Integrand size = 19, antiderivative size = 103 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {5 b c-2 a d}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {5 b c-2 a d}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {(5 b c-2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \]
1/3*(-2*a*d+5*b*c)/a^2/(a+b/x)^(3/2)+c*x/a/(a+b/x)^(3/2)-(-2*a*d+5*b*c)*ar ctanh((a+b/x)^(1/2)/a^(1/2))/a^(7/2)+(-2*a*d+5*b*c)/a^3/(a+b/x)^(1/2)
Time = 0.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} x \left (15 b^2 c+a^2 x (-8 d+3 c x)+a b (-6 d+20 c x)\right )}{3 a^3 (b+a x)^2}+\frac {(-5 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \]
(Sqrt[a + b/x]*x*(15*b^2*c + a^2*x*(-8*d + 3*c*x) + a*b*(-6*d + 20*c*x)))/ (3*a^3*(b + a*x)^2) + ((-5*b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^ (7/2)
Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {899, 87, 61, 61, 73, 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.
| \(\displaystyle \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\int \frac {\left (c+\frac {d}{x}\right ) x^2}{\left (a+\frac {b}{x}\right )^{5/2}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(5 b c-2 a d) \int \frac {x}{\left (a+\frac {b}{x}\right )^{5/2}}d\frac {1}{x}}{2 a}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(5 b c-2 a d) \left (\frac {\int \frac {x}{\left (a+\frac {b}{x}\right )^{3/2}}d\frac {1}{x}}{a}+\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(5 b c-2 a d) \left (\frac {\frac {\int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{a}+\frac {2}{a \sqrt {a+\frac {b}{x}}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(5 b c-2 a d) \left (\frac {\frac {2 \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{a b}+\frac {2}{a \sqrt {a+\frac {b}{x}}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (\frac {\frac {2}{a \sqrt {a+\frac {b}{x}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\right ) (5 b c-2 a d)}{2 a}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}\) |
(c*x)/(a*(a + b/x)^(3/2)) + ((5*b*c - 2*a*d)*(2/(3*a*(a + b/x)^(3/2)) + (2 /(a*Sqrt[a + b/x]) - (2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2))/a))/(2*a)
3.3.61.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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. \(253\) vs. \(2(89)=178\).
Time = 0.18 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.47
| method | result | size |
| risch | \(\frac {c \left (a x +b \right )}{a^{3} \sqrt {\frac {a x +b}{x}}}+\frac {\left (2 \sqrt {a}\, d \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )-\frac {5 b c \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{\sqrt {a}}+\frac {2 \left (a d -b c \right ) b^{2} \left (\frac {2 \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 b \left (x +\frac {b}{a}\right )^{2}}+\frac {4 a \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 b^{2} \left (x +\frac {b}{a}\right )}\right )}{a^{2}}-\frac {4 \left (2 a d -3 b c \right ) \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{a \left (x +\frac {b}{a}\right )}\right ) \sqrt {x \left (a x +b \right )}}{2 a^{3} x \sqrt {\frac {a x +b}{x}}}\) | \(254\) |
| default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, x \left (12 a^{\frac {9}{2}} \sqrt {x \left (a x +b \right )}\, d \,x^{3}-30 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b c \,x^{3}-12 a^{\frac {7}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} d x +36 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b d \,x^{2}+24 a^{\frac {5}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b c x -90 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} c \,x^{2}-8 a^{\frac {5}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b d +36 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} d x -6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b d \,x^{3}+15 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{2} c \,x^{3}+20 a^{\frac {3}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b^{2} c -90 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b^{3} c x -18 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{2} d \,x^{2}+45 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{3} c \,x^{2}+12 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b^{3} d -18 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{3} d x +45 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{4} c x -30 \sqrt {a}\, \sqrt {x \left (a x +b \right )}\, b^{4} c -6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{4} d +15 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{5} c \right )}{6 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b \left (a x +b \right )^{3}}\) | \(541\) |
1/a^3*c*(a*x+b)/((a*x+b)/x)^(1/2)+1/2/a^3*(2*a^(1/2)*d*ln((1/2*b+a*x)/a^(1 /2)+(a*x^2+b*x)^(1/2))-5*b*c*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))/a^( 1/2)+2*(a*d-b*c)*b^2/a^2*(2/3/b/(x+b/a)^2*(a*(x+b/a)^2-b*(x+b/a))^(1/2)+4/ 3*a/b^2/(x+b/a)*(a*(x+b/a)^2-b*(x+b/a))^(1/2))-4*(2*a*d-3*b*c)/a/(x+b/a)*( a*(x+b/a)^2-b*(x+b/a))^(1/2))/x/((a*x+b)/x)^(1/2)*(x*(a*x+b))^(1/2)
Time = 0.27 (sec) , antiderivative size = 331, normalized size of antiderivative = 3.21 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{3} c - 2 \, a b^{2} d + {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{2} + 2 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (3 \, a^{3} c x^{3} + 4 \, {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{2} + 3 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{6 \, {\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}}, \frac {3 \, {\left (5 \, b^{3} c - 2 \, a b^{2} d + {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{2} + 2 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (3 \, a^{3} c x^{3} + 4 \, {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{2} + 3 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}}\right ] \]
[-1/6*(3*(5*b^3*c - 2*a*b^2*d + (5*a^2*b*c - 2*a^3*d)*x^2 + 2*(5*a*b^2*c - 2*a^2*b*d)*x)*sqrt(a)*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2* (3*a^3*c*x^3 + 4*(5*a^2*b*c - 2*a^3*d)*x^2 + 3*(5*a*b^2*c - 2*a^2*b*d)*x)* sqrt((a*x + b)/x))/(a^6*x^2 + 2*a^5*b*x + a^4*b^2), 1/3*(3*(5*b^3*c - 2*a* b^2*d + (5*a^2*b*c - 2*a^3*d)*x^2 + 2*(5*a*b^2*c - 2*a^2*b*d)*x)*sqrt(-a)* arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (3*a^3*c*x^3 + 4*(5*a^2*b*c - 2*a^3 *d)*x^2 + 3*(5*a*b^2*c - 2*a^2*b*d)*x)*sqrt((a*x + b)/x))/(a^6*x^2 + 2*a^5 *b*x + a^4*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 1479 vs. \(2 (90) = 180\).
