3.464 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^{12}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{15 a^4 b^2}{7 x^7}-\frac{4 a^3 b^3}{x^5}-\frac{5 a^2 b^4}{x^3}-\frac{2 a^5 b}{3 x^9}-\frac{a^6}{11 x^{11}}-\frac{6 a b^5}{x}+b^6 x \]

[Out]

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

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Rubi [A]  time = 0.0370139, antiderivative size = 71, 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} \[ -\frac{15 a^4 b^2}{7 x^7}-\frac{4 a^3 b^3}{x^5}-\frac{5 a^2 b^4}{x^3}-\frac{2 a^5 b}{3 x^9}-\frac{a^6}{11 x^{11}}-\frac{6 a b^5}{x}+b^6 x \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^12,x]

[Out]

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

Mathematica [A]  time = 0.0055391, size = 71, normalized size = 1. \[ -\frac{15 a^4 b^2}{7 x^7}-\frac{4 a^3 b^3}{x^5}-\frac{5 a^2 b^4}{x^3}-\frac{2 a^5 b}{3 x^9}-\frac{a^6}{11 x^{11}}-\frac{6 a b^5}{x}+b^6 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^12,x]

[Out]

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

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Maple [A]  time = 0.049, size = 66, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{11\,{x}^{11}}}-{\frac{2\,{a}^{5}b}{3\,{x}^{9}}}-{\frac{15\,{a}^{4}{b}^{2}}{7\,{x}^{7}}}-4\,{\frac{{a}^{3}{b}^{3}}{{x}^{5}}}-5\,{\frac{{a}^{2}{b}^{4}}{{x}^{3}}}-6\,{\frac{a{b}^{5}}{x}}+{b}^{6}x \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^12,x)

[Out]

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

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Maxima [A]  time = 0.983274, size = 92, normalized size = 1.3 \begin{align*} b^{6} x - \frac{1386 \, a b^{5} x^{10} + 1155 \, a^{2} b^{4} x^{8} + 924 \, a^{3} b^{3} x^{6} + 495 \, a^{4} b^{2} x^{4} + 154 \, a^{5} b x^{2} + 21 \, a^{6}}{231 \, x^{11}} \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^12,x, algorithm="maxima")

[Out]

b^6*x - 1/231*(1386*a*b^5*x^10 + 1155*a^2*b^4*x^8 + 924*a^3*b^3*x^6 + 495*a^4*b^2*x^4 + 154*a^5*b*x^2 + 21*a^6
)/x^11

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Fricas [A]  time = 1.73128, size = 169, normalized size = 2.38 \begin{align*} \frac{231 \, b^{6} x^{12} - 1386 \, a b^{5} x^{10} - 1155 \, a^{2} b^{4} x^{8} - 924 \, a^{3} b^{3} x^{6} - 495 \, a^{4} b^{2} x^{4} - 154 \, a^{5} b x^{2} - 21 \, a^{6}}{231 \, x^{11}} \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^12,x, algorithm="fricas")

[Out]

1/231*(231*b^6*x^12 - 1386*a*b^5*x^10 - 1155*a^2*b^4*x^8 - 924*a^3*b^3*x^6 - 495*a^4*b^2*x^4 - 154*a^5*b*x^2 -
21*a^6)/x^11

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Sympy [A]  time = 0.636993, size = 70, normalized size = 0.99 \begin{align*} b^{6} x - \frac{21 a^{6} + 154 a^{5} b x^{2} + 495 a^{4} b^{2} x^{4} + 924 a^{3} b^{3} x^{6} + 1155 a^{2} b^{4} x^{8} + 1386 a b^{5} x^{10}}{231 x^{11}} \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**12,x)

[Out]

b**6*x - (21*a**6 + 154*a**5*b*x**2 + 495*a**4*b**2*x**4 + 924*a**3*b**3*x**6 + 1155*a**2*b**4*x**8 + 1386*a*b
**5*x**10)/(231*x**11)

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Giac [A]  time = 1.1375, size = 92, normalized size = 1.3 \begin{align*} b^{6} x - \frac{1386 \, a b^{5} x^{10} + 1155 \, a^{2} b^{4} x^{8} + 924 \, a^{3} b^{3} x^{6} + 495 \, a^{4} b^{2} x^{4} + 154 \, a^{5} b x^{2} + 21 \, a^{6}}{231 \, x^{11}} \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^12,x, algorithm="giac")

[Out]

b^6*x - 1/231*(1386*a*b^5*x^10 + 1155*a^2*b^4*x^8 + 924*a^3*b^3*x^6 + 495*a^4*b^2*x^4 + 154*a^5*b*x^2 + 21*a^6
)/x^11