3.439 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^2}{x^{12}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{6 a^2 b^2}{7 x^7}-\frac{4 a^3 b}{9 x^9}-\frac{a^4}{11 x^{11}}-\frac{4 a b^3}{5 x^5}-\frac{b^4}{3 x^3} \]

[Out]

-a^4/(11*x^11) - (4*a^3*b)/(9*x^9) - (6*a^2*b^2)/(7*x^7) - (4*a*b^3)/(5*x^5) - b^4/(3*x^3)

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Rubi [A]  time = 0.0263073, antiderivative size = 56, 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} \[ -\frac{6 a^2 b^2}{7 x^7}-\frac{4 a^3 b}{9 x^9}-\frac{a^4}{11 x^{11}}-\frac{4 a b^3}{5 x^5}-\frac{b^4}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^12,x]

[Out]

-a^4/(11*x^11) - (4*a^3*b)/(9*x^9) - (6*a^2*b^2)/(7*x^7) - (4*a*b^3)/(5*x^5) - b^4/(3*x^3)

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^{12}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^4}{x^{12}} \, dx}{b^4}\\ &=\frac{\int \left (\frac{a^4 b^4}{x^{12}}+\frac{4 a^3 b^5}{x^{10}}+\frac{6 a^2 b^6}{x^8}+\frac{4 a b^7}{x^6}+\frac{b^8}{x^4}\right ) \, dx}{b^4}\\ &=-\frac{a^4}{11 x^{11}}-\frac{4 a^3 b}{9 x^9}-\frac{6 a^2 b^2}{7 x^7}-\frac{4 a b^3}{5 x^5}-\frac{b^4}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0066583, size = 56, normalized size = 1. \[ -\frac{6 a^2 b^2}{7 x^7}-\frac{4 a^3 b}{9 x^9}-\frac{a^4}{11 x^{11}}-\frac{4 a b^3}{5 x^5}-\frac{b^4}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^12,x]

[Out]

-a^4/(11*x^11) - (4*a^3*b)/(9*x^9) - (6*a^2*b^2)/(7*x^7) - (4*a*b^3)/(5*x^5) - b^4/(3*x^3)

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Maple [A]  time = 0.049, size = 47, normalized size = 0.8 \begin{align*} -{\frac{{a}^{4}}{11\,{x}^{11}}}-{\frac{4\,{a}^{3}b}{9\,{x}^{9}}}-{\frac{6\,{b}^{2}{a}^{2}}{7\,{x}^{7}}}-{\frac{4\,a{b}^{3}}{5\,{x}^{5}}}-{\frac{{b}^{4}}{3\,{x}^{3}}} \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^12,x)

[Out]

-1/11*a^4/x^11-4/9*a^3*b/x^9-6/7*a^2*b^2/x^7-4/5*a*b^3/x^5-1/3*b^4/x^3

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Maxima [A]  time = 0.994372, size = 65, normalized size = 1.16 \begin{align*} -\frac{1155 \, b^{4} x^{8} + 2772 \, a b^{3} x^{6} + 2970 \, a^{2} b^{2} x^{4} + 1540 \, a^{3} b x^{2} + 315 \, a^{4}}{3465 \, x^{11}} \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^12,x, algorithm="maxima")

[Out]

-1/3465*(1155*b^4*x^8 + 2772*a*b^3*x^6 + 2970*a^2*b^2*x^4 + 1540*a^3*b*x^2 + 315*a^4)/x^11

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Fricas [A]  time = 1.71152, size = 124, normalized size = 2.21 \begin{align*} -\frac{1155 \, b^{4} x^{8} + 2772 \, a b^{3} x^{6} + 2970 \, a^{2} b^{2} x^{4} + 1540 \, a^{3} b x^{2} + 315 \, a^{4}}{3465 \, x^{11}} \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^12,x, algorithm="fricas")

[Out]

-1/3465*(1155*b^4*x^8 + 2772*a*b^3*x^6 + 2970*a^2*b^2*x^4 + 1540*a^3*b*x^2 + 315*a^4)/x^11

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Sympy [A]  time = 0.528373, size = 51, normalized size = 0.91 \begin{align*} - \frac{315 a^{4} + 1540 a^{3} b x^{2} + 2970 a^{2} b^{2} x^{4} + 2772 a b^{3} x^{6} + 1155 b^{4} x^{8}}{3465 x^{11}} \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**12,x)

[Out]

-(315*a**4 + 1540*a**3*b*x**2 + 2970*a**2*b**2*x**4 + 2772*a*b**3*x**6 + 1155*b**4*x**8)/(3465*x**11)

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Giac [A]  time = 1.15832, size = 65, normalized size = 1.16 \begin{align*} -\frac{1155 \, b^{4} x^{8} + 2772 \, a b^{3} x^{6} + 2970 \, a^{2} b^{2} x^{4} + 1540 \, a^{3} b x^{2} + 315 \, a^{4}}{3465 \, x^{11}} \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^12,x, algorithm="giac")

[Out]

-1/3465*(1155*b^4*x^8 + 2772*a*b^3*x^6 + 2970*a^2*b^2*x^4 + 1540*a^3*b*x^2 + 315*a^4)/x^11