\(\int \frac {x}{(a+\frac {b}{x^2})^{5/2}} \, dx\) [1945]

Optimal result

Integrand size = 13, antiderivative size = 88 \[ \int \frac {x}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=\frac {5 b}{6 a^2 \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {5 b}{2 a^3 \sqrt {a+\frac {b}{x^2}}}+\frac {x^2}{2 a \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {5 b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2 a^{7/2}} \]

[Out]

5/6*b/a^2/(a+b/x^2)^(3/2)+1/2*x^2/a/(a+b/x^2)^(3/2)-5/2*b*arctanh((a+b/x^2)^(1/2)/a^(1/2))/a^(7/2)+5/2*b/a^3/(
a+b/x^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {272, 44, 53, 65, 214} \[ \int \frac {x}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=-\frac {5 b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2 a^{7/2}}+\frac {5 b}{2 a^3 \sqrt {a+\frac {b}{x^2}}}+\frac {5 b}{6 a^2 \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {x^2}{2 a \left (a+\frac {b}{x^2}\right )^{3/2}} \]

[In]

Int[x/(a + b/x^2)^(5/2),x]

[Out]

(5*b)/(6*a^2*(a + b/x^2)^(3/2)) + (5*b)/(2*a^3*Sqrt[a + b/x^2]) + x^2/(2*a*(a + b/x^2)^(3/2)) - (5*b*ArcTanh[S
qrt[a + b/x^2]/Sqrt[a]])/(2*a^(7/2))

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}& = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^{5/2}} \, dx,x,\frac {1}{x^2}\right )\right ) \\ & = \frac {x^2}{2 a \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {(5 b) \text {Subst}\left (\int \frac {1}{x (a+b x)^{5/2}} \, dx,x,\frac {1}{x^2}\right )}{4 a} \\ & = \frac {5 b}{6 a^2 \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {x^2}{2 a \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {(5 b) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{x^2}\right )}{4 a^2} \\ & = \frac {5 b}{6 a^2 \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {5 b}{2 a^3 \sqrt {a+\frac {b}{x^2}}}+\frac {x^2}{2 a \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {(5 b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )}{4 a^3} \\ & = \frac {5 b}{6 a^2 \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {5 b}{2 a^3 \sqrt {a+\frac {b}{x^2}}}+\frac {x^2}{2 a \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {5 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right )}{2 a^3} \\ & = \frac {5 b}{6 a^2 \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {5 b}{2 a^3 \sqrt {a+\frac {b}{x^2}}}+\frac {x^2}{2 a \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2 a^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.18 \[ \int \frac {x}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=\frac {\sqrt {a} x \left (15 b^2+20 a b x^2+3 a^2 x^4\right )+30 b \left (b+a x^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {b}-\sqrt {b+a x^2}}\right )}{6 a^{7/2} \sqrt {a+\frac {b}{x^2}} x \left (b+a x^2\right )} \]

[In]

Integrate[x/(a + b/x^2)^(5/2),x]

[Out]

(Sqrt[a]*x*(15*b^2 + 20*a*b*x^2 + 3*a^2*x^4) + 30*b*(b + a*x^2)^(3/2)*ArcTanh[(Sqrt[a]*x)/(Sqrt[b] - Sqrt[b +
a*x^2])])/(6*a^(7/2)*Sqrt[a + b/x^2]*x*(b + a*x^2))

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {\left (a \,x^{2}+b \right ) \left (3 x^{5} a^{\frac {7}{2}}+20 a^{\frac {5}{2}} b \,x^{3}+15 a^{\frac {3}{2}} b^{2} x -15 \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right ) \left (a \,x^{2}+b \right )^{\frac {3}{2}} a b \right )}{6 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} x^{5} a^{\frac {9}{2}}}\)	\(85\)
risch	\(\frac {a \,x^{2}+b}{2 a^{3} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}+\frac {\left (-\frac {5 b \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )}{2 a^{\frac {7}{2}}}-\frac {b^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{a}\right )^{2} a +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{a}\right )}}{12 a^{4} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{a}\right )^{2}}+\frac {7 b \sqrt {\left (x -\frac {\sqrt {-a b}}{a}\right )^{2} a +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{a}\right )}}{6 a^{4} \left (x -\frac {\sqrt {-a b}}{a}\right )}+\frac {b^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{a}\right )^{2} a -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{a}\right )}}{12 a^{4} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{a}\right )^{2}}+\frac {7 b \sqrt {\left (x +\frac {\sqrt {-a b}}{a}\right )^{2} a -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{a}\right )}}{6 a^{4} \left (x +\frac {\sqrt {-a b}}{a}\right )}\right ) \sqrt {a \,x^{2}+b}}{\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}\)	\(334\)

