3.494 \(\int \frac{x^9}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\)

Optimal. Leaf size=77 \[ -\frac{a^4}{6 b^5 \left (a+b x^2\right )^3}+\frac{a^3}{b^5 \left (a+b x^2\right )^2}-\frac{3 a^2}{b^5 \left (a+b x^2\right )}-\frac{2 a \log \left (a+b x^2\right )}{b^5}+\frac{x^2}{2 b^4} \]

[Out]

x^2/(2*b^4) - a^4/(6*b^5*(a + b*x^2)^3) + a^3/(b^5*(a + b*x^2)^2) - (3*a^2)/(b^5
*(a + b*x^2)) - (2*a*Log[a + b*x^2])/b^5

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Rubi [A]  time = 0.167811, antiderivative size = 77, 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 \[ -\frac{a^4}{6 b^5 \left (a+b x^2\right )^3}+\frac{a^3}{b^5 \left (a+b x^2\right )^2}-\frac{3 a^2}{b^5 \left (a+b x^2\right )}-\frac{2 a \log \left (a+b x^2\right )}{b^5}+\frac{x^2}{2 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

x^2/(2*b^4) - a^4/(6*b^5*(a + b*x^2)^3) + a^3/(b^5*(a + b*x^2)^2) - (3*a^2)/(b^5
*(a + b*x^2)) - (2*a*Log[a + b*x^2])/b^5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{4}}{6 b^{5} \left (a + b x^{2}\right )^{3}} + \frac{a^{3}}{b^{5} \left (a + b x^{2}\right )^{2}} - \frac{3 a^{2}}{b^{5} \left (a + b x^{2}\right )} - \frac{2 a \log{\left (a + b x^{2} \right )}}{b^{5}} + \frac{b^{4} \int ^{x^{2}} \frac{1}{b^{8}}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

-a**4/(6*b**5*(a + b*x**2)**3) + a**3/(b**5*(a + b*x**2)**2) - 3*a**2/(b**5*(a +
b*x**2)) - 2*a*log(a + b*x**2)/b**5 + b**4*Integral(b**(-8), (x, x**2))/2

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Mathematica [A]  time = 0.0884273, size = 59, normalized size = 0.77 \[ -\frac{\frac{a^2 \left (13 a^2+30 a b x^2+18 b^2 x^4\right )}{\left (a+b x^2\right )^3}+12 a \log \left (a+b x^2\right )-3 b x^2}{6 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

-(-3*b*x^2 + (a^2*(13*a^2 + 30*a*b*x^2 + 18*b^2*x^4))/(a + b*x^2)^3 + 12*a*Log[a
+ b*x^2])/(6*b^5)

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Maple [A]  time = 0.016, size = 74, normalized size = 1. \[{\frac{{x}^{2}}{2\,{b}^{4}}}-{\frac{{a}^{4}}{6\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{{a}^{3}}{{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-3\,{\frac{{a}^{2}}{{b}^{5} \left ( b{x}^{2}+a \right ) }}-2\,{\frac{a\ln \left ( b{x}^{2}+a \right ) }{{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(b^2*x^4+2*a*b*x^2+a^2)^2,x)

[Out]

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

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Maxima [A]  time = 0.697124, size = 119, normalized size = 1.55 \[ -\frac{18 \, a^{2} b^{2} x^{4} + 30 \, a^{3} b x^{2} + 13 \, a^{4}}{6 \,{\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}} + \frac{x^{2}}{2 \, b^{4}} - \frac{2 \, a \log \left (b x^{2} + a\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="maxima")

[Out]

-1/6*(18*a^2*b^2*x^4 + 30*a^3*b*x^2 + 13*a^4)/(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6
*x^2 + a^3*b^5) + 1/2*x^2/b^4 - 2*a*log(b*x^2 + a)/b^5

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Fricas [A]  time = 0.256851, size = 167, normalized size = 2.17 \[ \frac{3 \, b^{4} x^{8} + 9 \, a b^{3} x^{6} - 9 \, a^{2} b^{2} x^{4} - 27 \, a^{3} b x^{2} - 13 \, a^{4} - 12 \,{\left (a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + 3 \, a^{3} b x^{2} + a^{4}\right )} \log \left (b x^{2} + a\right )}{6 \,{\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="fricas")

[Out]

1/6*(3*b^4*x^8 + 9*a*b^3*x^6 - 9*a^2*b^2*x^4 - 27*a^3*b*x^2 - 13*a^4 - 12*(a*b^3
*x^6 + 3*a^2*b^2*x^4 + 3*a^3*b*x^2 + a^4)*log(b*x^2 + a))/(b^8*x^6 + 3*a*b^7*x^4
+ 3*a^2*b^6*x^2 + a^3*b^5)

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Sympy [A]  time = 2.64225, size = 88, normalized size = 1.14 \[ - \frac{2 a \log{\left (a + b x^{2} \right )}}{b^{5}} - \frac{13 a^{4} + 30 a^{3} b x^{2} + 18 a^{2} b^{2} x^{4}}{6 a^{3} b^{5} + 18 a^{2} b^{6} x^{2} + 18 a b^{7} x^{4} + 6 b^{8} x^{6}} + \frac{x^{2}}{2 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

-2*a*log(a + b*x**2)/b**5 - (13*a**4 + 30*a**3*b*x**2 + 18*a**2*b**2*x**4)/(6*a*
*3*b**5 + 18*a**2*b**6*x**2 + 18*a*b**7*x**4 + 6*b**8*x**6) + x**2/(2*b**4)

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GIAC/XCAS [A]  time = 0.271643, size = 99, normalized size = 1.29 \[ \frac{x^{2}}{2 \, b^{4}} - \frac{2 \, a{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{b^{5}} + \frac{22 \, a b^{3} x^{6} + 48 \, a^{2} b^{2} x^{4} + 36 \, a^{3} b x^{2} + 9 \, a^{4}}{6 \,{\left (b x^{2} + a\right )}^{3} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="giac")

[Out]

1/2*x^2/b^4 - 2*a*ln(abs(b*x^2 + a))/b^5 + 1/6*(22*a*b^3*x^6 + 48*a^2*b^2*x^4 +
36*a^3*b*x^2 + 9*a^4)/((b*x^2 + a)^3*b^5)