3.1456 \(\int \frac{1}{x^5 \left (a+b x^8\right )} \, dx\)

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 = 0.0586388, 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 \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^4}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{1}{4 a x^4} \]

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))

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Rubi in Sympy [A]  time = 9.39739, size = 36, normalized size = 0.9 \[ - \frac{1}{4 a x^{4}} - \frac{\sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} x^{4}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b*x**8+a),x)

[Out]

-1/(4*a*x**4) - sqrt(b)*atan(sqrt(b)*x**4/sqrt(a))/(4*a**(3/2))

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Mathematica [B]  time = 0.272778, 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*ArcTan[(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 = 0.007, size = 32, normalized size = 0.8 \[ -{\frac{b}{4\,a}\arctan \left ({b{x}^{4}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{4\,a{x}^{4}}} \]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222933, size = 1, normalized size = 0.02 \[ \left [\frac{x^{4} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{8} - 2 \, a x^{4} \sqrt{-\frac{b}{a}} - a}{b x^{8} + a}\right ) - 2}{8 \, a x^{4}}, \frac{x^{4} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b x^{4}}\right ) - 1}{4 \, a x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^5),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 = 3.57367, size = 71, normalized size = 1.78 \[ \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (- \frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x^{4} \right )}}{8} - \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (\frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x^{4} \right )}}{8} - \frac{1}{4 a x^{4}} \]

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/XCAS [A]  time = 0.230718, size = 42, normalized size = 1.05 \[ -\frac{b \arctan \left (\frac{b x^{4}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a} - \frac{1}{4 \, a x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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