[In]

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

[Out]

1/6*(a*x^2+b)*(3*x^5*a^(7/2)+20*a^(5/2)*b*x^3+15*a^(3/2)*b^2*x-15*ln(a^(1/2)*x+(a*x^2+b)^(1/2))*(a*x^2+b)^(3/2
)*a*b)/((a*x^2+b)/x^2)^(5/2)/x^5/a^(9/2)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.95 \[ \int \frac {x}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=\left [\frac {15 \, {\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )} \sqrt {a} \log \left (-2 \, a x^{2} + 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) + 2 \, {\left (3 \, a^{3} x^{6} + 20 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{12 \, {\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}}, \frac {15 \, {\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (3 \, a^{3} x^{6} + 20 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{6 \, {\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}}\right ] \]

[In]

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

[Out]

[1/12*(15*(a^2*b*x^4 + 2*a*b^2*x^2 + b^3)*sqrt(a)*log(-2*a*x^2 + 2*sqrt(a)*x^2*sqrt((a*x^2 + b)/x^2) - b) + 2*
(3*a^3*x^6 + 20*a^2*b*x^4 + 15*a*b^2*x^2)*sqrt((a*x^2 + b)/x^2))/(a^6*x^4 + 2*a^5*b*x^2 + a^4*b^2), 1/6*(15*(a
^2*b*x^4 + 2*a*b^2*x^2 + b^3)*sqrt(-a)*arctan(sqrt(-a)*x^2*sqrt((a*x^2 + b)/x^2)/(a*x^2 + b)) + (3*a^3*x^6 + 2
0*a^2*b*x^4 + 15*a*b^2*x^2)*sqrt((a*x^2 + b)/x^2))/(a^6*x^4 + 2*a^5*b*x^2 + a^4*b^2)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 819 vs. \(2 (80) = 160\).

Time = 2.68 (sec) , antiderivative size = 819, normalized size of antiderivative = 9.31 \[ \int \frac {x}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=\frac {6 a^{17} x^{8} \sqrt {1 + \frac {b}{a x^{2}}}}{12 a^{\frac {39}{2}} x^{6} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{2} + 12 a^{\frac {33}{2}} b^{3}} + \frac {46 a^{16} b x^{6} \sqrt {1 + \frac {b}{a x^{2}}}}{12 a^{\frac {39}{2}} x^{6} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{2} + 12 a^{\frac {33}{2}} b^{3}} + \frac {15 a^{16} b x^{6} \log {\left (\frac {b}{a x^{2}} \right )}}{12 a^{\frac {39}{2}} x^{6} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{2} + 12 a^{\frac {33}{2}} b^{3}} - \frac {30 a^{16} b x^{6} \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{6} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{2} + 12 a^{\frac {33}{2}} b^{3}} + \frac {70 a^{15} b^{2} x^{4} \sqrt {1 + \frac {b}{a x^{2}}}}{12 a^{\frac {39}{2}} x^{6} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{2} + 12 a^{\frac {33}{2}} b^{3}} + \frac {45 a^{15} b^{2} x^{4} \log {\left (\frac {b}{a x^{2}} \right )}}{12 a^{\frac {39}{2}} x^{6} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{2} + 12 a^{\frac {33}{2}} b^{3}} - \frac {90 a^{15} b^{2} x^{4} \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{6} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{2} + 12 a^{\frac {33}{2}} b^{3}} + \frac {30 a^{14} b^{3} x^{2} \sqrt {1 + \frac {b}{a x^{2}}}}{12 a^{\frac {39}{2}} x^{6} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{2} + 12 a^{\frac {33}{2}} b^{3}} + \frac {45 a^{14} b^{3} x^{2} \log {\left (\frac {b}{a x^{2}} \right )}}{12 a^{\frac {39}{2}} x^{6} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{2} + 12 a^{\frac {33}{2}} b^{3}} - \frac {90 a^{14} b^{3} x^{2} \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{6} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{2} + 12 a^{\frac {33}{2}} b^{3}} + \frac {15 a^{13} b^{4} \log {\left (\frac {b}{a x^{2}} \right )}}{12 a^{\frac {39}{2}} x^{6} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{2} + 12 a^{\frac {33}{2}} b^{3}} - \frac {30 a^{13} b^{4} \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{6} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{2} + 12 a^{\frac {33}{2}} b^{3}} \]

