3.123 \(\int \frac{(a+b x^2)^8}{x^{20}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{28 a^6 b^2}{15 x^{15}}-\frac{56 a^5 b^3}{13 x^{13}}-\frac{70 a^4 b^4}{11 x^{11}}-\frac{56 a^3 b^5}{9 x^9}-\frac{4 a^2 b^6}{x^7}-\frac{8 a^7 b}{17 x^{17}}-\frac{a^8}{19 x^{19}}-\frac{8 a b^7}{5 x^5}-\frac{b^8}{3 x^3} \]

[Out]

-a^8/(19*x^19) - (8*a^7*b)/(17*x^17) - (28*a^6*b^2)/(15*x^15) - (56*a^5*b^3)/(13*x^13) - (70*a^4*b^4)/(11*x^11
) - (56*a^3*b^5)/(9*x^9) - (4*a^2*b^6)/x^7 - (8*a*b^7)/(5*x^5) - b^8/(3*x^3)

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Rubi [A]  time = 0.0395873, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ -\frac{28 a^6 b^2}{15 x^{15}}-\frac{56 a^5 b^3}{13 x^{13}}-\frac{70 a^4 b^4}{11 x^{11}}-\frac{56 a^3 b^5}{9 x^9}-\frac{4 a^2 b^6}{x^7}-\frac{8 a^7 b}{17 x^{17}}-\frac{a^8}{19 x^{19}}-\frac{8 a b^7}{5 x^5}-\frac{b^8}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^20,x]

[Out]

-a^8/(19*x^19) - (8*a^7*b)/(17*x^17) - (28*a^6*b^2)/(15*x^15) - (56*a^5*b^3)/(13*x^13) - (70*a^4*b^4)/(11*x^11
) - (56*a^3*b^5)/(9*x^9) - (4*a^2*b^6)/x^7 - (8*a*b^7)/(5*x^5) - b^8/(3*x^3)

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+b x^2\right )^8}{x^{20}} \, dx &=\int \left (\frac{a^8}{x^{20}}+\frac{8 a^7 b}{x^{18}}+\frac{28 a^6 b^2}{x^{16}}+\frac{56 a^5 b^3}{x^{14}}+\frac{70 a^4 b^4}{x^{12}}+\frac{56 a^3 b^5}{x^{10}}+\frac{28 a^2 b^6}{x^8}+\frac{8 a b^7}{x^6}+\frac{b^8}{x^4}\right ) \, dx\\ &=-\frac{a^8}{19 x^{19}}-\frac{8 a^7 b}{17 x^{17}}-\frac{28 a^6 b^2}{15 x^{15}}-\frac{56 a^5 b^3}{13 x^{13}}-\frac{70 a^4 b^4}{11 x^{11}}-\frac{56 a^3 b^5}{9 x^9}-\frac{4 a^2 b^6}{x^7}-\frac{8 a b^7}{5 x^5}-\frac{b^8}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0112797, size = 106, normalized size = 1. \[ -\frac{28 a^6 b^2}{15 x^{15}}-\frac{56 a^5 b^3}{13 x^{13}}-\frac{70 a^4 b^4}{11 x^{11}}-\frac{56 a^3 b^5}{9 x^9}-\frac{4 a^2 b^6}{x^7}-\frac{8 a^7 b}{17 x^{17}}-\frac{a^8}{19 x^{19}}-\frac{8 a b^7}{5 x^5}-\frac{b^8}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^20,x]

[Out]

-a^8/(19*x^19) - (8*a^7*b)/(17*x^17) - (28*a^6*b^2)/(15*x^15) - (56*a^5*b^3)/(13*x^13) - (70*a^4*b^4)/(11*x^11
) - (56*a^3*b^5)/(9*x^9) - (4*a^2*b^6)/x^7 - (8*a*b^7)/(5*x^5) - b^8/(3*x^3)

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Maple [A]  time = 0.007, size = 91, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{19\,{x}^{19}}}-{\frac{8\,{a}^{7}b}{17\,{x}^{17}}}-{\frac{28\,{a}^{6}{b}^{2}}{15\,{x}^{15}}}-{\frac{56\,{a}^{5}{b}^{3}}{13\,{x}^{13}}}-{\frac{70\,{a}^{4}{b}^{4}}{11\,{x}^{11}}}-{\frac{56\,{a}^{3}{b}^{5}}{9\,{x}^{9}}}-4\,{\frac{{a}^{2}{b}^{6}}{{x}^{7}}}-{\frac{8\,a{b}^{7}}{5\,{x}^{5}}}-{\frac{{b}^{8}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^20,x)

