Integrand size = 15, antiderivative size = 138 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {35 b^3}{8 a^4 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {105 b^3}{8 a^5 \sqrt {a+\frac {b}{x}}}+\frac {21 b^2 x}{8 a^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {3 b x^2}{4 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {105 b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{11/2}} \] Output:
35/8*b^3/a^4/(a+b/x)^(3/2)+105/8*b^3/a^5/(a+b/x)^(1/2)+21/8*b^2*x/a^3/(a+b /x)^(3/2)-3/4*b*x^2/a^2/(a+b/x)^(3/2)+1/3*x^3/a/(a+b/x)^(3/2)-105/8*b^3*ar ctanh((a+b/x)^(1/2)/a^(1/2))/a^(11/2)
Time = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} x \left (315 b^4+420 a b^3 x+63 a^2 b^2 x^2-18 a^3 b x^3+8 a^4 x^4\right )}{24 a^5 (b+a x)^2}-\frac {105 b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{11/2}} \] Input:
Integrate[x^2/(a + b/x)^(5/2),x]
Output:
(Sqrt[a + b/x]*x*(315*b^4 + 420*a*b^3*x + 63*a^2*b^2*x^2 - 18*a^3*b*x^3 + 8*a^4*x^4))/(24*a^5*(b + a*x)^2) - (105*b^3*ArcTanh[Sqrt[a + b/x]/Sqrt[a]] )/(8*a^(11/2))
Time = 0.37 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {798, 52, 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.
| \(\displaystyle \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle -\int \frac {x^4}{\left (a+\frac {b}{x}\right )^{5/2}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {3 b \int \frac {x^3}{\left (a+\frac {b}{x}\right )^{5/2}}d\frac {1}{x}}{2 a}+\frac {x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {3 b \left (-\frac {7 b \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}}d\frac {1}{x}}{4 a}-\frac {x^2}{2 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}+\frac {x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {3 b \left (-\frac {7 b \left (-\frac {5 b \int \frac {x}{\left (a+\frac {b}{x}\right )^{5/2}}d\frac {1}{x}}{2 a}-\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{4 a}-\frac {x^2}{2 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}+\frac {x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {3 b \left (-\frac {7 b \left (-\frac {5 b \left (\frac {\int \frac {x}{\left (a+\frac {b}{x}\right )^{3/2}}d\frac {1}{x}}{a}+\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}-\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{4 a}-\frac {x^2}{2 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}+\frac {x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {3 b \left (-\frac {7 b \left (-\frac {5 b \left (\frac {\frac {\int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{a}+\frac {2}{a \sqrt {a+\frac {b}{x}}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}-\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{4 a}-\frac {x^2}{2 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}+\frac {x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {3 b \left (-\frac {7 b \left (-\frac {5 b \left (\frac {\frac {2 \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{a b}+\frac {2}{a \sqrt {a+\frac {b}{x}}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}-\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{4 a}-\frac {x^2}{2 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}+\frac {x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {3 b \left (-\frac {7 b \left (-\frac {5 b \left (\frac {\frac {2}{a \sqrt {a+\frac {b}{x}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}-\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{4 a}-\frac {x^2}{2 a \left (a+\frac {b}{x}\right )^{3/2}}\right )}{2 a}+\frac {x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
Input:
Int[x^2/(a + b/x)^(5/2),x]
Output:
x^3/(3*a*(a + b/x)^(3/2)) + (3*b*(-1/2*x^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)))/(2*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 = 0.35 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.31
| method | result | size |
| risch | \(\frac {\left (8 a^{2} x^{2}-34 a b x +123 b^{2}\right ) \left (a x +b \right )}{24 a^{5} \sqrt {\frac {a x +b}{x}}}+\frac {\left (-\frac {105 b^{3} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{16 a^{\frac {11}{2}}}+\frac {26 b^{3} \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 a^{6} \left (x +\frac {b}{a}\right )}-\frac {2 b^{4} \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 a^{7} \left (x +\frac {b}{a}\right )^{2}}\right ) \sqrt {x \left (a x +b \right )}}{x \sqrt {\frac {a x +b}{x}}}\) | \(181\) |
