\(\int \frac {1}{(a+\frac {b}{x^2})^{5/2} x} \, dx\) [1946]

Optimal result

Integrand size = 15, antiderivative size = 59 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x} \, dx=-\frac {1}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {1}{a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 53, 65, 214} \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {1}{a^2 \sqrt {a+\frac {b}{x^2}}}-\frac {1}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}} \]

[In]

Int[1/((a + b/x^2)^(5/2)*x),x]

[Out]

-1/3*1/(a*(a + b/x^2)^(3/2)) - 1/(a^2*Sqrt[a + b/x^2]) + ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]]/a^(5/2)

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x} \, dx=\frac {-\sqrt {a} x \left (3 b+4 a x^2\right )+6 \left (b+a x^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {a} x}{-\sqrt {b}+\sqrt {b+a x^2}}\right )}{3 a^{5/2} \sqrt {a+\frac {b}{x^2}} x \left (b+a x^2\right )} \]

[In]

Integrate[1/((a + b/x^2)^(5/2)*x),x]

[Out]

(-(Sqrt[a]*x*(3*b + 4*a*x^2)) + 6*(b + a*x^2)^(3/2)*ArcTanh[(Sqrt[a]*x)/(-Sqrt[b] + Sqrt[b + a*x^2])])/(3*a^(5
/2)*Sqrt[a + b/x^2]*x*(b + a*x^2))

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24

method	result	size
default	\(-\frac {\left (a \,x^{2}+b \right ) \left (4 x^{3} a^{\frac {5}{2}}+3 a^{\frac {3}{2}} b x -3 \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right ) \left (a \,x^{2}+b \right )^{\frac {3}{2}} a \right )}{3 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} x^{5} a^{\frac {7}{2}}}\)	\(73\)

[In]

int(1/(a+b/x^2)^(5/2)/x,x,method=_RETURNVERBOSE)

[Out]

-1/3*(a*x^2+b)*(4*x^3*a^(5/2)+3*a^(3/2)*b*x-3*ln(a^(1/2)*x+(a*x^2+b)^(1/2))*(a*x^2+b)^(3/2)*a)/((a*x^2+b)/x^2)
^(5/2)/x^5/a^(7/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (47) = 94\).

Time = 0.45 (sec) , antiderivative size = 232, normalized size of antiderivative = 3.93 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x} \, dx=\left [\frac {3 \, {\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt {a} \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) - 2 \, {\left (4 \, a^{2} x^{4} + 3 \, a b x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{6 \, {\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}}, -\frac {3 \, {\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (4 \, a^{2} x^{4} + 3 \, a b x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}}\right ] \]

[In]

integrate(1/(a+b/x^2)^(5/2)/x,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (51) = 102\).

Time = 1.59 (sec) , antiderivative size = 743, normalized size of antiderivative = 12.59 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x} \, dx=- \frac {8 a^{7} x^{6} \sqrt {1 + \frac {b}{a x^{2}}}}{6 a^{\frac {19}{2}} x^{6} + 18 a^{\frac {17}{2}} b x^{4} + 18 a^{\frac {15}{2}} b^{2} x^{2} + 6 a^{\frac {13}{2}} b^{3}} - \frac {3 a^{7} x^{6} \log {\left (\frac {b}{a x^{2}} \right )}}{6 a^{\frac {19}{2}} x^{6} + 18 a^{\frac {17}{2}} b x^{4} + 18 a^{\frac {15}{2}} b^{2} x^{2} + 6 a^{\frac {13}{2}} b^{3}} + \frac {6 a^{7} x^{6} \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{6 a^{\frac {19}{2}} x^{6} + 18 a^{\frac {17}{2}} b x^{4} + 18 a^{\frac {15}{2}} b^{2} x^{2} + 6 a^{\frac {13}{2}} b^{3}} - \frac {14 a^{6} b x^{4} \sqrt {1 + \frac {b}{a x^{2}}}}{6 a^{\frac {19}{2}} x^{6} + 18 a^{\frac {17}{2}} b x^{4} + 18 a^{\frac {15}{2}} b^{2} x^{2} + 6 a^{\frac {13}{2}} b^{3}} - \frac {9 a^{6} b x^{4} \log {\left (\frac {b}{a x^{2}} \right )}}{6 a^{\frac {19}{2}} x^{6} + 18 a^{\frac {17}{2}} b x^{4} + 18 a^{\frac {15}{2}} b^{2} x^{2} + 6 a^{\frac {13}{2}} b^{3}} + \frac {18 a^{6} b x^{4} \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{6 a^{\frac {19}{2}} x^{6} + 18 a^{\frac {17}{2}} b x^{4} + 18 a^{\frac {15}{2}} b^{2} x^{2} + 6 a^{\frac {13}{2}} b^{3}} - \frac {6 a^{5} b^{2} x^{2} \sqrt {1 + \frac {b}{a x^{2}}}}{6 a^{\frac {19}{2}} x^{6} + 18 a^{\frac {17}{2}} b x^{4} + 18 a^{\frac {15}{2}} b^{2} x^{2} + 6 a^{\frac {13}{2}} b^{3}} - \frac {9 a^{5} b^{2} x^{2} \log {\left (\frac {b}{a x^{2}} \right )}}{6 a^{\frac {19}{2}} x^{6} + 18 a^{\frac {17}{2}} b x^{4} + 18 a^{\frac {15}{2}} b^{2} x^{2} + 6 a^{\frac {13}{2}} b^{3}} + \frac {18 a^{5} b^{2} x^{2} \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{6 a^{\frac {19}{2}} x^{6} + 18 a^{\frac {17}{2}} b x^{4} + 18 a^{\frac {15}{2}} b^{2} x^{2} + 6 a^{\frac {13}{2}} b^{3}} - \frac {3 a^{4} b^{3} \log {\left (\frac {b}{a x^{2}} \right )}}{6 a^{\frac {19}{2}} x^{6} + 18 a^{\frac {17}{2}} b x^{4} + 18 a^{\frac {15}{2}} b^{2} x^{2} + 6 a^{\frac {13}{2}} b^{3}} + \frac {6 a^{4} b^{3} \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{6 a^{\frac {19}{2}} x^{6} + 18 a^{\frac {17}{2}} b x^{4} + 18 a^{\frac {15}{2}} b^{2} x^{2} + 6 a^{\frac {13}{2}} b^{3}} \]

