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

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

[Out]

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

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Rubi [A]  time = 0.023734, 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, Rules used = {1108} \[ -\frac{a^2}{3 x^3}+x \left (2 a c+b^2\right )-\frac{2 a b}{x}+\frac{2}{3} b c x^3+\frac{c^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^4,x]

[Out]

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

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^4} \, dx &=\int \left (b^2 \left (1+\frac{2 a c}{b^2}\right )+\frac{a^2}{x^4}+\frac{2 a b}{x^2}+2 b c x^2+c^2 x^4\right ) \, dx\\ &=-\frac{a^2}{3 x^3}-\frac{2 a b}{x}+\left (b^2+2 a c\right ) x+\frac{2}{3} b c x^3+\frac{c^2 x^5}{5}\\ \end{align*}

Mathematica [A]  time = 0.0180374, size = 47, normalized size = 1. \[ -\frac{a^2}{3 x^3}+x \left (2 a c+b^2\right )-\frac{2 a b}{x}+\frac{2}{3} b c x^3+\frac{c^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.047, size = 42, normalized size = 0.9 \begin{align*}{\frac{{c}^{2}{x}^{5}}{5}}+{\frac{2\,bc{x}^{3}}{3}}+2\,acx+{b}^{2}x-{\frac{{a}^{2}}{3\,{x}^{3}}}-2\,{\frac{ab}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.966724, size = 57, normalized size = 1.21 \begin{align*} \frac{1}{5} \, c^{2} x^{5} + \frac{2}{3} \, b c x^{3} +{\left (b^{2} + 2 \, a c\right )} x - \frac{6 \, a b x^{2} + a^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^2*x^5 + 2/3*b*c*x^3 + (b^2 + 2*a*c)*x - 1/3*(6*a*b*x^2 + a^2)/x^3

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Fricas [A]  time = 1.41883, size = 107, normalized size = 2.28 \begin{align*} \frac{3 \, c^{2} x^{8} + 10 \, b c x^{6} + 15 \,{\left (b^{2} + 2 \, a c\right )} x^{4} - 30 \, a b x^{2} - 5 \, a^{2}}{15 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.341833, size = 44, normalized size = 0.94 \begin{align*} \frac{2 b c x^{3}}{3} + \frac{c^{2} x^{5}}{5} + x \left (2 a c + b^{2}\right ) - \frac{a^{2} + 6 a b x^{2}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*c*x**3/3 + c**2*x**5/5 + x*(2*a*c + b**2) - (a**2 + 6*a*b*x**2)/(3*x**3)

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Giac [A]  time = 1.09942, size = 57, normalized size = 1.21 \begin{align*} \frac{1}{5} \, c^{2} x^{5} + \frac{2}{3} \, b c x^{3} + b^{2} x + 2 \, a c x - \frac{6 \, a b x^{2} + a^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^2*x^5 + 2/3*b*c*x^3 + b^2*x + 2*a*c*x - 1/3*(6*a*b*x^2 + a^2)/x^3