3.119 \(\int \frac{\left (a+b x^2\right )^8}{x^{12}} \, dx\)

Optimal. Leaf size=100 \[ -\frac{a^8}{11 x^{11}}-\frac{8 a^7 b}{9 x^9}-\frac{4 a^6 b^2}{x^7}-\frac{56 a^5 b^3}{5 x^5}-\frac{70 a^4 b^4}{3 x^3}-\frac{56 a^3 b^5}{x}+28 a^2 b^6 x+\frac{8}{3} a b^7 x^3+\frac{b^8 x^5}{5} \]

[Out]

-a^8/(11*x^11) - (8*a^7*b)/(9*x^9) - (4*a^6*b^2)/x^7 - (56*a^5*b^3)/(5*x^5) - (7
0*a^4*b^4)/(3*x^3) - (56*a^3*b^5)/x + 28*a^2*b^6*x + (8*a*b^7*x^3)/3 + (b^8*x^5)
/5

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Rubi [A]  time = 0.105073, antiderivative size = 100, 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 \[ -\frac{a^8}{11 x^{11}}-\frac{8 a^7 b}{9 x^9}-\frac{4 a^6 b^2}{x^7}-\frac{56 a^5 b^3}{5 x^5}-\frac{70 a^4 b^4}{3 x^3}-\frac{56 a^3 b^5}{x}+28 a^2 b^6 x+\frac{8}{3} a b^7 x^3+\frac{b^8 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^8/x^12,x]

[Out]

-a^8/(11*x^11) - (8*a^7*b)/(9*x^9) - (4*a^6*b^2)/x^7 - (56*a^5*b^3)/(5*x^5) - (7
0*a^4*b^4)/(3*x^3) - (56*a^3*b^5)/x + 28*a^2*b^6*x + (8*a*b^7*x^3)/3 + (b^8*x^5)
/5

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Rubi in Sympy [A]  time = 19.6947, size = 99, normalized size = 0.99 \[ - \frac{a^{8}}{11 x^{11}} - \frac{8 a^{7} b}{9 x^{9}} - \frac{4 a^{6} b^{2}}{x^{7}} - \frac{56 a^{5} b^{3}}{5 x^{5}} - \frac{70 a^{4} b^{4}}{3 x^{3}} - \frac{56 a^{3} b^{5}}{x} + 28 a^{2} b^{6} x + \frac{8 a b^{7} x^{3}}{3} + \frac{b^{8} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**8/x**12,x)

[Out]

-a**8/(11*x**11) - 8*a**7*b/(9*x**9) - 4*a**6*b**2/x**7 - 56*a**5*b**3/(5*x**5)
- 70*a**4*b**4/(3*x**3) - 56*a**3*b**5/x + 28*a**2*b**6*x + 8*a*b**7*x**3/3 + b*
*8*x**5/5

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Mathematica [A]  time = 0.0151624, size = 100, normalized size = 1. \[ -\frac{a^8}{11 x^{11}}-\frac{8 a^7 b}{9 x^9}-\frac{4 a^6 b^2}{x^7}-\frac{56 a^5 b^3}{5 x^5}-\frac{70 a^4 b^4}{3 x^3}-\frac{56 a^3 b^5}{x}+28 a^2 b^6 x+\frac{8}{3} a b^7 x^3+\frac{b^8 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^8/x^12,x]

[Out]

-a^8/(11*x^11) - (8*a^7*b)/(9*x^9) - (4*a^6*b^2)/x^7 - (56*a^5*b^3)/(5*x^5) - (7
0*a^4*b^4)/(3*x^3) - (56*a^3*b^5)/x + 28*a^2*b^6*x + (8*a*b^7*x^3)/3 + (b^8*x^5)
/5

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Maple [A]  time = 0.009, size = 89, normalized size = 0.9 \[ -{\frac{{a}^{8}}{11\,{x}^{11}}}-{\frac{8\,{a}^{7}b}{9\,{x}^{9}}}-4\,{\frac{{a}^{6}{b}^{2}}{{x}^{7}}}-{\frac{56\,{a}^{5}{b}^{3}}{5\,{x}^{5}}}-{\frac{70\,{a}^{4}{b}^{4}}{3\,{x}^{3}}}-56\,{\frac{{a}^{3}{b}^{5}}{x}}+28\,{a}^{2}{b}^{6}x+{\frac{8\,a{b}^{7}{x}^{3}}{3}}+{\frac{{b}^{8}{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^8/x^12,x)

