3.1857 \(\int \frac{1}{(a+\frac{b}{x^2}) x^8} \, dx\)

Optimal. Leaf size=58 \[ -\frac{a^2}{b^3 x}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{7/2}}+\frac{a}{3 b^2 x^3}-\frac{1}{5 b x^5} \]

[Out]

-1/(5*b*x^5) + a/(3*b^2*x^3) - a^2/(b^3*x) - (a^(5/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/b^(7/2)

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Rubi [A]  time = 0.0261248, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 325, 205} \[ -\frac{a^2}{b^3 x}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{7/2}}+\frac{a}{3 b^2 x^3}-\frac{1}{5 b x^5} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)*x^8),x]

[Out]

-1/(5*b*x^5) + a/(3*b^2*x^3) - a^2/(b^3*x) - (a^(5/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/b^(7/2)

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^8} \, dx &=\int \frac{1}{x^6 \left (b+a x^2\right )} \, dx\\ &=-\frac{1}{5 b x^5}-\frac{a \int \frac{1}{x^4 \left (b+a x^2\right )} \, dx}{b}\\ &=-\frac{1}{5 b x^5}+\frac{a}{3 b^2 x^3}+\frac{a^2 \int \frac{1}{x^2 \left (b+a x^2\right )} \, dx}{b^2}\\ &=-\frac{1}{5 b x^5}+\frac{a}{3 b^2 x^3}-\frac{a^2}{b^3 x}-\frac{a^3 \int \frac{1}{b+a x^2} \, dx}{b^3}\\ &=-\frac{1}{5 b x^5}+\frac{a}{3 b^2 x^3}-\frac{a^2}{b^3 x}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0234124, size = 58, normalized size = 1. \[ -\frac{a^2}{b^3 x}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{7/2}}+\frac{a}{3 b^2 x^3}-\frac{1}{5 b x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)*x^8),x]

[Out]

-1/(5*b*x^5) + a/(3*b^2*x^3) - a^2/(b^3*x) - (a^(5/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/b^(7/2)

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Maple [A]  time = 0.006, size = 52, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{{b}^{3}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{5\,b{x}^{5}}}-{\frac{{a}^{2}}{{b}^{3}x}}+{\frac{a}{3\,{b}^{2}{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x^2*b)/x^8,x)

[Out]

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

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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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.7193, size = 296, normalized size = 5.1 \begin{align*} \left [\frac{15 \, a^{2} x^{5} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) - 30 \, a^{2} x^{4} + 10 \, a b x^{2} - 6 \, b^{2}}{30 \, b^{3} x^{5}}, -\frac{15 \, a^{2} x^{5} \sqrt{\frac{a}{b}} \arctan \left (x \sqrt{\frac{a}{b}}\right ) + 15 \, a^{2} x^{4} - 5 \, a b x^{2} + 3 \, b^{2}}{15 \, b^{3} x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)/x^8,x, algorithm="fricas")

[Out]

[1/30*(15*a^2*x^5*sqrt(-a/b)*log((a*x^2 - 2*b*x*sqrt(-a/b) - b)/(a*x^2 + b)) - 30*a^2*x^4 + 10*a*b*x^2 - 6*b^2
)/(b^3*x^5), -1/15*(15*a^2*x^5*sqrt(a/b)*arctan(x*sqrt(a/b)) + 15*a^2*x^4 - 5*a*b*x^2 + 3*b^2)/(b^3*x^5)]

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Sympy [B]  time = 0.587015, size = 100, normalized size = 1.72 \begin{align*} \frac{\sqrt{- \frac{a^{5}}{b^{7}}} \log{\left (x - \frac{b^{4} \sqrt{- \frac{a^{5}}{b^{7}}}}{a^{3}} \right )}}{2} - \frac{\sqrt{- \frac{a^{5}}{b^{7}}} \log{\left (x + \frac{b^{4} \sqrt{- \frac{a^{5}}{b^{7}}}}{a^{3}} \right )}}{2} - \frac{15 a^{2} x^{4} - 5 a b x^{2} + 3 b^{2}}{15 b^{3} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)/x**8,x)

[Out]

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

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Giac [A]  time = 1.16135, size = 70, normalized size = 1.21 \begin{align*} -\frac{a^{3} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} - \frac{15 \, a^{2} x^{4} - 5 \, a b x^{2} + 3 \, b^{2}}{15 \, b^{3} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)/x^8,x, algorithm="giac")

[Out]

-a^3*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*b^3) - 1/15*(15*a^2*x^4 - 5*a*b*x^2 + 3*b^2)/(b^3*x^5)