3.837 \(\int \frac{(a+b x^2+c x^4)^2}{x^{10}} \, dx\)

Optimal. Leaf size=52 \[ -\frac{a^2}{9 x^9}-\frac{2 a c+b^2}{5 x^5}-\frac{2 a b}{7 x^7}-\frac{2 b c}{3 x^3}-\frac{c^2}{x} \]

[Out]

-a^2/(9*x^9) - (2*a*b)/(7*x^7) - (b^2 + 2*a*c)/(5*x^5) - (2*b*c)/(3*x^3) - c^2/x

________________________________________________________________________________________

Rubi [A]  time = 0.0246766, antiderivative size = 52, 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} \[ -\frac{a^2}{9 x^9}-\frac{2 a c+b^2}{5 x^5}-\frac{2 a b}{7 x^7}-\frac{2 b c}{3 x^3}-\frac{c^2}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^10,x]

[Out]

-a^2/(9*x^9) - (2*a*b)/(7*x^7) - (b^2 + 2*a*c)/(5*x^5) - (2*b*c)/(3*x^3) - c^2/x

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^{10}} \, dx &=\int \left (\frac{a^2}{x^{10}}+\frac{2 a b}{x^8}+\frac{b^2+2 a c}{x^6}+\frac{2 b c}{x^4}+\frac{c^2}{x^2}\right ) \, dx\\ &=-\frac{a^2}{9 x^9}-\frac{2 a b}{7 x^7}-\frac{b^2+2 a c}{5 x^5}-\frac{2 b c}{3 x^3}-\frac{c^2}{x}\\ \end{align*}

Mathematica [A]  time = 0.0196876, size = 50, normalized size = 0.96 \[ -\frac{35 a^2+90 a b x^2+126 a c x^4+63 b^2 x^4+210 b c x^6+315 c^2 x^8}{315 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2/x^10,x]

[Out]

-(35*a^2 + 90*a*b*x^2 + 63*b^2*x^4 + 126*a*c*x^4 + 210*b*c*x^6 + 315*c^2*x^8)/(315*x^9)

________________________________________________________________________________________

Maple [A]  time = 0.047, size = 45, normalized size = 0.9 \begin{align*} -{\frac{2\,bc}{3\,{x}^{3}}}-{\frac{2\,ac+{b}^{2}}{5\,{x}^{5}}}-{\frac{2\,ab}{7\,{x}^{7}}}-{\frac{{c}^{2}}{x}}-{\frac{{a}^{2}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2/x^10,x)

[Out]

-2/3*b*c/x^3-1/5*(2*a*c+b^2)/x^5-2/7*a*b/x^7-c^2/x-1/9*a^2/x^9

________________________________________________________________________________________

Maxima [A]  time = 0.964756, size = 62, normalized size = 1.19 \begin{align*} -\frac{315 \, c^{2} x^{8} + 210 \, b c x^{6} + 63 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 90 \, a b x^{2} + 35 \, a^{2}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^10,x, algorithm="maxima")

[Out]

-1/315*(315*c^2*x^8 + 210*b*c*x^6 + 63*(b^2 + 2*a*c)*x^4 + 90*a*b*x^2 + 35*a^2)/x^9

________________________________________________________________________________________

Fricas [A]  time = 1.40219, size = 115, normalized size = 2.21 \begin{align*} -\frac{315 \, c^{2} x^{8} + 210 \, b c x^{6} + 63 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 90 \, a b x^{2} + 35 \, a^{2}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^10,x, algorithm="fricas")

[Out]

-1/315*(315*c^2*x^8 + 210*b*c*x^6 + 63*(b^2 + 2*a*c)*x^4 + 90*a*b*x^2 + 35*a^2)/x^9

________________________________________________________________________________________

Sympy [A]  time = 1.58361, size = 49, normalized size = 0.94 \begin{align*} - \frac{35 a^{2} + 90 a b x^{2} + 210 b c x^{6} + 315 c^{2} x^{8} + x^{4} \left (126 a c + 63 b^{2}\right )}{315 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/x**10,x)

[Out]

-(35*a**2 + 90*a*b*x**2 + 210*b*c*x**6 + 315*c**2*x**8 + x**4*(126*a*c + 63*b**2))/(315*x**9)

________________________________________________________________________________________

Giac [A]  time = 1.11946, size = 65, normalized size = 1.25 \begin{align*} -\frac{315 \, c^{2} x^{8} + 210 \, b c x^{6} + 63 \, b^{2} x^{4} + 126 \, a c x^{4} + 90 \, a b x^{2} + 35 \, a^{2}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^10,x, algorithm="giac")

[Out]

-1/315*(315*c^2*x^8 + 210*b*c*x^6 + 63*b^2*x^4 + 126*a*c*x^4 + 90*a*b*x^2 + 35*a^2)/x^9