3.190 \(\int \frac{1}{x^6 \left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=100 \[ -\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{11/2}}-\frac{63 b^2}{8 a^5 x}+\frac{21 b}{8 a^4 x^3}-\frac{63}{40 a^3 x^5}+\frac{9}{8 a^2 x^5 \left (a+b x^2\right )}+\frac{1}{4 a x^5 \left (a+b x^2\right )^2} \]

[Out]

-63/(40*a^3*x^5) + (21*b)/(8*a^4*x^3) - (63*b^2)/(8*a^5*x) + 1/(4*a*x^5*(a + b*x
^2)^2) + 9/(8*a^2*x^5*(a + b*x^2)) - (63*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8
*a^(11/2))

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Rubi [A]  time = 0.123551, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{11/2}}-\frac{63 b^2}{8 a^5 x}+\frac{21 b}{8 a^4 x^3}-\frac{63}{40 a^3 x^5}+\frac{9}{8 a^2 x^5 \left (a+b x^2\right )}+\frac{1}{4 a x^5 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a + b*x^2)^3),x]

[Out]

-63/(40*a^3*x^5) + (21*b)/(8*a^4*x^3) - (63*b^2)/(8*a^5*x) + 1/(4*a*x^5*(a + b*x
^2)^2) + 9/(8*a^2*x^5*(a + b*x^2)) - (63*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8
*a^(11/2))

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Rubi in Sympy [A]  time = 24.4599, size = 94, normalized size = 0.94 \[ \frac{1}{4 a x^{5} \left (a + b x^{2}\right )^{2}} + \frac{9}{8 a^{2} x^{5} \left (a + b x^{2}\right )} - \frac{63}{40 a^{3} x^{5}} + \frac{21 b}{8 a^{4} x^{3}} - \frac{63 b^{2}}{8 a^{5} x} - \frac{63 b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**2+a)**3,x)

[Out]

1/(4*a*x**5*(a + b*x**2)**2) + 9/(8*a**2*x**5*(a + b*x**2)) - 63/(40*a**3*x**5)
+ 21*b/(8*a**4*x**3) - 63*b**2/(8*a**5*x) - 63*b**(5/2)*atan(sqrt(b)*x/sqrt(a))/
(8*a**(11/2))

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Mathematica [A]  time = 0.102001, size = 90, normalized size = 0.9 \[ -\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{11/2}}-\frac{8 a^4-24 a^3 b x^2+168 a^2 b^2 x^4+525 a b^3 x^6+315 b^4 x^8}{40 a^5 x^5 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a + b*x^2)^3),x]

[Out]

-(8*a^4 - 24*a^3*b*x^2 + 168*a^2*b^2*x^4 + 525*a*b^3*x^6 + 315*b^4*x^8)/(40*a^5*
x^5*(a + b*x^2)^2) - (63*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(11/2))

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Maple [A]  time = 0.017, size = 89, normalized size = 0.9 \[ -{\frac{1}{5\,{a}^{3}{x}^{5}}}-6\,{\frac{{b}^{2}}{{a}^{5}x}}+{\frac{b}{{a}^{4}{x}^{3}}}-{\frac{15\,{b}^{4}{x}^{3}}{8\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{17\,{b}^{3}x}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{63\,{b}^{3}}{8\,{a}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^2+a)^3,x)

[Out]

-1/5/a^3/x^5-6*b^2/a^5/x+b/a^4/x^3-15/8/a^5*b^4/(b*x^2+a)^2*x^3-17/8/a^4*b^3/(b*
x^2+a)^2*x-63/8/a^5*b^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)^3*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222119, size = 1, normalized size = 0.01 \[ \left [-\frac{630 \, b^{4} x^{8} + 1050 \, a b^{3} x^{6} + 336 \, a^{2} b^{2} x^{4} - 48 \, a^{3} b x^{2} + 16 \, a^{4} - 315 \,{\left (b^{4} x^{9} + 2 \, a b^{3} x^{7} + a^{2} b^{2} x^{5}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{80 \,{\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}}, -\frac{315 \, b^{4} x^{8} + 525 \, a b^{3} x^{6} + 168 \, a^{2} b^{2} x^{4} - 24 \, a^{3} b x^{2} + 8 \, a^{4} + 315 \,{\left (b^{4} x^{9} + 2 \, a b^{3} x^{7} + a^{2} b^{2} x^{5}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{40 \,{\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^3*x^6),x, algorithm="fricas")

[Out]

[-1/80*(630*b^4*x^8 + 1050*a*b^3*x^6 + 336*a^2*b^2*x^4 - 48*a^3*b*x^2 + 16*a^4 -
315*(b^4*x^9 + 2*a*b^3*x^7 + a^2*b^2*x^5)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b
/a) - a)/(b*x^2 + a)))/(a^5*b^2*x^9 + 2*a^6*b*x^7 + a^7*x^5), -1/40*(315*b^4*x^8
+ 525*a*b^3*x^6 + 168*a^2*b^2*x^4 - 24*a^3*b*x^2 + 8*a^4 + 315*(b^4*x^9 + 2*a*b
^3*x^7 + a^2*b^2*x^5)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))))/(a^5*b^2*x^9 + 2*a^6*
b*x^7 + a^7*x^5)]

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Sympy [A]  time = 5.47564, size = 150, normalized size = 1.5 \[ \frac{63 \sqrt{- \frac{b^{5}}{a^{11}}} \log{\left (- \frac{a^{6} \sqrt{- \frac{b^{5}}{a^{11}}}}{b^{3}} + x \right )}}{16} - \frac{63 \sqrt{- \frac{b^{5}}{a^{11}}} \log{\left (\frac{a^{6} \sqrt{- \frac{b^{5}}{a^{11}}}}{b^{3}} + x \right )}}{16} - \frac{8 a^{4} - 24 a^{3} b x^{2} + 168 a^{2} b^{2} x^{4} + 525 a b^{3} x^{6} + 315 b^{4} x^{8}}{40 a^{7} x^{5} + 80 a^{6} b x^{7} + 40 a^{5} b^{2} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b*x**2+a)**3,x)

[Out]

63*sqrt(-b**5/a**11)*log(-a**6*sqrt(-b**5/a**11)/b**3 + x)/16 - 63*sqrt(-b**5/a*
*11)*log(a**6*sqrt(-b**5/a**11)/b**3 + x)/16 - (8*a**4 - 24*a**3*b*x**2 + 168*a*
*2*b**2*x**4 + 525*a*b**3*x**6 + 315*b**4*x**8)/(40*a**7*x**5 + 80*a**6*b*x**7 +
40*a**5*b**2*x**9)

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GIAC/XCAS [A]  time = 0.222618, size = 108, normalized size = 1.08 \[ -\frac{63 \, b^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{5}} - \frac{15 \, b^{4} x^{3} + 17 \, a b^{3} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{5}} - \frac{30 \, b^{2} x^{4} - 5 \, a b x^{2} + a^{2}}{5 \, a^{5} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^3*x^6),x, algorithm="giac")

[Out]

-63/8*b^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5) - 1/8*(15*b^4*x^3 + 17*a*b^3*x)/
((b*x^2 + a)^2*a^5) - 1/5*(30*b^2*x^4 - 5*a*b*x^2 + a^2)/(a^5*x^5)