3.191 \(\int \frac{1}{x^8 \left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=113 \[ \frac{99 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{13/2}}+\frac{99 b^3}{8 a^6 x}-\frac{33 b^2}{8 a^5 x^3}+\frac{99 b}{40 a^4 x^5}-\frac{99}{56 a^3 x^7}+\frac{11}{8 a^2 x^7 \left (a+b x^2\right )}+\frac{1}{4 a x^7 \left (a+b x^2\right )^2} \]

[Out]

-99/(56*a^3*x^7) + (99*b)/(40*a^4*x^5) - (33*b^2)/(8*a^5*x^3) + (99*b^3)/(8*a^6*
x) + 1/(4*a*x^7*(a + b*x^2)^2) + 11/(8*a^2*x^7*(a + b*x^2)) + (99*b^(7/2)*ArcTan
[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(13/2))

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Rubi [A]  time = 0.152835, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{99 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{13/2}}+\frac{99 b^3}{8 a^6 x}-\frac{33 b^2}{8 a^5 x^3}+\frac{99 b}{40 a^4 x^5}-\frac{99}{56 a^3 x^7}+\frac{11}{8 a^2 x^7 \left (a+b x^2\right )}+\frac{1}{4 a x^7 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*(a + b*x^2)^3),x]

[Out]

-99/(56*a^3*x^7) + (99*b)/(40*a^4*x^5) - (33*b^2)/(8*a^5*x^3) + (99*b^3)/(8*a^6*
x) + 1/(4*a*x^7*(a + b*x^2)^2) + 11/(8*a^2*x^7*(a + b*x^2)) + (99*b^(7/2)*ArcTan
[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(13/2))

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Rubi in Sympy [A]  time = 30.3363, size = 107, normalized size = 0.95 \[ \frac{1}{4 a x^{7} \left (a + b x^{2}\right )^{2}} + \frac{11}{8 a^{2} x^{7} \left (a + b x^{2}\right )} - \frac{99}{56 a^{3} x^{7}} + \frac{99 b}{40 a^{4} x^{5}} - \frac{33 b^{2}}{8 a^{5} x^{3}} + \frac{99 b^{3}}{8 a^{6} x} + \frac{99 b^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**8/(b*x**2+a)**3,x)

[Out]

1/(4*a*x**7*(a + b*x**2)**2) + 11/(8*a**2*x**7*(a + b*x**2)) - 99/(56*a**3*x**7)
+ 99*b/(40*a**4*x**5) - 33*b**2/(8*a**5*x**3) + 99*b**3/(8*a**6*x) + 99*b**(7/2
)*atan(sqrt(b)*x/sqrt(a))/(8*a**(13/2))

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Mathematica [A]  time = 0.0996398, size = 101, normalized size = 0.89 \[ \frac{99 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{13/2}}+\frac{-40 a^5+88 a^4 b x^2-264 a^3 b^2 x^4+1848 a^2 b^3 x^6+5775 a b^4 x^8+3465 b^5 x^{10}}{280 a^6 x^7 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*(a + b*x^2)^3),x]

[Out]

(-40*a^5 + 88*a^4*b*x^2 - 264*a^3*b^2*x^4 + 1848*a^2*b^3*x^6 + 5775*a*b^4*x^8 +
3465*b^5*x^10)/(280*a^6*x^7*(a + b*x^2)^2) + (99*b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt
[a]])/(8*a^(13/2))

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Maple [A]  time = 0.019, size = 101, normalized size = 0.9 \[ -{\frac{1}{7\,{a}^{3}{x}^{7}}}+10\,{\frac{{b}^{3}}{{a}^{6}x}}-2\,{\frac{{b}^{2}}{{a}^{5}{x}^{3}}}+{\frac{3\,b}{5\,{a}^{4}{x}^{5}}}+{\frac{19\,{b}^{5}{x}^{3}}{8\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{21\,{b}^{4}x}{8\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{99\,{b}^{4}}{8\,{a}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(b*x^2+a)^3,x)

[Out]

