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

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

[Out]

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

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Rubi [A]  time = 0.0245379, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {325, 205} \[ -\frac{b^2}{a^3 x}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2}}+\frac{b}{3 a^2 x^3}-\frac{1}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a + b*x^2)),x]

[Out]

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

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

Mathematica [A]  time = 0.0228291, size = 58, normalized size = 1. \[ -\frac{b^2}{a^3 x}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2}}+\frac{b}{3 a^2 x^3}-\frac{1}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a + b*x^2)),x]

[Out]

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

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Maple [A]  time = 0.007, size = 52, normalized size = 0.9 \begin{align*} -{\frac{1}{5\,a{x}^{5}}}-{\frac{{b}^{2}}{{a}^{3}x}}+{\frac{b}{3\,{a}^{2}{x}^{3}}}-{\frac{{b}^{3}}{{a}^{3}}\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^6/(b*x^2+a),x)

[Out]

-1/5/a/x^5-b^2/a^3/x+1/3*b/a^2/x^3-b^3/a^3/(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^6/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.29944, size = 296, normalized size = 5.1 \begin{align*} \left [\frac{15 \, b^{2} x^{5} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 30 \, b^{2} x^{4} + 10 \, a b x^{2} - 6 \, a^{2}}{30 \, a^{3} x^{5}}, -\frac{15 \, b^{2} x^{5} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 15 \, b^{2} x^{4} - 5 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{3} x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.433655, size = 100, normalized size = 1.72 \begin{align*} \frac{\sqrt{- \frac{b^{5}}{a^{7}}} \log{\left (- \frac{a^{4} \sqrt{- \frac{b^{5}}{a^{7}}}}{b^{3}} + x \right )}}{2} - \frac{\sqrt{- \frac{b^{5}}{a^{7}}} \log{\left (\frac{a^{4} \sqrt{- \frac{b^{5}}{a^{7}}}}{b^{3}} + x \right )}}{2} - \frac{3 a^{2} - 5 a b x^{2} + 15 b^{2} x^{4}}{15 a^{3} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.78794, size = 70, normalized size = 1.21 \begin{align*} -\frac{b^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} - \frac{15 \, b^{2} x^{4} - 5 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{3} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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