\(\int (a+\frac {b}{x})^{3/2} (c+\frac {d}{x})^2 \, dx\) [12]

Optimal result

Integrand size = 21, antiderivative size = 112 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=-2 c (b c+2 a d) \sqrt {a+\frac {b}{x}}-\frac {4}{3} c d \left (a+\frac {b}{x}\right )^{3/2}-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b}+a c^2 \sqrt {a+\frac {b}{x}} x+\sqrt {a} c (3 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.03 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=\frac {\sqrt {a+\frac {b}{x}} \left (-6 a^2 d^2 x^2+a b x \left (-12 d^2-80 c d x+15 c^2 x^2\right )-2 b^2 \left (3 d^2+10 c d x+15 c^2 x^2\right )\right )}{15 b x^2}+\sqrt {a} c (3 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \] Input:

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

Output:

(Sqrt[a + b/x]*(-6*a^2*d^2*x^2 + a*b*x*(-12*d^2 - 80*c*d*x + 15*c^2*x^2) - 
 2*b^2*(3*d^2 + 10*c*d*x + 15*c^2*x^2)))/(15*b*x^2) + Sqrt[a]*c*(3*b*c + 4 
*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {899, 100, 27, 90, 60, 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.

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(c^2*(a + b/x)^(5/2)*x)/a - ((4*a*d^2*(a + b/x)^(5/2))/(5*b) + c*(3*b*c + 
4*a*d)*((2*(a + b/x)^(3/2))/3 + a*(2*Sqrt[a + b/x] - 2*Sqrt[a]*ArcTanh[Sqr 
t[a + b/x]/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 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 90

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

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 [A] (verified)

Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.37

method	result	size
risch	\(-\frac {\left (-15 a b \,c^{2} x^{3}+6 a^{2} d^{2} x^{2}+80 a b c d \,x^{2}+30 b^{2} c^{2} x^{2}+12 x a b \,d^{2}+20 x \,b^{2} c d +6 b^{2} d^{2}\right ) \sqrt {\frac {a x +b}{x}}}{15 x^{2} b}+\frac {\left (4 a d +3 b c \right ) \sqrt {a}\, c \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 a x +2 b}\)	\(153\)
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (-120 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} c d \,x^{4}-90 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b \,c^{2} x^{4}-60 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b c d \,x^{4}-45 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{2} c^{2} x^{4}+120 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} c d \,x^{2}+60 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b \,c^{2} x^{2}+12 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} d^{2} x +40 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b c d x +12 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b \,d^{2}\right )}{30 x^{3} b \sqrt {x \left (a x +b \right )}\, \sqrt {a}}\)	\(260\)

Input:

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

Output:

-1/15*(-15*a*b*c^2*x^3+6*a^2*d^2*x^2+80*a*b*c*d*x^2+30*b^2*c^2*x^2+12*a*b* 
d^2*x+20*b^2*c*d*x+6*b^2*d^2)/x^2/b*((a*x+b)/x)^(1/2)+1/2*(4*a*d+3*b*c)*a^ 
(1/2)*c*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 = 0.13 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.44 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=\left [\frac {15 \, {\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt {a} x^{2} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (15 \, a b c^{2} x^{3} - 6 \, b^{2} d^{2} - 2 \, {\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} - 4 \, {\left (5 \, b^{2} c d + 3 \, a b d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{30 \, b x^{2}}, -\frac {15 \, {\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) - {\left (15 \, a b c^{2} x^{3} - 6 \, b^{2} d^{2} - 2 \, {\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} - 4 \, {\left (5 \, b^{2} c d + 3 \, a b d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{15 \, b x^{2}}\right ] \] Input:

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

Output:

