Integrand size = 21, antiderivative size = 134 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{c+\frac {d}{x}} \, dx=-\frac {b (2 b c+a d) \sqrt {a+\frac {b}{x}}}{c d}+\frac {a \left (a+\frac {b}{x}\right )^{3/2} x}{c}+\frac {2 (b c-a d)^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^2 d^{3/2}}+\frac {a^{3/2} (5 b c-2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^2} \]
a*(a+b/x)^(3/2)*x/c+2*(-a*d+b*c)^(5/2)*arctan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+ b*c)^(1/2))/c^2/d^(3/2)+a^(3/2)*(-2*a*d+5*b*c)*arctanh((a+b/x)^(1/2)/a^(1/ 2))/c^2-b*(a*d+2*b*c)*(a+b/x)^(1/2)/c/d
Time = 0.38 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{c+\frac {d}{x}} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} \left (-2 b^2 c+a^2 d x\right )}{d}+\frac {2 (b c-a d)^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{d^{3/2}}-a^{3/2} (-5 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^2} \]
((c*Sqrt[a + b/x]*(-2*b^2*c + a^2*d*x))/d + (2*(b*c - a*d)^(5/2)*ArcTan[(S qrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/d^(3/2) - a^(3/2)*(-5*b*c + 2*a*d) *ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/c^2
Time = 0.35 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {899, 109, 27, 171, 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.
| \(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{c+\frac {d}{x}} \, dx\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^{5/2} x^2}{c+\frac {d}{x}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\int -\frac {\sqrt {a+\frac {b}{x}} \left (a (5 b c-2 a d)+\frac {b (2 b c+a d)}{x}\right ) x}{2 \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}+\frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c}-\frac {\int \frac {\sqrt {a+\frac {b}{x}} \left (a (5 b c-2 a d)+\frac {b (2 b c+a d)}{x}\right ) x}{c+\frac {d}{x}}d\frac {1}{x}}{2 c}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c}-\frac {\frac {2 \int \frac {\left (a^2 d (5 b c-2 a d)-\frac {b \left (2 b^2 c^2-6 a b d c+a^2 d^2\right )}{x}\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{d}+\frac {2 b \sqrt {a+\frac {b}{x}} (a d+2 b c)}{d}}{2 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c}-\frac {\frac {\int \frac {\left (a^2 d (5 b c-2 a d)-\frac {b \left (2 b^2 c^2-6 a b d c+a^2 d^2\right )}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{d}+\frac {2 b \sqrt {a+\frac {b}{x}} (a d+2 b c)}{d}}{2 c}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c}-\frac {\frac {\frac {a^2 d (5 b c-2 a d) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}-\frac {2 (b c-a d)^3 \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}}{d}+\frac {2 b \sqrt {a+\frac {b}{x}} (a d+2 b c)}{d}}{2 c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c}-\frac {\frac {\frac {2 a^2 d (5 b c-2 a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {4 (b c-a d)^3 \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}}{d}+\frac {2 b \sqrt {a+\frac {b}{x}} (a d+2 b c)}{d}}{2 c}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c}-\frac {\frac {\frac {2 a^2 d (5 b c-2 a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {4 (b c-a d)^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {d}}}{d}+\frac {2 b \sqrt {a+\frac {b}{x}} (a d+2 b c)}{d}}{2 c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c}-\frac {\frac {-\frac {2 a^{3/2} d \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (5 b c-2 a d)}{c}-\frac {4 (b c-a d)^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {d}}}{d}+\frac {2 b \sqrt {a+\frac {b}{x}} (a d+2 b c)}{d}}{2 c}\) |
(a*(a + b/x)^(3/2)*x)/c - ((2*b*(2*b*c + a*d)*Sqrt[a + b/x])/d + ((-4*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(c*Sqrt[d]) - (2*a^(3/2)*d*(5*b*c - 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/c)/d)/(2*c )
3.3.42.3.1 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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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] && 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. \(278\) vs. \(2(114)=228\).