[Out]

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

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Maxima [A]  time = 1.3232, size = 123, normalized size = 1.23 \[ \frac{1}{5} \, b^{8} x^{5} + \frac{8}{3} \, a b^{7} x^{3} + 28 \, a^{2} b^{6} x - \frac{27720 \, a^{3} b^{5} x^{10} + 11550 \, a^{4} b^{4} x^{8} + 5544 \, a^{5} b^{3} x^{6} + 1980 \, a^{6} b^{2} x^{4} + 440 \, a^{7} b x^{2} + 45 \, a^{8}}{495 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^12,x, algorithm="maxima")

[Out]

1/5*b^8*x^5 + 8/3*a*b^7*x^3 + 28*a^2*b^6*x - 1/495*(27720*a^3*b^5*x^10 + 11550*a
^4*b^4*x^8 + 5544*a^5*b^3*x^6 + 1980*a^6*b^2*x^4 + 440*a^7*b*x^2 + 45*a^8)/x^11

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Fricas [A]  time = 0.198086, size = 124, normalized size = 1.24 \[ \frac{99 \, b^{8} x^{16} + 1320 \, a b^{7} x^{14} + 13860 \, a^{2} b^{6} x^{12} - 27720 \, a^{3} b^{5} x^{10} - 11550 \, a^{4} b^{4} x^{8} - 5544 \, a^{5} b^{3} x^{6} - 1980 \, a^{6} b^{2} x^{4} - 440 \, a^{7} b x^{2} - 45 \, a^{8}}{495 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^12,x, algorithm="fricas")

[Out]

1/495*(99*b^8*x^16 + 1320*a*b^7*x^14 + 13860*a^2*b^6*x^12 - 27720*a^3*b^5*x^10 -
11550*a^4*b^4*x^8 - 5544*a^5*b^3*x^6 - 1980*a^6*b^2*x^4 - 440*a^7*b*x^2 - 45*a^
8)/x^11

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Sympy [A]  time = 2.53334, size = 97, normalized size = 0.97 \[ 28 a^{2} b^{6} x + \frac{8 a b^{7} x^{3}}{3} + \frac{b^{8} x^{5}}{5} - \frac{45 a^{8} + 440 a^{7} b x^{2} + 1980 a^{6} b^{2} x^{4} + 5544 a^{5} b^{3} x^{6} + 11550 a^{4} b^{4} x^{8} + 27720 a^{3} b^{5} x^{10}}{495 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**8/x**12,x)

[Out]

28*a**2*b**6*x + 8*a*b**7*x**3/3 + b**8*x**5/5 - (45*a**8 + 440*a**7*b*x**2 + 19
80*a**6*b**2*x**4 + 5544*a**5*b**3*x**6 + 11550*a**4*b**4*x**8 + 27720*a**3*b**5
*x**10)/(495*x**11)

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GIAC/XCAS [A]  time = 0.219101, size = 123, normalized size = 1.23 \[ \frac{1}{5} \, b^{8} x^{5} + \frac{8}{3} \, a b^{7} x^{3} + 28 \, a^{2} b^{6} x - \frac{27720 \, a^{3} b^{5} x^{10} + 11550 \, a^{4} b^{4} x^{8} + 5544 \, a^{5} b^{3} x^{6} + 1980 \, a^{6} b^{2} x^{4} + 440 \, a^{7} b x^{2} + 45 \, a^{8}}{495 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^12,x, algorithm="giac")

[Out]

1/5*b^8*x^5 + 8/3*a*b^7*x^3 + 28*a^2*b^6*x - 1/495*(27720*a^3*b^5*x^10 + 11550*a
^4*b^4*x^8 + 5544*a^5*b^3*x^6 + 1980*a^6*b^2*x^4 + 440*a^7*b*x^2 + 45*a^8)/x^11