3.442 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{15}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{a^4}{14 x^{14}}-\frac{a^3 b}{3 x^{12}}-\frac{3 a^2 b^2}{5 x^{10}}-\frac{a b^3}{2 x^8}-\frac{b^4}{6 x^6} \]

[Out]

-a^4/(14*x^14) - (a^3*b)/(3*x^12) - (3*a^2*b^2)/(5*x^10) - (a*b^3)/(2*x^8) - b^4
/(6*x^6)

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Rubi [A]  time = 0.0877397, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{a^4}{14 x^{14}}-\frac{a^3 b}{3 x^{12}}-\frac{3 a^2 b^2}{5 x^{10}}-\frac{a b^3}{2 x^8}-\frac{b^4}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^15,x]

[Out]

-a^4/(14*x^14) - (a^3*b)/(3*x^12) - (3*a^2*b^2)/(5*x^10) - (a*b^3)/(2*x^8) - b^4
/(6*x^6)

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Rubi in Sympy [A]  time = 20.8273, size = 51, normalized size = 0.91 \[ - \frac{a^{4}}{14 x^{14}} - \frac{a^{3} b}{3 x^{12}} - \frac{3 a^{2} b^{2}}{5 x^{10}} - \frac{a b^{3}}{2 x^{8}} - \frac{b^{4}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**2/x**15,x)

[Out]

-a**4/(14*x**14) - a**3*b/(3*x**12) - 3*a**2*b**2/(5*x**10) - a*b**3/(2*x**8) -
b**4/(6*x**6)

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Mathematica [A]  time = 0.00711226, size = 56, normalized size = 1. \[ -\frac{a^4}{14 x^{14}}-\frac{a^3 b}{3 x^{12}}-\frac{3 a^2 b^2}{5 x^{10}}-\frac{a b^3}{2 x^8}-\frac{b^4}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^15,x]

[Out]

-a^4/(14*x^14) - (a^3*b)/(3*x^12) - (3*a^2*b^2)/(5*x^10) - (a*b^3)/(2*x^8) - b^4
/(6*x^6)

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Maple [A]  time = 0.008, size = 47, normalized size = 0.8 \[ -{\frac{{a}^{4}}{14\,{x}^{14}}}-{\frac{{a}^{3}b}{3\,{x}^{12}}}-{\frac{3\,{a}^{2}{b}^{2}}{5\,{x}^{10}}}-{\frac{a{b}^{3}}{2\,{x}^{8}}}-{\frac{{b}^{4}}{6\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^2/x^15,x)

[Out]

-1/14*a^4/x^14-1/3*a^3*b/x^12-3/5*a^2*b^2/x^10-1/2*a*b^3/x^8-1/6*b^4/x^6

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Maxima [A]  time = 0.694763, size = 65, normalized size = 1.16 \[ -\frac{35 \, b^{4} x^{8} + 105 \, a b^{3} x^{6} + 126 \, a^{2} b^{2} x^{4} + 70 \, a^{3} b x^{2} + 15 \, a^{4}}{210 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2/x^15,x, algorithm="maxima")

[Out]

-1/210*(35*b^4*x^8 + 105*a*b^3*x^6 + 126*a^2*b^2*x^4 + 70*a^3*b*x^2 + 15*a^4)/x^
14

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Fricas [A]  time = 0.257721, size = 65, normalized size = 1.16 \[ -\frac{35 \, b^{4} x^{8} + 105 \, a b^{3} x^{6} + 126 \, a^{2} b^{2} x^{4} + 70 \, a^{3} b x^{2} + 15 \, a^{4}}{210 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2/x^15,x, algorithm="fricas")

[Out]

-1/210*(35*b^4*x^8 + 105*a*b^3*x^6 + 126*a^2*b^2*x^4 + 70*a^3*b*x^2 + 15*a^4)/x^
14

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Sympy [A]  time = 2.07676, size = 51, normalized size = 0.91 \[ - \frac{15 a^{4} + 70 a^{3} b x^{2} + 126 a^{2} b^{2} x^{4} + 105 a b^{3} x^{6} + 35 b^{4} x^{8}}{210 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**2/x**15,x)

[Out]

-(15*a**4 + 70*a**3*b*x**2 + 126*a**2*b**2*x**4 + 105*a*b**3*x**6 + 35*b**4*x**8
)/(210*x**14)

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GIAC/XCAS [A]  time = 0.268165, size = 65, normalized size = 1.16 \[ -\frac{35 \, b^{4} x^{8} + 105 \, a b^{3} x^{6} + 126 \, a^{2} b^{2} x^{4} + 70 \, a^{3} b x^{2} + 15 \, a^{4}}{210 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2/x^15,x, algorithm="giac")

[Out]

-1/210*(35*b^4*x^8 + 105*a*b^3*x^6 + 126*a^2*b^2*x^4 + 70*a^3*b*x^2 + 15*a^4)/x^
14