Integrand size = 21, antiderivative size = 166 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^2} \, dx=\frac {(b c-2 a d) (b c-a d) \sqrt {a+\frac {b}{x}}}{c^2 d \left (c+\frac {d}{x}\right )}+\frac {a \left (a+\frac {b}{x}\right )^{3/2} x}{c \left (c+\frac {d}{x}\right )}-\frac {(b c-a d)^{3/2} (b c+4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 d^{3/2}}+\frac {a^{3/2} (5 b c-4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^3} \]
a*(a+b/x)^(3/2)*x/c/(c+d/x)-(-a*d+b*c)^(3/2)*(4*a*d+b*c)*arctan(d^(1/2)*(a +b/x)^(1/2)/(-a*d+b*c)^(1/2))/c^3/d^(3/2)+a^(3/2)*(-4*a*d+5*b*c)*arctanh(( a+b/x)^(1/2)/a^(1/2))/c^3+(-2*a*d+b*c)*(-a*d+b*c)*(a+b/x)^(1/2)/c^2/d/(c+d /x)
Time = 0.48 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^2} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x \left (b^2 c^2-2 a b c d+a^2 d (2 d+c x)\right )}{d (d+c x)}-\frac {(b c-a d)^{3/2} (b c+4 a d) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{d^{3/2}}-a^{3/2} (-5 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^3} \]
((c*Sqrt[a + b/x]*x*(b^2*c^2 - 2*a*b*c*d + a^2*d*(2*d + c*x)))/(d*(d + c*x )) - ((b*c - a*d)^(3/2)*(b*c + 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[ b*c - a*d]])/d^(3/2) - a^(3/2)*(-5*b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt [a]])/c^3
Time = 0.38 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.12, 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, 166, 25, 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.
| \(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^2} \, dx\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^{5/2} x^2}{\left (c+\frac {d}{x}\right )^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\int -\frac {\sqrt {a+\frac {b}{x}} \left (a (5 b c-4 a d)+\frac {b (2 b c-a d)}{x}\right ) x}{2 \left (c+\frac {d}{x}\right )^2}d\frac {1}{x}}{c}+\frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )}-\frac {\int \frac {\sqrt {a+\frac {b}{x}} \left (a (5 b c-4 a d)+\frac {b (2 b c-a d)}{x}\right ) x}{\left (c+\frac {d}{x}\right )^2}d\frac {1}{x}}{2 c}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )}-\frac {\frac {2 \sqrt {a+\frac {b}{x}} \left (-\frac {2 a^2 d}{c}+3 a b-\frac {b^2 c}{d}\right )}{c+\frac {d}{x}}-\frac {\int -\frac {\left (d (5 b c-4 a d) a^2+\frac {b \left (b^2 c^2+2 a b d c-2 a^2 d^2\right )}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c d}}{2 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )}-\frac {\frac {\int \frac {\left (d (5 b c-4 a d) a^2+\frac {b \left (b^2 c^2+2 a b d c-2 a^2 d^2\right )}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c d}+\frac {2 \sqrt {a+\frac {b}{x}} \left (-\frac {2 a^2 d}{c}+3 a b-\frac {b^2 c}{d}\right )}{c+\frac {d}{x}}}{2 c}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )}-\frac {\frac {\frac {a^2 d (5 b c-4 a d) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}+\frac {(b c-a d)^2 (4 a d+b