Integrand size = 13, antiderivative size = 114 \[ \int \frac {x}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {35 b^2}{12 a^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {35 b^2}{4 a^4 \sqrt {a+\frac {b}{x}}}-\frac {7 b x}{4 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {x^2}{2 a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {35 b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 a^{9/2}} \] Output:
-35/12*b^2/a^3/(a+b/x)^(3/2)-35/4*b^2/a^4/(a+b/x)^(1/2)-7/4*b*x/a^2/(a+b/x )^(3/2)+1/2*x^2/a/(a+b/x)^(3/2)+35/4*b^2*arctanh((a+b/x)^(1/2)/a^(1/2))/a^ (9/2)
Time = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.76 \[ \int \frac {x}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} x \left (-105 b^3-140 a b^2 x-21 a^2 b x^2+6 a^3 x^3\right )}{12 a^4 (b+a x)^2}+\frac {35 b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 a^{9/2}} \] Input:
Integrate[x/(a + b/x)^(5/2),x]
Output:
(Sqrt[a + b/x]*x*(-105*b^3 - 140*a*b^2*x - 21*a^2*b*x^2 + 6*a^3*x^3))/(12* a^4*(b + a*x)^2) + (35*b^2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(4*a^(9/2))
Time = 0.35 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {798, 52, 52, 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.
| \(\displaystyle \int \frac {x}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle -\int \frac {x^3}{\left (a+\frac {b}{x}\right )^{5/2}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {7 b \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}}d\frac {1}{x}}{4 a}+\frac {x^2}{2 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {7 b \left (-\frac {5 b \int \frac {x}{\left (a+\frac {b}{x}\right )^{5/2}}d\frac {1}{x}}{2 a}-\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{4 a}+\frac {x^2}{2 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {7 b \left (-\frac {5 b \left (\frac {\int \frac {x}{\left (a+\frac {b}{x}\right )^{3/2}}d\frac {1}{x}}{a}+\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}-\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{4 a}+\frac {x^2}{2 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {7 b \left (-\frac {5 b \left (\frac {\frac {\int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{a}+\frac {2}{a \sqrt {a+\frac {b}{x}}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}-\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{4 a}+\frac {x^2}{2 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {7 b \left (-\frac {5 b \left (\frac {\frac {2 \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{a b}+\frac {2}{a \sqrt {a+\frac {b}{x}}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}-\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{4 a}+\frac {x^2}{2 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {7 b \left (-\frac {5 b \left (\frac {\frac {2}{a \sqrt {a+\frac {b}{x}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}-\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{4 a}+\frac {x^2}{2 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
Input:
Int[x/(a + b/x)^(5/2),x]
Output:
x^2/(2*a*(a + b/x)^(3/2)) + (7*b*(-(x/(a*(a + b/x)^(3/2))) - (5*b*(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)))/(4*a)
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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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_)^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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.32 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.49
| method | result | size |
| risch | \(\frac {\left (2 a x -11 b \right ) \left (a x +b \right )}{4 a^{4} \sqrt {\frac {a x +b}{x}}}+\frac {\left (\frac {35 b^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{8 a^{\frac {9}{2}}}-\frac {20 b^{2} \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 a^{5} \left (x +\frac {b}{a}\right )}+\frac {2 b^{3} \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 a^{6} \left (x +\frac {b}{a}\right )^{2}}\right ) \sqrt {x \left (a x +b \right )}}{x \sqrt {\frac {a x +b}{x}}}\) | \(170\) |
| default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (12 \sqrt {a \,x^{2}+b x}\, a^{\frac {11}{2}} x^{4}-216 \sqrt {x \left (a x +b \right )}\, a^{\frac {9}{2}} b \,x^{3}+42 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} b \,x^{3}+108 a^{4} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{2} x^{3}+144 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {7}{2}} b x -648 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} x^{2}+54 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} b^{2} x^{2}+324 a^{3} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{3} x^{2}-3 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b^{2} x^{3}+128 a^{\frac {5}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b^{2}-648 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b^{3} x +30 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b^{3} x +324 a^{2} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{4} x -9 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{3} x^{2}-216 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b^{4}+6 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{4}+108 a \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{5}-9 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{4} x -3 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{5}\right )}{24 a^{\frac {11}{2}} \sqrt {x \left (a x +b \right )}\, \left (a x +b \right )^{3}}\) | \(531\) |
