\(\int \frac {1}{(a+\frac {b}{x^2}) x^6} \, dx\) [1855]

Optimal result

Integrand size = 13, antiderivative size = 43 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^6} \, dx=-\frac {1}{3 b x^3}+\frac {a}{b^2 x}+\frac {a^{3/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{b^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 331, 211} \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^6} \, dx=\frac {a^{3/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{b^{5/2}}+\frac {a}{b^2 x}-\frac {1}{3 b x^3} \]

[In]

Int[1/((a + b/x^2)*x^6),x]

[Out]

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

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 269

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

Rule 331

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^6} \, dx=-\frac {1}{3 b x^3}+\frac {a}{b^2 x}+\frac {a^{3/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{b^{5/2}} \]

[In]

Integrate[1/((a + b/x^2)*x^6),x]

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91

method	result	size
default	\(-\frac {1}{3 b \,x^{3}}+\frac {a}{b^{2} x}+\frac {a^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\)	\(39\)
risch	\(\frac {\frac {a \,x^{2}}{b^{2}}-\frac {1}{3 b}}{x^{3}}+\frac {\sqrt {-a b}\, a \ln \left (-a x -\sqrt {-a b}\right )}{2 b^{3}}-\frac {\sqrt {-a b}\, a \ln \left (-a x +\sqrt {-a b}\right )}{2 b^{3}}\)	\(70\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.47 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^6} \, dx=\left [\frac {3 \, a x^{3} \sqrt {-\frac {a}{b}} \log \left (\frac {a x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - b}{a x^{2} + b}\right ) + 6 \, a x^{2} - 2 \, b}{6 \, b^{2} x^{3}}, \frac {3 \, a x^{3} \sqrt {\frac {a}{b}} \arctan \left (x \sqrt {\frac {a}{b}}\right ) + 3 \, a x^{2} - b}{3 \, b^{2} x^{3}}\right ] \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (37) = 74\).

Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.02 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^6} \, dx=- \frac {\sqrt {- \frac {a^{3}}{b^{5}}} \log {\left (x - \frac {b^{3} \sqrt {- \frac {a^{3}}{b^{5}}}}{a^{2}} \right )}}{2} + \frac {\sqrt {- \frac {a^{3}}{b^{5}}} \log {\left (x + \frac {b^{3} \sqrt {- \frac {a^{3}}{b^{5}}}}{a^{2}} \right )}}{2} + \frac {3 a x^{2} - b}{3 b^{2} x^{3}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^6} \, dx=\frac {a^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {3 \, a x^{2} - b}{3 \, b^{2} x^{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^6} \, dx=\frac {a^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {3 \, a x^{2} - b}{3 \, b^{2} x^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.80 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^6} \, dx=\frac {a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{b^{5/2}}-\frac {\frac {1}{3\,b}-\frac {a\,x^2}{b^2}}{x^3} \]

[In]

int(1/(x^6*(a + b/x^2)),x)

[Out]

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