c) \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}}{c d}+\frac {2 \sqrt {a+\frac {b}{x}} \left (-\frac {2 a^2 d}{c}+3 a b-\frac {b^2 c}{d}\right )}{c+\frac {d}{x}}}{2 c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )}-\frac {\frac {\frac {2 a^2 d (5 b c-4 a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}+\frac {2 (b c-a d)^2 (4 a d+b c) \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}}{c d}+\frac {2 \sqrt {a+\frac {b}{x}} \left (-\frac {2 a^2 d}{c}+3 a b-\frac {b^2 c}{d}\right )}{c+\frac {d}{x}}}{2 c}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )}-\frac {\frac {\frac {2 a^2 d (5 b c-4 a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}+\frac {2 (b c-a d)^{3/2} (4 a d+b c) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {d}}}{c d}+\frac {2 \sqrt {a+\frac {b}{x}} \left (-\frac {2 a^2 d}{c}+3 a b-\frac {b^2 c}{d}\right )}{c+\frac {d}{x}}}{2 c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )}-\frac {\frac {\frac {2 (b c-a d)^{3/2} (4 a d+b c) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {d}}-\frac {2 a^{3/2} d \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (5 b c-4 a d)}{c}}{c d}+\frac {2 \sqrt {a+\frac {b}{x}} \left (-\frac {2 a^2 d}{c}+3 a b-\frac {b^2 c}{d}\right )}{c+\frac {d}{x}}}{2 c}\) |
(a*(a + b/x)^(3/2)*x)/(c*(c + d/x)) - ((2*(3*a*b - (b^2*c)/d - (2*a^2*d)/c )*Sqrt[a + b/x])/(c + d/x) + ((2*(b*c - a*d)^(3/2)*(b*c + 4*a*d)*ArcTan[(S qrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(c*Sqrt[d]) - (2*a^(3/2)*d*(5*b*c - 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/c)/(c*d))/(2*c)
3.3.43.3.1 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[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((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 - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
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. \(516\) vs. \(2(146)=292\).
Time = 0.28 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.11
| method | result | size |
| risch | \(\frac {a^{2} x \sqrt {\frac {a x +b}{x}}}{c^{2}}-\frac {\left (\frac {a^{\frac {3}{2}} \left (4 a d -5 b c \right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{c}+\frac {6 a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{c^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}+\frac {\left (2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) \left (-\frac {c^{2} \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{\left (a d -b c \right ) d \left (x +\frac {d}{c}\right )}-\frac {\left (2 a d -b c \right ) c \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{2 \left (a d -b c \right ) d \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right )}{c^{3}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 c^{2} \left (a x +b \right )}\) | \(517\) |
| default | \(\text {Expression too large to display}\) | \(1323\) |