Time = 0.25 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.08
| method | result | size |
| risch | \(\frac {\left (a^{2} d x -2 b^{2} c \right ) \sqrt {\frac {a x +b}{x}}}{d c}-\frac {\left (\frac {a^{\frac {3}{2}} d \left (2 a d -5 b c \right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{c}-\frac {\left (-2 a^{3} d^{3}+6 a^{2} b c \,d^{2}-6 a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{c^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 c d \left (a x +b \right )}\) | \(279\) |
| default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (2 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 ) d^{4} x^{2}-2 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, c^{2} d^{2} x^{2}-6 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 c \,d^{3} x^{2}+4 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b \,c^{3} d \,x^{2}+6 a^{\frac {3}{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} c^{2} d^{2} x^{2}-8 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b \,c^{3} d \,x^{2}-2 \sqrt {a}\, \sqrt {x \left (a x +b \right )}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b^{2} c^{4} x^{2}-2 \sqrt {a}\, \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^{3} c^{3} d \,x^{2}+2 \sqrt {a}\, \sqrt {a \,x^{2}+b x}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b^{2} c^{4} x^{2}-4 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a \,b^{2} c^{3} d \,x^{2}+\ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b^{3} c^{4} x^{2}+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} c \,d^{3} x^{2}-5 \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 \,c^{2} d^{2} x^{2}+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 \,b^{2} c^{3} d \,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}}}\, b^{3} c^{4} x^{2}+4 \sqrt {a}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b \,c^{3} d \right )}{2 x \sqrt {x \left (a x +b \right )}\, d^{2} \sqrt {a}\, c^{3} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\) | \(859\) |
(a^2*d*x-2*b^2*c)/d/c*((a*x+b)/x)^(1/2)-1/2/c/d*(a^(3/2)*d*(2*a*d-5*b*c)/c *ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))-(-2*a^3*d^3+6*a^2*b*c*d^2-6*a*b ^2*c^2*d+2*b^3*c^3)/c^2/((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)))*((a*x+b)/x)^(1/2)*(x*(a*x+b))^(1/2) /(a*x+b)
