Integrand size = 16, antiderivative size = 145 \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \, dx=-\frac {3}{4} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \left (3 b+\frac {2 c}{x}\right )+\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} x+\frac {3}{2} \sqrt {a} b \text {arctanh}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )-\frac {3 \left (b^2+4 a c\right ) \text {arctanh}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{8 \sqrt {c}} \]
(a+c/x^2+b/x)^(3/2)*x+3/2*b*arctanh(1/2*(2*a+b/x)/a^(1/2)/(a+c/x^2+b/x)^(1 /2))*a^(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)-3/4*(3*b+2*c/x)*(a+c/x^2+b/x)^(1/2)
Time = 0.58 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.09 \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \, dx=\frac {\sqrt {a+\frac {c+b x}{x^2}} \left (3 \left (b^2+4 a c\right ) x^2 \text {arctanh}\left (\frac {\sqrt {a} x-\sqrt {c+x (b+a x)}}{\sqrt {c}}\right )-\sqrt {c} \left ((2 c+x (5 b-4 a x)) \sqrt {c+x (b+a x)}+6 \sqrt {a} b x^2 \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+x (b+a x)}\right )\right )\right )}{4 \sqrt {c} x \sqrt {c+x (b+a x)}} \]
(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)])
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 |
\(\displaystyle x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}-\frac {3}{2} \left (\frac {1}{4} \left (\frac {\left (4 a c+b^2\right ) \text {arctanh}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )}{\sqrt {c}}-4 \sqrt {a} b \text {arctanh}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )\right )+\frac {1}{2} \left (3 b+\frac {2 c}{x}\right ) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}\right )\) |
(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
3.5.50.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
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])
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]
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]
Time = 0.07 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.23
method | result | size |
risch | \(-\frac {\left (5 b x +2 c \right ) \sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}}{4 x}+\frac {\left (a \sqrt {a \,x^{2}+b x +c}+\frac {3 \sqrt {a}\, b \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{2}-\frac {3 \sqrt {c}\, \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) a}{2}-\frac {3 \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) b^{2}}{8 \sqrt {c}}\right ) \sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}\, x}{\sqrt {a \,x^{2}+b x +c}}\) | \(179\) |
default | \(-\frac {\left (\frac {a \,x^{2}+b x +c}{x^{2}}\right )^{\frac {3}{2}} x \left (12 a^{\frac {5}{2}} c^{\frac {5}{2}} \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) x^{2}-2 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b \,x^{3}-4 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} c \,x^{2}-6 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x +c}\, b c \,x^{3}+3 a^{\frac {3}{2}} c^{\frac {3}{2}} \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) b^{2} x^{2}-12 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x +c}\, c^{2} x^{2}+2 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b x -2 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b^{2} x^{2}+4 \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} c \,a^{\frac {3}{2}}-6 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x +c}\, b^{2} c \,x^{2}-12 a^{2} \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b \,c^{2} x^{2}\right )}{8 \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} c^{2} a^{\frac {3}{2}}}\) | \(334\) |
-1/4*(5*b*x+2*c)/x*((a*x^2+b*x+c)/x^2)^(1/2)+(a*(a*x^2+b*x+c)^(1/2)+3/2*a^ (1/2)*b*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x+c)^(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*x^2+b*x+c)/x^2)^(1/2)*x/(a*x^2+b*x+c)^(1/2)
Time = 0.32 (sec) , antiderivative size = 709, normalized size of antiderivative = 4.89 \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \, dx=\left [\frac {12 \, \sqrt {a} b c x \log \left (-8 \, a^{2} x^{2} - 8 \, a b x - b^{2} - 4 \, a c - 4 \, {\left (2 \, a x^{2} + b x\right )} \sqrt {a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {c} x \log \left (-\frac {8 \, b c x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, c^{2} - 4 \, {\left (b x^{2} + 2 \, c x\right )} \sqrt {c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{x^{2}}\right ) + 4 \, {\left (4 \, a c x^{2} - 5 \, b c x - 2 \, c^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{16 \, c x}, -\frac {24 \, \sqrt {-a} b c x \arctan \left (\frac {{\left (2 \, a x^{2} + b x\right )} \sqrt {-a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a^{2} x^{2} + a b x + a c\right )}}\right ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {c} x \log \left (-\frac {8 \, b c x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, c^{2} - 4 \, {\left (b x^{2} + 2 \, c x\right )} \sqrt {c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{x^{2}}\right ) - 4 \, {\left (4 \, a c x^{2} - 5 \, b c x - 2 \, c^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{16 \, c x}, \frac {6 \, \sqrt {a} b c x \log \left (-8 \, a^{2} x^{2} - 8 \, a b x - b^{2} - 4 \, a c - 4 \, {\left (2 \, a x^{2} + b x\right )} \sqrt {a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-c} x \arctan \left (\frac {{\left (b x^{2} + 2 \, c x\right )} \sqrt {-c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a c x^{2} + b c x + c^{2}\right )}}\right ) + 2 \, {\left (4 \, a c x^{2} - 5 \, b c x - 2 \, c^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{8 \, c x}, -\frac {12 \, \sqrt {-a} b c x \arctan \left (\frac {{\left (2 \, a x^{2} + b x\right )} \sqrt {-a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a^{2} x^{2} + a b x + a c\right )}}\right ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-c} x \arctan \left (\frac {{\left (b x^{2} + 2 \, c x\right )} \sqrt {-c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a c x^{2} + b c x + c^{2}\right )}}\right ) - 2 \, {\left (4 \, a c x^{2} - 5 \, b c x - 2 \, c^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{8 \, c x}\right ] \]
[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)]
\[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \, dx=\int \left (a + \frac {b}{x} + \frac {c}{x^{2}}\right )^{\frac {3}{2}}\, dx \]
\[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \, dx=\int { {\left (a + \frac {b}{x} + \frac {c}{x^{2}}\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \, dx=\int {\left (a+\frac {b}{x}+\frac {c}{x^2}\right )}^{3/2} \,d x \]