3.833 \(\int \frac{(a+b x^2+c x^4)^2}{x^6} \, dx\)

Optimal. Leaf size=48 \[ -\frac{a^2}{5 x^5}-\frac{2 a c+b^2}{x}-\frac{2 a b}{3 x^3}+2 b c x+\frac{c^2 x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.0230556, antiderivative size = 48, 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, Rules used = {1108} \[ -\frac{a^2}{5 x^5}-\frac{2 a c+b^2}{x}-\frac{2 a b}{3 x^3}+2 b c x+\frac{c^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^6,x]

[Out]

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

Rule 1108

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a
+ b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] &&  !IntegerQ[(m + 1)/2]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^2}{x^6} \, dx &=\int \left (2 b c+\frac{a^2}{x^6}+\frac{2 a b}{x^4}+\frac{b^2+2 a c}{x^2}+c^2 x^2\right ) \, dx\\ &=-\frac{a^2}{5 x^5}-\frac{2 a b}{3 x^3}-\frac{b^2+2 a c}{x}+2 b c x+\frac{c^2 x^3}{3}\\ \end{align*}

Mathematica [A]  time = 0.0209539, size = 49, normalized size = 1.02 \[ -\frac{a^2}{5 x^5}+\frac{-2 a c-b^2}{x}-\frac{2 a b}{3 x^3}+2 b c x+\frac{c^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2/x^6,x]

[Out]

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

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Maple [A]  time = 0.047, size = 43, normalized size = 0.9 \begin{align*}{\frac{{c}^{2}{x}^{3}}{3}}+2\,bcx-{\frac{2\,ab}{3\,{x}^{3}}}-{\frac{{a}^{2}}{5\,{x}^{5}}}-{\frac{2\,ac+{b}^{2}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2/x^6,x)

[Out]

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

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Maxima [A]  time = 0.978461, size = 61, normalized size = 1.27 \begin{align*} \frac{1}{3} \, c^{2} x^{3} + 2 \, b c x - \frac{15 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 10 \, a b x^{2} + 3 \, a^{2}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^6,x, algorithm="maxima")

[Out]

1/3*c^2*x^3 + 2*b*c*x - 1/15*(15*(b^2 + 2*a*c)*x^4 + 10*a*b*x^2 + 3*a^2)/x^5

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Fricas [A]  time = 1.37401, size = 107, normalized size = 2.23 \begin{align*} \frac{5 \, c^{2} x^{8} + 30 \, b c x^{6} - 15 \,{\left (b^{2} + 2 \, a c\right )} x^{4} - 10 \, a b x^{2} - 3 \, a^{2}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^6,x, algorithm="fricas")

[Out]

1/15*(5*c^2*x^8 + 30*b*c*x^6 - 15*(b^2 + 2*a*c)*x^4 - 10*a*b*x^2 - 3*a^2)/x^5

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Sympy [A]  time = 0.596332, size = 46, normalized size = 0.96 \begin{align*} 2 b c x + \frac{c^{2} x^{3}}{3} - \frac{3 a^{2} + 10 a b x^{2} + x^{4} \left (30 a c + 15 b^{2}\right )}{15 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/x**6,x)

[Out]

2*b*c*x + c**2*x**3/3 - (3*a**2 + 10*a*b*x**2 + x**4*(30*a*c + 15*b**2))/(15*x**5)

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Giac [A]  time = 1.13228, size = 63, normalized size = 1.31 \begin{align*} \frac{1}{3} \, c^{2} x^{3} + 2 \, b c x - \frac{15 \, b^{2} x^{4} + 30 \, a c x^{4} + 10 \, a b x^{2} + 3 \, a^{2}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^6,x, algorithm="giac")

[Out]

1/3*c^2*x^3 + 2*b*c*x - 1/15*(15*b^2*x^4 + 30*a*c*x^4 + 10*a*b*x^2 + 3*a^2)/x^5