-1/7/a^3/x^7+10*b^3/a^6/x-2*b^2/a^5/x^3+3/5*b/a^4/x^5+19/8/a^6*b^5/(b*x^2+a)^2*x
^3+21/8/a^5*b^4/(b*x^2+a)^2*x+99/8/a^6*b^4/(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)^3*x^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221491, size = 1, normalized size = 0.01 \[ \left [\frac{6930 \, b^{5} x^{10} + 11550 \, a b^{4} x^{8} + 3696 \, a^{2} b^{3} x^{6} - 528 \, a^{3} b^{2} x^{4} + 176 \, a^{4} b x^{2} - 80 \, a^{5} + 3465 \,{\left (b^{5} x^{11} + 2 \, a b^{4} x^{9} + a^{2} b^{3} x^{7}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{560 \,{\left (a^{6} b^{2} x^{11} + 2 \, a^{7} b x^{9} + a^{8} x^{7}\right )}}, \frac{3465 \, b^{5} x^{10} + 5775 \, a b^{4} x^{8} + 1848 \, a^{2} b^{3} x^{6} - 264 \, a^{3} b^{2} x^{4} + 88 \, a^{4} b x^{2} - 40 \, a^{5} + 3465 \,{\left (b^{5} x^{11} + 2 \, a b^{4} x^{9} + a^{2} b^{3} x^{7}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{280 \,{\left (a^{6} b^{2} x^{11} + 2 \, a^{7} b x^{9} + a^{8} x^{7}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^3*x^8),x, algorithm="fricas")

[Out]

[1/560*(6930*b^5*x^10 + 11550*a*b^4*x^8 + 3696*a^2*b^3*x^6 - 528*a^3*b^2*x^4 + 1
76*a^4*b*x^2 - 80*a^5 + 3465*(b^5*x^11 + 2*a*b^4*x^9 + a^2*b^3*x^7)*sqrt(-b/a)*l
og((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^6*b^2*x^11 + 2*a^7*b*x^9 + a^
8*x^7), 1/280*(3465*b^5*x^10 + 5775*a*b^4*x^8 + 1848*a^2*b^3*x^6 - 264*a^3*b^2*x
^4 + 88*a^4*b*x^2 - 40*a^5 + 3465*(b^5*x^11 + 2*a*b^4*x^9 + a^2*b^3*x^7)*sqrt(b/
a)*arctan(b*x/(a*sqrt(b/a))))/(a^6*b^2*x^11 + 2*a^7*b*x^9 + a^8*x^7)]

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Sympy [A]  time = 10.188, size = 162, normalized size = 1.43 \[ - \frac{99 \sqrt{- \frac{b^{7}}{a^{13}}} \log{\left (- \frac{a^{7} \sqrt{- \frac{b^{7}}{a^{13}}}}{b^{4}} + x \right )}}{16} + \frac{99 \sqrt{- \frac{b^{7}}{a^{13}}} \log{\left (\frac{a^{7} \sqrt{- \frac{b^{7}}{a^{13}}}}{b^{4}} + x \right )}}{16} + \frac{- 40 a^{5} + 88 a^{4} b x^{2} - 264 a^{3} b^{2} x^{4} + 1848 a^{2} b^{3} x^{6} + 5775 a b^{4} x^{8} + 3465 b^{5} x^{10}}{280 a^{8} x^{7} + 560 a^{7} b x^{9} + 280 a^{6} b^{2} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(b*x**2+a)**3,x)

[Out]

-99*sqrt(-b**7/a**13)*log(-a**7*sqrt(-b**7/a**13)/b**4 + x)/16 + 99*sqrt(-b**7/a
**13)*log(a**7*sqrt(-b**7/a**13)/b**4 + x)/16 + (-40*a**5 + 88*a**4*b*x**2 - 264
*a**3*b**2*x**4 + 1848*a**2*b**3*x**6 + 5775*a*b**4*x**8 + 3465*b**5*x**10)/(280
*a**8*x**7 + 560*a**7*b*x**9 + 280*a**6*b**2*x**11)

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GIAC/XCAS [A]  time = 0.222218, size = 126, normalized size = 1.12 \[ \frac{99 \, b^{4} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{6}} + \frac{19 \, b^{5} x^{3} + 21 \, a b^{4} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{6}} + \frac{350 \, b^{3} x^{6} - 70 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} - 5 \, a^{3}}{35 \, a^{6} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^3*x^8),x, algorithm="giac")

[Out]

99/8*b^4*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^6) + 1/8*(19*b^5*x^3 + 21*a*b^4*x)/(
(b*x^2 + a)^2*a^6) + 1/35*(350*b^3*x^6 - 70*a*b^2*x^4 + 21*a^2*b*x^2 - 5*a^3)/(a
^6*x^7)