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

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

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Rubi [A]  time = 0.04, antiderivative size = 76, 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 {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} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 76, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/13*a^6/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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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 )^3}{x^{14}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.80, size = 70, normalized size = 0.92 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.18, size = 70, normalized size = 0.92 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.00, size = 69, normalized size = 0.91 \begin {gather*} -\frac {b^{6}}{x}-\frac {2 a \,b^{5}}{x^{3}}-\frac {3 a^{2} b^{4}}{x^{5}}-\frac {20 a^{3} b^{3}}{7 x^{7}}-\frac {5 a^{4} b^{2}}{3 x^{9}}-\frac {6 a^{5} b}{11 x^{11}}-\frac {a^{6}}{13 x^{13}} \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)^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 = 1.34, size = 70, normalized size = 0.92 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.05, size = 69, normalized size = 0.91 \begin {gather*} -\frac {\frac {a^6}{13}+\frac {6\,a^5\,b\,x^2}{11}+\frac {5\,a^4\,b^2\,x^4}{3}+\frac {20\,a^3\,b^3\,x^6}{7}+3\,a^2\,b^4\,x^8+2\,a\,b^5\,x^{10}+b^6\,x^{12}}{x^{13}} \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)^3/x^14,x)

[Out]

-(a^6/13 + b^6*x^12 + (6*a^5*b*x^2)/11 + 2*a*b^5*x^10 + (5*a^4*b^2*x^4)/3 + (20*a^3*b^3*x^6)/7 + 3*a^2*b^4*x^8
)/x^13

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sympy [A]  time = 0.60, size = 75, normalized size = 0.99 \begin {gather*} \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 {gather*}

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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