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

Optimal. Leaf size=48 \[ -\frac{a^2}{x}+\frac{1}{3} x^3 \left (2 a c+b^2\right )+2 a b x+\frac{2}{5} b c x^5+\frac{c^2 x^7}{7} \]

[Out]

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

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Rubi [A]  time = 0.021085, 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} \[ -\frac{a^2}{x}+\frac{1}{3} x^3 \left (2 a c+b^2\right )+2 a b x+\frac{2}{5} b c x^5+\frac{c^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^2,x]

[Out]

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

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^2} \, dx &=\int \left (2 a b+\frac{a^2}{x^2}+\left (b^2+2 a c\right ) x^2+2 b c x^4+c^2 x^6\right ) \, dx\\ &=-\frac{a^2}{x}+2 a b x+\frac{1}{3} \left (b^2+2 a c\right ) x^3+\frac{2}{5} b c x^5+\frac{c^2 x^7}{7}\\ \end{align*}

Mathematica [A]  time = 0.0169373, size = 48, normalized size = 1. \[ -\frac{a^2}{x}+\frac{1}{3} x^3 \left (2 a c+b^2\right )+2 a b x+\frac{2}{5} b c x^5+\frac{c^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 45, normalized size = 0.9 \begin{align*}{\frac{{c}^{2}{x}^{7}}{7}}+{\frac{2\,bc{x}^{5}}{5}}+{\frac{2\,ac{x}^{3}}{3}}+{\frac{{b}^{2}{x}^{3}}{3}}+2\,abx-{\frac{{a}^{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^2,x)

[Out]

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

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Maxima [A]  time = 0.951097, size = 57, normalized size = 1.19 \begin{align*} \frac{1}{7} \, c^{2} x^{7} + \frac{2}{5} \, b c x^{5} + \frac{1}{3} \,{\left (b^{2} + 2 \, a c\right )} x^{3} + 2 \, a b x - \frac{a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.42244, size = 111, normalized size = 2.31 \begin{align*} \frac{15 \, c^{2} x^{8} + 42 \, b c x^{6} + 35 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 210 \, a b x^{2} - 105 \, a^{2}}{105 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.32274, size = 44, normalized size = 0.92 \begin{align*} - \frac{a^{2}}{x} + 2 a b x + \frac{2 b c x^{5}}{5} + \frac{c^{2} x^{7}}{7} + x^{3} \left (\frac{2 a c}{3} + \frac{b^{2}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13804, size = 59, normalized size = 1.23 \begin{align*} \frac{1}{7} \, c^{2} x^{7} + \frac{2}{5} \, b c x^{5} + \frac{1}{3} \, b^{2} x^{3} + \frac{2}{3} \, a c x^{3} + 2 \, a b x - \frac{a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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