3.838 \(\int \frac{\left (a+b x^2+c x^4\right )^2}{x^{11}} \, dx\)

Optimal. Leaf size=54 \[ -\frac{a^2}{10 x^{10}}-\frac{2 a c+b^2}{6 x^6}-\frac{a b}{4 x^8}-\frac{b c}{2 x^4}-\frac{c^2}{2 x^2} \]

[Out]

-a^2/(10*x^10) - (a*b)/(4*x^8) - (b^2 + 2*a*c)/(6*x^6) - (b*c)/(2*x^4) - c^2/(2*
x^2)

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Rubi [A]  time = 0.088051, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{a^2}{10 x^{10}}-\frac{2 a c+b^2}{6 x^6}-\frac{a b}{4 x^8}-\frac{b c}{2 x^4}-\frac{c^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^2/x^11,x]

[Out]

-a^2/(10*x^10) - (a*b)/(4*x^8) - (b^2 + 2*a*c)/(6*x^6) - (b*c)/(2*x^4) - c^2/(2*
x^2)

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Rubi in Sympy [A]  time = 15.3318, size = 48, normalized size = 0.89 \[ - \frac{a^{2}}{10 x^{10}} - \frac{a b}{4 x^{8}} - \frac{b c}{2 x^{4}} - \frac{c^{2}}{2 x^{2}} - \frac{\frac{a c}{3} + \frac{b^{2}}{6}}{x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**2/x**11,x)

[Out]

-a**2/(10*x**10) - a*b/(4*x**8) - b*c/(2*x**4) - c**2/(2*x**2) - (a*c/3 + b**2/6
)/x**6

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Mathematica [A]  time = 0.0290196, size = 53, normalized size = 0.98 \[ -\frac{6 a^2+5 a \left (3 b x^2+4 c x^4\right )+10 x^4 \left (b^2+3 b c x^2+3 c^2 x^4\right )}{60 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^2/x^11,x]

[Out]

-(6*a^2 + 5*a*(3*b*x^2 + 4*c*x^4) + 10*x^4*(b^2 + 3*b*c*x^2 + 3*c^2*x^4))/(60*x^
10)

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Maple [A]  time = 0.007, size = 45, normalized size = 0.8 \[ -{\frac{ab}{4\,{x}^{8}}}-{\frac{2\,ac+{b}^{2}}{6\,{x}^{6}}}-{\frac{{c}^{2}}{2\,{x}^{2}}}-{\frac{{a}^{2}}{10\,{x}^{10}}}-{\frac{bc}{2\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^2/x^11,x)

[Out]

-1/4*a*b/x^8-1/6*(2*a*c+b^2)/x^6-1/2*c^2/x^2-1/10*a^2/x^10-1/2*b*c/x^4

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Maxima [A]  time = 0.685209, size = 62, normalized size = 1.15 \[ -\frac{30 \, c^{2} x^{8} + 30 \, b c x^{6} + 10 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 15 \, a b x^{2} + 6 \, a^{2}}{60 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^11,x, algorithm="maxima")

[Out]

-1/60*(30*c^2*x^8 + 30*b*c*x^6 + 10*(b^2 + 2*a*c)*x^4 + 15*a*b*x^2 + 6*a^2)/x^10

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Fricas [A]  time = 0.246775, size = 62, normalized size = 1.15 \[ -\frac{30 \, c^{2} x^{8} + 30 \, b c x^{6} + 10 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 15 \, a b x^{2} + 6 \, a^{2}}{60 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^11,x, algorithm="fricas")

[Out]

-1/60*(30*c^2*x^8 + 30*b*c*x^6 + 10*(b^2 + 2*a*c)*x^4 + 15*a*b*x^2 + 6*a^2)/x^10

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Sympy [A]  time = 6.52052, size = 49, normalized size = 0.91 \[ - \frac{6 a^{2} + 15 a b x^{2} + 30 b c x^{6} + 30 c^{2} x^{8} + x^{4} \left (20 a c + 10 b^{2}\right )}{60 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**2/x**11,x)

[Out]

-(6*a**2 + 15*a*b*x**2 + 30*b*c*x**6 + 30*c**2*x**8 + x**4*(20*a*c + 10*b**2))/(
60*x**10)

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GIAC/XCAS [A]  time = 0.262013, size = 65, normalized size = 1.2 \[ -\frac{30 \, c^{2} x^{8} + 30 \, b c x^{6} + 10 \, b^{2} x^{4} + 20 \, a c x^{4} + 15 \, a b x^{2} + 6 \, a^{2}}{60 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^11,x, algorithm="giac")

[Out]

-1/60*(30*c^2*x^8 + 30*b*c*x^6 + 10*b^2*x^4 + 20*a*c*x^4 + 15*a*b*x^2 + 6*a^2)/x
^10