3.474 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^{22}} \, dx\)

Optimal. Leaf size=82 \[ -\frac{15 a^4 b^2}{17 x^{17}}-\frac{4 a^3 b^3}{3 x^{15}}-\frac{15 a^2 b^4}{13 x^{13}}-\frac{6 a^5 b}{19 x^{19}}-\frac{a^6}{21 x^{21}}-\frac{6 a b^5}{11 x^{11}}-\frac{b^6}{9 x^9} \]

[Out]

-a^6/(21*x^21) - (6*a^5*b)/(19*x^19) - (15*a^4*b^2)/(17*x^17) - (4*a^3*b^3)/(3*x^15) - (15*a^2*b^4)/(13*x^13)
- (6*a*b^5)/(11*x^11) - b^6/(9*x^9)

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Rubi [A]  time = 0.0397837, antiderivative size = 82, 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} \[ -\frac{15 a^4 b^2}{17 x^{17}}-\frac{4 a^3 b^3}{3 x^{15}}-\frac{15 a^2 b^4}{13 x^{13}}-\frac{6 a^5 b}{19 x^{19}}-\frac{a^6}{21 x^{21}}-\frac{6 a b^5}{11 x^{11}}-\frac{b^6}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^22,x]

[Out]

-a^6/(21*x^21) - (6*a^5*b)/(19*x^19) - (15*a^4*b^2)/(17*x^17) - (4*a^3*b^3)/(3*x^15) - (15*a^2*b^4)/(13*x^13)
- (6*a*b^5)/(11*x^11) - b^6/(9*x^9)

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^{22}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^{22}} \, dx}{b^6}\\ &=\frac{\int \left (\frac{a^6 b^6}{x^{22}}+\frac{6 a^5 b^7}{x^{20}}+\frac{15 a^4 b^8}{x^{18}}+\frac{20 a^3 b^9}{x^{16}}+\frac{15 a^2 b^{10}}{x^{14}}+\frac{6 a b^{11}}{x^{12}}+\frac{b^{12}}{x^{10}}\right ) \, dx}{b^6}\\ &=-\frac{a^6}{21 x^{21}}-\frac{6 a^5 b}{19 x^{19}}-\frac{15 a^4 b^2}{17 x^{17}}-\frac{4 a^3 b^3}{3 x^{15}}-\frac{15 a^2 b^4}{13 x^{13}}-\frac{6 a b^5}{11 x^{11}}-\frac{b^6}{9 x^9}\\ \end{align*}

Mathematica [A]  time = 0.0097603, size = 82, normalized size = 1. \[ -\frac{15 a^4 b^2}{17 x^{17}}-\frac{4 a^3 b^3}{3 x^{15}}-\frac{15 a^2 b^4}{13 x^{13}}-\frac{6 a^5 b}{19 x^{19}}-\frac{a^6}{21 x^{21}}-\frac{6 a b^5}{11 x^{11}}-\frac{b^6}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^22,x]

[Out]

-a^6/(21*x^21) - (6*a^5*b)/(19*x^19) - (15*a^4*b^2)/(17*x^17) - (4*a^3*b^3)/(3*x^15) - (15*a^2*b^4)/(13*x^13)
- (6*a*b^5)/(11*x^11) - b^6/(9*x^9)

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Maple [A]  time = 0.047, size = 69, normalized size = 0.8 \begin{align*} -{\frac{{a}^{6}}{21\,{x}^{21}}}-{\frac{6\,{a}^{5}b}{19\,{x}^{19}}}-{\frac{15\,{a}^{4}{b}^{2}}{17\,{x}^{17}}}-{\frac{4\,{a}^{3}{b}^{3}}{3\,{x}^{15}}}-{\frac{15\,{a}^{2}{b}^{4}}{13\,{x}^{13}}}-{\frac{6\,a{b}^{5}}{11\,{x}^{11}}}-{\frac{{b}^{6}}{9\,{x}^{9}}} \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^22,x)

[Out]

-1/21*a^6/x^21-6/19*a^5*b/x^19-15/17*a^4*b^2/x^17-4/3*a^3*b^3/x^15-15/13*a^2*b^4/x^13-6/11*a*b^5/x^11-1/9*b^6/
x^9

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Maxima [A]  time = 1.23608, size = 95, normalized size = 1.16 \begin{align*} -\frac{323323 \, b^{6} x^{12} + 1587222 \, a b^{5} x^{10} + 3357585 \, a^{2} b^{4} x^{8} + 3879876 \, a^{3} b^{3} x^{6} + 2567565 \, a^{4} b^{2} x^{4} + 918918 \, a^{5} b x^{2} + 138567 \, a^{6}}{2909907 \, x^{21}} \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^22,x, algorithm="maxima")

[Out]

-1/2909907*(323323*b^6*x^12 + 1587222*a*b^5*x^10 + 3357585*a^2*b^4*x^8 + 3879876*a^3*b^3*x^6 + 2567565*a^4*b^2
*x^4 + 918918*a^5*b*x^2 + 138567*a^6)/x^21

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Fricas [A]  time = 1.66159, size = 208, normalized size = 2.54 \begin{align*} -\frac{323323 \, b^{6} x^{12} + 1587222 \, a b^{5} x^{10} + 3357585 \, a^{2} b^{4} x^{8} + 3879876 \, a^{3} b^{3} x^{6} + 2567565 \, a^{4} b^{2} x^{4} + 918918 \, a^{5} b x^{2} + 138567 \, a^{6}}{2909907 \, x^{21}} \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^22,x, algorithm="fricas")

[Out]

-1/2909907*(323323*b^6*x^12 + 1587222*a*b^5*x^10 + 3357585*a^2*b^4*x^8 + 3879876*a^3*b^3*x^6 + 2567565*a^4*b^2
*x^4 + 918918*a^5*b*x^2 + 138567*a^6)/x^21

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Sympy [A]  time = 0.896024, size = 75, normalized size = 0.91 \begin{align*} - \frac{138567 a^{6} + 918918 a^{5} b x^{2} + 2567565 a^{4} b^{2} x^{4} + 3879876 a^{3} b^{3} x^{6} + 3357585 a^{2} b^{4} x^{8} + 1587222 a b^{5} x^{10} + 323323 b^{6} x^{12}}{2909907 x^{21}} \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**22,x)

[Out]

-(138567*a**6 + 918918*a**5*b*x**2 + 2567565*a**4*b**2*x**4 + 3879876*a**3*b**3*x**6 + 3357585*a**2*b**4*x**8
+ 1587222*a*b**5*x**10 + 323323*b**6*x**12)/(2909907*x**21)

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Giac [A]  time = 1.12113, size = 95, normalized size = 1.16 \begin{align*} -\frac{323323 \, b^{6} x^{12} + 1587222 \, a b^{5} x^{10} + 3357585 \, a^{2} b^{4} x^{8} + 3879876 \, a^{3} b^{3} x^{6} + 2567565 \, a^{4} b^{2} x^{4} + 918918 \, a^{5} b x^{2} + 138567 \, a^{6}}{2909907 \, x^{21}} \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^22,x, algorithm="giac")

[Out]

-1/2909907*(323323*b^6*x^12 + 1587222*a*b^5*x^10 + 3357585*a^2*b^4*x^8 + 3879876*a^3*b^3*x^6 + 2567565*a^4*b^2
*x^4 + 918918*a^5*b*x^2 + 138567*a^6)/x^21