Input:
int(x/(a+b/x)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/4*(2*a*x-11*b)*(a*x+b)/a^4/((a*x+b)/x)^(1/2)+(35/8/a^(9/2)*b^2*ln((1/2*b +a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))-20/3/a^5*b^2/(x+b/a)*(a*(x+b/a)^2-b*(x+b/ a))^(1/2)+2/3/a^6*b^3/(x+b/a)^2*(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 = 0.13 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.25 \[ \int \frac {x}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\left [\frac {105 \, {\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (6 \, a^{4} x^{4} - 21 \, a^{3} b x^{3} - 140 \, a^{2} b^{2} x^{2} - 105 \, a b^{3} x\right )} \sqrt {\frac {a x + b}{x}}}{24 \, {\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}}, -\frac {105 \, {\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) - {\left (6 \, a^{4} x^{4} - 21 \, a^{3} b x^{3} - 140 \, a^{2} b^{2} x^{2} - 105 \, a b^{3} x\right )} \sqrt {\frac {a x + b}{x}}}{12 \, {\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}}\right ] \] Input:
integrate(x/(a+b/x)^(5/2),x, algorithm="fricas")
Output:
[1/24*(105*(a^2*b^2*x^2 + 2*a*b^3*x + b^4)*sqrt(a)*log(2*a*x + 2*sqrt(a)*x *sqrt((a*x + b)/x) + b) + 2*(6*a^4*x^4 - 21*a^3*b*x^3 - 140*a^2*b^2*x^2 - 105*a*b^3*x)*sqrt((a*x + b)/x))/(a^7*x^2 + 2*a^6*b*x + a^5*b^2), -1/12*(10 5*(a^2*b^2*x^2 + 2*a*b^3*x + b^4)*sqrt(-a)*arctan(sqrt(-a)*x*sqrt((a*x + b )/x)/(a*x + b)) - (6*a^4*x^4 - 21*a^3*b*x^3 - 140*a^2*b^2*x^2 - 105*a*b^3* x)*sqrt((a*x + b)/x))/(a^7*x^2 + 2*a^6*b*x + a^5*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (97) = 194\).
Time = 7.53 (sec) , antiderivative size = 464, normalized size of antiderivative = 4.07 \[ \int \frac {x}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {6 a^{\frac {89}{2}} b^{75} x^{49}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {93}{2}} \sqrt {\frac {a x}{b} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {91}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {21 a^{\frac {87}{2}} b^{76} x^{48}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {93}{2}} \sqrt {\frac {a x}{b} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {91}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {140 a^{\frac {85}{2}} b^{77} x^{47}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {93}{2}} \sqrt {\frac {a x}{b} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {91}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {105 a^{\frac {83}{2}} b^{78} x^{46}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {93}{2}} \sqrt {\frac {a x}{b} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {91}{2}} \sqrt {\frac {a x}{b} + 1}} + \frac {105 a^{42} b^{\frac {155}{2}} x^{\frac {93}{2}} \sqrt {\frac {a x}{b} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {93}{2}} \sqrt {\frac {a x}{b} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {91}{2}} \sqrt {\frac {a x}{b} + 1}} + \frac {105 a^{41} b^{\frac {157}{2}} x^{\frac {91}{2}} \sqrt {\frac {a x}{b} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {93}{2}} \sqrt {\frac {a x}{b} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {91}{2}} \sqrt {\frac {a x}{b} + 1}} \] Input:
integrate(x/(a+b/x)**(5/2),x)
Output:
6*a**(89/2)*b**75*x**49/(12*a**(93/2)*b**(151/2)*x**(93/2)*sqrt(a*x/b + 1) + 12*a**(91/2)*b**(153/2)*x**(91/2)*sqrt(a*x/b + 1)) - 21*a**(87/2)*b**76 *x**48/(12*a**(93/2)*b**(151/2)*x**(93/2)*sqrt(a*x/b + 1) + 12*a**(91/2)*b **(153/2)*x**(91/2)*sqrt(a*x/b + 1)) - 140*a**(85/2)*b**77*x**47/(12*a**(9 3/2)*b**(151/2)*x**(93/2)*sqrt(a*x/b + 1) + 12*a**(91/2)*b**(153/2)*x**(91 /2)*sqrt(a*x/b + 1)) - 105*a**(83/2)*b**78*x**46/(12*a**(93/2)*b**(151/2)* x**(93/2)*sqrt(a*x/b + 1) + 12*a**(91/2)*b**(153/2)*x**(91/2)*sqrt(a*x/b + 1)) + 105*a**42*b**(155/2)*x**(93/2)*sqrt(a*x/b + 1)*asinh(sqrt(a)*sqrt(x )/sqrt(b))/(12*a**(93/2)*b**(151/2)*x**(93/2)*sqrt(a*x/b + 1) + 12*a**(91/ 2)*b**(153/2)*x**(91/2)*sqrt(a*x/b + 1)) + 105*a**41*b**(157/2)*x**(91/2)* sqrt(a*x/b + 1)*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(12*a**(93/2)*b**(151/2)*x* *(93/2)*sqrt(a*x/b + 1) + 12*a**(91/2)*b**(153/2)*x**(91/2)*sqrt(a*x/b + 1 ))
Time = 0.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.22 \[ \int \frac {x}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {105 \, {\left (a + \frac {b}{x}\right )}^{3} b^{2} - 175 \, {\left (a + \frac {b}{x}\right )}^{2} a b^{2} + 56 \, {\left (a + \frac {b}{x}\right )} a^{2} b^{2} + 8 \, a^{3} b^{2}}{12 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{4} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{5} + {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{6}\right )}} - \frac {35 \, b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{8 \, a^{\frac {9}{2}}} \] Input:
integrate(x/(a+b/x)^(5/2),x, algorithm="maxima")
Output:
-1/12*(105*(a + b/x)^3*b^2 - 175*(a + b/x)^2*a*b^2 + 56*(a + b/x)*a^2*b^2 + 8*a^3*b^2)/((a + b/x)^(7/2)*a^4 - 2*(a + b/x)^(5/2)*a^5 + (a + b/x)^(3/2 )*a^6) - 35/8*b^2*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a))) /a^(9/2)
Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (90) = 180\).
