Integrand size = 21, antiderivative size = 147 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )} \, dx=\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+\frac {b}{x}}}+\frac {x}{a c \sqrt {a+\frac {b}{x}}}+\frac {2 d^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^2 (b c-a d)^{3/2}}-\frac {(3 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2} c^2} \]
2*d^(5/2)*arctan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+b*c)^(1/2))/c^2/(-a*d+b*c)^(3 /2)-(2*a*d+3*b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(5/2)/c^2+b*(-a*d+3*b*c )/a^2/c/(-a*d+b*c)/(a+b/x)^(1/2)+x/a/c/(a+b/x)^(1/2)
Time = 0.68 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x \left (-3 b^2 c+a^2 d x+a b (d-c x)\right )}{a^2 (-b c+a d) (b+a x)}+\frac {2 d^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}-\frac {(3 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}}{c^2} \]
((c*Sqrt[a + b/x]*x*(-3*b^2*c + a^2*d*x + a*b*(d - c*x)))/(a^2*(-(b*c) + a *d)*(b + a*x)) + (2*d^(5/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d] ])/(b*c - a*d)^(3/2) - ((3*b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^ (5/2))/c^2
Time = 0.34 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.24, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {899, 114, 27, 169, 27, 174, 73, 218, 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.
| \(\displaystyle \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )} \, dx\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\int \frac {x^2}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {\int \frac {\left (3 b c+2 a d+\frac {3 b d}{x}\right ) x}{2 \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (3 b c+2 a d+\frac {3 b d}{x}\right ) x}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\frac {2 \int \frac {\left (\frac {b d (3 b c-a d)}{x}+(b c-a d) (3 b c+2 a d)\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a (b c-a d)}+\frac {2 b (3 b c-a d)}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (\frac {b d (3 b c-a d)}{x}+(b c-a d) (3 b c+2 a d)\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a (b c-a d)}+\frac {2 b (3 b c-a d)}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}}}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {\frac {\frac {2 a^2 d^3 \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}+\frac {(b c-a d) (2 a d+3 b c) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}}{a (b c-a d)}+\frac {2 b (3 b c-a d)}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {4 a^2 d^3 \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}+\frac {2 (b c-a d) (2 a d+3 b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}}{a (b c-a d)}+\frac {2 b (3 b c-a d)}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {2 (b c-a d) (2 a d+3 b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}+\frac {4 a^2 d^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}}{a (b c-a d)}+\frac {2 b (3 b c-a d)}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {4 a^2 d^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (b c-a d) (2 a d+3 b c)}{\sqrt {a} c}}{a (b c-a d)}+\frac {2 b (3 b c-a d)}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{2 a c}+\frac {x}{a c \sqrt {a+\frac {b}{x}}}\) |
x/(a*c*Sqrt[a + b/x]) + ((2*b*(3*b*c - a*d))/(a*(b*c - a*d)*Sqrt[a + b/x]) + ((4*a^2*d^(5/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(c*Sqr t[b*c - a*d]) - (2*(b*c - a*d)*(3*b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[ a]])/(Sqrt[a]*c))/(a*(b*c - a*d)))/(2*a*c)
3.3.56.3.1 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_))^(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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(296\) vs. \(2(127)=254\).
Time = 0.29 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.02
| method | result | size |
| risch | \(\frac {a x +b}{a^{2} c \sqrt {\frac {a x +b}{x}}}-\frac {\left (\frac {\left (2 a d +3 b c \right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{c \sqrt {a}}+\frac {4 c \,b^{2} \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{\left (a d -b c \right ) a \left (x +\frac {b}{a}\right )}+\frac {2 a^{2} d^{3} \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{c^{2} \left (a d -b c \right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right ) \sqrt {x \left (a x +b \right )}}{2 a^{2} c x \sqrt {\frac {a x +b}{x}}}\) | \(297\) |
| default | \(-\frac {\left (2 a^{\frac {9}{2}} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c -2 a d x +b c x -b d}{c x +d}\right ) d^{3} x^{2}-2 a^{\frac {7}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c^{2} d \,x^{2}+4 a^{\frac {7}{2}} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c -2 a d x +b c x -b d}{c x +d}\right ) b \,d^{3} x +6 a^{\frac {5}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, b \,c^{3} x^{2}-4 a^{\frac {5}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, b \,c^{2} d x +2 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c -2 a d x +b c x -b d}{c x +d}\right ) b^{2} d^{3}-4 a^{\frac {3}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b \,c^{3}+12 a^{\frac {3}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, b^{2} c^{3} x +2 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{4} c \,d^{2} x^{2}+\ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{3} b \,c^{2} d \,x^{2}-3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{2} b^{2} c^{3} x^{2}-2 a^{\frac {3}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, b^{2} c^{2} d +4 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{3} b c \,d^{2} x +2 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{2} b^{2} c^{2} d x -6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a \,b^{3} c^{3} x +6 \sqrt {a}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, b^{3} c^{3}+2 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{2} b^{2} c \,d^{2}+\ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a \,b^{3} c^{2} d -3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b^{4} c^{3}\right ) x \sqrt {\frac {a x +b}{x}}}{2 a^{\frac {5}{2}} \left (a x +b \right )^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c^{3} \left (a d -b c \right ) \sqrt {x \left (a x +b \right )}}\) | \(962\) |
1/a^2/c*(a*x+b)/((a*x+b)/x)^(1/2)-1/2/a^2/c*((2*a*d+3*b*c)/c*ln((1/2*b+a*x )/a^(1/2)+(a*x^2+b*x)^(1/2))/a^(1/2)+4*c*b^2/(a*d-b*c)/a/(x+b/a)*(a*(x+b/a )^2-b*(x+b/a))^(1/2)+2/c^2*a^2*d^3/(a*d-b*c)/((a*d-b*c)*d/c^2)^(1/2)*ln((2 *(a*d-b*c)*d/c^2-(2*a*d-b*c)/c*(x+d/c)+2*((a*d-b*c)*d/c^2)^(1/2)*(a*(x+d/c )^2-(2*a*d-b*c)/c*(x+d/c)+(a*d-b*c)*d/c^2)^(1/2))/(x+d/c)))/x/((a*x+b)/x)^ (1/2)*(x*(a*x+b))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (127) = 254\).
