a + b/x]/Sqrt[a]])/a^(5/2)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.36, 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, 163, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 163

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(c*(c + d/x)^2*x)/(a*Sqrt[a + b/x]) + ((2*((b*c - 2*a*d)*(3*b^2*c^2 - 2*a* 
b*c*d + 2*a^2*d^2) - (a*b*d^2*(b*c + 2*a*d))/x))/(a*b^2*Sqrt[a + b/x]) - ( 
6*c^2*(b*c - 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2))/(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 163

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

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. \(225\) vs. \(2(92)=184\).

Time = 0.36 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.17

method	result	size
risch	\(-\frac {\left (a x +b \right ) \left (-b^{2} x \,c^{3}+2 a^{2} d^{3}\right )}{b^{2} a^{2} x \sqrt {\frac {a x +b}{x}}}+\frac {\left (\frac {2 \left (-2 a^{3} d^{3}+6 a^{2} b c \,d^{2}-6 a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{a b \left (x +\frac {b}{a}\right )}-\frac {3 b^{2} c^{3} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{\sqrt {a}}+6 \sqrt {a}\, b \,c^{2} d \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )\right ) \sqrt {x \left (a x +b \right )}}{2 a^{2} b x \sqrt {\frac {a x +b}{x}}}\)	\(226\)
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (-6 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b^{3} c \,d^{2} x^{2}+12 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b^{4} c^{2} d \,x^{2}+24 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b^{3} c^{2} d \,x^{3}-6 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b^{3} c \,d^{2} x^{2}-6 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} b c \,d^{2} x^{4}-6 a^{\frac {9}{2}} \sqrt {x \left (a x +b \right )}\, b c \,d^{2} x^{4}+12 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} c^{2} d \,x^{4}-12 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} b^{2} c \,d^{2} x^{3}+12 a^{\frac {7}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b c \,d^{2} x^{2}-12 a^{\frac {5}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b^{2} c^{2} d \,x^{2}-12 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} c \,d^{2} x^{3}-6 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b^{3} c^{3} x^{4}-6 \sqrt {a}\, \sqrt {x \left (a x +b \right )}\, b^{5} c^{3} x^{2}+4 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2} d^{3}-3 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b^{2} c \,d^{2} x^{4}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{4} c \,d^{2} x^{2}-6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{5} c^{2} d \,x^{2}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b^{2} c \,d^{2} x^{4}-6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{3} c^{2} d \,x^{4}-6 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{3} c \,d^{2} x^{3}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{6} c^{3} x^{2}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{4} c^{3} x^{4}+4 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} d^{3} x^{2}-4 a^{\frac {9}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} d^{3} x^{2}+6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{5} c^{3} x^{3}+4 a^{\frac {3}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b^{3} c^{3} x^{2}-12 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b^{4} c^{3} x^{3}+6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{3} c \,d^{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^{2} b^{4} c^{2} d \,x^{3}+8 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b \,d^{3} x -3 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{4} c \,d^{2} x^{2}\right )}{2 x \,a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b^{3} \left (a x +b \right )^{2}}\)	\(969\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 341, normalized size of antiderivative = 3.28 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (b^{4} c^{3} - 2 \, a b^{3} c^{2} d + {\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d\right )} x\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (a^{2} b^{2} c^{3} x^{2} - 2 \, a^{3} b d^{3} + {\left (3 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - 4 \, a^{4} d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a^{4} b^{2} x + a^{3} b^{3}\right )}}, \frac {3 \, {\left (b^{4} c^{3} - 2 \, a b^{3} c^{2} d + {\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + {\left (a^{2} b^{2} c^{3} x^{2} - 2 \, a^{3} b d^{3} + {\left (3 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - 4 \, a^{4} d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{a^{4} b^{2} x + a^{3} b^{3}}\right ] \] Input:

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

Output:

[-1/2*(3*(b^4*c^3 - 2*a*b^3*c^2*d + (a*b^3*c^3 - 2*a^2*b^2*c^2*d)*x)*sqrt( 
a)*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2*(a^2*b^2*c^3*x^2 - 2 
*a^3*b*d^3 + (3*a*b^3*c^3 - 6*a^2*b^2*c^2*d + 6*a^3*b*c*d^2 - 4*a^4*d^3)*x 
)*sqrt((a*x + b)/x))/(a^4*b^2*x + a^3*b^3), (3*(b^4*c^3 - 2*a*b^3*c^2*d + 
(a*b^3*c^3 - 2*a^2*b^2*c^2*d)*x)*sqrt(-a)*arctan(sqrt(-a)*x*sqrt((a*x + b) 
/x)/(a*x + b)) + (a^2*b^2*c^3*x^2 - 2*a^3*b*d^3 + (3*a*b^3*c^3 - 6*a^2*b^2 
*c^2*d + 6*a^3*b*c*d^2 - 4*a^4*d^3)*x)*sqrt((a*x + b)/x))/(a^4*b^2*x + a^3 
*b^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (92) = 184\).

