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

Optimal. Leaf size=69 \[ -\frac{b^2}{3 a^3 x^3}+\frac{b^3}{a^4 x}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}+\frac{b}{5 a^2 x^5}-\frac{1}{7 a x^7} \]

[Out]

-1/(7*a*x^7) + b/(5*a^2*x^5) - b^2/(3*a^3*x^3) + b^3/(a^4*x) + (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)

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Rubi [A]  time = 0.0346157, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {325, 205} \[ -\frac{b^2}{3 a^3 x^3}+\frac{b^3}{a^4 x}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}+\frac{b}{5 a^2 x^5}-\frac{1}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(a + b*x^2)),x]

[Out]

-1/(7*a*x^7) + b/(5*a^2*x^5) - b^2/(3*a^3*x^3) + b^3/(a^4*x) + (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)

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}{x^8 \left (a+b x^2\right )} \, dx &=-\frac{1}{7 a x^7}-\frac{b \int \frac{1}{x^6 \left (a+b x^2\right )} \, dx}{a}\\ &=-\frac{1}{7 a x^7}+\frac{b}{5 a^2 x^5}+\frac{b^2 \int \frac{1}{x^4 \left (a+b x^2\right )} \, dx}{a^2}\\ &=-\frac{1}{7 a x^7}+\frac{b}{5 a^2 x^5}-\frac{b^2}{3 a^3 x^3}-\frac{b^3 \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{a^3}\\ &=-\frac{1}{7 a x^7}+\frac{b}{5 a^2 x^5}-\frac{b^2}{3 a^3 x^3}+\frac{b^3}{a^4 x}+\frac{b^4 \int \frac{1}{a+b x^2} \, dx}{a^4}\\ &=-\frac{1}{7 a x^7}+\frac{b}{5 a^2 x^5}-\frac{b^2}{3 a^3 x^3}+\frac{b^3}{a^4 x}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.0246361, size = 69, normalized size = 1. \[ -\frac{b^2}{3 a^3 x^3}+\frac{b^3}{a^4 x}+\frac{b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}+\frac{b}{5 a^2 x^5}-\frac{1}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(a + b*x^2)),x]

[Out]

-1/(7*a*x^7) + b/(5*a^2*x^5) - b^2/(3*a^3*x^3) + b^3/(a^4*x) + (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)

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Maple [A]  time = 0.005, size = 61, normalized size = 0.9 \begin{align*} -{\frac{1}{7\,a{x}^{7}}}-{\frac{{b}^{2}}{3\,{a}^{3}{x}^{3}}}+{\frac{b}{5\,{a}^{2}{x}^{5}}}+{\frac{{b}^{3}}{{a}^{4}x}}+{\frac{{b}^{4}}{{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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/x^8/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.17482, size = 352, normalized size = 5.1 \begin{align*} \left [\frac{105 \, b^{3} x^{7} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 210 \, b^{3} x^{6} - 70 \, a b^{2} x^{4} + 42 \, a^{2} b x^{2} - 30 \, a^{3}}{210 \, a^{4} x^{7}}, \frac{105 \, b^{3} x^{7} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 105 \, b^{3} x^{6} - 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} - 15 \, a^{3}}{105 \, a^{4} x^{7}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/210*(105*b^3*x^7*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 210*b^3*x^6 - 70*a*b^2*x^4 +
42*a^2*b*x^2 - 30*a^3)/(a^4*x^7), 1/105*(105*b^3*x^7*sqrt(b/a)*arctan(x*sqrt(b/a)) + 105*b^3*x^6 - 35*a*b^2*x^
4 + 21*a^2*b*x^2 - 15*a^3)/(a^4*x^7)]

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Sympy [A]  time = 0.53454, size = 112, normalized size = 1.62 \begin{align*} - \frac{\sqrt{- \frac{b^{7}}{a^{9}}} \log{\left (- \frac{a^{5} \sqrt{- \frac{b^{7}}{a^{9}}}}{b^{4}} + x \right )}}{2} + \frac{\sqrt{- \frac{b^{7}}{a^{9}}} \log{\left (\frac{a^{5} \sqrt{- \frac{b^{7}}{a^{9}}}}{b^{4}} + x \right )}}{2} + \frac{- 15 a^{3} + 21 a^{2} b x^{2} - 35 a b^{2} x^{4} + 105 b^{3} x^{6}}{105 a^{4} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-b**7/a**9)*log(-a**5*sqrt(-b**7/a**9)/b**4 + x)/2 + sqrt(-b**7/a**9)*log(a**5*sqrt(-b**7/a**9)/b**4 + x
)/2 + (-15*a**3 + 21*a**2*b*x**2 - 35*a*b**2*x**4 + 105*b**3*x**6)/(105*a**4*x**7)

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Giac [A]  time = 1.99494, size = 84, normalized size = 1.22 \begin{align*} \frac{b^{4} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} + \frac{105 \, b^{3} x^{6} - 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} - 15 \, a^{3}}{105 \, a^{4} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^4*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) + 1/105*(105*b^3*x^6 - 35*a*b^2*x^4 + 21*a^2*b*x^2 - 15*a^3)/(a^4*x^
7)