1/c^2*a^2*x*((a*x+b)/x)^(1/2)-1/2/c^2*(a^(3/2)*(4*a*d-5*b*c)/c*ln((1/2*b+a *x)/a^(1/2)+(a*x^2+b*x)^(1/2))+6*a/c^2*(a^2*d^2-2*a*b*c*d+b^2*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))+1/c^3*(2*a^3*d^3-6*a^2*b*c*d^2+6*a*b^2*c^2*d-2*b^3*c^3)*(-1/(a*d-b* c)/d*c^2/(x+d/c)*(a*(x+d/c)^2-(2*a*d-b*c)/c*(x+d/c)+(a*d-b*c)*d/c^2)^(1/2) -1/2*(2*a*d-b*c)*c/(a*d-b*c)/d/((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 = 0.33 (sec) , antiderivative size = 1001, normalized size of antiderivative = 6.03 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^2} \, dx=\left [-\frac {{\left (5 \, a b c d^{2} - 4 \, a^{2} d^{3} + {\left (5 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + {\left (b^{2} c^{2} d + 3 \, a b c d^{2} - 4 \, a^{2} d^{3} + {\left (b^{2} c^{3} + 3 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\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^{2} d x^{2} + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (c^{4} d x + c^{3} d^{2}\right )}}, -\frac {2 \, {\left (5 \, a b c d^{2} - 4 \, a^{2} d^{3} + {\left (5 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (b^{2} c^{2} d + 3 \, a b c d^{2} - 4 \, a^{2} d^{3} + {\left (b^{2} c^{3} + 3 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\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^{2} d x^{2} + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (c^{4} d x + c^{3} d^{2}\right )}}, \frac {2 \, {\left (b^{2} c^{2} d + 3 \, a b c d^{2} - 4 \, a^{2} d^{3} + {\left (b^{2} c^{3} + 3 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\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} - 4 \, a^{2} d^{3} + {\left (5 \, a b c^{2} d - 4 \, a^{2} c 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^{2} c^{2} d x^{2} + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (c^{4} d x + c^{3} d^{2}\right )}}, \frac {{\left (b^{2} c^{2} d + 3 \, a b c d^{2} - 4 \, a^{2} d^{3} + {\left (b^{2} c^{3} + 3 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\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} - 4 \, a^{2} d^{3} + {\left (5 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (a^{2} c^{2} d x^{2} + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{c^{4} d x + c^{3} d^{2}}\right ] \]
[-1/2*((5*a*b*c*d^2 - 4*a^2*d^3 + (5*a*b*c^2*d - 4*a^2*c*d^2)*x)*sqrt(a)*l og(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a^2*c*d^2)*x)*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^2*d*x^2 + (b^2*c^3 - 2*a*b*c^2*d + 2*a^2*c*d^2)*x) *sqrt((a*x + b)/x))/(c^4*d*x + c^3*d^2), -1/2*(2*(5*a*b*c*d^2 - 4*a^2*d^3 + (5*a*b*c^2*d - 4*a^2*c*d^2)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x )/a) + (b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a ^2*c*d^2)*x)*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^2*d*x^2 + (b^2*c^ 3 - 2*a*b*c^2*d + 2*a^2*c*d^2)*x)*sqrt((a*x + b)/x))/(c^4*d*x + c^3*d^2), 1/2*(2*(b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a ^2*c*d^2)*x)*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 - 4*a^2*d^3 + (5*a*b*c^2*d - 4*a^2*c*d^ 2)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(a^2*c^2* d*x^2 + (b^2*c^3 - 2*a*b*c^2*d + 2*a^2*c*d^2)*x)*sqrt((a*x + b)/x))/(c^4*d *x + c^3*d^2), ((b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^ 2*d - 4*a^2*c*d^2)*x)*sqrt((b*c - a*d)/d)*arctan(-d*sqrt((b*c - a*d)/d)*sq rt((a*x + b)/x)/(b*c - a*d)) - (5*a*b*c*d^2 - 4*a^2*d^3 + (5*a*b*c^2*d - 4 *a^2*c*d^2)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (a^2*c^2...
\[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^2} \, dx=\int \frac {x^{2} \left (a + \frac {b}{x}\right )^{\frac {5}{2}}}{\left (c x + d\right )^{2}}\, dx \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^2} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}}}{{\left (c + \frac {d}{x}\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 667 vs. \(2 (146) = 292\).
Time = 0.34 (sec) , antiderivative size = 667, normalized size of antiderivative = 4.02 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^2} \, dx=\frac {\sqrt {a x^{2} + b x} a^{2} \mathrm {sgn}\left (x\right )}{c^{2}} - \frac {{\left (5 \, a^{2} b c \mathrm {sgn}\left (x\right ) - 4 \, a^{3} d \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} - b \right |}\right )}{2 \, \sqrt {a} c^{3}} + \frac {{\left (b^{3} c^{3} \mathrm {sgn}\left (x\right ) + 2 \, a b^{2} c^{2} d \mathrm {sgn}\left (x\right ) - 7 \, a^{2} b c d^{2} \mathrm {sgn}\left (x\right ) + 4 \, a^{3} d^{3} \mathrm {sgn}\left (x\right )\right )} \arctan \left (-\frac {{\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} c + \sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right )}{\sqrt {b c d - a d^{2}} c^{3} d} + \frac {{\left (2 \, \sqrt {a} b^{3} c^{3} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) + 4 \, a^{\frac {3}{2}} b^{2} c^{2} d \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - 14 \, a^{\frac {5}{2}} b c d^{2} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) + 8 \, a^{\frac {7}{2}} d^{3} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) + 5 \, \sqrt {b c d - a d^{2}} a^{2} b c d \log \left ({\left | b \right |}\right ) - 4 \, \sqrt {b c d - a d^{2}} a^{3} d^{2} \log \left ({\left | b \right |}\right ) - 2 \, \sqrt {b c d - a d^{2}} a b^{2} c^{2} + 4 \, \sqrt {b c d - a d^{2}} a^{2} b c d - 2 \, \sqrt {b c d - a d^{2}} a^{3} d^{2}\right )} \mathrm {sgn}\left (x\right )}{2 \, \sqrt {b c d - a d^{2}} \sqrt {a} c^{3} d} - \frac {{\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{3} c^{3} \mathrm {sgn}\left (x\right ) - 4 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{\frac {3}{2}} b^{2} c^{2} d \mathrm {sgn}\left (x\right ) + 5 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{\frac {5}{2}} b c d^{2} \mathrm {sgn}\left (x\right ) - 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{\frac {7}{2}} d^{3} \mathrm {sgn}\left (x\right ) - a b^{3} c^{2} d \mathrm {sgn}\left (x\right ) + 2 \, a^{2} b^{2} c d^{2} \mathrm {sgn}\left (x\right ) - a^{3} b d^{3} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} c + 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} d + b d\right )} \sqrt {a} c^{3} d} \]