| default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, x \left (-16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {11}{2}} x^{3}+84 \sqrt {a \,x^{2}+b x}\, a^{\frac {11}{2}} b \,x^{4}-672 \sqrt {x \left (a x +b \right )}\, a^{\frac {9}{2}} b^{2} x^{3}-48 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} b \,x^{2}+294 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} b^{2} x^{3}+336 a^{4} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{3} x^{3}+384 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2} x -2016 \sqrt {x \left (a x +b \right )}\, a^{\frac {7}{2}} b^{3} x^{2}-48 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2} x +378 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} b^{3} x^{2}+1008 a^{3} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{4} x^{2}-21 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b^{3} x^{3}+352 b^{3} a^{\frac {5}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}}-2016 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}} b^{4} x -16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3}+210 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b^{4} x +1008 a^{2} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{5} x -63 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{4} x^{2}-672 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b^{5}+42 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{5}+336 a \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{6}-63 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{5} x -21 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{6}\right )}{48 a^{\frac {13}{2}} \sqrt {x \left (a x +b \right )}\, \left (a x +b \right )^{3}}\) | \(616\) |
Input:
int(x^2/(a+b/x)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/24*(8*a^2*x^2-34*a*b*x+123*b^2)*(a*x+b)/a^5/((a*x+b)/x)^(1/2)+(-105/16/a ^(11/2)*b^3*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))+26/3/a^6*b^3/(x+b/a) *(a*(x+b/a)^2-b*(x+b/a))^(1/2)-2/3/a^7*b^4/(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 = 0.08 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.01 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\left [\frac {315 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (8 \, a^{5} x^{5} - 18 \, a^{4} b x^{4} + 63 \, a^{3} b^{2} x^{3} + 420 \, a^{2} b^{3} x^{2} + 315 \, a b^{4} x\right )} \sqrt {\frac {a x + b}{x}}}{48 \, {\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}}, \frac {315 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + {\left (8 \, a^{5} x^{5} - 18 \, a^{4} b x^{4} + 63 \, a^{3} b^{2} x^{3} + 420 \, a^{2} b^{3} x^{2} + 315 \, a b^{4} x\right )} \sqrt {\frac {a x + b}{x}}}{24 \, {\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}}\right ] \] Input:
integrate(x^2/(a+b/x)^(5/2),x, algorithm="fricas")
Output:
[1/48*(315*(a^2*b^3*x^2 + 2*a*b^4*x + b^5)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x *sqrt((a*x + b)/x) + b) + 2*(8*a^5*x^5 - 18*a^4*b*x^4 + 63*a^3*b^2*x^3 + 4 20*a^2*b^3*x^2 + 315*a*b^4*x)*sqrt((a*x + b)/x))/(a^8*x^2 + 2*a^7*b*x + a^ 6*b^2), 1/24*(315*(a^2*b^3*x^2 + 2*a*b^4*x + b^5)*sqrt(-a)*arctan(sqrt(-a) *x*sqrt((a*x + b)/x)/(a*x + b)) + (8*a^5*x^5 - 18*a^4*b*x^4 + 63*a^3*b^2*x ^3 + 420*a^2*b^3*x^2 + 315*a*b^4*x)*sqrt((a*x + b)/x))/(a^8*x^2 + 2*a^7*b* x + a^6*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (119) = 238\).
Time = 33.42 (sec) , antiderivative size = 532, normalized size of antiderivative = 3.86 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {8 a^{\frac {133}{2}} b^{128} x^{72}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {18 a^{\frac {131}{2}} b^{129} x^{71}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} + \frac {63 a^{\frac {129}{2}} b^{130} x^{70}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} + \frac {420 a^{\frac {127}{2}} b^{131} x^{69}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} + \frac {315 a^{\frac {125}{2}} b^{132} x^{68}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {315 a^{63} b^{\frac {263}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {315 a^{62} b^{\frac {265}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} \] Input:
integrate(x**2/(a+b/x)**(5/2),x)
Output:
8*a**(133/2)*b**128*x**72/(24*a**(137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/2)*x**(135/2)*sqrt(a*x/b + 1)) - 18*a**(131/2 )*b**129*x**71/(24*a**(137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a **(135/2)*b**(259/2)*x**(135/2)*sqrt(a*x/b + 1)) + 63*a**(129/2)*b**130*x* *70/(24*a**(137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a**(135/2)*b **(259/2)*x**(135/2)*sqrt(a*x/b + 1)) + 420*a**(127/2)*b**131*x**69/(24*a* *(137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/2)* x**(135/2)*sqrt(a*x/b + 1)) + 315*a**(125/2)*b**132*x**68/(24*a**(137/2)*b **(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/2)*x**(135/2) *sqrt(a*x/b + 1)) - 315*a**63*b**(263/2)*x**(137/2)*sqrt(a*x/b + 1)*asinh( sqrt(a)*sqrt(x)/sqrt(b))/(24*a**(137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/2)*x**(135/2)*sqrt(a*x/b + 1)) - 315*a**62*b** (265/2)*x**(135/2)*sqrt(a*x/b + 1)*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(24*a**( 137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/2)*x* *(135/2)*sqrt(a*x/b + 1))
