\(\int \frac {(c+\frac {d}{x})^3}{\sqrt {a+\frac {b}{x}}} \, dx\) [25]

Optimal result

Integrand size = 21, antiderivative size = 105 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\sqrt {a+\frac {b}{x}}} \, dx=-\frac {2 d^2 (3 b c-a d) \sqrt {a+\frac {b}{x}}}{b^2}-\frac {2 d^3 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^2}+\frac {c^3 \sqrt {a+\frac {b}{x}} x}{a}-\frac {c^2 (b c-6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.90 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \left (4 a^2 d^3 x+3 b^2 c^3 x^2-2 a b d^2 (d+9 c x)\right )}{3 a b^2 x}+\frac {c^2 (-b c+6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.25, 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, 164, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 164

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(c*Sqrt[a + b/x]*(c + d/x)^2*x)/a + ((-2*d*Sqrt[a + b/x]*(2*(3*b^2*c^2 + 9 
*a*b*c*d - 2*a^2*d^2) + (b*d*(3*b*c + 2*a*d))/x))/(3*b^2) - (2*c^2*(b*c - 
6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[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 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 164

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

Time = 0.33 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.24

method	result	size
risch	\(\frac {\left (a x +b \right ) \left (3 b^{2} c^{3} x^{2}+4 a^{2} d^{3} x -18 a b c \,d^{2} x -2 a \,d^{3} b \right )}{3 b^{2} x^{2} a \sqrt {\frac {a x +b}{x}}}+\frac {\left (6 a d -b c \right ) c^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {x \left (a x +b \right )}}{2 a^{\frac {3}{2}} x \sqrt {\frac {a x +b}{x}}}\)	\(130\)
default	\(\frac {\sqrt {\frac {a x +b}{x}}\, \left (-6 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} d^{3} x^{3}+18 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b c \,d^{2} x^{3}+18 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{2} c^{2} d \,x^{3}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b \,d^{3} x^{3}-9 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} c \,d^{2} x^{3}+9 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3} c^{2} d \,x^{3}-3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{4} c^{3} x^{3}-6 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, d^{3} x^{3}+18 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b c \,d^{2} x^{3}-18 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} c^{2} d \,x^{3}+6 \sqrt {a}\, \sqrt {x \left (a x +b \right )}\, b^{3} c^{3} x^{3}-3 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b \,d^{3} x^{3}+9 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} c \,d^{2} x^{3}+9 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3} c^{2} d \,x^{3}+12 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} d^{3} x -36 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b c \,d^{2} x -4 d^{3} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b \,a^{\frac {3}{2}}\right )}{6 x^{2} \sqrt {x \left (a x +b \right )}\, b^{3} a^{\frac {3}{2}}}\)	\(535\)

Input:

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

Output:

1/3*(a*x+b)*(3*b^2*c^3*x^2+4*a^2*d^3*x-18*a*b*c*d^2*x-2*a*b*d^3)/b^2/x^2/a 
/((a*x+b)/x)^(1/2)+1/2*(6*a*d-b*c)*c^2/a^(3/2)*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 = 0.12 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.27 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\sqrt {a+\frac {b}{x}}} \, dx=\left [-\frac {3 \, {\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d\right )} \sqrt {a} x \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (3 \, a b^{2} c^{3} x^{2} - 2 \, a^{2} b d^{3} - 2 \, {\left (9 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{6 \, a^{2} b^{2} x}, \frac {3 \, {\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d\right )} \sqrt {-a} x \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + {\left (3 \, a b^{2} c^{3} x^{2} - 2 \, a^{2} b d^{3} - 2 \, {\left (9 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{3 \, a^{2} b^{2} x}\right ] \] Input:

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

Output:

[-1/6*(3*(b^3*c^3 - 6*a*b^2*c^2*d)*sqrt(a)*x*log(2*a*x + 2*sqrt(a)*x*sqrt( 
(a*x + b)/x) + b) - 2*(3*a*b^2*c^3*x^2 - 2*a^2*b*d^3 - 2*(9*a^2*b*c*d^2 - 
2*a^3*d^3)*x)*sqrt((a*x + b)/x))/(a^2*b^2*x), 1/3*(3*(b^3*c^3 - 6*a*b^2*c^ 
2*d)*sqrt(-a)*x*arctan(sqrt(-a)*x*sqrt((a*x + b)/x)/(a*x + b)) + (3*a*b^2* 
c^3*x^2 - 2*a^2*b*d^3 - 2*(9*a^2*b*c*d^2 - 2*a^3*d^3)*x)*sqrt((a*x + b)/x) 
)/(a^2*b^2*x)]
 

Sympy [A] (verification not implemented)

Time = 18.03 (sec) , antiderivative size = 391, normalized size of antiderivative = 3.72 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {4 a^{\frac {7}{2}} b^{\frac {3}{2}} d^{3} x^{2} \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} + \frac {2 a^{\frac {5}{2}} b^{\frac {5}{2}} d^{3} x \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} - \frac {2 a^{\frac {3}{2}} b^{\frac {7}{2}} d^{3} \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} - \frac {4 a^{4} b d^{3} x^{\frac {5}{2}}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} - \frac {4 a^{3} b^{2} d^{3} x^{\frac {3}{2}}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} - 3 c^{2} d \left (\begin {cases} \frac {2 \operatorname {atan}{\left (\frac {\sqrt {a + \frac {b}{x}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} & \text {for}\: b \neq 0 \\- \frac {\log {\left (x \right )}}{\sqrt {a}} & \text {otherwise} \end {cases}\right ) + 3 c d^{2} \left (\begin {cases} - \frac {1}{\sqrt {a} x} & \text {for}\: b = 0 \\- \frac {2 \sqrt {a + \frac {b}{x}}}{b} & \text {otherwise} \end {cases}\right ) + \frac {\sqrt {b} c^{3} \sqrt {x} \sqrt {\frac {a x}{b} + 1}}{a} - \frac {b c^{3} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{a^{\frac {3}{2}}} \] Input:

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

Output:

4*a**(7/2)*b**(3/2)*d**3*x**2*sqrt(a*x/b + 1)/(3*a**(5/2)*b**3*x**(5/2) + 
3*a**(3/2)*b**4*x**(3/2)) + 2*a**(5/2)*b**(5/2)*d**3*x*sqrt(a*x/b + 1)/(3* 
a**(5/2)*b**3*x**(5/2) + 3*a**(3/2)*b**4*x**(3/2)) - 2*a**(3/2)*b**(7/2)*d 
**3*sqrt(a*x/b + 1)/(3*a**(5/2)*b**3*x**(5/2) + 3*a**(3/2)*b**4*x**(3/2)) 
- 4*a**4*b*d**3*x**(5/2)/(3*a**(5/2)*b**3*x**(5/2) + 3*a**(3/2)*b**4*x**(3 
/2)) - 4*a**3*b**2*d**3*x**(3/2)/(3*a**(5/2)*b**3*x**(5/2) + 3*a**(3/2)*b* 
*4*x**(3/2)) - 3*c**2*d*Piecewise((2*atan(sqrt(a + b/x)/sqrt(-a))/sqrt(-a) 
, Ne(b, 0)), (-log(x)/sqrt(a), True)) + 3*c*d**2*Piecewise((-1/(sqrt(a)*x) 
, Eq(b, 0)), (-2*sqrt(a + b/x)/b, True)) + sqrt(b)*c**3*sqrt(x)*sqrt(a*x/b 
 + 1)/a - b*c**3*asinh(sqrt(a)*sqrt(x)/sqrt(b))/a**(3/2)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.58 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {1}{2} \, c^{3} {\left (\frac {2 \, \sqrt {a + \frac {b}{x}} b}{{\left (a + \frac {b}{x}\right )} a - a^{2}} + \frac {b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {3}{2}}}\right )} - \frac {2}{3} \, d^{3} {\left (\frac {{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}}}{b^{2}} - \frac {3 \, \sqrt {a + \frac {b}{x}} a}{b^{2}}\right )} - \frac {3 \, c^{2} d \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{\sqrt {a}} - \frac {6 \, \sqrt {a + \frac {b}{x}} c d^{2}}{b} \] Input:

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

Output:

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

Giac [F(-2)]

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

integrate((c+d/x)^3/(a+b/x)^(1/2),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 = 1.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.02 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\sqrt {a+\frac {b}{x}}} \, dx=\sqrt {a+\frac {b}{x}}\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{b^2}-\frac {4\,a\,d^3}{b^2}\right )-\frac {2\,d^3\,{\left (a+\frac {b}{x}\right )}^{3/2}}{3\,b^2}+\frac {c^3\,x\,\sqrt {a+\frac {b}{x}}}{a}-\frac {c^2\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,\left (6\,a\,d-b\,c\right )\,1{}\mathrm {i}}{a^{3/2}} \] Input:

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

Output:

(a + b/x)^(1/2)*((6*a*d^3 - 6*b*c*d^2)/b^2 - (4*a*d^3)/b^2) - (2*d^3*(a + 
b/x)^(3/2))/(3*b^2) + (c^3*x*(a + b/x)^(1/2))/a - (c^2*atan(((a + b/x)^(1/ 
2)*1i)/a^(1/2))*(6*a*d - b*c)*1i)/a^(3/2)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.80 \[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {8 \sqrt {x}\, \sqrt {a x +b}\, a^{3} d^{3} x -36 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b c \,d^{2} x -4 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b \,d^{3}+6 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{2} c^{3} x^{2}+36 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a \,b^{2} c^{2} d \,x^{2}-6 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b^{3} c^{3} x^{2}-8 \sqrt {a}\, a^{3} d^{3} x^{2}+12 \sqrt {a}\, a^{2} b c \,d^{2} x^{2}+\sqrt {a}\, b^{3} c^{3} x^{2}}{6 a^{2} b^{2} x^{2}} \] Input:

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

Output:

(8*sqrt(x)*sqrt(a*x + b)*a**3*d**3*x - 36*sqrt(x)*sqrt(a*x + b)*a**2*b*c*d 
**2*x - 4*sqrt(x)*sqrt(a*x + b)*a**2*b*d**3 + 6*sqrt(x)*sqrt(a*x + b)*a*b* 
*2*c**3*x**2 + 36*sqrt(a)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*a 
*b**2*c**2*d*x**2 - 6*sqrt(a)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b 
))*b**3*c**3*x**2 - 8*sqrt(a)*a**3*d**3*x**2 + 12*sqrt(a)*a**2*b*c*d**2*x* 
*2 + sqrt(a)*b**3*c**3*x**2)/(6*a**2*b**2*x**2)