3.830 \(\int \frac{\left (a+b x^2+c x^4\right )^2}{x^3} \, dx\)

Optimal. Leaf size=51 \[ -\frac{a^2}{2 x^2}+\frac{1}{2} x^2 \left (2 a c+b^2\right )+2 a b \log (x)+\frac{1}{2} b c x^4+\frac{c^2 x^6}{6} \]

[Out]

-a^2/(2*x^2) + ((b^2 + 2*a*c)*x^2)/2 + (b*c*x^4)/2 + (c^2*x^6)/6 + 2*a*b*Log[x]

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Rubi [A]  time = 0.0963491, antiderivative size = 51, 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 \[ -\frac{a^2}{2 x^2}+\frac{1}{2} x^2 \left (2 a c+b^2\right )+2 a b \log (x)+\frac{1}{2} b c x^4+\frac{c^2 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^2/x^3,x]

[Out]

-a^2/(2*x^2) + ((b^2 + 2*a*c)*x^2)/2 + (b*c*x^4)/2 + (c^2*x^6)/6 + 2*a*b*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{2 x^{2}} + a b \log{\left (x^{2} \right )} + b c \int ^{x^{2}} x\, dx + \frac{c^{2} x^{6}}{6} + \frac{\left (2 a c + b^{2}\right ) \int ^{x^{2}} b^{2}\, dx}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**2/x**3,x)

[Out]

-a**2/(2*x**2) + a*b*log(x**2) + b*c*Integral(x, (x, x**2)) + c**2*x**6/6 + (2*a
*c + b**2)*Integral(b**2, (x, x**2))/(2*b**2)

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Mathematica [A]  time = 0.0253631, size = 46, normalized size = 0.9 \[ \frac{1}{6} \left (-\frac{3 a^2}{x^2}+3 x^2 \left (2 a c+b^2\right )+12 a b \log (x)+3 b c x^4+c^2 x^6\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^2/x^3,x]

[Out]

((-3*a^2)/x^2 + 3*(b^2 + 2*a*c)*x^2 + 3*b*c*x^4 + c^2*x^6 + 12*a*b*Log[x])/6

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Maple [A]  time = 0.009, size = 45, normalized size = 0.9 \[{\frac{{c}^{2}{x}^{6}}{6}}+{\frac{bc{x}^{4}}{2}}+{x}^{2}ac+{\frac{{b}^{2}{x}^{2}}{2}}+2\,ab\ln \left ( x \right ) -{\frac{{a}^{2}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^2/x^3,x)

[Out]

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

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Maxima [A]  time = 0.687854, size = 59, normalized size = 1.16 \[ \frac{1}{6} \, c^{2} x^{6} + \frac{1}{2} \, b c x^{4} + \frac{1}{2} \,{\left (b^{2} + 2 \, a c\right )} x^{2} + a b \log \left (x^{2}\right ) - \frac{a^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.253825, size = 63, normalized size = 1.24 \[ \frac{c^{2} x^{8} + 3 \, b c x^{6} + 3 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 12 \, a b x^{2} \log \left (x\right ) - 3 \, a^{2}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.156, size = 44, normalized size = 0.86 \[ - \frac{a^{2}}{2 x^{2}} + 2 a b \log{\left (x \right )} + \frac{b c x^{4}}{2} + \frac{c^{2} x^{6}}{6} + x^{2} \left (a c + \frac{b^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**2/x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.265301, size = 72, normalized size = 1.41 \[ \frac{1}{6} \, c^{2} x^{6} + \frac{1}{2} \, b c x^{4} + \frac{1}{2} \, b^{2} x^{2} + a c x^{2} + a b{\rm ln}\left (x^{2}\right ) - \frac{2 \, a b x^{2} + a^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^3,x, algorithm="giac")

[Out]

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