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

Optimal. Leaf size=54 \[ -\frac{6 a^2 b^2}{5 x^5}-\frac{4 a^3 b}{7 x^7}-\frac{a^4}{9 x^9}-\frac{4 a b^3}{3 x^3}-\frac{b^4}{x} \]

[Out]

-a^4/(9*x^9) - (4*a^3*b)/(7*x^7) - (6*a^2*b^2)/(5*x^5) - (4*a*b^3)/(3*x^3) - b^4/x

________________________________________________________________________________________

Rubi [A]  time = 0.0261705, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {28, 270} \[ -\frac{6 a^2 b^2}{5 x^5}-\frac{4 a^3 b}{7 x^7}-\frac{a^4}{9 x^9}-\frac{4 a b^3}{3 x^3}-\frac{b^4}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^4/(9*x^9) - (4*a^3*b)/(7*x^7) - (6*a^2*b^2)/(5*x^5) - (4*a*b^3)/(3*x^3) - b^4/x

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0084107, size = 54, normalized size = 1. \[ -\frac{6 a^2 b^2}{5 x^5}-\frac{4 a^3 b}{7 x^7}-\frac{a^4}{9 x^9}-\frac{4 a b^3}{3 x^3}-\frac{b^4}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^4/(9*x^9) - (4*a^3*b)/(7*x^7) - (6*a^2*b^2)/(5*x^5) - (4*a*b^3)/(3*x^3) - b^4/x

________________________________________________________________________________________

Maple [A]  time = 0.047, size = 47, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}}{9\,{x}^{9}}}-{\frac{4\,{a}^{3}b}{7\,{x}^{7}}}-{\frac{6\,{b}^{2}{a}^{2}}{5\,{x}^{5}}}-{\frac{4\,a{b}^{3}}{3\,{x}^{3}}}-{\frac{{b}^{4}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a^4/x^9-4/7*a^3*b/x^7-6/5*a^2*b^2/x^5-4/3*a*b^3/x^3-b^4/x

________________________________________________________________________________________

Maxima [A]  time = 0.984635, size = 65, normalized size = 1.2 \begin{align*} -\frac{315 \, b^{4} x^{8} + 420 \, a b^{3} x^{6} + 378 \, a^{2} b^{2} x^{4} + 180 \, a^{3} b x^{2} + 35 \, a^{4}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(315*b^4*x^8 + 420*a*b^3*x^6 + 378*a^2*b^2*x^4 + 180*a^3*b*x^2 + 35*a^4)/x^9

________________________________________________________________________________________

Fricas [A]  time = 1.68739, size = 115, normalized size = 2.13 \begin{align*} -\frac{315 \, b^{4} x^{8} + 420 \, a b^{3} x^{6} + 378 \, a^{2} b^{2} x^{4} + 180 \, a^{3} b x^{2} + 35 \, a^{4}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(315*b^4*x^8 + 420*a*b^3*x^6 + 378*a^2*b^2*x^4 + 180*a^3*b*x^2 + 35*a^4)/x^9

________________________________________________________________________________________

Sympy [A]  time = 0.489036, size = 51, normalized size = 0.94 \begin{align*} - \frac{35 a^{4} + 180 a^{3} b x^{2} + 378 a^{2} b^{2} x^{4} + 420 a b^{3} x^{6} + 315 b^{4} x^{8}}{315 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(35*a**4 + 180*a**3*b*x**2 + 378*a**2*b**2*x**4 + 420*a*b**3*x**6 + 315*b**4*x**8)/(315*x**9)

________________________________________________________________________________________

Giac [A]  time = 1.1258, size = 65, normalized size = 1.2 \begin{align*} -\frac{315 \, b^{4} x^{8} + 420 \, a b^{3} x^{6} + 378 \, a^{2} b^{2} x^{4} + 180 \, a^{3} b x^{2} + 35 \, a^{4}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(315*b^4*x^8 + 420*a*b^3*x^6 + 378*a^2*b^2*x^4 + 180*a^3*b*x^2 + 35*a^4)/x^9