[In]

integrate(1/(a+b/x**2)**(5/2)/x,x)

[Out]

-8*a**7*x**6*sqrt(1 + b/(a*x**2))/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/2)*b**2*x**2 + 6*a**(13/
2)*b**3) - 3*a**7*x**6*log(b/(a*x**2))/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/2)*b**2*x**2 + 6*a*
*(13/2)*b**3) + 6*a**7*x**6*log(sqrt(1 + b/(a*x**2)) + 1)/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/
2)*b**2*x**2 + 6*a**(13/2)*b**3) - 14*a**6*b*x**4*sqrt(1 + b/(a*x**2))/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4
+ 18*a**(15/2)*b**2*x**2 + 6*a**(13/2)*b**3) - 9*a**6*b*x**4*log(b/(a*x**2))/(6*a**(19/2)*x**6 + 18*a**(17/2)
*b*x**4 + 18*a**(15/2)*b**2*x**2 + 6*a**(13/2)*b**3) + 18*a**6*b*x**4*log(sqrt(1 + b/(a*x**2)) + 1)/(6*a**(19/
2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/2)*b**2*x**2 + 6*a**(13/2)*b**3) - 6*a**5*b**2*x**2*sqrt(1 + b/(a*x*
*2))/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/2)*b**2*x**2 + 6*a**(13/2)*b**3) - 9*a**5*b**2*x**2*l
og(b/(a*x**2))/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/2)*b**2*x**2 + 6*a**(13/2)*b**3) + 18*a**5*
b**2*x**2*log(sqrt(1 + b/(a*x**2)) + 1)/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/2)*b**2*x**2 + 6*a
**(13/2)*b**3) - 3*a**4*b**3*log(b/(a*x**2))/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/2)*b**2*x**2
+ 6*a**(13/2)*b**3) + 6*a**4*b**3*log(sqrt(1 + b/(a*x**2)) + 1)/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a
**(15/2)*b**2*x**2 + 6*a**(13/2)*b**3)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x} \, dx=-\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{2 \, a^{\frac {5}{2}}} - \frac {4 \, a + \frac {3 \, b}{x^{2}}}{3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{2}} \]

[In]

integrate(1/(a+b/x^2)^(5/2)/x,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x} \, dx=-\frac {x {\left (\frac {4 \, x^{2}}{a \mathrm {sgn}\left (x\right )} + \frac {3 \, b}{a^{2} \mathrm {sgn}\left (x\right )}\right )}}{3 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}}} + \frac {\log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{2 \, a^{\frac {5}{2}}} - \frac {\log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \]

[In]

integrate(1/(a+b/x^2)^(5/2)/x,x, algorithm="giac")

[Out]

-1/3*x*(4*x^2/(a*sgn(x)) + 3*b/(a^2*sgn(x)))/(a*x^2 + b)^(3/2) + 1/2*log(abs(b))*sgn(x)/a^(5/2) - log(abs(-sqr
t(a)*x + sqrt(a*x^2 + b)))/(a^(5/2)*sgn(x))

Mupad [B] (verification not implemented)

Time = 6.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {a+\frac {b}{x^2}}{a^2}+\frac {1}{3\,a}}{{\left (a+\frac {b}{x^2}\right )}^{3/2}} \]

[In]

int(1/(x*(a + b/x^2)^(5/2)),x)

[Out]

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