Time = 0.38 (sec) , antiderivative size = 659, normalized size of antiderivative = 4.92 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{c+\frac {d}{x}} \, dx=\left [-\frac {{\left (5 \, a b c d - 2 \, a^{2} d^{2}\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-\frac {b c - a d}{d}} \log \left (\frac {2 \, d x \sqrt {-\frac {b c - a d}{d}} \sqrt {\frac {a x + b}{x}} + b d - {\left (b c - 2 \, a d\right )} x}{c x + d}\right ) - 2 \, {\left (a^{2} c d x - 2 \, b^{2} c^{2}\right )} \sqrt {\frac {a x + b}{x}}}{2 \, c^{2} d}, -\frac {{\left (5 \, a b c d - 2 \, a^{2} d^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-\frac {b c - a d}{d}} \log \left (\frac {2 \, d x \sqrt {-\frac {b c - a d}{d}} \sqrt {\frac {a x + b}{x}} + b d - {\left (b c - 2 \, a d\right )} x}{c x + d}\right ) - {\left (a^{2} c d x - 2 \, b^{2} c^{2}\right )} \sqrt {\frac {a x + b}{x}}}{c^{2} d}, -\frac {4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {\frac {b c - a d}{d}} \arctan \left (-\frac {d \sqrt {\frac {b c - a d}{d}} \sqrt {\frac {a x + b}{x}}}{b c - a d}\right ) + {\left (5 \, a b c d - 2 \, a^{2} d^{2}\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (a^{2} c d x - 2 \, b^{2} c^{2}\right )} \sqrt {\frac {a x + b}{x}}}{2 \, c^{2} d}, -\frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {\frac {b c - a d}{d}} \arctan \left (-\frac {d \sqrt {\frac {b c - a d}{d}} \sqrt {\frac {a x + b}{x}}}{b c - a d}\right ) + {\left (5 \, a b c d - 2 \, a^{2} d^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) - {\left (a^{2} c d x - 2 \, b^{2} c^{2}\right )} \sqrt {\frac {a x + b}{x}}}{c^{2} d}\right ] \]
[-1/2*((5*a*b*c*d - 2*a^2*d^2)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-(b*c - a*d)/d)*log(( 2*d*x*sqrt(-(b*c - a*d)/d)*sqrt((a*x + b)/x) + b*d - (b*c - 2*a*d)*x)/(c*x + d)) - 2*(a^2*c*d*x - 2*b^2*c^2)*sqrt((a*x + b)/x))/(c^2*d), -((5*a*b*c* d - 2*a^2*d^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-(b*c - a*d)/d)*log((2*d*x*sqrt(-(b*c - a*d)/d)* sqrt((a*x + b)/x) + b*d - (b*c - 2*a*d)*x)/(c*x + d)) - (a^2*c*d*x - 2*b^2 *c^2)*sqrt((a*x + b)/x))/(c^2*d), -1/2*(4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)* sqrt((b*c - a*d)/d)*arctan(-d*sqrt((b*c - a*d)/d)*sqrt((a*x + b)/x)/(b*c - a*d)) + (5*a*b*c*d - 2*a^2*d^2)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2*(a^2*c*d*x - 2*b^2*c^2)*sqrt((a*x + b)/x))/(c^2*d), -(2* (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt((b*c - a*d)/d)*arctan(-d*sqrt((b*c - a*d)/d)*sqrt((a*x + b)/x)/(b*c - a*d)) + (5*a*b*c*d - 2*a^2*d^2)*sqrt(-a)* arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) - (a^2*c*d*x - 2*b^2*c^2)*sqrt((a*x + b)/x))/(c^2*d)]
\[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{c+\frac {d}{x}} \, dx=\int \frac {x \left (a + \frac {b}{x}\right )^{\frac {5}{2}}}{c x + d}\, dx \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{c+\frac {d}{x}} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}}}{c + \frac {d}{x}} \,d x } \]
