3.462 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{10}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{a^6}{9 x^9}-\frac{6 a^5 b}{7 x^7}-\frac{3 a^4 b^2}{x^5}-\frac{20 a^3 b^3}{3 x^3}-\frac{15 a^2 b^4}{x}+6 a b^5 x+\frac{b^6 x^3}{3} \]

[Out]

-a^6/(9*x^9) - (6*a^5*b)/(7*x^7) - (3*a^4*b^2)/x^5 - (20*a^3*b^3)/(3*x^3) - (15*
a^2*b^4)/x + 6*a*b^5*x + (b^6*x^3)/3

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Rubi [A]  time = 0.0937105, antiderivative size = 74, 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 \[ -\frac{a^6}{9 x^9}-\frac{6 a^5 b}{7 x^7}-\frac{3 a^4 b^2}{x^5}-\frac{20 a^3 b^3}{3 x^3}-\frac{15 a^2 b^4}{x}+6 a b^5 x+\frac{b^6 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^10,x]

[Out]

-a^6/(9*x^9) - (6*a^5*b)/(7*x^7) - (3*a^4*b^2)/x^5 - (20*a^3*b^3)/(3*x^3) - (15*
a^2*b^4)/x + 6*a*b^5*x + (b^6*x^3)/3

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Rubi in Sympy [A]  time = 22.4119, size = 71, normalized size = 0.96 \[ - \frac{a^{6}}{9 x^{9}} - \frac{6 a^{5} b}{7 x^{7}} - \frac{3 a^{4} b^{2}}{x^{5}} - \frac{20 a^{3} b^{3}}{3 x^{3}} - \frac{15 a^{2} b^{4}}{x} + 6 a b^{5} x + \frac{b^{6} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**3/x**10,x)

[Out]

-a**6/(9*x**9) - 6*a**5*b/(7*x**7) - 3*a**4*b**2/x**5 - 20*a**3*b**3/(3*x**3) -
15*a**2*b**4/x + 6*a*b**5*x + b**6*x**3/3

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Mathematica [A]  time = 0.0154171, size = 74, normalized size = 1. \[ -\frac{a^6}{9 x^9}-\frac{6 a^5 b}{7 x^7}-\frac{3 a^4 b^2}{x^5}-\frac{20 a^3 b^3}{3 x^3}-\frac{15 a^2 b^4}{x}+6 a b^5 x+\frac{b^6 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^10,x]

[Out]

-a^6/(9*x^9) - (6*a^5*b)/(7*x^7) - (3*a^4*b^2)/x^5 - (20*a^3*b^3)/(3*x^3) - (15*
a^2*b^4)/x + 6*a*b^5*x + (b^6*x^3)/3

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Maple [A]  time = 0.009, size = 67, normalized size = 0.9 \[ -{\frac{{a}^{6}}{9\,{x}^{9}}}-{\frac{6\,{a}^{5}b}{7\,{x}^{7}}}-3\,{\frac{{a}^{4}{b}^{2}}{{x}^{5}}}-{\frac{20\,{a}^{3}{b}^{3}}{3\,{x}^{3}}}-15\,{\frac{{a}^{2}{b}^{4}}{x}}+6\,a{b}^{5}x+{\frac{{b}^{6}{x}^{3}}{3}} \]

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^10,x)

[Out]

-1/9*a^6/x^9-6/7*a^5*b/x^7-3*a^4*b^2/x^5-20/3*a^3*b^3/x^3-15*a^2*b^4/x+6*a*b^5*x
+1/3*b^6*x^3

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Maxima [A]  time = 0.69522, size = 93, normalized size = 1.26 \[ \frac{1}{3} \, b^{6} x^{3} + 6 \, a b^{5} x - \frac{945 \, a^{2} b^{4} x^{8} + 420 \, a^{3} b^{3} x^{6} + 189 \, a^{4} b^{2} x^{4} + 54 \, a^{5} b x^{2} + 7 \, a^{6}}{63 \, x^{9}} \]

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^10,x, algorithm="maxima")

[Out]

1/3*b^6*x^3 + 6*a*b^5*x - 1/63*(945*a^2*b^4*x^8 + 420*a^3*b^3*x^6 + 189*a^4*b^2*
x^4 + 54*a^5*b*x^2 + 7*a^6)/x^9

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Fricas [A]  time = 0.246904, size = 95, normalized size = 1.28 \[ \frac{21 \, b^{6} x^{12} + 378 \, a b^{5} x^{10} - 945 \, a^{2} b^{4} x^{8} - 420 \, a^{3} b^{3} x^{6} - 189 \, a^{4} b^{2} x^{4} - 54 \, a^{5} b x^{2} - 7 \, a^{6}}{63 \, x^{9}} \]

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^10,x, algorithm="fricas")

[Out]

1/63*(21*b^6*x^12 + 378*a*b^5*x^10 - 945*a^2*b^4*x^8 - 420*a^3*b^3*x^6 - 189*a^4
*b^2*x^4 - 54*a^5*b*x^2 - 7*a^6)/x^9

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Sympy [A]  time = 1.96051, size = 71, normalized size = 0.96 \[ 6 a b^{5} x + \frac{b^{6} x^{3}}{3} - \frac{7 a^{6} + 54 a^{5} b x^{2} + 189 a^{4} b^{2} x^{4} + 420 a^{3} b^{3} x^{6} + 945 a^{2} b^{4} x^{8}}{63 x^{9}} \]

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**10,x)

[Out]

6*a*b**5*x + b**6*x**3/3 - (7*a**6 + 54*a**5*b*x**2 + 189*a**4*b**2*x**4 + 420*a
**3*b**3*x**6 + 945*a**2*b**4*x**8)/(63*x**9)

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GIAC/XCAS [A]  time = 0.269932, size = 93, normalized size = 1.26 \[ \frac{1}{3} \, b^{6} x^{3} + 6 \, a b^{5} x - \frac{945 \, a^{2} b^{4} x^{8} + 420 \, a^{3} b^{3} x^{6} + 189 \, a^{4} b^{2} x^{4} + 54 \, a^{5} b x^{2} + 7 \, a^{6}}{63 \, x^{9}} \]

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^10,x, algorithm="giac")

[Out]

1/3*b^6*x^3 + 6*a*b^5*x - 1/63*(945*a^2*b^4*x^8 + 420*a^3*b^3*x^6 + 189*a^4*b^2*
x^4 + 54*a^5*b*x^2 + 7*a^6)/x^9