3.437 \(\int \frac {(a^2+2 a b x^2+b^2 x^4)^2}{x^{10}} \, dx\)

Optimal. Leaf size=54 \[ -\frac {a^4}{9 x^9}-\frac {4 a^3 b}{7 x^7}-\frac {6 a^2 b^2}{5 x^5}-\frac {4 a b^3}{3 x^3}-\frac {b^4}{x} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 54, 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} \[ -\frac {6 a^2 b^2}{5 x^5}-\frac {4 a^3 b}{7 x^7}-\frac {a^4}{9 x^9}-\frac {4 a b^3}{3 x^3}-\frac {b^4}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^10,x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 54, normalized size = 1.00 \[ -\frac {a^4}{9 x^9}-\frac {4 a^3 b}{7 x^7}-\frac {6 a^2 b^2}{5 x^5}-\frac {4 a b^3}{3 x^3}-\frac {b^4}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^10,x]

[Out]

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

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fricas [A]  time = 0.75, size = 48, normalized size = 0.89 \[ -\frac {315 \, b^{4} x^{8} + 420 \, a b^{3} x^{6} + 378 \, a^{2} b^{2} x^{4} + 180 \, a^{3} b x^{2} + 35 \, a^{4}}{315 \, x^{9}} \]

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^10,x, algorithm="fricas")

[Out]

-1/315*(315*b^4*x^8 + 420*a*b^3*x^6 + 378*a^2*b^2*x^4 + 180*a^3*b*x^2 + 35*a^4)/x^9

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giac [A]  time = 0.17, size = 48, normalized size = 0.89 \[ -\frac {315 \, b^{4} x^{8} + 420 \, a b^{3} x^{6} + 378 \, a^{2} b^{2} x^{4} + 180 \, a^{3} b x^{2} + 35 \, a^{4}}{315 \, x^{9}} \]

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^10,x, algorithm="giac")

[Out]

-1/315*(315*b^4*x^8 + 420*a*b^3*x^6 + 378*a^2*b^2*x^4 + 180*a^3*b*x^2 + 35*a^4)/x^9

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maple [A]  time = 0.01, size = 47, normalized size = 0.87 \[ -\frac {b^{4}}{x}-\frac {4 a \,b^{3}}{3 x^{3}}-\frac {6 a^{2} b^{2}}{5 x^{5}}-\frac {4 a^{3} b}{7 x^{7}}-\frac {a^{4}}{9 x^{9}} \]

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^10,x)

[Out]

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

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maxima [A]  time = 1.43, size = 48, normalized size = 0.89 \[ -\frac {315 \, b^{4} x^{8} + 420 \, a b^{3} x^{6} + 378 \, a^{2} b^{2} x^{4} + 180 \, a^{3} b x^{2} + 35 \, a^{4}}{315 \, x^{9}} \]

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^10,x, algorithm="maxima")

[Out]

-1/315*(315*b^4*x^8 + 420*a*b^3*x^6 + 378*a^2*b^2*x^4 + 180*a^3*b*x^2 + 35*a^4)/x^9

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mupad [B]  time = 0.03, size = 47, normalized size = 0.87 \[ -\frac {\frac {a^4}{9}+\frac {4\,a^3\,b\,x^2}{7}+\frac {6\,a^2\,b^2\,x^4}{5}+\frac {4\,a\,b^3\,x^6}{3}+b^4\,x^8}{x^9} \]

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^10,x)

[Out]

-(a^4/9 + b^4*x^8 + (4*a^3*b*x^2)/7 + (4*a*b^3*x^6)/3 + (6*a^2*b^2*x^4)/5)/x^9

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sympy [A]  time = 0.37, size = 51, normalized size = 0.94 \[ \frac {- 35 a^{4} - 180 a^{3} b x^{2} - 378 a^{2} b^{2} x^{4} - 420 a b^{3} x^{6} - 315 b^{4} x^{8}}{315 x^{9}} \]

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**10,x)

[Out]

(-35*a**4 - 180*a**3*b*x**2 - 378*a**2*b**2*x**4 - 420*a*b**3*x**6 - 315*b**4*x**8)/(315*x**9)

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