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

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

[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 = 0.0754309, 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 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2}}-\frac{b^2}{a^3 x}+\frac{b}{3 a^2 x^3}-\frac{1}{5 a x^5} \]

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)

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Rubi in Sympy [A]  time = 14.989, size = 49, normalized size = 0.84 \[ - \frac{1}{5 a x^{5}} + \frac{b}{3 a^{2} x^{3}} - \frac{b^{2}}{a^{3} x} - \frac{b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**2+a),x)

[Out]

-1/(5*a*x**5) + b/(3*a**2*x**3) - b**2/(a**3*x) - b**(5/2)*atan(sqrt(b)*x/sqrt(a
))/a**(7/2)

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Mathematica [A]  time = 0.0455841, size = 58, normalized size = 1. \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2}}-\frac{b^2}{a^3 x}+\frac{b}{3 a^2 x^3}-\frac{1}{5 a x^5} \]

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 = 0.009, size = 52, normalized size = 0.9 \[ -{\frac{1}{5\,a{x}^{5}}}-{\frac{{b}^{2}}{{a}^{3}x}}+{\frac{b}{3\,{a}^{2}{x}^{3}}}-{\frac{{b}^{3}}{{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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

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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^2 + a)*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.207747, size = 1, normalized size = 0.02 \[ \left [\frac{15 \, b^{2} x^{5} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 30 \, b^{2} x^{4} + 10 \, a b x^{2} - 6 \, a^{2}}{30 \, a^{3} x^{5}}, -\frac{15 \, b^{2} x^{5} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + 15 \, b^{2} x^{4} - 5 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*x^6),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)) - 3
0*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(b*
x/(a*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 [A]  time = 1.74768, size = 100, normalized size = 1.72 \[ \frac{\sqrt{- \frac{b^{5}}{a^{7}}} \log{\left (- \frac{a^{4} \sqrt{- \frac{b^{5}}{a^{7}}}}{b^{3}} + x \right )}}{2} - \frac{\sqrt{- \frac{b^{5}}{a^{7}}} \log{\left (\frac{a^{4} \sqrt{- \frac{b^{5}}{a^{7}}}}{b^{3}} + x \right )}}{2} - \frac{3 a^{2} - 5 a b x^{2} + 15 b^{2} x^{4}}{15 a^{3} x^{5}} \]

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/XCAS [A]  time = 0.207905, size = 70, normalized size = 1.21 \[ -\frac{b^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} - \frac{15 \, b^{2} x^{4} - 5 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*x^6),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)