3.492 \(\int \frac{1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx\)

Optimal. Leaf size=81 \[ -\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2}}-\frac{7 b^2}{2 a^4 x}+\frac{7 b}{6 a^3 x^3}-\frac{7}{10 a^2 x^5}+\frac{1}{2 a x^5 \left (a+b x^2\right )} \]

[Out]

-7/(10*a^2*x^5) + (7*b)/(6*a^3*x^3) - (7*b^2)/(2*a^4*x) + 1/(2*a*x^5*(a + b*x^2)
) - (7*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2))

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Rubi [A]  time = 0.11887, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2}}-\frac{7 b^2}{2 a^4 x}+\frac{7 b}{6 a^3 x^3}-\frac{7}{10 a^2 x^5}+\frac{1}{2 a x^5 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]

[Out]

-7/(10*a^2*x^5) + (7*b)/(6*a^3*x^3) - (7*b^2)/(2*a^4*x) + 1/(2*a*x^5*(a + b*x^2)
) - (7*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2))

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Rubi in Sympy [A]  time = 33.9936, size = 75, normalized size = 0.93 \[ \frac{1}{2 a x^{5} \left (a + b x^{2}\right )} - \frac{7}{10 a^{2} x^{5}} + \frac{7 b}{6 a^{3} x^{3}} - \frac{7 b^{2}}{2 a^{4} x} - \frac{7 b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

1/(2*a*x**5*(a + b*x**2)) - 7/(10*a**2*x**5) + 7*b/(6*a**3*x**3) - 7*b**2/(2*a**
4*x) - 7*b**(5/2)*atan(sqrt(b)*x/sqrt(a))/(2*a**(9/2))

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Mathematica [A]  time = 0.0800348, size = 80, normalized size = 0.99 \[ -\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2}}-\frac{b^3 x}{2 a^4 \left (a+b x^2\right )}-\frac{3 b^2}{a^4 x}+\frac{2 b}{3 a^3 x^3}-\frac{1}{5 a^2 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]

[Out]

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

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Maple [A]  time = 0.017, size = 70, normalized size = 0.9 \[ -{\frac{{b}^{3}x}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{7\,{b}^{3}}{2\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{5\,{a}^{2}{x}^{5}}}-3\,{\frac{{b}^{2}}{{a}^{4}x}}+{\frac{2\,b}{3\,{a}^{3}{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

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

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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^2*x^4 + 2*a*b*x^2 + a^2)*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2652, size = 1, normalized size = 0.01 \[ \left [-\frac{210 \, b^{3} x^{6} + 140 \, a b^{2} x^{4} - 28 \, a^{2} b x^{2} + 12 \, a^{3} - 105 \,{\left (b^{3} x^{7} + a b^{2} x^{5}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{60 \,{\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}, -\frac{105 \, b^{3} x^{6} + 70 \, a b^{2} x^{4} - 14 \, a^{2} b x^{2} + 6 \, a^{3} + 105 \,{\left (b^{3} x^{7} + a b^{2} x^{5}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{30 \,{\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*x^6),x, algorithm="fricas")

[Out]

[-1/60*(210*b^3*x^6 + 140*a*b^2*x^4 - 28*a^2*b*x^2 + 12*a^3 - 105*(b^3*x^7 + a*b
^2*x^5)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^4*b*x^7 +
a^5*x^5), -1/30*(105*b^3*x^6 + 70*a*b^2*x^4 - 14*a^2*b*x^2 + 6*a^3 + 105*(b^3*x
^7 + a*b^2*x^5)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))))/(a^4*b*x^7 + a^5*x^5)]

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Sympy [A]  time = 2.62633, size = 126, normalized size = 1.56 \[ \frac{7 \sqrt{- \frac{b^{5}}{a^{9}}} \log{\left (- \frac{a^{5} \sqrt{- \frac{b^{5}}{a^{9}}}}{b^{3}} + x \right )}}{4} - \frac{7 \sqrt{- \frac{b^{5}}{a^{9}}} \log{\left (\frac{a^{5} \sqrt{- \frac{b^{5}}{a^{9}}}}{b^{3}} + x \right )}}{4} - \frac{6 a^{3} - 14 a^{2} b x^{2} + 70 a b^{2} x^{4} + 105 b^{3} x^{6}}{30 a^{5} x^{5} + 30 a^{4} b x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

7*sqrt(-b**5/a**9)*log(-a**5*sqrt(-b**5/a**9)/b**3 + x)/4 - 7*sqrt(-b**5/a**9)*l
og(a**5*sqrt(-b**5/a**9)/b**3 + x)/4 - (6*a**3 - 14*a**2*b*x**2 + 70*a*b**2*x**4
+ 105*b**3*x**6)/(30*a**5*x**5 + 30*a**4*b*x**7)

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GIAC/XCAS [A]  time = 0.269998, size = 95, normalized size = 1.17 \[ -\frac{7 \, b^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{4}} - \frac{b^{3} x}{2 \,{\left (b x^{2} + a\right )} a^{4}} - \frac{45 \, b^{2} x^{4} - 10 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*x^6),x, algorithm="giac")

[Out]

-7/2*b^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) - 1/2*b^3*x/((b*x^2 + a)*a^4) - 1
/15*(45*b^2*x^4 - 10*a*b*x^2 + 3*a^2)/(a^4*x^5)