3.1458 \(\int \frac{1}{x^5 (a+b x^8)} \, dx\)

Optimal. Leaf size=40 \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^4}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{1}{4 a x^4} \]

[Out]

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

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Rubi [A]  time = 0.0207813, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 325, 205} \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^4}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x^8)),x]

[Out]

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

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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

Mathematica [B]  time = 0.0368631, size = 164, normalized size = 4.1 \[ \frac{\sqrt{b} x^4 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )-\sqrt{b} x^4 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+\sqrt{b} x^4 \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+\sqrt{b} x^4 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )-\sqrt{a}}{4 a^{3/2} x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x^8)),x]

[Out]

(-Sqrt[a] + Sqrt[b]*x^4*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)] + Sqrt[b]*x^4*ArcTan[Cot[Pi/8] + (b^
(1/8)*x*Csc[Pi/8])/a^(1/8)] + Sqrt[b]*x^4*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]] - Sqrt[b]*x^4*ArcT
an[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]])/(4*a^(3/2)*x^4)

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Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \begin{align*} -{\frac{b}{4\,a}\arctan \left ({b{x}^{4}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{4\,a{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b*x^8+a),x)

[Out]

-1/4*b/a/(a*b)^(1/2)*arctan(b*x^4/(a*b)^(1/2))-1/4/a/x^4

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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^5/(b*x^8+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.32202, size = 201, normalized size = 5.02 \begin{align*} \left [\frac{x^{4} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{8} - 2 \, a x^{4} \sqrt{-\frac{b}{a}} - a}{b x^{8} + a}\right ) - 2}{8 \, a x^{4}}, \frac{x^{4} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b x^{4}}\right ) - 1}{4 \, a x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^8+a),x, algorithm="fricas")

[Out]

[1/8*(x^4*sqrt(-b/a)*log((b*x^8 - 2*a*x^4*sqrt(-b/a) - a)/(b*x^8 + a)) - 2)/(a*x^4), 1/4*(x^4*sqrt(b/a)*arctan
(a*sqrt(b/a)/(b*x^4)) - 1)/(a*x^4)]

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Sympy [A]  time = 0.786061, size = 71, normalized size = 1.78 \begin{align*} \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (- \frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x^{4} \right )}}{8} - \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (\frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x^{4} \right )}}{8} - \frac{1}{4 a x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b*x**8+a),x)

[Out]

sqrt(-b/a**3)*log(-a**2*sqrt(-b/a**3)/b + x**4)/8 - sqrt(-b/a**3)*log(a**2*sqrt(-b/a**3)/b + x**4)/8 - 1/(4*a*
x**4)

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Giac [A]  time = 1.2237, size = 42, normalized size = 1.05 \begin{align*} -\frac{b \arctan \left (\frac{b x^{4}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a} - \frac{1}{4 \, a x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^8+a),x, algorithm="giac")

[Out]

-1/4*b*arctan(b*x^4/sqrt(a*b))/(sqrt(a*b)*a) - 1/4/(a*x^4)