Time = 0.12 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.92 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {1}{2} \, c^{3} {\left (\frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )} b - 2 \, a b\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} - \sqrt {a + \frac {b}{x}} a^{3}} + \frac {3 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )} - 3 \, c^{2} d {\left (\frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2}{\sqrt {a + \frac {b}{x}} a}\right )} - 2 \, d^{3} {\left (\frac {\sqrt {a + \frac {b}{x}}}{b^{2}} + \frac {a}{\sqrt {a + \frac {b}{x}} b^{2}}\right )} + \frac {6 \, c d^{2}}{\sqrt {a + \frac {b}{x}} b} \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable 
to make series expansion Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 1.23 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.65 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {\frac {2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{a}-\frac {\left (a+\frac {b}{x}\right )\,\left (2\,a^3\,d^3-6\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d-3\,b^3\,c^3\right )}{a^2}}{b^2\,{\left (a+\frac {b}{x}\right )}^{3/2}-a\,b^2\,\sqrt {a+\frac {b}{x}}}-\frac {2\,d^3\,\sqrt {a+\frac {b}{x}}}{b^2}+\frac {3\,c^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\,\left (2\,a\,d-b\,c\right )}{a^{5/2}} \] Input:

int((c + d/x)^3/(a + b/x)^(3/2),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))/a - ((a + b/x)*(2 
*a^3*d^3 - 3*b^3*c^3 + 6*a*b^2*c^2*d - 6*a^2*b*c*d^2))/a^2)/(b^2*(a + b/x) 
^(3/2) - a*b^2*(a + b/x)^(1/2)) - (2*d^3*(a + b/x)^(1/2))/b^2 + (3*c^2*ata 
nh((a + b/x)^(1/2)/a^(1/2))*(2*a*d - b*c))/a^(5/2)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.31 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {6 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a \,b^{2} c^{2} d x -3 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b^{3} c^{3} x -4 \sqrt {a}\, \sqrt {a x +b}\, a^{3} d^{3} x +6 \sqrt {a}\, \sqrt {a x +b}\, a^{2} b c \,d^{2} x -6 \sqrt {a}\, \sqrt {a x +b}\, a \,b^{2} c^{2} d x +2 \sqrt {a}\, \sqrt {a x +b}\, b^{3} c^{3} x -4 \sqrt {x}\, a^{4} d^{3} x +6 \sqrt {x}\, a^{3} b c \,d^{2} x -2 \sqrt {x}\, a^{3} b \,d^{3}+\sqrt {x}\, a^{2} b^{2} c^{3} x^{2}-6 \sqrt {x}\, a^{2} b^{2} c^{2} d x +3 \sqrt {x}\, a \,b^{3} c^{3} x}{\sqrt {a x +b}\, a^{3} b^{2} x} \] Input:

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

Output:

(6*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*a* 
b**2*c**2*d*x - 3*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + sqrt(x)*sqrt( 
a))/sqrt(b))*b**3*c**3*x - 4*sqrt(a)*sqrt(a*x + b)*a**3*d**3*x + 6*sqrt(a) 
*sqrt(a*x + b)*a**2*b*c*d**2*x - 6*sqrt(a)*sqrt(a*x + b)*a*b**2*c**2*d*x + 
 2*sqrt(a)*sqrt(a*x + b)*b**3*c**3*x - 4*sqrt(x)*a**4*d**3*x + 6*sqrt(x)*a 
**3*b*c*d**2*x - 2*sqrt(x)*a**3*b*d**3 + sqrt(x)*a**2*b**2*c**3*x**2 - 6*s 
qrt(x)*a**2*b**2*c**2*d*x + 3*sqrt(x)*a*b**3*c**3*x)/(sqrt(a*x + b)*a**3*b 
**2*x)