sqrt(a*x^2 + b*x)*a^2*sgn(x)/c^2 - 1/2*(5*a^2*b*c*sgn(x) - 4*a^3*d*sgn(x)) *log(abs(-2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) - b))/(sqrt(a)*c^3) + (b^3*c^3*sgn(x) + 2*a*b^2*c^2*d*sgn(x) - 7*a^2*b*c*d^2*sgn(x) + 4*a^3*d^3* sgn(x))*arctan(-((sqrt(a)*x - sqrt(a*x^2 + b*x))*c + sqrt(a)*d)/sqrt(b*c*d - a*d^2))/(sqrt(b*c*d - a*d^2)*c^3*d) + 1/2*(2*sqrt(a)*b^3*c^3*arctan(sqr t(a)*d/sqrt(b*c*d - a*d^2)) + 4*a^(3/2)*b^2*c^2*d*arctan(sqrt(a)*d/sqrt(b* c*d - a*d^2)) - 14*a^(5/2)*b*c*d^2*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 8*a^(7/2)*d^3*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 5*sqrt(b*c*d - a*d^ 2)*a^2*b*c*d*log(abs(b)) - 4*sqrt(b*c*d - a*d^2)*a^3*d^2*log(abs(b)) - 2*s qrt(b*c*d - a*d^2)*a*b^2*c^2 + 4*sqrt(b*c*d - a*d^2)*a^2*b*c*d - 2*sqrt(b* c*d - a*d^2)*a^3*d^2)*sgn(x)/(sqrt(b*c*d - a*d^2)*sqrt(a)*c^3*d) - ((sqrt( a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^3*c^3*sgn(x) - 4*(sqrt(a)*x - sqrt(a*x ^2 + b*x))*a^(3/2)*b^2*c^2*d*sgn(x) + 5*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^ (5/2)*b*c*d^2*sgn(x) - 2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^(7/2)*d^3*sgn(x ) - a*b^3*c^2*d*sgn(x) + 2*a^2*b^2*c*d^2*sgn(x) - a^3*b*d^3*sgn(x))/(((sqr t(a)*x - sqrt(a*x^2 + b*x))^2*c + 2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a )*d + b*d)*sqrt(a)*c^3*d)
Time = 6.59 (sec) , antiderivative size = 1153, normalized size of antiderivative = 6.95 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^2} \, dx=\frac {\frac {\sqrt {a+\frac {b}{x}}\,\left (2\,a^3\,b\,d^2-3\,a^2\,b^2\,c\,d+a\,b^3\,c^2\right )}{c^2\,d}-\frac {b\,{\left (a+\frac {b}{x}\right )}^{3/2}\,\left (2\,a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{c^2\,d}}{\left (a+\frac {b}{x}\right )\,\left (2\,a\,d-b\,c\right )-d\,{\left (a+\frac {b}{x}\right )}^2-a^2\,d+a\,b\,c}-\frac {\mathrm {atanh}\left (\frac {10\,b^9\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}}{10\,a^2\,b^9+\frac {32\,a^3\,b^8\,d}{c}-\frac {132\,a^4\,b^7\,d^2}{c^2}+\frac {130\,a^5\,b^6\,d^3}{c^3}-\frac {40\,a^6\,b^5\,d^4}{c^4}}+\frac {32\,a\,b^8\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}}{32\,a^3\,b^8+\frac {10\,a^2\,b^9\,c}{d}-\frac {132\,a^4\,b^7\,d}{c}+\frac {130\,a^5\,b^6\,d^2}{c^2}-\frac {40\,a^6\,b^5\,d^3}{c^3}}-\frac {132\,a^2\,b^7\,d\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}}{32\,a^3\,b^8\,c-132\,a^4\,b^7\,d+\frac {10\,a^2\,b^9\,c^2}{d}+\frac {130\,a^5\,b^6\,d^2}{c}-\frac {40\,a^6\,b^5\,d^3}{c^2}}+\frac {130\,a^3\,b^6\,d^2\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}}{32\,a^3\,b^8\,c^2+130\,a^5\,b^6\,d^2+\frac {10\,a^2\,b^9\,c^3}{d}-\frac {40\,a^6\,b^5\,d^3}{c}-132\,a^4\,b^7\,c\,d}-\frac {40\,a^4\,b^5\,d^3\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}}{32\,a^3\,b^8\,c^3-40\,a^6\,b^5\,d^3-132\,a^4\,b^7\,c^2\,d+130\,a^5\,b^6\,c\,d^2+\frac {10\,a^2\,b^9\,c^4}{d}}\right )\,\left (4\,a\,d-5\,b\,c\right )\,\sqrt {a^3}}{c^3}+\frac {\mathrm {atanh}\left (\frac {30\,a^3\,b^6\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3\,d^6-3\,a^2\,b\,c\,d^5+3\,a\,b^2\,c^2\,d^4-b^3\,c^3\,d^3}}{14\,a^2\,b^9\,c^3+110\,a^5\,b^6\,d^3-4\,a^3\,b^8\,c^2\,d-82\,a^4\,b^7\,c\,d^2+\frac {2\,a\,b^{10}\,c^4}{d}-\frac {40\,a^6\,b^5\,d^4}{c}}+\frac {18\,a^2\,b^7\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3\,d^6-3\,a^2\,b\,c\,d^5+3\,a\,b^2\,c^2\,d^4-b^3\,c^3\,d^3}}{2\,a\,b^{10}\,c^3-82\,a^4\,b^7\,d^3+14\,a^2\,b^9\,c^2\,d-4\,a^3\,b^8\,c\,d^2+\frac {110\,a^5\,b^6\,d^4}{c}-\frac {40\,a^6\,b^5\,d^5}{c^2}}+\frac {40\,a^4\,b^5\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3\,d^6-3\,a^2\,b\,c\,d^5+3\,a\,b^2\,c^2\,d^4-b^3\,c^3\,d^3}}{4\,a^3\,b^8\,c^3+40\,a^6\,b^5\,d^3+82\,a^4\,b^7\,c^2\,d-110\,a^5\,b^6\,c\,d^2-\frac {2\,a\,b^{10}\,c^5}{d^2}-\frac {14\,a^2\,b^9\,c^4}{d}}-\frac {2\,a\,b^8\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3\,d^6-3\,a^2\,b\,c\,d^5+3\,a\,b^2\,c^2\,d^4-b^3\,c^3\,d^3}}{4\,a^3\,b^8\,d^3-14\,a^2\,b^9\,c\,d^2+\frac {82\,a^4\,b^7\,d^4}{c}-\frac {110\,a^5\,b^6\,d^5}{c^2}+\frac {40\,a^6\,b^5\,d^6}{c^3}-2\,a\,b^{10}\,c^2\,d}\right )\,\sqrt {d^3\,{\left (a\,d-b\,c\right )}^3}\,\left (4\,a\,d+b\,c\right )}{c^3\,d^3} \]
(((a + b/x)^(1/2)*(a*b^3*c^2 + 2*a^3*b*d^2 - 3*a^2*b^2*c*d))/(c^2*d) - (b* (a + b/x)^(3/2)*(2*a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(c^2*d))/((a + b/x)*(2* a*d - b*c) - d*(a + b/x)^2 - a^2*d + a*b*c) - (atanh((10*b^9*(a + b/x)^(1/ 2)*(a^3)^(1/2))/(10*a^2*b^9 + (32*a^3*b^8*d)/c - (132*a^4*b^7*d^2)/c^2 + ( 130*a^5*b^6*d^3)/c^3 - (40*a^6*b^5*d^4)/c^4) + (32*a*b^8*(a + b/x)^(1/2)*( a^3)^(1/2))/(32*a^3*b^8 + (10*a^2*b^9*c)/d - (132*a^4*b^7*d)/c + (130*a^5* b^6*d^2)/c^2 - (40*a^6*b^5*d^3)/c^3) - (132*a^2*b^7*d*(a + b/x)^(1/2)*(a^3 )^(1/2))/(32*a^3*b^8*c - 132*a^4*b^7*d + (10*a^2*b^9*c^2)/d + (130*a^5*b^6 *d^2)/c - (40*a^6*b^5*d^3)/c^2) + (130*a^3*b^6*d^2*(a + b/x)^(1/2)*(a^3)^( 1/2))/(32*a^3*b^8*c^2 + 130*a^5*b^6*d^2 + (10*a^2*b^9*c^3)/d - (40*a^6*b^5 *d^3)/c - 132*a^4*b^7*c*d) - (40*a^4*b^5*d^3*(a + b/x)^(1/2)*(a^3)^(1/2))/ (32*a^3*b^8*c^3 - 40*a^6*b^5*d^3 - 132*a^4*b^7*c^2*d + 130*a^5*b^6*c*d^2 + (10*a^2*b^9*c^4)/d))*(4*a*d - 5*b*c)*(a^3)^(1/2))/c^3 + (atanh((30*a^3*b^ 6*(a + b/x)^(1/2)*(a^3*d^6 - b^3*c^3*d^3 + 3*a*b^2*c^2*d^4 - 3*a^2*b*c*d^5 )^(1/2))/(14*a^2*b^9*c^3 + 110*a^5*b^6*d^3 - 4*a^3*b^8*c^2*d - 82*a^4*b^7* c*d^2 + (2*a*b^10*c^4)/d - (40*a^6*b^5*d^4)/c) + (18*a^2*b^7*(a + b/x)^(1/ 2)*(a^3*d^6 - b^3*c^3*d^3 + 3*a*b^2*c^2*d^4 - 3*a^2*b*c*d^5)^(1/2))/(2*a*b ^10*c^3 - 82*a^4*b^7*d^3 + 14*a^2*b^9*c^2*d - 4*a^3*b^8*c*d^2 + (110*a^5*b ^6*d^4)/c - (40*a^6*b^5*d^5)/c^2) + (40*a^4*b^5*(a + b/x)^(1/2)*(a^3*d^6 - b^3*c^3*d^3 + 3*a*b^2*c^2*d^4 - 3*a^2*b*c*d^5)^(1/2))/(4*a^3*b^8*c^3 +...