3.466 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^{14}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{5 a^4 b^2}{3 x^9}-\frac{20 a^3 b^3}{7 x^7}-\frac{3 a^2 b^4}{x^5}-\frac{6 a^5 b}{11 x^{11}}-\frac{a^6}{13 x^{13}}-\frac{2 a b^5}{x^3}-\frac{b^6}{x} \]

[Out]

-a^6/(13*x^13) - (6*a^5*b)/(11*x^11) - (5*a^4*b^2)/(3*x^9) - (20*a^3*b^3)/(7*x^7) - (3*a^2*b^4)/x^5 - (2*a*b^5
)/x^3 - b^6/x

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Rubi [A]  time = 0.0401458, antiderivative size = 76, 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} \[ -\frac{5 a^4 b^2}{3 x^9}-\frac{20 a^3 b^3}{7 x^7}-\frac{3 a^2 b^4}{x^5}-\frac{6 a^5 b}{11 x^{11}}-\frac{a^6}{13 x^{13}}-\frac{2 a b^5}{x^3}-\frac{b^6}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^14,x]

[Out]

-a^6/(13*x^13) - (6*a^5*b)/(11*x^11) - (5*a^4*b^2)/(3*x^9) - (20*a^3*b^3)/(7*x^7) - (3*a^2*b^4)/x^5 - (2*a*b^5
)/x^3 - 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^{14}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^{14}} \, dx}{b^6}\\ &=\frac{\int \left (\frac{a^6 b^6}{x^{14}}+\frac{6 a^5 b^7}{x^{12}}+\frac{15 a^4 b^8}{x^{10}}+\frac{20 a^3 b^9}{x^8}+\frac{15 a^2 b^{10}}{x^6}+\frac{6 a b^{11}}{x^4}+\frac{b^{12}}{x^2}\right ) \, dx}{b^6}\\ &=-\frac{a^6}{13 x^{13}}-\frac{6 a^5 b}{11 x^{11}}-\frac{5 a^4 b^2}{3 x^9}-\frac{20 a^3 b^3}{7 x^7}-\frac{3 a^2 b^4}{x^5}-\frac{2 a b^5}{x^3}-\frac{b^6}{x}\\ \end{align*}

Mathematica [A]  time = 0.0089975, size = 76, normalized size = 1. \[ -\frac{5 a^4 b^2}{3 x^9}-\frac{20 a^3 b^3}{7 x^7}-\frac{3 a^2 b^4}{x^5}-\frac{6 a^5 b}{11 x^{11}}-\frac{a^6}{13 x^{13}}-\frac{2 a b^5}{x^3}-\frac{b^6}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^14,x]

[Out]

-a^6/(13*x^13) - (6*a^5*b)/(11*x^11) - (5*a^4*b^2)/(3*x^9) - (20*a^3*b^3)/(7*x^7) - (3*a^2*b^4)/x^5 - (2*a*b^5
)/x^3 - b^6/x

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Maple [A]  time = 0.048, size = 69, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{13\,{x}^{13}}}-{\frac{6\,{a}^{5}b}{11\,{x}^{11}}}-{\frac{5\,{a}^{4}{b}^{2}}{3\,{x}^{9}}}-{\frac{20\,{a}^{3}{b}^{3}}{7\,{x}^{7}}}-3\,{\frac{{a}^{2}{b}^{4}}{{x}^{5}}}-2\,{\frac{a{b}^{5}}{{x}^{3}}}-{\frac{{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^14,x)

[Out]

-1/13*a^6/x^13-6/11*a^5*b/x^11-5/3*a^4*b^2/x^9-20/7*a^3*b^3/x^7-3*a^2*b^4/x^5-2*a*b^5/x^3-b^6/x

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Maxima [A]  time = 1.01139, size = 95, normalized size = 1.25 \begin{align*} -\frac{3003 \, b^{6} x^{12} + 6006 \, a b^{5} x^{10} + 9009 \, a^{2} b^{4} x^{8} + 8580 \, a^{3} b^{3} x^{6} + 5005 \, a^{4} b^{2} x^{4} + 1638 \, a^{5} b x^{2} + 231 \, a^{6}}{3003 \, x^{13}} \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^14,x, algorithm="maxima")

[Out]

-1/3003*(3003*b^6*x^12 + 6006*a*b^5*x^10 + 9009*a^2*b^4*x^8 + 8580*a^3*b^3*x^6 + 5005*a^4*b^2*x^4 + 1638*a^5*b
*x^2 + 231*a^6)/x^13

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Fricas [A]  time = 1.61484, size = 178, normalized size = 2.34 \begin{align*} -\frac{3003 \, b^{6} x^{12} + 6006 \, a b^{5} x^{10} + 9009 \, a^{2} b^{4} x^{8} + 8580 \, a^{3} b^{3} x^{6} + 5005 \, a^{4} b^{2} x^{4} + 1638 \, a^{5} b x^{2} + 231 \, a^{6}}{3003 \, x^{13}} \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^14,x, algorithm="fricas")

[Out]

-1/3003*(3003*b^6*x^12 + 6006*a*b^5*x^10 + 9009*a^2*b^4*x^8 + 8580*a^3*b^3*x^6 + 5005*a^4*b^2*x^4 + 1638*a^5*b
*x^2 + 231*a^6)/x^13

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Sympy [A]  time = 0.736059, size = 75, normalized size = 0.99 \begin{align*} - \frac{231 a^{6} + 1638 a^{5} b x^{2} + 5005 a^{4} b^{2} x^{4} + 8580 a^{3} b^{3} x^{6} + 9009 a^{2} b^{4} x^{8} + 6006 a b^{5} x^{10} + 3003 b^{6} x^{12}}{3003 x^{13}} \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**14,x)

[Out]

-(231*a**6 + 1638*a**5*b*x**2 + 5005*a**4*b**2*x**4 + 8580*a**3*b**3*x**6 + 9009*a**2*b**4*x**8 + 6006*a*b**5*
x**10 + 3003*b**6*x**12)/(3003*x**13)

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Giac [A]  time = 1.13566, size = 95, normalized size = 1.25 \begin{align*} -\frac{3003 \, b^{6} x^{12} + 6006 \, a b^{5} x^{10} + 9009 \, a^{2} b^{4} x^{8} + 8580 \, a^{3} b^{3} x^{6} + 5005 \, a^{4} b^{2} x^{4} + 1638 \, a^{5} b x^{2} + 231 \, a^{6}}{3003 \, x^{13}} \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^14,x, algorithm="giac")

[Out]

-1/3003*(3003*b^6*x^12 + 6006*a*b^5*x^10 + 9009*a^2*b^4*x^8 + 8580*a^3*b^3*x^6 + 5005*a^4*b^2*x^4 + 1638*a^5*b
*x^2 + 231*a^6)/x^13