3.1855 \(\int \frac{1}{(a+\frac{b}{x^2}) x^6} \, dx\)

Optimal. Leaf size=43 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{5/2}}+\frac{a}{b^2 x}-\frac{1}{3 b x^3} \]

[Out]

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

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Rubi [A]  time = 0.0174545, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 325, 205} \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{5/2}}+\frac{a}{b^2 x}-\frac{1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)*x^6),x]

[Out]

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

Rule 263

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 325

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]

Rule 205

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

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right ) x^6} \, dx &=\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]  time = 0.0205476, size = 43, normalized size = 1. \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{5/2}}+\frac{a}{b^2 x}-\frac{1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)*x^6),x]

[Out]

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

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Maple [A]  time = 0.005, size = 39, normalized size = 0.9 \begin{align*}{\frac{{a}^{2}}{{b}^{2}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{3\,b{x}^{3}}}+{\frac{a}{{b}^{2}x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.4676, size = 234, normalized size = 5.44 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [B]  time = 0.493538, size = 87, normalized size = 2.02 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.15485, size = 54, normalized size = 1.26 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)