3.517 \(\int \frac{x^5}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=53 \[ -\frac{a^2}{10 b^3 \left (a+b x^2\right )^5}+\frac{a}{4 b^3 \left (a+b x^2\right )^4}-\frac{1}{6 b^3 \left (a+b x^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.112587, antiderivative size = 53, 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 \[ -\frac{a^2}{10 b^3 \left (a+b x^2\right )^5}+\frac{a}{4 b^3 \left (a+b x^2\right )^4}-\frac{1}{6 b^3 \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 19.4327, size = 46, normalized size = 0.87 \[ - \frac{a^{2}}{10 b^{3} \left (a + b x^{2}\right )^{5}} + \frac{a}{4 b^{3} \left (a + b x^{2}\right )^{4}} - \frac{1}{6 b^{3} \left (a + b x^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0243526, size = 35, normalized size = 0.66 \[ -\frac{a^2+5 a b x^2+10 b^2 x^4}{60 b^3 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(a^2 + 5*a*b*x^2 + 10*b^2*x^4)/(60*b^3*(a + b*x^2)^5)

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Maple [A]  time = 0.011, size = 48, normalized size = 0.9 \[ -{\frac{{a}^{2}}{10\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{a}{4\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{4}}}-{\frac{1}{6\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

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

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Maxima [A]  time = 0.700852, size = 108, normalized size = 2.04 \[ -\frac{10 \, b^{2} x^{4} + 5 \, a b x^{2} + a^{2}}{60 \,{\left (b^{8} x^{10} + 5 \, a b^{7} x^{8} + 10 \, a^{2} b^{6} x^{6} + 10 \, a^{3} b^{5} x^{4} + 5 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="maxima")

[Out]

-1/60*(10*b^2*x^4 + 5*a*b*x^2 + a^2)/(b^8*x^10 + 5*a*b^7*x^8 + 10*a^2*b^6*x^6 +
10*a^3*b^5*x^4 + 5*a^4*b^4*x^2 + a^5*b^3)

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Fricas [A]  time = 0.26238, size = 108, normalized size = 2.04 \[ -\frac{10 \, b^{2} x^{4} + 5 \, a b x^{2} + a^{2}}{60 \,{\left (b^{8} x^{10} + 5 \, a b^{7} x^{8} + 10 \, a^{2} b^{6} x^{6} + 10 \, a^{3} b^{5} x^{4} + 5 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="fricas")

[Out]

-1/60*(10*b^2*x^4 + 5*a*b*x^2 + a^2)/(b^8*x^10 + 5*a*b^7*x^8 + 10*a^2*b^6*x^6 +
10*a^3*b^5*x^4 + 5*a^4*b^4*x^2 + a^5*b^3)

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Sympy [A]  time = 3.96229, size = 83, normalized size = 1.57 \[ - \frac{a^{2} + 5 a b x^{2} + 10 b^{2} x^{4}}{60 a^{5} b^{3} + 300 a^{4} b^{4} x^{2} + 600 a^{3} b^{5} x^{4} + 600 a^{2} b^{6} x^{6} + 300 a b^{7} x^{8} + 60 b^{8} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

-(a**2 + 5*a*b*x**2 + 10*b**2*x**4)/(60*a**5*b**3 + 300*a**4*b**4*x**2 + 600*a**
3*b**5*x**4 + 600*a**2*b**6*x**6 + 300*a*b**7*x**8 + 60*b**8*x**10)

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GIAC/XCAS [A]  time = 0.270173, size = 45, normalized size = 0.85 \[ -\frac{10 \, b^{2} x^{4} + 5 \, a b x^{2} + a^{2}}{60 \,{\left (b x^{2} + a\right )}^{5} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="giac")

[Out]

-1/60*(10*b^2*x^4 + 5*a*b*x^2 + a^2)/((b*x^2 + a)^5*b^3)