\(\int \frac {1}{(a+\frac {b}{x^2})^{5/2} x^6} \, dx\) [1955]

Optimal result

Integrand size = 15, antiderivative size = 68 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=\frac {1}{3 b \left (a+\frac {b}{x^2}\right )^{3/2} x^3}+\frac {1}{b^2 \sqrt {a+\frac {b}{x^2}} x}-\frac {\text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{b^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 68, 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 = {342, 294, 223, 212} \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{b^{5/2}}+\frac {1}{b^2 x \sqrt {a+\frac {b}{x^2}}}+\frac {1}{3 b x^3 \left (a+\frac {b}{x^2}\right )^{3/2}} \]

[In]

Int[1/((a + b/x^2)^(5/2)*x^6),x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 342

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=\frac {\sqrt {b} \left (4 b+3 a x^2\right )-3 \left (b+a x^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )}{3 b^{5/2} \sqrt {a+\frac {b}{x^2}} x \left (b+a x^2\right )} \]

[In]

Integrate[1/((a + b/x^2)^(5/2)*x^6),x]

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.13

method	result	size
default	\(\frac {\left (a \,x^{2}+b \right ) \left (3 x^{2} a \,b^{\frac {3}{2}}+4 b^{\frac {5}{2}}-3 \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) \left (a \,x^{2}+b \right )^{\frac {3}{2}} b \right )}{3 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} x^{5} b^{\frac {7}{2}}}\)	\(77\)

[In]

int(1/(a+b/x^2)^(5/2)/x^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 226, normalized size of antiderivative = 3.32 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=\left [\frac {3 \, {\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt {b} \log \left (-\frac {a x^{2} - 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) + 2 \, {\left (3 \, a b x^{3} + 4 \, b^{2} x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{6 \, {\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )}}, \frac {3 \, {\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (3 \, a b x^{3} + 4 \, b^{2} x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )}}\right ] \]

[In]

integrate(1/(a+b/x^2)^(5/2)/x^6,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 740 vs. \(2 (58) = 116\).

Time = 1.78 (sec) , antiderivative size = 740, normalized size of antiderivative = 10.88 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=\frac {3 a^{3} b^{4} x^{6} \log {\left (\frac {a x^{2}}{b} \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} - \frac {6 a^{3} b^{4} x^{6} \log {\left (\sqrt {\frac {a x^{2}}{b} + 1} + 1 \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} + \frac {6 a^{2} b^{5} x^{4} \sqrt {\frac {a x^{2}}{b} + 1}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} + \frac {9 a^{2} b^{5} x^{4} \log {\left (\frac {a x^{2}}{b} \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} - \frac {18 a^{2} b^{5} x^{4} \log {\left (\sqrt {\frac {a x^{2}}{b} + 1} + 1 \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} + \frac {14 a b^{6} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} + \frac {9 a b^{6} x^{2} \log {\left (\frac {a x^{2}}{b} \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} - \frac {18 a b^{6} x^{2} \log {\left (\sqrt {\frac {a x^{2}}{b} + 1} + 1 \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} + \frac {8 b^{7} \sqrt {\frac {a x^{2}}{b} + 1}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} + \frac {3 b^{7} \log {\left (\frac {a x^{2}}{b} \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} - \frac {6 b^{7} \log {\left (\sqrt {\frac {a x^{2}}{b} + 1} + 1 \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} \]

[In]

integrate(1/(a+b/x**2)**(5/2)/x**6,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right )}{2 \, b^{\frac {5}{2}}} + \frac {3 \, {\left (a + \frac {b}{x^{2}}\right )} x^{2} + b}{3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b^{2} x^{3}} \]

[In]

integrate(1/(a+b/x^2)^(5/2)/x^6,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=-\frac {{\left (3 \, \sqrt {b} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 4 \, \sqrt {-b}\right )} \mathrm {sgn}\left (x\right )}{3 \, \sqrt {-b} b^{\frac {5}{2}}} + \frac {\arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2} \mathrm {sgn}\left (x\right )} + \frac {3 \, a x^{2} + 4 \, b}{3 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} b^{2} \mathrm {sgn}\left (x\right )} \]

[In]

integrate(1/(a+b/x^2)^(5/2)/x^6,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=\int \frac {1}{x^6\,{\left (a+\frac {b}{x^2}\right )}^{5/2}} \,d x \]

[In]

int(1/(x^6*(a + b/x^2)^(5/2)),x)

[Out]

int(1/(x^6*(a + b/x^2)^(5/2)), x)