3.835 \(\int \frac{\left (a+b x^2+c x^4\right )^2}{x^8} \, dx\)

Optimal. Leaf size=47 \[ -\frac{a^2}{7 x^7}-\frac{2 a c+b^2}{3 x^3}-\frac{2 a b}{5 x^5}-\frac{2 b c}{x}+c^2 x \]

[Out]

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

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Rubi [A]  time = 0.0622405, antiderivative size = 47, 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 \[ -\frac{a^2}{7 x^7}-\frac{2 a c+b^2}{3 x^3}-\frac{2 a b}{5 x^5}-\frac{2 b c}{x}+c^2 x \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^2/x^8,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{7 x^{7}} - \frac{2 a b}{5 x^{5}} - \frac{2 b c}{x} + \int c^{2}\, dx - \frac{\frac{2 a c}{3} + \frac{b^{2}}{3}}{x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**2/x**8,x)

[Out]

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

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Mathematica [A]  time = 0.042591, size = 49, normalized size = 1.04 \[ -\frac{a^2}{7 x^7}+\frac{-2 a c-b^2}{3 x^3}-\frac{2 a b}{5 x^5}-\frac{2 b c}{x}+c^2 x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^2/x^8,x]

[Out]

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

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Maple [A]  time = 0.008, size = 42, normalized size = 0.9 \[{c}^{2}x-{\frac{2\,ac+{b}^{2}}{3\,{x}^{3}}}-2\,{\frac{bc}{x}}-{\frac{2\,ab}{5\,{x}^{5}}}-{\frac{{a}^{2}}{7\,{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^2/x^8,x)

[Out]

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

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Maxima [A]  time = 0.691059, size = 59, normalized size = 1.26 \[ c^{2} x - \frac{210 \, b c x^{6} + 35 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^8,x, algorithm="maxima")

[Out]

c^2*x - 1/105*(210*b*c*x^6 + 35*(b^2 + 2*a*c)*x^4 + 42*a*b*x^2 + 15*a^2)/x^7

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Fricas [A]  time = 0.248692, size = 62, normalized size = 1.32 \[ \frac{105 \, c^{2} x^{8} - 210 \, b c x^{6} - 35 \,{\left (b^{2} + 2 \, a c\right )} x^{4} - 42 \, a b x^{2} - 15 \, a^{2}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^8,x, algorithm="fricas")

[Out]

1/105*(105*c^2*x^8 - 210*b*c*x^6 - 35*(b^2 + 2*a*c)*x^4 - 42*a*b*x^2 - 15*a^2)/x
^7

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Sympy [A]  time = 2.89466, size = 44, normalized size = 0.94 \[ c^{2} x - \frac{15 a^{2} + 42 a b x^{2} + 210 b c x^{6} + x^{4} \left (70 a c + 35 b^{2}\right )}{105 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**2/x**8,x)

[Out]

c**2*x - (15*a**2 + 42*a*b*x**2 + 210*b*c*x**6 + x**4*(70*a*c + 35*b**2))/(105*x
**7)

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GIAC/XCAS [A]  time = 0.261715, size = 62, normalized size = 1.32 \[ c^{2} x - \frac{210 \, b c x^{6} + 35 \, b^{2} x^{4} + 70 \, a c x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^8,x, algorithm="giac")

[Out]

c^2*x - 1/105*(210*b*c*x^6 + 35*b^2*x^4 + 70*a*c*x^4 + 42*a*b*x^2 + 15*a^2)/x^7