3.436 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^9} \, dx\)

Optimal. Leaf size=50 \[ -\frac{a^4}{8 x^8}-\frac{2 a^3 b}{3 x^6}-\frac{3 a^2 b^2}{2 x^4}-\frac{2 a b^3}{x^2}+b^4 \log (x) \]

[Out]

-a^4/(8*x^8) - (2*a^3*b)/(3*x^6) - (3*a^2*b^2)/(2*x^4) - (2*a*b^3)/x^2 + b^4*Log
[x]

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Rubi [A]  time = 0.0835066, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{a^4}{8 x^8}-\frac{2 a^3 b}{3 x^6}-\frac{3 a^2 b^2}{2 x^4}-\frac{2 a b^3}{x^2}+b^4 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^9,x]

[Out]

-a^4/(8*x^8) - (2*a^3*b)/(3*x^6) - (3*a^2*b^2)/(2*x^4) - (2*a*b^3)/x^2 + b^4*Log
[x]

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Rubi in Sympy [A]  time = 20.1615, size = 53, normalized size = 1.06 \[ - \frac{a^{4}}{8 x^{8}} - \frac{2 a^{3} b}{3 x^{6}} - \frac{3 a^{2} b^{2}}{2 x^{4}} - \frac{2 a b^{3}}{x^{2}} + \frac{b^{4} \log{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**2/x**9,x)

[Out]

-a**4/(8*x**8) - 2*a**3*b/(3*x**6) - 3*a**2*b**2/(2*x**4) - 2*a*b**3/x**2 + b**4
*log(x**2)/2

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Mathematica [A]  time = 0.00791222, size = 50, normalized size = 1. \[ -\frac{a^4}{8 x^8}-\frac{2 a^3 b}{3 x^6}-\frac{3 a^2 b^2}{2 x^4}-\frac{2 a b^3}{x^2}+b^4 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^9,x]

[Out]

-a^4/(8*x^8) - (2*a^3*b)/(3*x^6) - (3*a^2*b^2)/(2*x^4) - (2*a*b^3)/x^2 + b^4*Log
[x]

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Maple [A]  time = 0.009, size = 45, normalized size = 0.9 \[ -{\frac{{a}^{4}}{8\,{x}^{8}}}-{\frac{2\,{a}^{3}b}{3\,{x}^{6}}}-{\frac{3\,{a}^{2}{b}^{2}}{2\,{x}^{4}}}-2\,{\frac{a{b}^{3}}{{x}^{2}}}+{b}^{4}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^2/x^9,x)

[Out]

-1/8*a^4/x^8-2/3*a^3*b/x^6-3/2*a^2*b^2/x^4-2*a*b^3/x^2+b^4*ln(x)

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Maxima [A]  time = 0.694109, size = 68, normalized size = 1.36 \[ \frac{1}{2} \, b^{4} \log \left (x^{2}\right ) - \frac{48 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 16 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2/x^9,x, algorithm="maxima")

[Out]

1/2*b^4*log(x^2) - 1/24*(48*a*b^3*x^6 + 36*a^2*b^2*x^4 + 16*a^3*b*x^2 + 3*a^4)/x
^8

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Fricas [A]  time = 0.255078, size = 68, normalized size = 1.36 \[ \frac{24 \, b^{4} x^{8} \log \left (x\right ) - 48 \, a b^{3} x^{6} - 36 \, a^{2} b^{2} x^{4} - 16 \, a^{3} b x^{2} - 3 \, a^{4}}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2/x^9,x, algorithm="fricas")

[Out]

1/24*(24*b^4*x^8*log(x) - 48*a*b^3*x^6 - 36*a^2*b^2*x^4 - 16*a^3*b*x^2 - 3*a^4)/
x^8

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Sympy [A]  time = 1.74038, size = 48, normalized size = 0.96 \[ b^{4} \log{\left (x \right )} - \frac{3 a^{4} + 16 a^{3} b x^{2} + 36 a^{2} b^{2} x^{4} + 48 a b^{3} x^{6}}{24 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**2/x**9,x)

[Out]

b**4*log(x) - (3*a**4 + 16*a**3*b*x**2 + 36*a**2*b**2*x**4 + 48*a*b**3*x**6)/(24
*x**8)

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GIAC/XCAS [A]  time = 0.269035, size = 78, normalized size = 1.56 \[ \frac{1}{2} \, b^{4}{\rm ln}\left (x^{2}\right ) - \frac{25 \, b^{4} x^{8} + 48 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 16 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2/x^9,x, algorithm="giac")

[Out]

1/2*b^4*ln(x^2) - 1/24*(25*b^4*x^8 + 48*a*b^3*x^6 + 36*a^2*b^2*x^4 + 16*a^3*b*x^
2 + 3*a^4)/x^8