Time = 0.35 (sec) , antiderivative size = 1075, normalized size of antiderivative = 7.31 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )} \, dx=\left [\frac {{\left (3 \, b^{3} c^{2} - a b^{2} c d - 2 \, a^{2} b d^{2} + {\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (a^{4} d^{2} x + a^{3} b d^{2}\right )} \sqrt {-\frac {d}{b c - a d}} \log \left (-\frac {2 \, {\left (b c - a d\right )} x \sqrt {-\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}} - b d + {\left (b c - 2 \, a d\right )} x}{c x + d}\right ) + 2 \, {\left ({\left (a^{2} b c^{2} - a^{3} c d\right )} x^{2} + {\left (3 \, a b^{2} c^{2} - a^{2} b c d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x\right )}}, \frac {{\left (3 \, b^{3} c^{2} - a b^{2} c d - 2 \, a^{2} b d^{2} + {\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) - {\left (a^{4} d^{2} x + a^{3} b d^{2}\right )} \sqrt {-\frac {d}{b c - a d}} \log \left (-\frac {2 \, {\left (b c - a d\right )} x \sqrt {-\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}} - b d + {\left (b c - 2 \, a d\right )} x}{c x + d}\right ) + {\left ({\left (a^{2} b c^{2} - a^{3} c d\right )} x^{2} + {\left (3 \, a b^{2} c^{2} - a^{2} b c d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{a^{3} b^{2} c^{3} - a^{4} b c^{2} d + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x}, \frac {4 \, {\left (a^{4} d^{2} x + a^{3} b d^{2}\right )} \sqrt {\frac {d}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} x \sqrt {\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}}}{a d x + b d}\right ) + {\left (3 \, b^{3} c^{2} - a b^{2} c d - 2 \, a^{2} b d^{2} + {\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left ({\left (a^{2} b c^{2} - a^{3} c d\right )} x^{2} + {\left (3 \, a b^{2} c^{2} - a^{2} b c d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x\right )}}, \frac {2 \, {\left (a^{4} d^{2} x + a^{3} b d^{2}\right )} \sqrt {\frac {d}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} x \sqrt {\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}}}{a d x + b d}\right ) + {\left (3 \, b^{3} c^{2} - a b^{2} c d - 2 \, a^{2} b d^{2} + {\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left ({\left (a^{2} b c^{2} - a^{3} c d\right )} x^{2} + {\left (3 \, a b^{2} c^{2} - a^{2} b c d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{a^{3} b^{2} c^{3} - a^{4} b c^{2} d + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x}\right ] \]
[1/2*((3*b^3*c^2 - a*b^2*c*d - 2*a^2*b*d^2 + (3*a*b^2*c^2 - a^2*b*c*d - 2* a^3*d^2)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2*(a^ 4*d^2*x + a^3*b*d^2)*sqrt(-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sqrt(-d/(b *c - a*d))*sqrt((a*x + b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x + d)) + 2*((a^2 *b*c^2 - a^3*c*d)*x^2 + (3*a*b^2*c^2 - a^2*b*c*d)*x)*sqrt((a*x + b)/x))/(a ^3*b^2*c^3 - a^4*b*c^2*d + (a^4*b*c^3 - a^5*c^2*d)*x), ((3*b^3*c^2 - a*b^2 *c*d - 2*a^2*b*d^2 + (3*a*b^2*c^2 - a^2*b*c*d - 2*a^3*d^2)*x)*sqrt(-a)*arc tan(sqrt(-a)*sqrt((a*x + b)/x)/a) - (a^4*d^2*x + a^3*b*d^2)*sqrt(-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sqrt(-d/(b*c - a*d))*sqrt((a*x + b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x + d)) + ((a^2*b*c^2 - a^3*c*d)*x^2 + (3*a*b^2*c^2 - a^2*b*c*d)*x)*sqrt((a*x + b)/x))/(a^3*b^2*c^3 - a^4*b*c^2*d + (a^4*b*c^3 - a^5*c^2*d)*x), 1/2*(4*(a^4*d^2*x + a^3*b*d^2)*sqrt(d/(b*c - a*d))*arcta n(-(b*c - a*d)*x*sqrt(d/(b*c - a*d))*sqrt((a*x + b)/x)/(a*d*x + b*d)) + (3 *b^3*c^2 - a*b^2*c*d - 2*a^2*b*d^2 + (3*a*b^2*c^2 - a^2*b*c*d - 2*a^3*d^2) *x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*((a^2*b*c^2 - a^3*c*d)*x^2 + (3*a*b^2*c^2 - a^2*b*c*d)*x)*sqrt((a*x + b)/x))/(a^3*b^2 *c^3 - a^4*b*c^2*d + (a^4*b*c^3 - a^5*c^2*d)*x), (2*(a^4*d^2*x + a^3*b*d^2 )*sqrt(d/(b*c - a*d))*arctan(-(b*c - a*d)*x*sqrt(d/(b*c - a*d))*sqrt((a*x + b)/x)/(a*d*x + b*d)) + (3*b^3*c^2 - a*b^2*c*d - 2*a^2*b*d^2 + (3*a*b^2*c ^2 - a^2*b*c*d - 2*a^3*d^2)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/...