[Out]

-1/19*a^8/x^19-8/17*a^7*b/x^17-28/15*a^6*b^2/x^15-56/13*a^5*b^3/x^13-70/11*a^4*b^4/x^11-56/9*a^3*b^5/x^9-4*a^2
*b^6/x^7-8/5*a*b^7/x^5-1/3*b^8/x^3

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Maxima [A]  time = 2.64907, size = 124, normalized size = 1.17 \begin{align*} -\frac{692835 \, b^{8} x^{16} + 3325608 \, a b^{7} x^{14} + 8314020 \, a^{2} b^{6} x^{12} + 12932920 \, a^{3} b^{5} x^{10} + 13226850 \, a^{4} b^{4} x^{8} + 8953560 \, a^{5} b^{3} x^{6} + 3879876 \, a^{6} b^{2} x^{4} + 978120 \, a^{7} b x^{2} + 109395 \, a^{8}}{2078505 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^20,x, algorithm="maxima")

[Out]

-1/2078505*(692835*b^8*x^16 + 3325608*a*b^7*x^14 + 8314020*a^2*b^6*x^12 + 12932920*a^3*b^5*x^10 + 13226850*a^4
*b^4*x^8 + 8953560*a^5*b^3*x^6 + 3879876*a^6*b^2*x^4 + 978120*a^7*b*x^2 + 109395*a^8)/x^19

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Fricas [A]  time = 1.17586, size = 273, normalized size = 2.58 \begin{align*} -\frac{692835 \, b^{8} x^{16} + 3325608 \, a b^{7} x^{14} + 8314020 \, a^{2} b^{6} x^{12} + 12932920 \, a^{3} b^{5} x^{10} + 13226850 \, a^{4} b^{4} x^{8} + 8953560 \, a^{5} b^{3} x^{6} + 3879876 \, a^{6} b^{2} x^{4} + 978120 \, a^{7} b x^{2} + 109395 \, a^{8}}{2078505 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^20,x, algorithm="fricas")

[Out]

-1/2078505*(692835*b^8*x^16 + 3325608*a*b^7*x^14 + 8314020*a^2*b^6*x^12 + 12932920*a^3*b^5*x^10 + 13226850*a^4
*b^4*x^8 + 8953560*a^5*b^3*x^6 + 3879876*a^6*b^2*x^4 + 978120*a^7*b*x^2 + 109395*a^8)/x^19

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Sympy [A]  time = 1.05334, size = 99, normalized size = 0.93 \begin{align*} - \frac{109395 a^{8} + 978120 a^{7} b x^{2} + 3879876 a^{6} b^{2} x^{4} + 8953560 a^{5} b^{3} x^{6} + 13226850 a^{4} b^{4} x^{8} + 12932920 a^{3} b^{5} x^{10} + 8314020 a^{2} b^{6} x^{12} + 3325608 a b^{7} x^{14} + 692835 b^{8} x^{16}}{2078505 x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**20,x)

[Out]

-(109395*a**8 + 978120*a**7*b*x**2 + 3879876*a**6*b**2*x**4 + 8953560*a**5*b**3*x**6 + 13226850*a**4*b**4*x**8
+ 12932920*a**3*b**5*x**10 + 8314020*a**2*b**6*x**12 + 3325608*a*b**7*x**14 + 692835*b**8*x**16)/(2078505*x**
19)

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Giac [A]  time = 1.57089, size = 124, normalized size = 1.17 \begin{align*} -\frac{692835 \, b^{8} x^{16} + 3325608 \, a b^{7} x^{14} + 8314020 \, a^{2} b^{6} x^{12} + 12932920 \, a^{3} b^{5} x^{10} + 13226850 \, a^{4} b^{4} x^{8} + 8953560 \, a^{5} b^{3} x^{6} + 3879876 \, a^{6} b^{2} x^{4} + 978120 \, a^{7} b x^{2} + 109395 \, a^{8}}{2078505 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^20,x, algorithm="giac")

[Out]

-1/2078505*(692835*b^8*x^16 + 3325608*a*b^7*x^14 + 8314020*a^2*b^6*x^12 + 12932920*a^3*b^5*x^10 + 13226850*a^4
*b^4*x^8 + 8953560*a^5*b^3*x^6 + 3879876*a^6*b^2*x^4 + 978120*a^7*b*x^2 + 109395*a^8)/x^19