\(\int (a+\frac {c}{x^2}+\frac {b}{x})^{3/2} \, dx\) [15]

Optimal result

Integrand size = 16, antiderivative size = 145 \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \, dx=-\frac {3}{4} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \left (3 b+\frac {2 c}{x}\right )+\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} x+\frac {3}{2} \sqrt {a} b \text {arctanh}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )-\frac {3 \left (b^2+4 a c\right ) \text {arctanh}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{8 \sqrt {c}} \] Output:

-3/4*(a+c/x^2+b/x)^(1/2)*(3*b+2*c/x)+(a+c/x^2+b/x)^(3/2)*x+3/2*a^(1/2)*b*a 
rctanh(1/2*(2*a+b/x)/a^(1/2)/(a+c/x^2+b/x)^(1/2))-3/8*(4*a*c+b^2)*arctanh( 
1/2*(b+2*c/x)/c^(1/2)/(a+c/x^2+b/x)^(1/2))/c^(1/2)
 

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.09 \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \, dx=\frac {\sqrt {a+\frac {c+b x}{x^2}} \left (3 \left (b^2+4 a c\right ) x^2 \text {arctanh}\left (\frac {\sqrt {a} x-\sqrt {c+x (b+a x)}}{\sqrt {c}}\right )-\sqrt {c} \left ((2 c+x (5 b-4 a x)) \sqrt {c+x (b+a x)}+6 \sqrt {a} b x^2 \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+x (b+a x)}\right )\right )\right )}{4 \sqrt {c} x \sqrt {c+x (b+a x)}} \] Input:

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

Output:

(Sqrt[a + (c + b*x)/x^2]*(3*(b^2 + 4*a*c)*x^2*ArcTanh[(Sqrt[a]*x - Sqrt[c 
+ x*(b + a*x)])/Sqrt[c]] - Sqrt[c]*((2*c + x*(5*b - 4*a*x))*Sqrt[c + x*(b 
+ a*x)] + 6*Sqrt[a]*b*x^2*Log[b + 2*a*x - 2*Sqrt[a]*Sqrt[c + x*(b + a*x)]] 
)))/(4*Sqrt[c]*x*Sqrt[c + x*(b + a*x)])
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {1681, 1161, 1231, 25, 27, 1269, 1092, 219, 1154, 219}

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.

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

\(\Big \downarrow \) 1681

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

\(\Big \downarrow \) 1161

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

\(\Big \downarrow \) 1231

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

\(\displaystyle x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}-\frac {3}{2} \left (\frac {1}{4} \left (\frac {\left (4 a c+b^2\right ) \text {arctanh}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )}{\sqrt {c}}-8 a b \int \frac {1}{4 a-\frac {1}{x^2}}d\frac {2 a+\frac {b}{x}}{\sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )+\frac {1}{2} \left (3 b+\frac {2 c}{x}\right ) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}\right )\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

(a + c/x^2 + b/x)^(3/2)*x - (3*((Sqrt[a + c/x^2 + b/x]*(3*b + (2*c)/x))/2 
+ (-4*Sqrt[a]*b*ArcTanh[(2*a + b/x)/(2*Sqrt[a]*Sqrt[a + c/x^2 + b/x])] + ( 
(b^2 + 4*a*c)*ArcTanh[(b + (2*c)/x)/(2*Sqrt[c]*Sqrt[a + c/x^2 + b/x])])/Sq 
rt[c])/4))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1161

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si 
mp[p/(e*(m + 1))   Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 
 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || 
 LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, 
 c, d, e, m, p, x]
 

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1681

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[ 
Int[(a + b/x^n + c/x^(2*n))^p/x^2, x], x, 1/x] /; FreeQ[{a, b, c, p}, x] && 
 EqQ[n2, 2*n] && ILtQ[n, 0]
 

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.23

method	result	size
risch	\(-\frac {\left (5 b x +2 c \right ) \sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}}{4 x}+\frac {\left (-\frac {3 \sqrt {c}\, \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) a}{2}-\frac {3 \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) b^{2}}{8 \sqrt {c}}+a \sqrt {a \,x^{2}+b x +c}+\frac {3 b \sqrt {a}\, \ln \left (\frac {\frac {b}{2}+x a}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{2}\right ) \sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}\, x}{\sqrt {a \,x^{2}+b x +c}}\)	\(179\)
default	\(-\frac {\left (\frac {a \,x^{2}+b x +c}{x^{2}}\right )^{\frac {3}{2}} x \left (12 a^{\frac {5}{2}} c^{\frac {5}{2}} \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) x^{2}-2 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b \,x^{3}-4 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} c \,x^{2}-6 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x +c}\, b c \,x^{3}+3 a^{\frac {3}{2}} c^{\frac {3}{2}} \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) b^{2} x^{2}-12 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x +c}\, c^{2} x^{2}+2 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b x -2 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b^{2} x^{2}+4 \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} c \,a^{\frac {3}{2}}-6 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x +c}\, b^{2} c \,x^{2}-12 a^{2} \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 x a +b}{2 \sqrt {a}}\right ) b \,c^{2} x^{2}\right )}{8 \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} c^{2} a^{\frac {3}{2}}}\)	\(334\)

