3.443 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^2}{x^{16}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{6 a^2 b^2}{11 x^{11}}-\frac{4 a^3 b}{13 x^{13}}-\frac{a^4}{15 x^{15}}-\frac{4 a b^3}{9 x^9}-\frac{b^4}{7 x^7} \]

[Out]

-a^4/(15*x^15) - (4*a^3*b)/(13*x^13) - (6*a^2*b^2)/(11*x^11) - (4*a*b^3)/(9*x^9) - b^4/(7*x^7)

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Rubi [A]  time = 0.0252742, 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} \[ -\frac{6 a^2 b^2}{11 x^{11}}-\frac{4 a^3 b}{13 x^{13}}-\frac{a^4}{15 x^{15}}-\frac{4 a b^3}{9 x^9}-\frac{b^4}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^16,x]

[Out]

-a^4/(15*x^15) - (4*a^3*b)/(13*x^13) - (6*a^2*b^2)/(11*x^11) - (4*a*b^3)/(9*x^9) - b^4/(7*x^7)

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^{16}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^4}{x^{16}} \, dx}{b^4}\\ &=\frac{\int \left (\frac{a^4 b^4}{x^{16}}+\frac{4 a^3 b^5}{x^{14}}+\frac{6 a^2 b^6}{x^{12}}+\frac{4 a b^7}{x^{10}}+\frac{b^8}{x^8}\right ) \, dx}{b^4}\\ &=-\frac{a^4}{15 x^{15}}-\frac{4 a^3 b}{13 x^{13}}-\frac{6 a^2 b^2}{11 x^{11}}-\frac{4 a b^3}{9 x^9}-\frac{b^4}{7 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0068133, size = 56, normalized size = 1. \[ -\frac{6 a^2 b^2}{11 x^{11}}-\frac{4 a^3 b}{13 x^{13}}-\frac{a^4}{15 x^{15}}-\frac{4 a b^3}{9 x^9}-\frac{b^4}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^16,x]

[Out]

-a^4/(15*x^15) - (4*a^3*b)/(13*x^13) - (6*a^2*b^2)/(11*x^11) - (4*a*b^3)/(9*x^9) - b^4/(7*x^7)

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Maple [A]  time = 0.047, size = 47, normalized size = 0.8 \begin{align*} -{\frac{{a}^{4}}{15\,{x}^{15}}}-{\frac{4\,{a}^{3}b}{13\,{x}^{13}}}-{\frac{6\,{b}^{2}{a}^{2}}{11\,{x}^{11}}}-{\frac{4\,a{b}^{3}}{9\,{x}^{9}}}-{\frac{{b}^{4}}{7\,{x}^{7}}} \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^16,x)

[Out]

-1/15*a^4/x^15-4/13*a^3*b/x^13-6/11*a^2*b^2/x^11-4/9*a*b^3/x^9-1/7*b^4/x^7

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Maxima [A]  time = 0.985168, size = 65, normalized size = 1.16 \begin{align*} -\frac{6435 \, b^{4} x^{8} + 20020 \, a b^{3} x^{6} + 24570 \, a^{2} b^{2} x^{4} + 13860 \, a^{3} b x^{2} + 3003 \, a^{4}}{45045 \, x^{15}} \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^16,x, algorithm="maxima")

[Out]

-1/45045*(6435*b^4*x^8 + 20020*a*b^3*x^6 + 24570*a^2*b^2*x^4 + 13860*a^3*b*x^2 + 3003*a^4)/x^15

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Fricas [A]  time = 1.87557, size = 131, normalized size = 2.34 \begin{align*} -\frac{6435 \, b^{4} x^{8} + 20020 \, a b^{3} x^{6} + 24570 \, a^{2} b^{2} x^{4} + 13860 \, a^{3} b x^{2} + 3003 \, a^{4}}{45045 \, x^{15}} \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^16,x, algorithm="fricas")

[Out]

-1/45045*(6435*b^4*x^8 + 20020*a*b^3*x^6 + 24570*a^2*b^2*x^4 + 13860*a^3*b*x^2 + 3003*a^4)/x^15

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Sympy [A]  time = 0.575622, size = 51, normalized size = 0.91 \begin{align*} - \frac{3003 a^{4} + 13860 a^{3} b x^{2} + 24570 a^{2} b^{2} x^{4} + 20020 a b^{3} x^{6} + 6435 b^{4} x^{8}}{45045 x^{15}} \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**16,x)

[Out]

-(3003*a**4 + 13860*a**3*b*x**2 + 24570*a**2*b**2*x**4 + 20020*a*b**3*x**6 + 6435*b**4*x**8)/(45045*x**15)

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Giac [A]  time = 1.29041, size = 65, normalized size = 1.16 \begin{align*} -\frac{6435 \, b^{4} x^{8} + 20020 \, a b^{3} x^{6} + 24570 \, a^{2} b^{2} x^{4} + 13860 \, a^{3} b x^{2} + 3003 \, a^{4}}{45045 \, x^{15}} \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^16,x, algorithm="giac")

[Out]

-1/45045*(6435*b^4*x^8 + 20020*a*b^3*x^6 + 24570*a^2*b^2*x^4 + 13860*a^3*b*x^2 + 3003*a^4)/x^15