3.468 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^{16}} \, dx\)

Optimal. Leaf size=82 \[ -\frac{15 a^4 b^2}{11 x^{11}}-\frac{20 a^3 b^3}{9 x^9}-\frac{15 a^2 b^4}{7 x^7}-\frac{6 a^5 b}{13 x^{13}}-\frac{a^6}{15 x^{15}}-\frac{6 a b^5}{5 x^5}-\frac{b^6}{3 x^3} \]

[Out]

-a^6/(15*x^15) - (6*a^5*b)/(13*x^13) - (15*a^4*b^2)/(11*x^11) - (20*a^3*b^3)/(9*x^9) - (15*a^2*b^4)/(7*x^7) -
(6*a*b^5)/(5*x^5) - b^6/(3*x^3)

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Rubi [A]  time = 0.0383924, antiderivative size = 82, 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} \[ -\frac{15 a^4 b^2}{11 x^{11}}-\frac{20 a^3 b^3}{9 x^9}-\frac{15 a^2 b^4}{7 x^7}-\frac{6 a^5 b}{13 x^{13}}-\frac{a^6}{15 x^{15}}-\frac{6 a b^5}{5 x^5}-\frac{b^6}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^16,x]

[Out]

-a^6/(15*x^15) - (6*a^5*b)/(13*x^13) - (15*a^4*b^2)/(11*x^11) - (20*a^3*b^3)/(9*x^9) - (15*a^2*b^4)/(7*x^7) -
(6*a*b^5)/(5*x^5) - b^6/(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 )^3}{x^{16}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^{16}} \, dx}{b^6}\\ &=\frac{\int \left (\frac{a^6 b^6}{x^{16}}+\frac{6 a^5 b^7}{x^{14}}+\frac{15 a^4 b^8}{x^{12}}+\frac{20 a^3 b^9}{x^{10}}+\frac{15 a^2 b^{10}}{x^8}+\frac{6 a b^{11}}{x^6}+\frac{b^{12}}{x^4}\right ) \, dx}{b^6}\\ &=-\frac{a^6}{15 x^{15}}-\frac{6 a^5 b}{13 x^{13}}-\frac{15 a^4 b^2}{11 x^{11}}-\frac{20 a^3 b^3}{9 x^9}-\frac{15 a^2 b^4}{7 x^7}-\frac{6 a b^5}{5 x^5}-\frac{b^6}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0095794, size = 82, normalized size = 1. \[ -\frac{15 a^4 b^2}{11 x^{11}}-\frac{20 a^3 b^3}{9 x^9}-\frac{15 a^2 b^4}{7 x^7}-\frac{6 a^5 b}{13 x^{13}}-\frac{a^6}{15 x^{15}}-\frac{6 a b^5}{5 x^5}-\frac{b^6}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^16,x]

[Out]

-a^6/(15*x^15) - (6*a^5*b)/(13*x^13) - (15*a^4*b^2)/(11*x^11) - (20*a^3*b^3)/(9*x^9) - (15*a^2*b^4)/(7*x^7) -
(6*a*b^5)/(5*x^5) - b^6/(3*x^3)

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Maple [A]  time = 0.05, size = 69, normalized size = 0.8 \begin{align*} -{\frac{{a}^{6}}{15\,{x}^{15}}}-{\frac{6\,{a}^{5}b}{13\,{x}^{13}}}-{\frac{15\,{a}^{4}{b}^{2}}{11\,{x}^{11}}}-{\frac{20\,{a}^{3}{b}^{3}}{9\,{x}^{9}}}-{\frac{15\,{a}^{2}{b}^{4}}{7\,{x}^{7}}}-{\frac{6\,a{b}^{5}}{5\,{x}^{5}}}-{\frac{{b}^{6}}{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)^3/x^16,x)

[Out]

-1/15*a^6/x^15-6/13*a^5*b/x^13-15/11*a^4*b^2/x^11-20/9*a^3*b^3/x^9-15/7*a^2*b^4/x^7-6/5*a*b^5/x^5-1/3*b^6/x^3

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Maxima [A]  time = 1.01108, size = 95, normalized size = 1.16 \begin{align*} -\frac{15015 \, b^{6} x^{12} + 54054 \, a b^{5} x^{10} + 96525 \, a^{2} b^{4} x^{8} + 100100 \, a^{3} b^{3} x^{6} + 61425 \, a^{4} b^{2} x^{4} + 20790 \, a^{5} b x^{2} + 3003 \, a^{6}}{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)^3/x^16,x, algorithm="maxima")

[Out]

-1/45045*(15015*b^6*x^12 + 54054*a*b^5*x^10 + 96525*a^2*b^4*x^8 + 100100*a^3*b^3*x^6 + 61425*a^4*b^2*x^4 + 207
90*a^5*b*x^2 + 3003*a^6)/x^15

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Fricas [A]  time = 1.6635, size = 190, normalized size = 2.32 \begin{align*} -\frac{15015 \, b^{6} x^{12} + 54054 \, a b^{5} x^{10} + 96525 \, a^{2} b^{4} x^{8} + 100100 \, a^{3} b^{3} x^{6} + 61425 \, a^{4} b^{2} x^{4} + 20790 \, a^{5} b x^{2} + 3003 \, a^{6}}{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)^3/x^16,x, algorithm="fricas")

[Out]

-1/45045*(15015*b^6*x^12 + 54054*a*b^5*x^10 + 96525*a^2*b^4*x^8 + 100100*a^3*b^3*x^6 + 61425*a^4*b^2*x^4 + 207
90*a^5*b*x^2 + 3003*a^6)/x^15

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Sympy [A]  time = 0.750857, size = 75, normalized size = 0.91 \begin{align*} - \frac{3003 a^{6} + 20790 a^{5} b x^{2} + 61425 a^{4} b^{2} x^{4} + 100100 a^{3} b^{3} x^{6} + 96525 a^{2} b^{4} x^{8} + 54054 a b^{5} x^{10} + 15015 b^{6} x^{12}}{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)**3/x**16,x)

[Out]

-(3003*a**6 + 20790*a**5*b*x**2 + 61425*a**4*b**2*x**4 + 100100*a**3*b**3*x**6 + 96525*a**2*b**4*x**8 + 54054*
a*b**5*x**10 + 15015*b**6*x**12)/(45045*x**15)

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Giac [A]  time = 1.18698, size = 95, normalized size = 1.16 \begin{align*} -\frac{15015 \, b^{6} x^{12} + 54054 \, a b^{5} x^{10} + 96525 \, a^{2} b^{4} x^{8} + 100100 \, a^{3} b^{3} x^{6} + 61425 \, a^{4} b^{2} x^{4} + 20790 \, a^{5} b x^{2} + 3003 \, a^{6}}{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)^3/x^16,x, algorithm="giac")

[Out]

-1/45045*(15015*b^6*x^12 + 54054*a*b^5*x^10 + 96525*a^2*b^4*x^8 + 100100*a^3*b^3*x^6 + 61425*a^4*b^2*x^4 + 207
90*a^5*b*x^2 + 3003*a^6)/x^15