Exception generated. \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{c+\frac {d}{x}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Time = 6.26 (sec) , antiderivative size = 1427, normalized size of antiderivative = 10.65 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{c+\frac {d}{x}} \, dx=\frac {a^2\,b\,d\,\sqrt {a+\frac {b}{x}}}{c\,\left (d\,\left (a+\frac {b}{x}\right )-a\,d\right )}-\frac {2\,b^2\,\sqrt {a+\frac {b}{x}}}{d}+\frac {\mathrm {atan}\left (\frac {a^3\,b^5\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^5\,d^8-5\,a^4\,b\,c\,d^7+10\,a^3\,b^2\,c^2\,d^6-10\,a^2\,b^3\,c^3\,d^5+5\,a\,b^4\,c^4\,d^4-b^5\,c^5\,d^3}\,160{}\mathrm {i}}{448\,a^3\,b^8\,c^3\,d-340\,a^6\,b^5\,d^4-128\,a^2\,b^9\,c^4+740\,a^5\,b^6\,c\,d^3+\frac {16\,a\,b^{10}\,c^5}{d}-796\,a^4\,b^7\,c^2\,d^2+\frac {60\,a^7\,b^4\,d^5}{c}}-\frac {a^2\,b^6\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^5\,d^8-5\,a^4\,b\,c\,d^7+10\,a^3\,b^2\,c^2\,d^6-10\,a^2\,b^3\,c^3\,d^5+5\,a\,b^4\,c^4\,d^4-b^5\,c^5\,d^3}\,80{}\mathrm {i}}{16\,a\,b^{10}\,c^4+740\,a^5\,b^6\,d^4-128\,a^2\,b^9\,c^3\,d-796\,a^4\,b^7\,c\,d^3+448\,a^3\,b^8\,c^2\,d^2-\frac {340\,a^6\,b^5\,d^5}{c}+\frac {60\,a^7\,b^4\,d^6}{c^2}}-\frac {a^4\,b^4\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^5\,d^8-5\,a^4\,b\,c\,d^7+10\,a^3\,b^2\,c^2\,d^6-10\,a^2\,b^3\,c^3\,d^5+5\,a\,b^4\,c^4\,d^4-b^5\,c^5\,d^3}\,60{}\mathrm {i}}{448\,a^3\,b^8\,c^4+60\,a^7\,b^4\,d^4-796\,a^4\,b^7\,c^3\,d-340\,a^6\,b^5\,c\,d^3+\frac {16\,a\,b^{10}\,c^6}{d^2}+740\,a^5\,b^6\,c^2\,d^2-\frac {128\,a^2\,b^9\,c^5}{d}}+\frac {a\,b^7\,c\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^5\,d^8-5\,a^4\,b\,c\,d^7+10\,a^3\,b^2\,c^2\,d^6-10\,a^2\,b^3\,c^3\,d^5+5\,a\,b^4\,c^4\,d^4-b^5\,c^5\,d^3}\,16{}\mathrm {i}}{740\,a^5\,b^6\,d^5-796\,a^4\,b^7\,c\,d^4-128\,a^2\,b^9\,c^3\,d^2+448\,a^3\,b^8\,c^2\,d^3-\frac {340\,a^6\,b^5\,d^6}{c}+\frac {60\,a^7\,b^4\,d^7}{c^2}+16\,a\,b^{10}\,c^4\,d}\right )\,\sqrt {d^3\,{\left (a\,d-b\,c\right )}^5}\,2{}\mathrm {i}}{c^2\,d^3}+\frac {\mathrm {atan}\left (\frac {b^9\,c^3\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}\,40{}\mathrm {i}}{40\,a^2\,b^9\,c^3-790\,a^5\,b^6\,d^3-256\,a^3\,b^8\,c^2\,d+696\,a^4\,b^7\,c\,d^2+\frac {370\,a^6\,b^5\,d^4}{c}-\frac {60\,a^7\,b^4\,d^5}{c^2}}+\frac {a\,b^8\,c^2\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}\,256{}\mathrm {i}}{256\,a^3\,b^8\,c^2+790\,a^5\,b^6\,d^2-\frac {40\,a^2\,b^9\,c^3}{d}-\frac {370\,a^6\,b^5\,d^3}{c}+\frac {60\,a^7\,b^4\,d^4}{c^2}-696\,a^4\,b^7\,c\,d}+\frac {a^3\,b^6\,d^2\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}\,790{}\mathrm {i}}{256\,a^3\,b^8\,c^2+790\,a^5\,b^6\,d^2-\frac {40\,a^2\,b^9\,c^3}{d}-\frac {370\,a^6\,b^5\,d^3}{c}+\frac {60\,a^7\,b^4\,d^4}{c^2}-696\,a^4\,b^7\,c\,d}-\frac {a^4\,b^5\,d^3\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}\,370{}\mathrm {i}}{256\,a^3\,b^8\,c^3-370\,a^6\,b^5\,d^3-696\,a^4\,b^7\,c^2\,d+790\,a^5\,b^6\,c\,d^2-\frac {40\,a^2\,b^9\,c^4}{d}+\frac {60\,a^7\,b^4\,d^4}{c}}+\frac {a^5\,b^4\,d^4\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}\,60{}\mathrm {i}}{256\,a^3\,b^8\,c^4+60\,a^7\,b^4\,d^4-696\,a^4\,b^7\,c^3\,d-370\,a^6\,b^5\,c\,d^3+790\,a^5\,b^6\,c^2\,d^2-\frac {40\,a^2\,b^9\,c^5}{d}}-\frac {a^2\,b^7\,c\,d\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}\,696{}\mathrm {i}}{256\,a^3\,b^8\,c^2+790\,a^5\,b^6\,d^2-\frac {40\,a^2\,b^9\,c^3}{d}-\frac {370\,a^6\,b^5\,d^3}{c}+\frac {60\,a^7\,b^4\,d^4}{c^2}-696\,a^4\,b^7\,c\,d}\right )\,\left (2\,a\,d-5\,b\,c\right )\,\sqrt {a^3}\,1{}\mathrm {i}}{c^2} \]
(atan((a^3*b^5*(a + b/x)^(1/2)*(a^5*d^8 - b^5*c^5*d^3 + 5*a*b^4*c^4*d^4 - 10*a^2*b^3*c^3*d^5 + 10*a^3*b^2*c^2*d^6 - 5*a^4*b*c*d^7)^(1/2)*160i)/(448* a^3*b^8*c^3*d - 340*a^6*b^5*d^4 - 128*a^2*b^9*c^4 + 740*a^5*b^6*c*d^3 + (1 6*a*b^10*c^5)/d - 796*a^4*b^7*c^2*d^2 + (60*a^7*b^4*d^5)/c) - (a^2*b^6*(a + b/x)^(1/2)*(a^5*d^8 - b^5*c^5*d^3 + 5*a*b^4*c^4*d^4 - 10*a^2*b^3*c^3*d^5 + 10*a^3*b^2*c^2*d^6 - 5*a^4*b*c*d^7)^(1/2)*80i)/(16*a*b^10*c^4 + 740*a^5 *b^6*d^4 - 128*a^2*b^9*c^3*d - 796*a^4*b^7*c*d^3 + 448*a^3*b^8*c^2*d^2 - ( 340*a^6*b^5*d^5)/c + (60*a^7*b^4*d^6)/c^2) - (a^4*b^4*(a + b/x)^(1/2)*(a^5 *d^8 - b^5*c^5*d^3 + 5*a*b^4*c^4*d^4 - 10*a^2*b^3*c^3*d^5 + 10*a^3*b^2*c^2 *d^6 - 5*a^4*b*c*d^7)^(1/2)*60i)/(448*a^3*b^8*c^4 + 60*a^7*b^4*d^4 - 796*a ^4*b^7*c^3*d - 340*a^6*b^5*c*d^3 + (16*a*b^10*c^6)/d^2 + 740*a^5*b^6*c^2*d ^2 - (128*a^2*b^9*c^5)/d) + (a*b^7*c*(a + b/x)^(1/2)*(a^5*d^8 - b^5*c^5*d^ 3 + 5*a*b^4*c^4*d^4 - 10*a^2*b^3*c^3*d^5 + 10*a^3*b^2*c^2*d^6 - 5*a^4*b*c* d^7)^(1/2)*16i)/(740*a^5*b^6*d^5 - 796*a^4*b^7*c*d^4 - 128*a^2*b^9*c^3*d^2 + 448*a^3*b^8*c^2*d^3 - (340*a^6*b^5*d^6)/c + (60*a^7*b^4*d^7)/c^2 + 16*a *b^10*c^4*d))*(d^3*(a*d - b*c)^5)^(1/2)*2i)/(c^2*d^3) - (2*b^2*(a + b/x)^( 1/2))/d + (atan((b^9*c^3*(a + b/x)^(1/2)*(a^3)^(1/2)*40i)/(40*a^2*b^9*c^3 - 790*a^5*b^6*d^3 - 256*a^3*b^8*c^2*d + 696*a^4*b^7*c*d^2 + (370*a^6*b^5*d ^4)/c - (60*a^7*b^4*d^5)/c^2) + (a*b^8*c^2*(a + b/x)^(1/2)*(a^3)^(1/2)*256 i)/(256*a^3*b^8*c^2 + 790*a^5*b^6*d^2 - (40*a^2*b^9*c^3)/d - (370*a^6*b...