[1/30*(15*(3*b^2*c^2 + 4*a*b*c*d)*sqrt(a)*x^2*log(2*a*x + 2*sqrt(a)*x*sqrt 
((a*x + b)/x) + b) + 2*(15*a*b*c^2*x^3 - 6*b^2*d^2 - 2*(15*b^2*c^2 + 40*a* 
b*c*d + 3*a^2*d^2)*x^2 - 4*(5*b^2*c*d + 3*a*b*d^2)*x)*sqrt((a*x + b)/x))/( 
b*x^2), -1/15*(15*(3*b^2*c^2 + 4*a*b*c*d)*sqrt(-a)*x^2*arctan(sqrt(-a)*x*s 
qrt((a*x + b)/x)/(a*x + b)) - (15*a*b*c^2*x^3 - 6*b^2*d^2 - 2*(15*b^2*c^2 
+ 40*a*b*c*d + 3*a^2*d^2)*x^2 - 4*(5*b^2*c*d + 3*a*b*d^2)*x)*sqrt((a*x + b 
)/x))/(b*x^2)]
 

Sympy [A] (verification not implemented)

Time = 28.07 (sec) , antiderivative size = 546, normalized size of antiderivative = 4.88 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=\frac {4 a^{\frac {11}{2}} b^{\frac {5}{2}} d^{2} x^{3} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} + \frac {2 a^{\frac {9}{2}} b^{\frac {7}{2}} d^{2} x^{2} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {8 a^{\frac {7}{2}} b^{\frac {9}{2}} d^{2} x \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {6 a^{\frac {5}{2}} b^{\frac {11}{2}} d^{2} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} + \sqrt {a} b c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )} - \frac {4 a^{6} b^{2} d^{2} x^{\frac {7}{2}}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {4 a^{5} b^{3} d^{2} x^{\frac {5}{2}}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} + a \sqrt {b} c^{2} \sqrt {x} \sqrt {\frac {a x}{b} + 1} - 2 a c d \left (\begin {cases} \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {a + \frac {b}{x}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 \sqrt {a + \frac {b}{x}} & \text {for}\: b \neq 0 \\- \sqrt {a} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + a d^{2} \left (\begin {cases} - \frac {\sqrt {a}}{x} & \text {for}\: b = 0 \\- \frac {2 \left (a + \frac {b}{x}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) - b c^{2} \left (\begin {cases} \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {a + \frac {b}{x}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 \sqrt {a + \frac {b}{x}} & \text {for}\: b \neq 0 \\- \sqrt {a} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + 2 b c d \left (\begin {cases} - \frac {\sqrt {a}}{x} & \text {for}\: b = 0 \\- \frac {2 \left (a + \frac {b}{x}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

4*a**(11/2)*b**(5/2)*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**(9/2)*b**(7/2)*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**(7/2)*b* 
*(9/2)*d**2*x*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b** 
4*x**(5/2)) - 6*a**(5/2)*b**(11/2)*d**2*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3* 
x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) + sqrt(a)*b*c**2*asinh(sqrt(a)*sqrt( 
x)/sqrt(b)) - 4*a**6*b**2*d**2*x**(7/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a* 
*(5/2)*b**4*x**(5/2)) - 4*a**5*b**3*d**2*x**(5/2)/(15*a**(7/2)*b**3*x**(7/ 
2) + 15*a**(5/2)*b**4*x**(5/2)) + a*sqrt(b)*c**2*sqrt(x)*sqrt(a*x/b + 1) - 
 2*a*c*d*Piecewise((2*a*atan(sqrt(a + b/x)/sqrt(-a))/sqrt(-a) + 2*sqrt(a + 
 b/x), Ne(b, 0)), (-sqrt(a)*log(x), True)) + a*d**2*Piecewise((-sqrt(a)/x, 
 Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), True)) - b*c**2*Piecewise((2*a*ata 
n(sqrt(a + b/x)/sqrt(-a))/sqrt(-a) + 2*sqrt(a + b/x), Ne(b, 0)), (-sqrt(a) 
*log(x), True)) + 2*b*c*d*Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)* 
*(3/2)/(3*b), True))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.36 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} d^{2}}{5 \, b} + \frac {1}{2} \, {\left (2 \, \sqrt {a + \frac {b}{x}} a x - 3 \, \sqrt {a} b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) - 4 \, \sqrt {a + \frac {b}{x}} b\right )} c^{2} - \frac {2}{3} \, {\left (3 \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} + 6 \, \sqrt {a + \frac {b}{x}} a\right )} c d \] Input:

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

Output:

-2/5*(a + b/x)^(5/2)*d^2/b + 1/2*(2*sqrt(a + b/x)*a*x - 3*sqrt(a)*b*log((s 
qrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a))) - 4*sqrt(a + b/x)*b)*c^ 
2 - 2/3*(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*d
 

Giac [F(-2)]

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

integrate((a+b/x)^(3/2)*(c+d/x)^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 = 2.03 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.76 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=\sqrt {a+\frac {b}{x}}\,\left (2\,a\,\left (\frac {4\,a\,d^2-4\,b\,c\,d}{b}-\frac {4\,a\,d^2}{b}\right )-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b}+\frac {2\,a^2\,d^2}{b}\right )+\left (\frac {4\,a\,d^2-4\,b\,c\,d}{3\,b}-\frac {4\,a\,d^2}{3\,b}\right )\,{\left (a+\frac {b}{x}\right )}^{3/2}-\frac {2\,d^2\,{\left (a+\frac {b}{x}\right )}^{5/2}}{5\,b}+a\,c^2\,x\,\sqrt {a+\frac {b}{x}}-2\,c\,\mathrm {atan}\left (\frac {2\,c\,\sqrt {a+\frac {b}{x}}\,\left (4\,a\,d+3\,b\,c\right )\,\sqrt {-\frac {a}{4}}}{4\,d\,a^2\,c+3\,b\,a\,c^2}\right )\,\left (4\,a\,d+3\,b\,c\right )\,\sqrt {-\frac {a}{4}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.03 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=\frac {-24 \sqrt {x}\, \sqrt {a x +b}\, a^{2} d^{2} x^{2}+60 \sqrt {x}\, \sqrt {a x +b}\, a b \,c^{2} x^{3}-320 \sqrt {x}\, \sqrt {a x +b}\, a b c d \,x^{2}-48 \sqrt {x}\, \sqrt {a x +b}\, a b \,d^{2} x -120 \sqrt {x}\, \sqrt {a x +b}\, b^{2} c^{2} x^{2}-80 \sqrt {x}\, \sqrt {a x +b}\, b^{2} c d x -24 \sqrt {x}\, \sqrt {a x +b}\, b^{2} d^{2}+240 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a b c d \,x^{3}+180 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b^{2} c^{2} x^{3}-24 \sqrt {a}\, a^{2} d^{2} x^{3}+128 \sqrt {a}\, a b c d \,x^{3}+99 \sqrt {a}\, b^{2} c^{2} x^{3}}{60 b \,x^{3}} \] Input:

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

Output:

( - 24*sqrt(x)*sqrt(a*x + b)*a**2*d**2*x**2 + 60*sqrt(x)*sqrt(a*x + b)*a*b 
*c**2*x**3 - 320*sqrt(x)*sqrt(a*x + b)*a*b*c*d*x**2 - 48*sqrt(x)*sqrt(a*x 
+ b)*a*b*d**2*x - 120*sqrt(x)*sqrt(a*x + b)*b**2*c**2*x**2 - 80*sqrt(x)*sq 
rt(a*x + b)*b**2*c*d*x - 24*sqrt(x)*sqrt(a*x + b)*b**2*d**2 + 240*sqrt(a)* 
log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*a*b*c*d*x**3 + 180*sqrt(a)* 
log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*b**2*c**2*x**3 - 24*sqrt(a) 
*a**2*d**2*x**3 + 128*sqrt(a)*a*b*c*d*x**3 + 99*sqrt(a)*b**2*c**2*x**3)/(6 
0*b*x**3)