\(\int \frac {x^2}{(a+\frac {b}{x})^{5/2}} \, dx\) [1742]

Optimal result

Integrand size = 15, antiderivative size = 138 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {35 b^3}{8 a^4 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {105 b^3}{8 a^5 \sqrt {a+\frac {b}{x}}}+\frac {21 b^2 x}{8 a^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {3 b x^2}{4 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {105 b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{11/2}} \]

[Out]

35/8*b^3/a^4/(a+b/x)^(3/2)+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)-10
5/8*b^3*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(11/2)+105/8*b^3/a^5/(a+b/x)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 44, 53, 65, 214} \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {105 b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{11/2}}+\frac {105 b^3}{8 a^5 \sqrt {a+\frac {b}{x}}}+\frac {35 b^3}{8 a^4 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {21 b^2 x}{8 a^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {3 b x^2}{4 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}} \]

[In]

Int[x^2/(a + b/x)^(5/2),x]

[Out]

(35*b^3)/(8*a^4*(a + b/x)^(3/2)) + (105*b^3)/(8*a^5*Sqrt[a + b/x]) + (21*b^2*x)/(8*a^3*(a + b/x)^(3/2)) - (3*b
*x^2)/(4*a^2*(a + b/x)^(3/2)) + x^3/(3*a*(a + b/x)^(3/2)) - (105*b^3*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(8*a^(11/
2))

Rule 44

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] - Dist[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] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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] - Dist[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] && NeQ[b*c - a*d, 0] && 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]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{x^4 (a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{x^3 (a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = -\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 {\left (21 b^2\right ) \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right )}{8 a^2} \\ & = \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 {\left (105 b^3\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right )}{16 a^3} \\ & = \frac {35 b^3}{8 a^4 \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 {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 {\left (105 b^3\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )}{16 a^4} \\ & = \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 {\left (105 b^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{16 a^5} \\ & = \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 {\left (105 b^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{8 a^5} \\ & = \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 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} x \left (315 b^4+420 a b^3 x+63 a^2 b^2 x^2-18 a^3 b x^3+8 a^4 x^4\right )}{24 a^5 (b+a x)^2}-\frac {105 b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{11/2}} \]

[In]

Integrate[x^2/(a + b/x)^(5/2),x]

[Out]

(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))

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.31

method	result	size
risch	\(\frac {\left (8 a^{2} x^{2}-34 a b x +123 b^{2}\right ) \left (a x +b \right )}{24 a^{5} \sqrt {\frac {a x +b}{x}}}+\frac {\left (-\frac {105 b^{3} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{16 a^{\frac {11}{2}}}-\frac {2 b^{4} \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 a^{7} \left (x +\frac {b}{a}\right )^{2}}+\frac {26 b^{3} \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 a^{6} \left (x +\frac {b}{a}\right )}\right ) \sqrt {x \left (a x +b \right )}}{x \sqrt {\frac {a x +b}{x}}}\)	\(181\)
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}+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}+672 a^{\frac {9}{2}} \sqrt {x \left (a x +b \right )}\, 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}+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}-384 a^{\frac {7}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b^{2} x +2016 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, 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}+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 -352 b^{3} a^{\frac {5}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}}+2016 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, 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}-42 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{5}+672 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, 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}}\)	\(616\)

[In]

int(x^2/(a+b/x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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))-2/3/a^7*b^4/(x+b/a)^2*(a*(x+b/a)^2-b*(x+b/a))^(1/2)+26/3/a^6*b^3/(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)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.98 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\left [\frac {315 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (8 \, a^{5} x^{5} - 18 \, a^{4} b x^{4} + 63 \, a^{3} b^{2} x^{3} + 420 \, a^{2} b^{3} x^{2} + 315 \, a b^{4} x\right )} \sqrt {\frac {a x + b}{x}}}{48 \, {\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}}, \frac {315 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (8 \, a^{5} x^{5} - 18 \, a^{4} b x^{4} + 63 \, a^{3} b^{2} x^{3} + 420 \, a^{2} b^{3} x^{2} + 315 \, a b^{4} x\right )} \sqrt {\frac {a x + b}{x}}}{24 \, {\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}}\right ] \]

[In]

integrate(x^2/(a+b/x)^(5/2),x, algorithm="fricas")

[Out]

[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 + 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), 1/24*(315*(a^2*b^3*x^2 + 2*a*b^4*x + b^5)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (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)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (119) = 238\).

Time = 87.59 (sec) , antiderivative size = 532, normalized size of antiderivative = 3.86 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {8 a^{\frac {133}{2}} b^{128} x^{72}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {18 a^{\frac {131}{2}} b^{129} x^{71}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} + \frac {63 a^{\frac {129}{2}} b^{130} x^{70}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} + \frac {420 a^{\frac {127}{2}} b^{131} x^{69}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} + \frac {315 a^{\frac {125}{2}} b^{132} x^{68}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {315 a^{63} b^{\frac {263}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {315 a^{62} b^{\frac {265}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} \]

[In]

integrate(x**2/(a+b/x)**(5/2),x)

[Out]

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/(2
4*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)) + 31
5*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))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.24 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {315 \, {\left (a + \frac {b}{x}\right )}^{4} b^{3} - 840 \, {\left (a + \frac {b}{x}\right )}^{3} a b^{3} + 693 \, {\left (a + \frac {b}{x}\right )}^{2} a^{2} b^{3} - 144 \, {\left (a + \frac {b}{x}\right )} a^{3} b^{3} - 16 \, a^{4} b^{3}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} a^{5} - 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{6} + 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{7} - {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{8}\right )}} + \frac {105 \, b^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{16 \, a^{\frac {11}{2}}} \]

[In]

integrate(x^2/(a+b/x)^(5/2),x, algorithm="maxima")

[Out]

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*l
og((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(11/2)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.51 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {1}{24} \, \sqrt {a x^{2} + b x} {\left (2 \, x {\left (\frac {4 \, x}{a^{3} \mathrm {sgn}\left (x\right )} - \frac {17 \, b}{a^{4} \mathrm {sgn}\left (x\right )}\right )} + \frac {123 \, b^{2}}{a^{5} \mathrm {sgn}\left (x\right )}\right )} + \frac {105 \, b^{3} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{16 \, a^{\frac {11}{2}} \mathrm {sgn}\left (x\right )} - \frac {{\left (315 \, b^{3} \log \left ({\left | b \right |}\right ) + 416 \, b^{3}\right )} \mathrm {sgn}\left (x\right )}{48 \, a^{\frac {11}{2}}} + \frac {2 \, {\left (15 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{4} + 27 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{5} + 13 \, b^{6}\right )}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )}^{3} a^{\frac {11}{2}} \mathrm {sgn}\left (x\right )} \]

[In]

integrate(x^2/(a+b/x)^(5/2),x, algorithm="giac")

[Out]

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(ab
s(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(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))

Mupad [B] (verification not implemented)

Time = 6.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.82 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {35\,b^3}{2\,a^4\,{\left (a+\frac {b}{x}\right )}^{3/2}}-\frac {105\,b^3\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8\,a^{11/2}}+\frac {x^3}{3\,a\,{\left (a+\frac {b}{x}\right )}^{3/2}}-\frac {3\,b\,x^2}{4\,a^2\,{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {21\,b^2\,x}{8\,a^3\,{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {105\,b^4}{8\,a^5\,x\,{\left (a+\frac {b}{x}\right )}^{3/2}} \]

[In]

int(x^2/(a + b/x)^(5/2),x)

[Out]

(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))