\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )} \, dx=\int \frac {x}{\left (a + \frac {b}{x}\right )^{\frac {3}{2}} \left (c x + d\right )}\, dx \]
\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} {\left (c + \frac {d}{x}\right )}} \,d x } \]
Exception generated. \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Time = 6.75 (sec) , antiderivative size = 3000, normalized size of antiderivative = 20.41 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )} \, dx=\text {Too large to display} \]
(atan((((d^5*(a*d - b*c)^3)^(1/2)*((a + b/x)^(1/2)*(18*a^6*b^9*c^10*d^3 - 66*a^7*b^8*c^9*d^4 + 68*a^8*b^7*c^8*d^5 + 20*a^9*b^6*c^7*d^6 - 62*a^10*b^5 *c^6*d^7 - 2*a^11*b^4*c^5*d^8 + 40*a^12*b^3*c^4*d^9 - 16*a^13*b^2*c^3*d^10 ) + ((d^5*(a*d - b*c)^3)^(1/2)*(64*a^9*b^8*c^11*d^3 - 12*a^8*b^9*c^12*d^2 - 132*a^10*b^7*c^10*d^4 + 128*a^11*b^6*c^9*d^5 - 52*a^12*b^5*c^8*d^6 + 4*a ^14*b^3*c^6*d^8 + ((d^5*(a*d - b*c)^3)^(1/2)*(a + b/x)^(1/2)*(8*a^10*b^8*c ^13*d^2 - 56*a^11*b^7*c^12*d^3 + 160*a^12*b^6*c^11*d^4 - 240*a^13*b^5*c^10 *d^5 + 200*a^14*b^4*c^9*d^6 - 88*a^15*b^3*c^8*d^7 + 16*a^16*b^2*c^7*d^8))/ (c^2*(a*d - b*c)^3)))/(c^2*(a*d - b*c)^3))*1i)/(c^2*(a*d - b*c)^3) + ((d^5 *(a*d - b*c)^3)^(1/2)*((a + b/x)^(1/2)*(18*a^6*b^9*c^10*d^3 - 66*a^7*b^8*c ^9*d^4 + 68*a^8*b^7*c^8*d^5 + 20*a^9*b^6*c^7*d^6 - 62*a^10*b^5*c^6*d^7 - 2 *a^11*b^4*c^5*d^8 + 40*a^12*b^3*c^4*d^9 - 16*a^13*b^2*c^3*d^10) + ((d^5*(a *d - b*c)^3)^(1/2)*(12*a^8*b^9*c^12*d^2 - 64*a^9*b^8*c^11*d^3 + 132*a^10*b ^7*c^10*d^4 - 128*a^11*b^6*c^9*d^5 + 52*a^12*b^5*c^8*d^6 - 4*a^14*b^3*c^6* d^8 + ((d^5*(a*d - b*c)^3)^(1/2)*(a + b/x)^(1/2)*(8*a^10*b^8*c^13*d^2 - 56 *a^11*b^7*c^12*d^3 + 160*a^12*b^6*c^11*d^4 - 240*a^13*b^5*c^10*d^5 + 200*a ^14*b^4*c^9*d^6 - 88*a^15*b^3*c^8*d^7 + 16*a^16*b^2*c^7*d^8))/(c^2*(a*d - b*c)^3)))/(c^2*(a*d - b*c)^3))*1i)/(c^2*(a*d - b*c)^3))/(36*a^6*b^8*c^7*d^ 5 - 96*a^7*b^7*c^6*d^6 + 64*a^8*b^6*c^5*d^7 + 24*a^9*b^5*c^4*d^8 - 36*a^10 *b^4*c^3*d^9 + 8*a^11*b^3*c^2*d^10 - ((d^5*(a*d - b*c)^3)^(1/2)*((a + b...