\(\int (a+\frac {b}{x})^{3/2} (c+\frac {d}{x})^3 \, dx\) [11]

Optimal result

Integrand size = 21, antiderivative size = 146 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx=-2 c^2 (b c+3 a d) \sqrt {a+\frac {b}{x}}-2 c^2 d \left (a+\frac {b}{x}\right )^{3/2}-\frac {2 d^2 (3 b c-a d) \left (a+\frac {b}{x}\right )^{5/2}}{5 b^2}-\frac {2 d^3 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^2}+a c^3 \sqrt {a+\frac {b}{x}} x+3 \sqrt {a} c^2 (b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.09 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx=\frac {\sqrt {a+\frac {b}{x}} \left (4 a^3 d^3 x^3-2 a^2 b d^2 x^2 (d+21 c x)+a b^2 x \left (-16 d^3-84 c d^2 x-280 c^2 d x^2+35 c^3 x^3\right )-2 b^3 \left (5 d^3+21 c d^2 x+35 c^2 d x^2+35 c^3 x^3\right )\right )}{35 b^2 x^3}+3 \sqrt {a} c^2 (b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \] Input:

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

Output:

(Sqrt[a + b/x]*(4*a^3*d^3*x^3 - 2*a^2*b*d^2*x^2*(d + 21*c*x) + a*b^2*x*(-1 
6*d^3 - 84*c*d^2*x - 280*c^2*d*x^2 + 35*c^3*x^3) - 2*b^3*(5*d^3 + 21*c*d^2 
*x + 35*c^2*d*x^2 + 35*c^3*x^3)))/(35*b^2*x^3) + 3*Sqrt[a]*c^2*(b*c + 2*a* 
d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {899, 108, 27, 170, 27, 164, 60, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 170

\(\displaystyle x \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3-\frac {3}{2} \left (\frac {2 \int \frac {1}{2} b \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) \left (7 c (b c+2 a d)+\frac {d (19 b c+2 a d)}{x}\right ) xd\frac {1}{x}}{7 b}+\frac {6}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2\right )\)

\(\Big \downarrow \) 27

\(\displaystyle x \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3-\frac {3}{2} \left (\frac {1}{7} \int \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) \left (7 c (b c+2 a d)+\frac {d (19 b c+2 a d)}{x}\right ) xd\frac {1}{x}+\frac {6}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2\right )\)

\(\Big \downarrow \) 164

\(\displaystyle x \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3-\frac {3}{2} \left (\frac {1}{7} \left (7 c^2 (2 a d+b c) \int \sqrt {a+\frac {b}{x}} xd\frac {1}{x}+\frac {2 d \left (a+\frac {b}{x}\right )^{3/2} \left (\frac {3 b d (2 a d+19 b c)}{x}+2 (13 b c-a d) (2 a d+5 b c)\right )}{15 b^2}\right )+\frac {6}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2\right )\)

\(\Big \downarrow \) 60

\(\displaystyle x \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3-\frac {3}{2} \left (\frac {1}{7} \left (7 c^2 (2 a d+b c) \left (a \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}+2 \sqrt {a+\frac {b}{x}}\right )+\frac {2 d \left (a+\frac {b}{x}\right )^{3/2} \left (\frac {3 b d (2 a d+19 b c)}{x}+2 (13 b c-a d) (2 a d+5 b c)\right )}{15 b^2}\right )+\frac {6}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2\right )\)

\(\Big \downarrow \) 73

\(\displaystyle x \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3-\frac {3}{2} \left (\frac {1}{7} \left (7 c^2 (2 a d+b c) \left (\frac {2 a \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b}+2 \sqrt {a+\frac {b}{x}}\right )+\frac {2 d \left (a+\frac {b}{x}\right )^{3/2} \left (\frac {3 b d (2 a d+19 b c)}{x}+2 (13 b c-a d) (2 a d+5 b c)\right )}{15 b^2}\right )+\frac {6}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2\right )\)

\(\Big \downarrow \) 221

\(\displaystyle x \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3-\frac {3}{2} \left (\frac {1}{7} \left (7 c^2 \left (2 \sqrt {a+\frac {b}{x}}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\right ) (2 a d+b c)+\frac {2 d \left (a+\frac {b}{x}\right )^{3/2} \left (\frac {3 b d (2 a d+19 b c)}{x}+2 (13 b c-a d) (2 a d+5 b c)\right )}{15 b^2}\right )+\frac {6}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2\right )\)

Input:

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

Output:

(a + b/x)^(3/2)*(c + d/x)^3*x - (3*((6*d*(a + b/x)^(3/2)*(c + d/x)^2)/7 + 
((2*d*(a + b/x)^(3/2)*(2*(13*b*c - a*d)*(5*b*c + 2*a*d) + (3*b*d*(19*b*c + 
 2*a*d))/x))/(15*b^2) + 7*c^2*(b*c + 2*a*d)*(2*Sqrt[a + b/x] - 2*Sqrt[a]*A 
rcTanh[Sqrt[a + b/x]/Sqrt[a]]))/7))/2
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[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 108

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) 
, x] - Simp[1/(b*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* 
x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (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 170

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

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 = 0.19 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.47

method	result	size
risch	\(\frac {\left (35 a \,b^{2} c^{3} x^{4}+4 x^{3} a^{3} d^{3}-42 x^{3} a^{2} b c \,d^{2}-280 x^{3} a \,b^{2} c^{2} d -70 x^{3} b^{3} c^{3}-2 x^{2} a^{2} b \,d^{3}-84 x^{2} a \,b^{2} c \,d^{2}-70 x^{2} b^{3} c^{2} d -16 a \,b^{2} d^{3} x -42 b^{3} c \,d^{2} x -10 b^{3} d^{3}\right ) \sqrt {\frac {a x +b}{x}}}{35 x^{3} b^{2}}+\frac {3 \left (2 a d +b c \right ) \sqrt {a}\, c^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 \left (a x +b \right )}\)	\(214\)
default	\(\frac {\sqrt {\frac {a x +b}{x}}\, \left (420 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b \,c^{2} d \,x^{5}+210 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{2} c^{3} x^{5}+210 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} c^{2} d \,x^{5}+105 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3} c^{3} x^{5}-420 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b \,c^{2} d \,x^{3}-140 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} c^{3} x^{3}+8 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} d^{3} x^{2}-84 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b c \,d^{2} x^{2}-140 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} c^{2} d \,x^{2}-12 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b \,d^{3} x -84 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} c \,d^{2} x -20 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} d^{3}\right )}{70 x^{4} b^{2} \sqrt {x \left (a x +b \right )}\, \sqrt {a}}\)	\(353\)

Input:

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

Output:

1/35*(35*a*b^2*c^3*x^4+4*a^3*d^3*x^3-42*a^2*b*c*d^2*x^3-280*a*b^2*c^2*d*x^ 
3-70*b^3*c^3*x^3-2*a^2*b*d^3*x^2-84*a*b^2*c*d^2*x^2-70*b^3*c^2*d*x^2-16*a* 
b^2*d^3*x-42*b^3*c*d^2*x-10*b^3*d^3)/x^3/b^2*((a*x+b)/x)^(1/2)+3/2*(2*a*d+ 
b*c)*a^(1/2)*c^2*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))*((a*x+b)/x)^(1/ 
2)*(x*(a*x+b))^(1/2)/(a*x+b)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.64 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx=\left [\frac {105 \, {\left (b^{3} c^{3} + 2 \, a b^{2} c^{2} d\right )} \sqrt {a} x^{3} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (35 \, a b^{2} c^{3} x^{4} - 10 \, b^{3} d^{3} - 2 \, {\left (35 \, b^{3} c^{3} + 140 \, a b^{2} c^{2} d + 21 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x^{3} - 2 \, {\left (35 \, b^{3} c^{2} d + 42 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} - 2 \, {\left (21 \, b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{70 \, b^{2} x^{3}}, -\frac {105 \, {\left (b^{3} c^{3} + 2 \, a b^{2} c^{2} d\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) - {\left (35 \, a b^{2} c^{3} x^{4} - 10 \, b^{3} d^{3} - 2 \, {\left (35 \, b^{3} c^{3} + 140 \, a b^{2} c^{2} d + 21 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x^{3} - 2 \, {\left (35 \, b^{3} c^{2} d + 42 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} - 2 \, {\left (21 \, b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{35 \, b^{2} x^{3}}\right ] \] Input:

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

Output:

[1/70*(105*(b^3*c^3 + 2*a*b^2*c^2*d)*sqrt(a)*x^3*log(2*a*x + 2*sqrt(a)*x*s 
qrt((a*x + b)/x) + b) + 2*(35*a*b^2*c^3*x^4 - 10*b^3*d^3 - 2*(35*b^3*c^3 + 
 140*a*b^2*c^2*d + 21*a^2*b*c*d^2 - 2*a^3*d^3)*x^3 - 2*(35*b^3*c^2*d + 42* 
a*b^2*c*d^2 + a^2*b*d^3)*x^2 - 2*(21*b^3*c*d^2 + 8*a*b^2*d^3)*x)*sqrt((a*x 
 + b)/x))/(b^2*x^3), -1/35*(105*(b^3*c^3 + 2*a*b^2*c^2*d)*sqrt(-a)*x^3*arc 
tan(sqrt(-a)*x*sqrt((a*x + b)/x)/(a*x + b)) - (35*a*b^2*c^3*x^4 - 10*b^3*d 
^3 - 2*(35*b^3*c^3 + 140*a*b^2*c^2*d + 21*a^2*b*c*d^2 - 2*a^3*d^3)*x^3 - 2 
*(35*b^3*c^2*d + 42*a*b^2*c*d^2 + a^2*b*d^3)*x^2 - 2*(21*b^3*c*d^2 + 8*a*b 
^2*d^3)*x)*sqrt((a*x + b)/x))/(b^2*x^3)]
 

