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

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

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Rubi [A]  time = 0.03, antiderivative size = 56, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 56, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15*a^4/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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{16}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.87, size = 48, normalized size = 0.86 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.15, size = 48, normalized size = 0.86 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.00, size = 47, normalized size = 0.84 \begin {gather*} -\frac {b^{4}}{7 x^{7}}-\frac {4 a \,b^{3}}{9 x^{9}}-\frac {6 a^{2} b^{2}}{11 x^{11}}-\frac {4 a^{3} b}{13 x^{13}}-\frac {a^{4}}{15 x^{15}} \end {gather*}

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 = 1.38, size = 48, normalized size = 0.86 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 4.35, size = 48, normalized size = 0.86 \begin {gather*} -\frac {\frac {a^4}{15}+\frac {4\,a^3\,b\,x^2}{13}+\frac {6\,a^2\,b^2\,x^4}{11}+\frac {4\,a\,b^3\,x^6}{9}+\frac {b^4\,x^8}{7}}{x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.46, size = 51, normalized size = 0.91 \begin {gather*} \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 {gather*}

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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