Time = 33.93 (sec) , antiderivative size = 1479, normalized size of antiderivative = 14.36 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\text {Too large to display} \]
c*(6*a**17*x**4*sqrt(1 + b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) + 46*a**16*b*x**3*sqrt(1 + b/(a* x))/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**( 33/2)*b**3) + 15*a**16*b*x**3*log(b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2 )*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) - 30*a**16*b*x**3*log(s qrt(1 + b/(a*x)) + 1)/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/ 2)*b**2*x + 6*a**(33/2)*b**3) + 70*a**15*b**2*x**2*sqrt(1 + b/(a*x))/(6*a* *(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b** 3) + 45*a**15*b**2*x**2*log(b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x* *2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) - 90*a**15*b**2*x**2*log(sqrt (1 + b/(a*x)) + 1)/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)* b**2*x + 6*a**(33/2)*b**3) + 30*a**14*b**3*x*sqrt(1 + b/(a*x))/(6*a**(39/2 )*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) + 4 5*a**14*b**3*x*log(b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a **(35/2)*b**2*x + 6*a**(33/2)*b**3) - 90*a**14*b**3*x*log(sqrt(1 + b/(a*x) ) + 1)/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a **(33/2)*b**3) + 15*a**13*b**4*log(b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/ 2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) - 30*a**13*b**4*log(sq rt(1 + b/(a*x)) + 1)/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2 )*b**2*x + 6*a**(33/2)*b**3)) + d*(-8*a**7*x**3*sqrt(1 + b/(a*x))/(3*a*...
Time = 0.34 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.65 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {1}{6} \, c {\left (\frac {2 \, {\left (15 \, {\left (a + \frac {b}{x}\right )}^{2} b - 10 \, {\left (a + \frac {b}{x}\right )} a b - 2 \, a^{2} b\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3} - {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{4}} + \frac {15 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )} - \frac {1}{3} \, d {\left (\frac {3 \, \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (4 \, a + \frac {3 \, b}{x}\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2}}\right )} \]
1/6*c*(2*(15*(a + b/x)^2*b - 10*(a + b/x)*a*b - 2*a^2*b)/((a + b/x)^(5/2)* a^3 - (a + b/x)^(3/2)*a^4) + 15*b*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(7/2)) - 1/3*d*(3*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(5/2) + 2*(4*a + 3*b/x)/((a + b/x)^(3/2)*a^2))
Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (89) = 178\).
Time = 0.31 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.51 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {{\left (15 \, b c \log \left ({\left | b \right |}\right ) - 6 \, a d \log \left ({\left | b \right |}\right ) + 28 \, b c - 16 \, a d\right )} \mathrm {sgn}\left (x\right )}{6 \, a^{\frac {7}{2}}} + \frac {\sqrt {a x^{2} + b x} c}{a^{3} \mathrm {sgn}\left (x\right )} + \frac {{\left (5 \, b c - 2 \, a d\right )} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{2 \, a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{2} c - 6 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{2} b d + 15 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{3} c - 9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{\frac {3}{2}} b^{2} d + 7 \, b^{4} c - 4 \, a b^{3} d\right )}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )}^{3} a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \]
-1/6*(15*b*c*log(abs(b)) - 6*a*d*log(abs(b)) + 28*b*c - 16*a*d)*sgn(x)/a^( 7/2) + sqrt(a*x^2 + b*x)*c/(a^3*sgn(x)) + 1/2*(5*b*c - 2*a*d)*log(abs(2*(s qrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b))/(a^(7/2)*sgn(x)) + 2/3*(9*(sqr t(a)*x - sqrt(a*x^2 + b*x))^2*a*b^2*c - 6*(sqrt(a)*x - sqrt(a*x^2 + b*x))^ 2*a^2*b*d + 15*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^3*c - 9*(sqrt(a)* x - sqrt(a*x^2 + b*x))*a^(3/2)*b^2*d + 7*b^4*c - 4*a*b^3*d)/(((sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b)^3*a^(7/2)*sgn(x))
Time = 7.00 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {2\,d\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {2\,d}{3\,a}+\frac {2\,d\,\left (a+\frac {b}{x}\right )}{a^2}}{{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {2\,c\,x\,{\left (\frac {a\,x}{b}+1\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {5}{2},\frac {7}{2};\ \frac {9}{2};\ -\frac {a\,x}{b}\right )}{7\,{\left (a+\frac {b}{x}\right )}^{5/2}} \]