3.472 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^{20}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{a^4 b^2}{x^{15}}-\frac{20 a^3 b^3}{13 x^{13}}-\frac{15 a^2 b^4}{11 x^{11}}-\frac{6 a^5 b}{17 x^{17}}-\frac{a^6}{19 x^{19}}-\frac{2 a b^5}{3 x^9}-\frac{b^6}{7 x^7} \]

[Out]

-a^6/(19*x^19) - (6*a^5*b)/(17*x^17) - (a^4*b^2)/x^15 - (20*a^3*b^3)/(13*x^13) - (15*a^2*b^4)/(11*x^11) - (2*a
*b^5)/(3*x^9) - b^6/(7*x^7)

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Rubi [A]  time = 0.0375387, antiderivative size = 80, 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} \[ -\frac{a^4 b^2}{x^{15}}-\frac{20 a^3 b^3}{13 x^{13}}-\frac{15 a^2 b^4}{11 x^{11}}-\frac{6 a^5 b}{17 x^{17}}-\frac{a^6}{19 x^{19}}-\frac{2 a b^5}{3 x^9}-\frac{b^6}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^20,x]

[Out]

-a^6/(19*x^19) - (6*a^5*b)/(17*x^17) - (a^4*b^2)/x^15 - (20*a^3*b^3)/(13*x^13) - (15*a^2*b^4)/(11*x^11) - (2*a
*b^5)/(3*x^9) - b^6/(7*x^7)

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

Mathematica [A]  time = 0.0095767, size = 80, normalized size = 1. \[ -\frac{a^4 b^2}{x^{15}}-\frac{20 a^3 b^3}{13 x^{13}}-\frac{15 a^2 b^4}{11 x^{11}}-\frac{6 a^5 b}{17 x^{17}}-\frac{a^6}{19 x^{19}}-\frac{2 a b^5}{3 x^9}-\frac{b^6}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^20,x]

[Out]

-a^6/(19*x^19) - (6*a^5*b)/(17*x^17) - (a^4*b^2)/x^15 - (20*a^3*b^3)/(13*x^13) - (15*a^2*b^4)/(11*x^11) - (2*a
*b^5)/(3*x^9) - b^6/(7*x^7)

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Maple [A]  time = 0.047, size = 69, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{19\,{x}^{19}}}-{\frac{6\,{a}^{5}b}{17\,{x}^{17}}}-{\frac{{a}^{4}{b}^{2}}{{x}^{15}}}-{\frac{20\,{a}^{3}{b}^{3}}{13\,{x}^{13}}}-{\frac{15\,{a}^{2}{b}^{4}}{11\,{x}^{11}}}-{\frac{2\,a{b}^{5}}{3\,{x}^{9}}}-{\frac{{b}^{6}}{7\,{x}^{7}}} \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^20,x)

[Out]

-1/19*a^6/x^19-6/17*a^5*b/x^17-a^4*b^2/x^15-20/13*a^3*b^3/x^13-15/11*a^2*b^4/x^11-2/3*a*b^5/x^9-1/7*b^6/x^7

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Maxima [A]  time = 0.999935, size = 95, normalized size = 1.19 \begin{align*} -\frac{138567 \, b^{6} x^{12} + 646646 \, a b^{5} x^{10} + 1322685 \, a^{2} b^{4} x^{8} + 1492260 \, a^{3} b^{3} x^{6} + 969969 \, a^{4} b^{2} x^{4} + 342342 \, a^{5} b x^{2} + 51051 \, a^{6}}{969969 \, x^{19}} \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^20,x, algorithm="maxima")

[Out]

-1/969969*(138567*b^6*x^12 + 646646*a*b^5*x^10 + 1322685*a^2*b^4*x^8 + 1492260*a^3*b^3*x^6 + 969969*a^4*b^2*x^
4 + 342342*a^5*b*x^2 + 51051*a^6)/x^19

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Fricas [A]  time = 1.63399, size = 203, normalized size = 2.54 \begin{align*} -\frac{138567 \, b^{6} x^{12} + 646646 \, a b^{5} x^{10} + 1322685 \, a^{2} b^{4} x^{8} + 1492260 \, a^{3} b^{3} x^{6} + 969969 \, a^{4} b^{2} x^{4} + 342342 \, a^{5} b x^{2} + 51051 \, a^{6}}{969969 \, x^{19}} \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^20,x, algorithm="fricas")

[Out]

-1/969969*(138567*b^6*x^12 + 646646*a*b^5*x^10 + 1322685*a^2*b^4*x^8 + 1492260*a^3*b^3*x^6 + 969969*a^4*b^2*x^
4 + 342342*a^5*b*x^2 + 51051*a^6)/x^19

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Sympy [A]  time = 0.862294, size = 75, normalized size = 0.94 \begin{align*} - \frac{51051 a^{6} + 342342 a^{5} b x^{2} + 969969 a^{4} b^{2} x^{4} + 1492260 a^{3} b^{3} x^{6} + 1322685 a^{2} b^{4} x^{8} + 646646 a b^{5} x^{10} + 138567 b^{6} x^{12}}{969969 x^{19}} \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**20,x)

[Out]

-(51051*a**6 + 342342*a**5*b*x**2 + 969969*a**4*b**2*x**4 + 1492260*a**3*b**3*x**6 + 1322685*a**2*b**4*x**8 +
646646*a*b**5*x**10 + 138567*b**6*x**12)/(969969*x**19)

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Giac [A]  time = 1.14533, size = 95, normalized size = 1.19 \begin{align*} -\frac{138567 \, b^{6} x^{12} + 646646 \, a b^{5} x^{10} + 1322685 \, a^{2} b^{4} x^{8} + 1492260 \, a^{3} b^{3} x^{6} + 969969 \, a^{4} b^{2} x^{4} + 342342 \, a^{5} b x^{2} + 51051 \, a^{6}}{969969 \, x^{19}} \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^20,x, algorithm="giac")

[Out]

-1/969969*(138567*b^6*x^12 + 646646*a*b^5*x^10 + 1322685*a^2*b^4*x^8 + 1492260*a^3*b^3*x^6 + 969969*a^4*b^2*x^
4 + 342342*a^5*b*x^2 + 51051*a^6)/x^19