Input:

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

Output:

-1/4*(5*b*x+2*c)/x*((a*x^2+b*x+c)/x^2)^(1/2)+(-3/2*c^(1/2)*ln((2*c+b*x+2*c 
^(1/2)*(a*x^2+b*x+c)^(1/2))/x)*a-3/8/c^(1/2)*ln((2*c+b*x+2*c^(1/2)*(a*x^2+ 
b*x+c)^(1/2))/x)*b^2+a*(a*x^2+b*x+c)^(1/2)+3/2*b*a^(1/2)*ln((1/2*b+x*a)/a^ 
(1/2)+(a*x^2+b*x+c)^(1/2)))*((a*x^2+b*x+c)/x^2)^(1/2)*x/(a*x^2+b*x+c)^(1/2 
)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 709, normalized size of antiderivative = 4.89 \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/16*(12*sqrt(a)*b*c*x*log(-8*a^2*x^2 - 8*a*b*x - b^2 - 4*a*c - 4*(2*a*x^ 
2 + b*x)*sqrt(a)*sqrt((a*x^2 + b*x + c)/x^2)) + 3*(b^2 + 4*a*c)*sqrt(c)*x* 
log(-(8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2 - 4*(b*x^2 + 2*c*x)*sqrt(c)*sqrt 
((a*x^2 + b*x + c)/x^2))/x^2) + 4*(4*a*c*x^2 - 5*b*c*x - 2*c^2)*sqrt((a*x^ 
2 + b*x + c)/x^2))/(c*x), -1/16*(24*sqrt(-a)*b*c*x*arctan(1/2*(2*a*x^2 + b 
*x)*sqrt(-a)*sqrt((a*x^2 + b*x + c)/x^2)/(a^2*x^2 + a*b*x + a*c)) - 3*(b^2 
 + 4*a*c)*sqrt(c)*x*log(-(8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2 - 4*(b*x^2 + 
 2*c*x)*sqrt(c)*sqrt((a*x^2 + b*x + c)/x^2))/x^2) - 4*(4*a*c*x^2 - 5*b*c*x 
 - 2*c^2)*sqrt((a*x^2 + b*x + c)/x^2))/(c*x), 1/8*(6*sqrt(a)*b*c*x*log(-8* 
a^2*x^2 - 8*a*b*x - b^2 - 4*a*c - 4*(2*a*x^2 + b*x)*sqrt(a)*sqrt((a*x^2 + 
b*x + c)/x^2)) + 3*(b^2 + 4*a*c)*sqrt(-c)*x*arctan(1/2*(b*x^2 + 2*c*x)*sqr 
t(-c)*sqrt((a*x^2 + b*x + c)/x^2)/(a*c*x^2 + b*c*x + c^2)) + 2*(4*a*c*x^2 
- 5*b*c*x - 2*c^2)*sqrt((a*x^2 + b*x + c)/x^2))/(c*x), -1/8*(12*sqrt(-a)*b 
*c*x*arctan(1/2*(2*a*x^2 + b*x)*sqrt(-a)*sqrt((a*x^2 + b*x + c)/x^2)/(a^2* 
x^2 + a*b*x + a*c)) - 3*(b^2 + 4*a*c)*sqrt(-c)*x*arctan(1/2*(b*x^2 + 2*c*x 
)*sqrt(-c)*sqrt((a*x^2 + b*x + c)/x^2)/(a*c*x^2 + b*c*x + c^2)) - 2*(4*a*c 
*x^2 - 5*b*c*x - 2*c^2)*sqrt((a*x^2 + b*x + c)/x^2))/(c*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.26 \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \, dx=\frac {8 \sqrt {a \,x^{2}+b x +c}\, a c \,x^{2}-10 \sqrt {a \,x^{2}+b x +c}\, b c x -4 \sqrt {a \,x^{2}+b x +c}\, c^{2}+12 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {a \,x^{2}+b x +c}-2 a x -b \right ) b c \,x^{2}+12 \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}-b x -2 c \right ) a c \,x^{2}+3 \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}-b x -2 c \right ) b^{2} x^{2}-12 \sqrt {c}\, \mathrm {log}\left (x \right ) a c \,x^{2}-3 \sqrt {c}\, \mathrm {log}\left (x \right ) b^{2} x^{2}}{8 c \,x^{2}} \] Input:

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

Output:

(8*sqrt(a*x**2 + b*x + c)*a*c*x**2 - 10*sqrt(a*x**2 + b*x + c)*b*c*x - 4*s 
qrt(a*x**2 + b*x + c)*c**2 + 12*sqrt(a)*log( - 2*sqrt(a)*sqrt(a*x**2 + b*x 
 + c) - 2*a*x - b)*b*c*x**2 + 12*sqrt(c)*log(2*sqrt(c)*sqrt(a*x**2 + b*x + 
 c) - b*x - 2*c)*a*c*x**2 + 3*sqrt(c)*log(2*sqrt(c)*sqrt(a*x**2 + b*x + c) 
 - b*x - 2*c)*b**2*x**2 - 12*sqrt(c)*log(x)*a*c*x**2 - 3*sqrt(c)*log(x)*b* 
*2*x**2)/(8*c*x**2)