Time = 0.15 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.68 \[ \int \frac {x}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {1}{4} \, \sqrt {a x^{2} + b x} {\left (\frac {2 \, x}{a^{3} \mathrm {sgn}\left (x\right )} - \frac {11 \, b}{a^{4} \mathrm {sgn}\left (x\right )}\right )} - \frac {35 \, b^{2} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{8 \, a^{\frac {9}{2}} \mathrm {sgn}\left (x\right )} + \frac {5 \, {\left (21 \, b^{2} \log \left ({\left | b \right |}\right ) + 32 \, b^{2}\right )} \mathrm {sgn}\left (x\right )}{24 \, a^{\frac {9}{2}}} - \frac {2 \, {\left (12 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{3} + 21 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{4} + 10 \, b^{5}\right )}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )}^{3} a^{\frac {9}{2}} \mathrm {sgn}\left (x\right )} \] Input:
integrate(x/(a+b/x)^(5/2),x, algorithm="giac")
Output:
1/4*sqrt(a*x^2 + b*x)*(2*x/(a^3*sgn(x)) - 11*b/(a^4*sgn(x))) - 35/8*b^2*lo g(abs(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b))/(a^(9/2)*sgn(x)) + 5 /24*(21*b^2*log(abs(b)) + 32*b^2)*sgn(x)/a^(9/2) - 2/3*(12*(sqrt(a)*x - sq rt(a*x^2 + b*x))^2*a*b^3 + 21*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^4 + 10*b^5)/(((sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b)^3*a^(9/2)*sgn(x))
Time = 0.83 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {35\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4\,a^{9/2}}-\frac {35\,b^2}{3\,a^3\,{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {x^2}{2\,a\,{\left (a+\frac {b}{x}\right )}^{3/2}}-\frac {35\,b^3}{4\,a^4\,x\,{\left (a+\frac {b}{x}\right )}^{3/2}}-\frac {7\,b\,x}{4\,a^2\,{\left (a+\frac {b}{x}\right )}^{3/2}} \] Input:
int(x/(a + b/x)^(5/2),x)
Output:
(35*b^2*atanh((a + b/x)^(1/2)/a^(1/2)))/(4*a^(9/2)) - (35*b^2)/(3*a^3*(a + b/x)^(3/2)) + x^2/(2*a*(a + b/x)^(3/2)) - (35*b^3)/(4*a^4*x*(a + b/x)^(3/ 2)) - (7*b*x)/(4*a^2*(a + b/x)^(3/2))
Time = 0.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.34 \[ \int \frac {x}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {840 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a \,b^{2} x +840 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b^{3}+175 \sqrt {a}\, \sqrt {a x +b}\, a \,b^{2} x +175 \sqrt {a}\, \sqrt {a x +b}\, b^{3}+48 \sqrt {x}\, a^{4} x^{3}-168 \sqrt {x}\, a^{3} b \,x^{2}-1120 \sqrt {x}\, a^{2} b^{2} x -840 \sqrt {x}\, a \,b^{3}}{96 \sqrt {a x +b}\, a^{5} \left (a x +b \right )} \] Input:
int(x/(a+b/x)^(5/2),x)
Output:
(840*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))* a*b**2*x + 840*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a)) /sqrt(b))*b**3 + 175*sqrt(a)*sqrt(a*x + b)*a*b**2*x + 175*sqrt(a)*sqrt(a*x + b)*b**3 + 48*sqrt(x)*a**4*x**3 - 168*sqrt(x)*a**3*b*x**2 - 1120*sqrt(x) *a**2*b**2*x - 840*sqrt(x)*a*b**3)/(96*sqrt(a*x + b)*a**5*(a*x + b))