3.832 \(\int \frac{\left (a+b x^2+c x^4\right )^2}{x^5} \, dx\)

Optimal. Leaf size=45 \[ -\frac{a^2}{4 x^4}+\log (x) \left (2 a c+b^2\right )-\frac{a b}{x^2}+b c x^2+\frac{c^2 x^4}{4} \]

[Out]

-a^2/(4*x^4) - (a*b)/x^2 + b*c*x^2 + (c^2*x^4)/4 + (b^2 + 2*a*c)*Log[x]

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Rubi [A]  time = 0.111425, antiderivative size = 45, 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 \[ -\frac{a^2}{4 x^4}+\log (x) \left (2 a c+b^2\right )-\frac{a b}{x^2}+b c x^2+\frac{c^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^2/x^5,x]

[Out]

-a^2/(4*x^4) - (a*b)/x^2 + b*c*x^2 + (c^2*x^4)/4 + (b^2 + 2*a*c)*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{4 x^{4}} - \frac{a b}{x^{2}} + b c x^{2} + \frac{c^{2} \int ^{x^{2}} x\, dx}{2} + \left (a c + \frac{b^{2}}{2}\right ) \log{\left (x^{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**2/x**5,x)

[Out]

-a**2/(4*x**4) - a*b/x**2 + b*c*x**2 + c**2*Integral(x, (x, x**2))/2 + (a*c + b*
*2/2)*log(x**2)

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Mathematica [A]  time = 0.0324623, size = 41, normalized size = 0.91 \[ \log (x) \left (2 a c+b^2\right )+\frac{\left (c x^4-a\right ) \left (a+4 b x^2+c x^4\right )}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^2/x^5,x]

[Out]

((-a + c*x^4)*(a + 4*b*x^2 + c*x^4))/(4*x^4) + (b^2 + 2*a*c)*Log[x]

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Maple [A]  time = 0.009, size = 43, normalized size = 1. \[{\frac{{c}^{2}{x}^{4}}{4}}+bc{x}^{2}+2\,\ln \left ( x \right ) ac+{b}^{2}\ln \left ( x \right ) -{\frac{ab}{{x}^{2}}}-{\frac{{a}^{2}}{4\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^2/x^5,x)

[Out]

1/4*c^2*x^4+b*c*x^2+2*ln(x)*a*c+b^2*ln(x)-a*b/x^2-1/4*a^2/x^4

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Maxima [A]  time = 0.691901, size = 61, normalized size = 1.36 \[ \frac{1}{4} \, c^{2} x^{4} + b c x^{2} + \frac{1}{2} \,{\left (b^{2} + 2 \, a c\right )} \log \left (x^{2}\right ) - \frac{4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^5,x, algorithm="maxima")

[Out]

1/4*c^2*x^4 + b*c*x^2 + 1/2*(b^2 + 2*a*c)*log(x^2) - 1/4*(4*a*b*x^2 + a^2)/x^4

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Fricas [A]  time = 0.260574, size = 63, normalized size = 1.4 \[ \frac{c^{2} x^{8} + 4 \, b c x^{6} + 4 \,{\left (b^{2} + 2 \, a c\right )} x^{4} \log \left (x\right ) - 4 \, a b x^{2} - a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.78518, size = 42, normalized size = 0.93 \[ b c x^{2} + \frac{c^{2} x^{4}}{4} + \left (2 a c + b^{2}\right ) \log{\left (x \right )} - \frac{a^{2} + 4 a b x^{2}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**2/x**5,x)

[Out]

b*c*x**2 + c**2*x**4/4 + (2*a*c + b**2)*log(x) - (a**2 + 4*a*b*x**2)/(4*x**4)

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GIAC/XCAS [A]  time = 0.262837, size = 81, normalized size = 1.8 \[ \frac{1}{4} \, c^{2} x^{4} + b c x^{2} + \frac{1}{2} \,{\left (b^{2} + 2 \, a c\right )}{\rm ln}\left (x^{2}\right ) - \frac{3 \, b^{2} x^{4} + 6 \, a c x^{4} + 4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^5,x, algorithm="giac")

[Out]

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