3.840 \(\int \frac{\left (a+b x^2+c x^4\right )^2}{x^{13}} \, dx\)

Optimal. Leaf size=54 \[ -\frac{a^2}{12 x^{12}}-\frac{2 a c+b^2}{8 x^8}-\frac{a b}{5 x^{10}}-\frac{b c}{3 x^6}-\frac{c^2}{4 x^4} \]

[Out]

-a^2/(12*x^12) - (a*b)/(5*x^10) - (b^2 + 2*a*c)/(8*x^8) - (b*c)/(3*x^6) - c^2/(4
*x^4)

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Rubi [A]  time = 0.0892001, 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 \[ -\frac{a^2}{12 x^{12}}-\frac{2 a c+b^2}{8 x^8}-\frac{a b}{5 x^{10}}-\frac{b c}{3 x^6}-\frac{c^2}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^2/x^13,x]

[Out]

-a^2/(12*x^12) - (a*b)/(5*x^10) - (b^2 + 2*a*c)/(8*x^8) - (b*c)/(3*x^6) - c^2/(4
*x^4)

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Rubi in Sympy [A]  time = 15.3126, size = 48, normalized size = 0.89 \[ - \frac{a^{2}}{12 x^{12}} - \frac{a b}{5 x^{10}} - \frac{b c}{3 x^{6}} - \frac{c^{2}}{4 x^{4}} - \frac{\frac{a c}{4} + \frac{b^{2}}{8}}{x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**2/x**13,x)

[Out]

-a**2/(12*x**12) - a*b/(5*x**10) - b*c/(3*x**6) - c**2/(4*x**4) - (a*c/4 + b**2/
8)/x**8

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Mathematica [A]  time = 0.0275253, size = 50, normalized size = 0.93 \[ -\frac{10 a^2+24 a b x^2+30 a c x^4+15 b^2 x^4+40 b c x^6+30 c^2 x^8}{120 x^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^2/x^13,x]

[Out]

-(10*a^2 + 24*a*b*x^2 + 15*b^2*x^4 + 30*a*c*x^4 + 40*b*c*x^6 + 30*c^2*x^8)/(120*
x^12)

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Maple [A]  time = 0.009, size = 45, normalized size = 0.8 \[ -{\frac{{a}^{2}}{12\,{x}^{12}}}-{\frac{2\,ac+{b}^{2}}{8\,{x}^{8}}}-{\frac{bc}{3\,{x}^{6}}}-{\frac{ab}{5\,{x}^{10}}}-{\frac{{c}^{2}}{4\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^2/x^13,x)

[Out]

-1/12*a^2/x^12-1/8*(2*a*c+b^2)/x^8-1/3*b*c/x^6-1/5*a*b/x^10-1/4*c^2/x^4

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Maxima [A]  time = 0.692199, size = 62, normalized size = 1.15 \[ -\frac{30 \, c^{2} x^{8} + 40 \, b c x^{6} + 15 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 24 \, a b x^{2} + 10 \, a^{2}}{120 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^13,x, algorithm="maxima")

[Out]

-1/120*(30*c^2*x^8 + 40*b*c*x^6 + 15*(b^2 + 2*a*c)*x^4 + 24*a*b*x^2 + 10*a^2)/x^
12

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Fricas [A]  time = 0.248639, size = 62, normalized size = 1.15 \[ -\frac{30 \, c^{2} x^{8} + 40 \, b c x^{6} + 15 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 24 \, a b x^{2} + 10 \, a^{2}}{120 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^13,x, algorithm="fricas")

[Out]

-1/120*(30*c^2*x^8 + 40*b*c*x^6 + 15*(b^2 + 2*a*c)*x^4 + 24*a*b*x^2 + 10*a^2)/x^
12

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Sympy [A]  time = 8.50504, size = 49, normalized size = 0.91 \[ - \frac{10 a^{2} + 24 a b x^{2} + 40 b c x^{6} + 30 c^{2} x^{8} + x^{4} \left (30 a c + 15 b^{2}\right )}{120 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**2/x**13,x)

[Out]

-(10*a**2 + 24*a*b*x**2 + 40*b*c*x**6 + 30*c**2*x**8 + x**4*(30*a*c + 15*b**2))/
(120*x**12)

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GIAC/XCAS [A]  time = 0.262737, size = 65, normalized size = 1.2 \[ -\frac{30 \, c^{2} x^{8} + 40 \, b c x^{6} + 15 \, b^{2} x^{4} + 30 \, a c x^{4} + 24 \, a b x^{2} + 10 \, a^{2}}{120 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^13,x, algorithm="giac")

[Out]

-1/120*(30*c^2*x^8 + 40*b*c*x^6 + 15*b^2*x^4 + 30*a*c*x^4 + 24*a*b*x^2 + 10*a^2)
/x^12