[In]

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

[Out]

6*a**17*x**8*sqrt(1 + b/(a*x**2))/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**2 + 12*a**(3
3/2)*b**3) + 46*a**16*b*x**6*sqrt(1 + b/(a*x**2))/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2
*x**2 + 12*a**(33/2)*b**3) + 15*a**16*b*x**6*log(b/(a*x**2))/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**
(35/2)*b**2*x**2 + 12*a**(33/2)*b**3) - 30*a**16*b*x**6*log(sqrt(1 + b/(a*x**2)) + 1)/(12*a**(39/2)*x**6 + 36*
a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**2 + 12*a**(33/2)*b**3) + 70*a**15*b**2*x**4*sqrt(1 + b/(a*x**2))/(12*a
**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**2 + 12*a**(33/2)*b**3) + 45*a**15*b**2*x**4*log(b/(
a*x**2))/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**2 + 12*a**(33/2)*b**3) - 90*a**15*b**
2*x**4*log(sqrt(1 + b/(a*x**2)) + 1)/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**2 + 12*a*
*(33/2)*b**3) + 30*a**14*b**3*x**2*sqrt(1 + b/(a*x**2))/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2
)*b**2*x**2 + 12*a**(33/2)*b**3) + 45*a**14*b**3*x**2*log(b/(a*x**2))/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4
+ 36*a**(35/2)*b**2*x**2 + 12*a**(33/2)*b**3) - 90*a**14*b**3*x**2*log(sqrt(1 + b/(a*x**2)) + 1)/(12*a**(39/2
)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**2 + 12*a**(33/2)*b**3) + 15*a**13*b**4*log(b/(a*x**2))/(12
*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**2 + 12*a**(33/2)*b**3) - 30*a**13*b**4*log(sqrt(1
+ b/(a*x**2)) + 1)/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**2 + 12*a**(33/2)*b**3)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.15 \[ \int \frac {x}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=\frac {15 \, {\left (a + \frac {b}{x^{2}}\right )}^{2} b - 10 \, {\left (a + \frac {b}{x^{2}}\right )} a b - 2 \, a^{2} b}{6 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}} a^{3} - {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{4}\right )}} + \frac {5 \, b \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{4 \, a^{\frac {7}{2}}} \]

[In]

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

[Out]

1/6*(15*(a + b/x^2)^2*b - 10*(a + b/x^2)*a*b - 2*a^2*b)/((a + b/x^2)^(5/2)*a^3 - (a + b/x^2)^(3/2)*a^4) + 5/4*
b*log((sqrt(a + b/x^2) - sqrt(a))/(sqrt(a + b/x^2) + sqrt(a)))/a^(7/2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05 \[ \int \frac {x}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=\frac {{\left (x^{2} {\left (\frac {3 \, x^{2}}{a \mathrm {sgn}\left (x\right )} + \frac {20 \, b}{a^{2} \mathrm {sgn}\left (x\right )}\right )} + \frac {15 \, b^{2}}{a^{3} \mathrm {sgn}\left (x\right )}\right )} x}{6 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}}} - \frac {5 \, b \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{4 \, a^{\frac {7}{2}}} + \frac {5 \, b \log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right )}{2 \, a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

1/6*(x^2*(3*x^2/(a*sgn(x)) + 20*b/(a^2*sgn(x))) + 15*b^2/(a^3*sgn(x)))*x/(a*x^2 + b)^(3/2) - 5/4*b*log(abs(b))
*sgn(x)/a^(7/2) + 5/2*b*log(abs(-sqrt(a)*x + sqrt(a*x^2 + b)))/(a^(7/2)*sgn(x))

Mupad [B] (verification not implemented)

Time = 6.45 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.83 \[ \int \frac {x}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=\frac {10\,b}{3\,a^2\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}+\frac {x^2}{2\,a\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}-\frac {5\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2\,a^{7/2}}+\frac {5\,b^2}{2\,a^3\,x^2\,{\left (a+\frac {b}{x^2}\right )}^{3/2}} \]

[In]

int(x/(a + b/x^2)^(5/2),x)

[Out]

(10*b)/(3*a^2*(a + b/x^2)^(3/2)) + x^2/(2*a*(a + b/x^2)^(3/2)) - (5*b*atanh((a + b/x^2)^(1/2)/a^(1/2)))/(2*a^(
7/2)) + (5*b^2)/(2*a^3*x^2*(a + b/x^2)^(3/2))