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]

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

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Rubi [A]  time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, 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]

-1/5*a^2/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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fricas [A]  time = 0.87, size = 46, normalized size = 0.96 \[ \frac {5 \, c^{2} x^{8} + 30 \, b c x^{6} - 15 \, {\left (b^{2} + 2 \, a c\right )} x^{4} - 10 \, a b x^{2} - 3 \, a^{2}}{15 \, x^{5}} \]

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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giac [A]  time = 0.15, size = 47, normalized size = 0.98 \[ \frac {1}{3} \, c^{2} x^{3} + 2 \, b c x - \frac {15 \, b^{2} x^{4} + 30 \, a c x^{4} + 10 \, a b x^{2} + 3 \, a^{2}}{15 \, x^{5}} \]

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

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maple [A]  time = 0.01, size = 43, normalized size = 0.90 \[ \frac {c^{2} x^{3}}{3}+2 b c x -\frac {2 a b}{3 x^{3}}-\frac {2 a c +b^{2}}{x}-\frac {a^{2}}{5 x^{5}} \]

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

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maxima [A]  time = 1.32, size = 45, normalized size = 0.94 \[ \frac {1}{3} \, c^{2} x^{3} + 2 \, b c x - \frac {15 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 10 \, a b x^{2} + 3 \, a^{2}}{15 \, x^{5}} \]

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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mupad [B]  time = 0.04, size = 44, normalized size = 0.92 \[ \frac {c^2\,x^3}{3}-\frac {x^4\,\left (b^2+2\,a\,c\right )+\frac {a^2}{5}+\frac {2\,a\,b\,x^2}{3}}{x^5}+2\,b\,c\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^2/x^6,x)

[Out]

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

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sympy [A]  time = 0.43, size = 48, normalized size = 1.00 \[ 2 b c x + \frac {c^{2} x^{3}}{3} + \frac {- 3 a^{2} - 10 a b x^{2} + x^{4} \left (- 30 a c - 15 b^{2}\right )}{15 x^{5}} \]

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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