3.837 \(\int \frac{\left (a+b x^2+c x^4\right )^2}{x^{10}} \, dx\)

Optimal. Leaf size=52 \[ -\frac{a^2}{9 x^9}-\frac{2 a c+b^2}{5 x^5}-\frac{2 a b}{7 x^7}-\frac{2 b c}{3 x^3}-\frac{c^2}{x} \]

[Out]

-a^2/(9*x^9) - (2*a*b)/(7*x^7) - (b^2 + 2*a*c)/(5*x^5) - (2*b*c)/(3*x^3) - c^2/x

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Rubi [A]  time = 0.0625679, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{a^2}{9 x^9}-\frac{2 a c+b^2}{5 x^5}-\frac{2 a b}{7 x^7}-\frac{2 b c}{3 x^3}-\frac{c^2}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^2/x^10,x]

[Out]

-a^2/(9*x^9) - (2*a*b)/(7*x^7) - (b^2 + 2*a*c)/(5*x^5) - (2*b*c)/(3*x^3) - c^2/x

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Rubi in Sympy [A]  time = 11.2097, size = 49, normalized size = 0.94 \[ - \frac{a^{2}}{9 x^{9}} - \frac{2 a b}{7 x^{7}} - \frac{2 b c}{3 x^{3}} - \frac{c^{2}}{x} - \frac{\frac{2 a c}{5} + \frac{b^{2}}{5}}{x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**2/x**10,x)

[Out]

-a**2/(9*x**9) - 2*a*b/(7*x**7) - 2*b*c/(3*x**3) - c**2/x - (2*a*c/5 + b**2/5)/x
**5

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Mathematica [A]  time = 0.0351783, size = 50, normalized size = 0.96 \[ -\frac{35 a^2+90 a b x^2+126 a c x^4+63 b^2 x^4+210 b c x^6+315 c^2 x^8}{315 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^2/x^10,x]

[Out]

-(35*a^2 + 90*a*b*x^2 + 63*b^2*x^4 + 126*a*c*x^4 + 210*b*c*x^6 + 315*c^2*x^8)/(3
15*x^9)

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Maple [A]  time = 0.008, size = 45, normalized size = 0.9 \[ -{\frac{2\,bc}{3\,{x}^{3}}}-{\frac{{a}^{2}}{9\,{x}^{9}}}-{\frac{{c}^{2}}{x}}-{\frac{2\,ac+{b}^{2}}{5\,{x}^{5}}}-{\frac{2\,ab}{7\,{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^2/x^10,x)

[Out]

-2/3*b*c/x^3-1/9*a^2/x^9-c^2/x-1/5*(2*a*c+b^2)/x^5-2/7*a*b/x^7

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Maxima [A]  time = 0.691133, size = 62, normalized size = 1.19 \[ -\frac{315 \, c^{2} x^{8} + 210 \, b c x^{6} + 63 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 90 \, a b x^{2} + 35 \, a^{2}}{315 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^10,x, algorithm="maxima")

[Out]

-1/315*(315*c^2*x^8 + 210*b*c*x^6 + 63*(b^2 + 2*a*c)*x^4 + 90*a*b*x^2 + 35*a^2)/
x^9

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Fricas [A]  time = 0.247836, size = 62, normalized size = 1.19 \[ -\frac{315 \, c^{2} x^{8} + 210 \, b c x^{6} + 63 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 90 \, a b x^{2} + 35 \, a^{2}}{315 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^10,x, algorithm="fricas")

[Out]

-1/315*(315*c^2*x^8 + 210*b*c*x^6 + 63*(b^2 + 2*a*c)*x^4 + 90*a*b*x^2 + 35*a^2)/
x^9

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Sympy [A]  time = 4.93467, size = 49, normalized size = 0.94 \[ - \frac{35 a^{2} + 90 a b x^{2} + 210 b c x^{6} + 315 c^{2} x^{8} + x^{4} \left (126 a c + 63 b^{2}\right )}{315 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**2/x**10,x)

[Out]

-(35*a**2 + 90*a*b*x**2 + 210*b*c*x**6 + 315*c**2*x**8 + x**4*(126*a*c + 63*b**2
))/(315*x**9)

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GIAC/XCAS [A]  time = 0.261027, size = 65, normalized size = 1.25 \[ -\frac{315 \, c^{2} x^{8} + 210 \, b c x^{6} + 63 \, b^{2} x^{4} + 126 \, a c x^{4} + 90 \, a b x^{2} + 35 \, a^{2}}{315 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^10,x, algorithm="giac")

[Out]

-1/315*(315*c^2*x^8 + 210*b*c*x^6 + 63*b^2*x^4 + 126*a*c*x^4 + 90*a*b*x^2 + 35*a
^2)/x^9