Sympy [A] (verification not implemented)

Time = 40.28 (sec) , antiderivative size = 1828, normalized size of antiderivative = 12.52 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

-16*a**(19/2)*b**(11/2)*d**3*x**6*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**( 
13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a* 
*(7/2)*b**10*x**(7/2)) - 40*a**(17/2)*b**(13/2)*d**3*x**5*sqrt(a*x/b + 1)/ 
(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2 
)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 30*a**(15/2)*b**(15/2)*d* 
*3*x**4*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8 
*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 4 
0*a**(13/2)*b**(17/2)*d**3*x**3*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13 
/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**( 
7/2)*b**10*x**(7/2)) + 4*a**(13/2)*b**(3/2)*d**3*x**3*sqrt(a*x/b + 1)/(15* 
a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 100*a**(11/2)*b**(19 
/2)*d**3*x**2*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2 
)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2 
)) + 12*a**(11/2)*b**(5/2)*c*d**2*x**3*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x 
**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) + 2*a**(11/2)*b**(5/2)*d**3*x**2*sqrt 
(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 96*a 
**(9/2)*b**(21/2)*d**3*x*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 3 
15*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b* 
*10*x**(7/2)) + 6*a**(9/2)*b**(7/2)*c*d**2*x**2*sqrt(a*x/b + 1)/(15*a**(7/ 
2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 8*a**(9/2)*b**(7/2)*d**...
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.30 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx=-\frac {6 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} c d^{2}}{5 \, b} + \frac {1}{2} \, {\left (2 \, \sqrt {a + \frac {b}{x}} a x - 3 \, \sqrt {a} b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) - 4 \, \sqrt {a + \frac {b}{x}} b\right )} c^{3} - {\left (3 \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} + 6 \, \sqrt {a + \frac {b}{x}} a\right )} c^{2} d - \frac {2}{35} \, {\left (\frac {5 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}}}{b^{2}} - \frac {7 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a}{b^{2}}\right )} d^{3} \] Input:

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

Output:

-6/5*(a + b/x)^(5/2)*c*d^2/b + 1/2*(2*sqrt(a + b/x)*a*x - 3*sqrt(a)*b*log( 
(sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a))) - 4*sqrt(a + b/x)*b)* 
c^3 - (3*a^(3/2)*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a))) 
+ 2*(a + b/x)^(3/2) + 6*sqrt(a + b/x)*a)*c^2*d - 2/35*(5*(a + b/x)^(7/2)/b 
^2 - 7*(a + b/x)^(5/2)*a/b^2)*d^3
 

Giac [F(-2)]

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

integrate((a+b/x)^(3/2)*(c+d/x)^3,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 = 3.60 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.24 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx={\left (a+\frac {b}{x}\right )}^{5/2}\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{5\,b^2}-\frac {4\,a\,d^3}{5\,b^2}\right )+\sqrt {a+\frac {b}{x}}\,\left (\frac {2\,{\left (a\,d-b\,c\right )}^3}{b^2}+2\,a\,\left (2\,a\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{b^2}-\frac {4\,a\,d^3}{b^2}\right )-\frac {6\,d\,{\left (a\,d-b\,c\right )}^2}{b^2}+\frac {2\,a^2\,d^3}{b^2}\right )-a^2\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{b^2}-\frac {4\,a\,d^3}{b^2}\right )\right )+{\left (a+\frac {b}{x}\right )}^{3/2}\,\left (\frac {2\,a\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{b^2}-\frac {4\,a\,d^3}{b^2}\right )}{3}-\frac {2\,d\,{\left (a\,d-b\,c\right )}^2}{b^2}+\frac {2\,a^2\,d^3}{3\,b^2}\right )-\frac {2\,d^3\,{\left (a+\frac {b}{x}\right )}^{7/2}}{7\,b^2}+a\,c^3\,x\,\sqrt {a+\frac {b}{x}}-2\,c^2\,\mathrm {atan}\left (\frac {2\,c^2\,\sqrt {a+\frac {b}{x}}\,\left (2\,a\,d+b\,c\right )\,\sqrt {-\frac {9\,a}{4}}}{6\,d\,a^2\,c^2+3\,b\,a\,c^3}\right )\,\left (2\,a\,d+b\,c\right )\,\sqrt {-\frac {9\,a}{4}} \] Input:

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

Output:

(a + b/x)^(5/2)*((6*a*d^3 - 6*b*c*d^2)/(5*b^2) - (4*a*d^3)/(5*b^2)) + (a + 
 b/x)^(1/2)*((2*(a*d - b*c)^3)/b^2 + 2*a*(2*a*((6*a*d^3 - 6*b*c*d^2)/b^2 - 
 (4*a*d^3)/b^2) - (6*d*(a*d - b*c)^2)/b^2 + (2*a^2*d^3)/b^2) - a^2*((6*a*d 
^3 - 6*b*c*d^2)/b^2 - (4*a*d^3)/b^2)) + (a + b/x)^(3/2)*((2*a*((6*a*d^3 - 
6*b*c*d^2)/b^2 - (4*a*d^3)/b^2))/3 - (2*d*(a*d - b*c)^2)/b^2 + (2*a^2*d^3) 
/(3*b^2)) - (2*d^3*(a + b/x)^(7/2))/(7*b^2) + a*c^3*x*(a + b/x)^(1/2) - 2* 
c^2*atan((2*c^2*(a + b/x)^(1/2)*(2*a*d + b*c)*(-(9*a)/4)^(1/2))/(6*a^2*c^2 
*d + 3*a*b*c^3))*(2*a*d + b*c)*(-(9*a)/4)^(1/2)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.34 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx=\frac {4 \sqrt {x}\, \sqrt {a x +b}\, a^{3} d^{3} x^{3}-42 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b c \,d^{2} x^{3}-2 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b \,d^{3} x^{2}+35 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{2} c^{3} x^{4}-280 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{2} c^{2} d \,x^{3}-84 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{2} c \,d^{2} x^{2}-16 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{2} d^{3} x -70 \sqrt {x}\, \sqrt {a x +b}\, b^{3} c^{3} x^{3}-70 \sqrt {x}\, \sqrt {a x +b}\, b^{3} c^{2} d \,x^{2}-42 \sqrt {x}\, \sqrt {a x +b}\, b^{3} c \,d^{2} x -10 \sqrt {x}\, \sqrt {a x +b}\, b^{3} d^{3}+210 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a \,b^{2} c^{2} d \,x^{4}+105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b^{3} c^{3} x^{4}-4 \sqrt {a}\, a^{3} d^{3} x^{4}-18 \sqrt {a}\, a^{2} b c \,d^{2} x^{4}+160 \sqrt {a}\, a \,b^{2} c^{2} d \,x^{4}+75 \sqrt {a}\, b^{3} c^{3} x^{4}}{35 b^{2} x^{4}} \] Input:

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

Output:

(4*sqrt(x)*sqrt(a*x + b)*a**3*d**3*x**3 - 42*sqrt(x)*sqrt(a*x + b)*a**2*b* 
c*d**2*x**3 - 2*sqrt(x)*sqrt(a*x + b)*a**2*b*d**3*x**2 + 35*sqrt(x)*sqrt(a 
*x + b)*a*b**2*c**3*x**4 - 280*sqrt(x)*sqrt(a*x + b)*a*b**2*c**2*d*x**3 - 
84*sqrt(x)*sqrt(a*x + b)*a*b**2*c*d**2*x**2 - 16*sqrt(x)*sqrt(a*x + b)*a*b 
**2*d**3*x - 70*sqrt(x)*sqrt(a*x + b)*b**3*c**3*x**3 - 70*sqrt(x)*sqrt(a*x 
 + b)*b**3*c**2*d*x**2 - 42*sqrt(x)*sqrt(a*x + b)*b**3*c*d**2*x - 10*sqrt( 
x)*sqrt(a*x + b)*b**3*d**3 + 210*sqrt(a)*log((sqrt(a*x + b) + sqrt(x)*sqrt 
(a))/sqrt(b))*a*b**2*c**2*d*x**4 + 105*sqrt(a)*log((sqrt(a*x + b) + sqrt(x 
)*sqrt(a))/sqrt(b))*b**3*c**3*x**4 - 4*sqrt(a)*a**3*d**3*x**4 - 18*sqrt(a) 
*a**2*b*c*d**2*x**4 + 160*sqrt(a)*a*b**2*c**2*d*x**4 + 75*sqrt(a)*b**3*c** 
3*x**4)/(35*b**2*x**4)