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

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

[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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Rubi [A]  time = 0.04, antiderivative size = 82, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 82, normalized size = 1.00 \[ -\frac {a^6}{15 x^{15}}-\frac {6 a^5 b}{13 x^{13}}-\frac {15 a^4 b^2}{11 x^{11}}-\frac {20 a^3 b^3}{9 x^9}-\frac {15 a^2 b^4}{7 x^7}-\frac {6 a b^5}{5 x^5}-\frac {b^6}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15*a^6/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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fricas [A]  time = 0.74, size = 70, normalized size = 0.85 \[ -\frac {15015 \, b^{6} x^{12} + 54054 \, a b^{5} x^{10} + 96525 \, a^{2} b^{4} x^{8} + 100100 \, a^{3} b^{3} x^{6} + 61425 \, a^{4} b^{2} x^{4} + 20790 \, a^{5} b x^{2} + 3003 \, a^{6}}{45045 \, x^{15}} \]

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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giac [A]  time = 0.15, size = 70, normalized size = 0.85 \[ -\frac {15015 \, b^{6} x^{12} + 54054 \, a b^{5} x^{10} + 96525 \, a^{2} b^{4} x^{8} + 100100 \, a^{3} b^{3} x^{6} + 61425 \, a^{4} b^{2} x^{4} + 20790 \, a^{5} b x^{2} + 3003 \, a^{6}}{45045 \, x^{15}} \]

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

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maple [A]  time = 0.00, size = 69, normalized size = 0.84 \[ -\frac {b^{6}}{3 x^{3}}-\frac {6 a \,b^{5}}{5 x^{5}}-\frac {15 a^{2} b^{4}}{7 x^{7}}-\frac {20 a^{3} b^{3}}{9 x^{9}}-\frac {15 a^{4} b^{2}}{11 x^{11}}-\frac {6 a^{5} b}{13 x^{13}}-\frac {a^{6}}{15 x^{15}} \]

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 = 1.37, size = 70, normalized size = 0.85 \[ -\frac {15015 \, b^{6} x^{12} + 54054 \, a b^{5} x^{10} + 96525 \, a^{2} b^{4} x^{8} + 100100 \, a^{3} b^{3} x^{6} + 61425 \, a^{4} b^{2} x^{4} + 20790 \, a^{5} b x^{2} + 3003 \, a^{6}}{45045 \, x^{15}} \]

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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mupad [B]  time = 0.05, size = 70, normalized size = 0.85 \[ -\frac {\frac {a^6}{15}+\frac {6\,a^5\,b\,x^2}{13}+\frac {15\,a^4\,b^2\,x^4}{11}+\frac {20\,a^3\,b^3\,x^6}{9}+\frac {15\,a^2\,b^4\,x^8}{7}+\frac {6\,a\,b^5\,x^{10}}{5}+\frac {b^6\,x^{12}}{3}}{x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)^3/x^16,x)

[Out]

-(a^6/15 + (b^6*x^12)/3 + (6*a^5*b*x^2)/13 + (6*a*b^5*x^10)/5 + (15*a^4*b^2*x^4)/11 + (20*a^3*b^3*x^6)/9 + (15
*a^2*b^4*x^8)/7)/x^15

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sympy [A]  time = 0.62, size = 75, normalized size = 0.91 \[ \frac {- 3003 a^{6} - 20790 a^{5} b x^{2} - 61425 a^{4} b^{2} x^{4} - 100100 a^{3} b^{3} x^{6} - 96525 a^{2} b^{4} x^{8} - 54054 a b^{5} x^{10} - 15015 b^{6} x^{12}}{45045 x^{15}} \]

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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