Time = 0.11 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.24 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {315 \, {\left (a + \frac {b}{x}\right )}^{4} b^{3} - 840 \, {\left (a + \frac {b}{x}\right )}^{3} a b^{3} + 693 \, {\left (a + \frac {b}{x}\right )}^{2} a^{2} b^{3} - 144 \, {\left (a + \frac {b}{x}\right )} a^{3} b^{3} - 16 \, a^{4} b^{3}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} a^{5} - 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{6} + 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{7} - {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{8}\right )}} + \frac {105 \, b^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{16 \, a^{\frac {11}{2}}} \] Input:
integrate(x^2/(a+b/x)^(5/2),x, algorithm="maxima")
Output:
1/24*(315*(a + b/x)^4*b^3 - 840*(a + b/x)^3*a*b^3 + 693*(a + b/x)^2*a^2*b^ 3 - 144*(a + b/x)*a^3*b^3 - 16*a^4*b^3)/((a + b/x)^(9/2)*a^5 - 3*(a + b/x) ^(7/2)*a^6 + 3*(a + b/x)^(5/2)*a^7 - (a + b/x)^(3/2)*a^8) + 105/16*b^3*log ((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(11/2)
Time = 0.15 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.51 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {1}{24} \, \sqrt {a x^{2} + b x} {\left (2 \, x {\left (\frac {4 \, x}{a^{3} \mathrm {sgn}\left (x\right )} - \frac {17 \, b}{a^{4} \mathrm {sgn}\left (x\right )}\right )} + \frac {123 \, b^{2}}{a^{5} \mathrm {sgn}\left (x\right )}\right )} + \frac {105 \, b^{3} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{16 \, a^{\frac {11}{2}} \mathrm {sgn}\left (x\right )} - \frac {{\left (315 \, b^{3} \log \left ({\left | b \right |}\right ) + 416 \, b^{3}\right )} \mathrm {sgn}\left (x\right )}{48 \, a^{\frac {11}{2}}} + \frac {2 \, {\left (15 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{4} + 27 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{5} + 13 \, b^{6}\right )}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )}^{3} a^{\frac {11}{2}} \mathrm {sgn}\left (x\right )} \] Input:
integrate(x^2/(a+b/x)^(5/2),x, algorithm="giac")
Output:
1/24*sqrt(a*x^2 + b*x)*(2*x*(4*x/(a^3*sgn(x)) - 17*b/(a^4*sgn(x))) + 123*b ^2/(a^5*sgn(x))) + 105/16*b^3*log(abs(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sq rt(a) + b))/(a^(11/2)*sgn(x)) - 1/48*(315*b^3*log(abs(b)) + 416*b^3)*sgn(x )/a^(11/2) + 2/3*(15*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^4 + 27*(sqrt(a) *x - sqrt(a*x^2 + b*x))*sqrt(a)*b^5 + 13*b^6)/(((sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b)^3*a^(11/2)*sgn(x))
Time = 1.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.82 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {35\,b^3}{2\,a^4\,{\left (a+\frac {b}{x}\right )}^{3/2}}-\frac {105\,b^3\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8\,a^{11/2}}+\frac {x^3}{3\,a\,{\left (a+\frac {b}{x}\right )}^{3/2}}-\frac {3\,b\,x^2}{4\,a^2\,{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {21\,b^2\,x}{8\,a^3\,{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {105\,b^4}{8\,a^5\,x\,{\left (a+\frac {b}{x}\right )}^{3/2}} \] Input:
int(x^2/(a + b/x)^(5/2),x)
Output:
(35*b^3)/(2*a^4*(a + b/x)^(3/2)) - (105*b^3*atanh((a + b/x)^(1/2)/a^(1/2)) )/(8*a^(11/2)) + x^3/(3*a*(a + b/x)^(3/2)) - (3*b*x^2)/(4*a^2*(a + b/x)^(3 /2)) + (21*b^2*x)/(8*a^3*(a + b/x)^(3/2)) + (105*b^4)/(8*a^5*x*(a + b/x)^( 3/2))
Time = 0.23 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.20 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {-2520 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a \,b^{3} x -2520 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b^{4}-567 \sqrt {a}\, \sqrt {a x +b}\, a \,b^{3} x -567 \sqrt {a}\, \sqrt {a x +b}\, b^{4}+64 \sqrt {x}\, a^{5} x^{4}-144 \sqrt {x}\, a^{4} b \,x^{3}+504 \sqrt {x}\, a^{3} b^{2} x^{2}+3360 \sqrt {x}\, a^{2} b^{3} x +2520 \sqrt {x}\, a \,b^{4}}{192 \sqrt {a x +b}\, a^{6} \left (a x +b \right )} \] Input:
int(x^2/(a+b/x)^(5/2),x)
Output:
( - 2520*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt( b))*a*b**3*x - 2520*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + sqrt(x)*sqr t(a))/sqrt(b))*b**4 - 567*sqrt(a)*sqrt(a*x + b)*a*b**3*x - 567*sqrt(a)*sqr t(a*x + b)*b**4 + 64*sqrt(x)*a**5*x**4 - 144*sqrt(x)*a**4*b*x**3 + 504*sqr t(x)*a**3*b**2*x**2 + 3360*sqrt(x)*a**2*b**3*x + 2520*sqrt(x)*a*b**4)/(192 *sqrt(a*x + b)*a**6*(a*x + b))