3.458 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^6} \, dx\)

Optimal. Leaf size=72 \[ -\frac{a^6}{5 x^5}-\frac{2 a^5 b}{x^3}-\frac{15 a^4 b^2}{x}+20 a^3 b^3 x+5 a^2 b^4 x^3+\frac{6}{5} a b^5 x^5+\frac{b^6 x^7}{7} \]

[Out]

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

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Rubi [A]  time = 0.095626, antiderivative size = 72, 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 \[ -\frac{a^6}{5 x^5}-\frac{2 a^5 b}{x^3}-\frac{15 a^4 b^2}{x}+20 a^3 b^3 x+5 a^2 b^4 x^3+\frac{6}{5} a b^5 x^5+\frac{b^6 x^7}{7} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 22.1085, size = 70, normalized size = 0.97 \[ - \frac{a^{6}}{5 x^{5}} - \frac{2 a^{5} b}{x^{3}} - \frac{15 a^{4} b^{2}}{x} + 20 a^{3} b^{3} x + 5 a^{2} b^{4} x^{3} + \frac{6 a b^{5} x^{5}}{5} + \frac{b^{6} x^{7}}{7} \]

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**6,x)

[Out]

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

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Mathematica [A]  time = 0.0105662, size = 72, normalized size = 1. \[ -\frac{a^6}{5 x^5}-\frac{2 a^5 b}{x^3}-\frac{15 a^4 b^2}{x}+20 a^3 b^3 x+5 a^2 b^4 x^3+\frac{6}{5} a b^5 x^5+\frac{b^6 x^7}{7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^6,x]

[Out]

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

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Maple [A]  time = 0.008, size = 67, normalized size = 0.9 \[ -{\frac{{a}^{6}}{5\,{x}^{5}}}-2\,{\frac{{a}^{5}b}{{x}^{3}}}-15\,{\frac{{a}^{4}{b}^{2}}{x}}+20\,{a}^{3}{b}^{3}x+5\,{a}^{2}{b}^{4}{x}^{3}+{\frac{6\,a{b}^{5}{x}^{5}}{5}}+{\frac{{b}^{6}{x}^{7}}{7}} \]

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^6,x)

[Out]

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

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Maxima [A]  time = 0.692387, size = 90, normalized size = 1.25 \[ \frac{1}{7} \, b^{6} x^{7} + \frac{6}{5} \, a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{3} + 20 \, a^{3} b^{3} x - \frac{75 \, a^{4} b^{2} x^{4} + 10 \, a^{5} b x^{2} + a^{6}}{5 \, x^{5}} \]

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^6,x, algorithm="maxima")

[Out]

1/7*b^6*x^7 + 6/5*a*b^5*x^5 + 5*a^2*b^4*x^3 + 20*a^3*b^3*x - 1/5*(75*a^4*b^2*x^4
+ 10*a^5*b*x^2 + a^6)/x^5

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Fricas [A]  time = 0.250349, size = 95, normalized size = 1.32 \[ \frac{5 \, b^{6} x^{12} + 42 \, a b^{5} x^{10} + 175 \, a^{2} b^{4} x^{8} + 700 \, a^{3} b^{3} x^{6} - 525 \, a^{4} b^{2} x^{4} - 70 \, a^{5} b x^{2} - 7 \, a^{6}}{35 \, x^{5}} \]

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^6,x, algorithm="fricas")

[Out]

1/35*(5*b^6*x^12 + 42*a*b^5*x^10 + 175*a^2*b^4*x^8 + 700*a^3*b^3*x^6 - 525*a^4*b
^2*x^4 - 70*a^5*b*x^2 - 7*a^6)/x^5

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Sympy [A]  time = 1.53811, size = 71, normalized size = 0.99 \[ 20 a^{3} b^{3} x + 5 a^{2} b^{4} x^{3} + \frac{6 a b^{5} x^{5}}{5} + \frac{b^{6} x^{7}}{7} - \frac{a^{6} + 10 a^{5} b x^{2} + 75 a^{4} b^{2} x^{4}}{5 x^{5}} \]

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**6,x)

[Out]

20*a**3*b**3*x + 5*a**2*b**4*x**3 + 6*a*b**5*x**5/5 + b**6*x**7/7 - (a**6 + 10*a
**5*b*x**2 + 75*a**4*b**2*x**4)/(5*x**5)

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GIAC/XCAS [A]  time = 0.267685, size = 90, normalized size = 1.25 \[ \frac{1}{7} \, b^{6} x^{7} + \frac{6}{5} \, a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{3} + 20 \, a^{3} b^{3} x - \frac{75 \, a^{4} b^{2} x^{4} + 10 \, a^{5} b x^{2} + a^{6}}{5 \, x^{5}} \]

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^6,x, algorithm="giac")

[Out]

1/7*b^6*x^7 + 6/5*a*b^5*x^5 + 5*a^2*b^4*x^3 + 20*a^3*b^3*x - 1/5*(75*a^4*b^2*x^4
+ 10*a^5*b*x^2 + a^6)/x^5