3.462 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^{10}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{3 a^4 b^2}{x^5}-\frac{20 a^3 b^3}{3 x^3}-\frac{15 a^2 b^4}{x}-\frac{6 a^5 b}{7 x^7}-\frac{a^6}{9 x^9}+6 a b^5 x+\frac{b^6 x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.0393374, antiderivative size = 74, 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} \[ -\frac{3 a^4 b^2}{x^5}-\frac{20 a^3 b^3}{3 x^3}-\frac{15 a^2 b^4}{x}-\frac{6 a^5 b}{7 x^7}-\frac{a^6}{9 x^9}+6 a b^5 x+\frac{b^6 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^10,x]

[Out]

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

Mathematica [A]  time = 0.0090245, size = 74, normalized size = 1. \[ -\frac{3 a^4 b^2}{x^5}-\frac{20 a^3 b^3}{3 x^3}-\frac{15 a^2 b^4}{x}-\frac{6 a^5 b}{7 x^7}-\frac{a^6}{9 x^9}+6 a b^5 x+\frac{b^6 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^10,x]

[Out]

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

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Maple [A]  time = 0.048, size = 67, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{9\,{x}^{9}}}-{\frac{6\,{a}^{5}b}{7\,{x}^{7}}}-3\,{\frac{{a}^{4}{b}^{2}}{{x}^{5}}}-{\frac{20\,{a}^{3}{b}^{3}}{3\,{x}^{3}}}-15\,{\frac{{a}^{2}{b}^{4}}{x}}+6\,a{b}^{5}x+{\frac{{b}^{6}{x}^{3}}{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^10,x)

[Out]

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

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Maxima [A]  time = 0.972736, size = 93, normalized size = 1.26 \begin{align*} \frac{1}{3} \, b^{6} x^{3} + 6 \, a b^{5} x - \frac{945 \, a^{2} b^{4} x^{8} + 420 \, a^{3} b^{3} x^{6} + 189 \, a^{4} b^{2} x^{4} + 54 \, a^{5} b x^{2} + 7 \, a^{6}}{63 \, x^{9}} \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^10,x, algorithm="maxima")

[Out]

1/3*b^6*x^3 + 6*a*b^5*x - 1/63*(945*a^2*b^4*x^8 + 420*a^3*b^3*x^6 + 189*a^4*b^2*x^4 + 54*a^5*b*x^2 + 7*a^6)/x^
9

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Fricas [A]  time = 1.64801, size = 159, normalized size = 2.15 \begin{align*} \frac{21 \, b^{6} x^{12} + 378 \, a b^{5} x^{10} - 945 \, a^{2} b^{4} x^{8} - 420 \, a^{3} b^{3} x^{6} - 189 \, a^{4} b^{2} x^{4} - 54 \, a^{5} b x^{2} - 7 \, a^{6}}{63 \, x^{9}} \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^10,x, algorithm="fricas")

[Out]

1/63*(21*b^6*x^12 + 378*a*b^5*x^10 - 945*a^2*b^4*x^8 - 420*a^3*b^3*x^6 - 189*a^4*b^2*x^4 - 54*a^5*b*x^2 - 7*a^
6)/x^9

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Sympy [A]  time = 0.550926, size = 71, normalized size = 0.96 \begin{align*} 6 a b^{5} x + \frac{b^{6} x^{3}}{3} - \frac{7 a^{6} + 54 a^{5} b x^{2} + 189 a^{4} b^{2} x^{4} + 420 a^{3} b^{3} x^{6} + 945 a^{2} b^{4} x^{8}}{63 x^{9}} \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**10,x)

[Out]

6*a*b**5*x + b**6*x**3/3 - (7*a**6 + 54*a**5*b*x**2 + 189*a**4*b**2*x**4 + 420*a**3*b**3*x**6 + 945*a**2*b**4*
x**8)/(63*x**9)

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Giac [A]  time = 1.15072, size = 93, normalized size = 1.26 \begin{align*} \frac{1}{3} \, b^{6} x^{3} + 6 \, a b^{5} x - \frac{945 \, a^{2} b^{4} x^{8} + 420 \, a^{3} b^{3} x^{6} + 189 \, a^{4} b^{2} x^{4} + 54 \, a^{5} b x^{2} + 7 \, a^{6}}{63 \, x^{9}} \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^10,x, algorithm="giac")

[Out]

1/3*b^6*x^3 + 6*a*b^5*x - 1/63*(945*a^2*b^4*x^8 + 420*a^3*b^3*x^6 + 189*a^4*b^2*x^4 + 54*a^5*b*x^2 + 7*a^6)/x^
9