3.514 \(\int \frac{x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=109 \[ \frac{a^5}{10 b^6 \left (a+b x^2\right )^5}-\frac{5 a^4}{8 b^6 \left (a+b x^2\right )^4}+\frac{5 a^3}{3 b^6 \left (a+b x^2\right )^3}-\frac{5 a^2}{2 b^6 \left (a+b x^2\right )^2}+\frac{5 a}{2 b^6 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^6} \]

[Out]

a^5/(10*b^6*(a + b*x^2)^5) - (5*a^4)/(8*b^6*(a + b*x^2)^4) + (5*a^3)/(3*b^6*(a +
b*x^2)^3) - (5*a^2)/(2*b^6*(a + b*x^2)^2) + (5*a)/(2*b^6*(a + b*x^2)) + Log[a +
b*x^2]/(2*b^6)

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Rubi [A]  time = 0.22197, antiderivative size = 109, 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 \[ \frac{a^5}{10 b^6 \left (a+b x^2\right )^5}-\frac{5 a^4}{8 b^6 \left (a+b x^2\right )^4}+\frac{5 a^3}{3 b^6 \left (a+b x^2\right )^3}-\frac{5 a^2}{2 b^6 \left (a+b x^2\right )^2}+\frac{5 a}{2 b^6 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^6} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

a^5/(10*b^6*(a + b*x^2)^5) - (5*a^4)/(8*b^6*(a + b*x^2)^4) + (5*a^3)/(3*b^6*(a +
b*x^2)^3) - (5*a^2)/(2*b^6*(a + b*x^2)^2) + (5*a)/(2*b^6*(a + b*x^2)) + Log[a +
b*x^2]/(2*b^6)

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Rubi in Sympy [A]  time = 36.1144, size = 100, normalized size = 0.92 \[ \frac{a^{5}}{10 b^{6} \left (a + b x^{2}\right )^{5}} - \frac{5 a^{4}}{8 b^{6} \left (a + b x^{2}\right )^{4}} + \frac{5 a^{3}}{3 b^{6} \left (a + b x^{2}\right )^{3}} - \frac{5 a^{2}}{2 b^{6} \left (a + b x^{2}\right )^{2}} + \frac{5 a}{2 b^{6} \left (a + b x^{2}\right )} + \frac{\log{\left (a + b x^{2} \right )}}{2 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

a**5/(10*b**6*(a + b*x**2)**5) - 5*a**4/(8*b**6*(a + b*x**2)**4) + 5*a**3/(3*b**
6*(a + b*x**2)**3) - 5*a**2/(2*b**6*(a + b*x**2)**2) + 5*a/(2*b**6*(a + b*x**2))
+ log(a + b*x**2)/(2*b**6)

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Mathematica [A]  time = 0.0428848, size = 72, normalized size = 0.66 \[ \frac{\frac{a \left (137 a^4+625 a^3 b x^2+1100 a^2 b^2 x^4+900 a b^3 x^6+300 b^4 x^8\right )}{\left (a+b x^2\right )^5}+60 \log \left (a+b x^2\right )}{120 b^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

((a*(137*a^4 + 625*a^3*b*x^2 + 1100*a^2*b^2*x^4 + 900*a*b^3*x^6 + 300*b^4*x^8))/
(a + b*x^2)^5 + 60*Log[a + b*x^2])/(120*b^6)

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Maple [A]  time = 0.016, size = 98, normalized size = 0.9 \[{\frac{{a}^{5}}{10\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{5\,{a}^{4}}{8\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{5\,{a}^{3}}{3\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{5\,{a}^{2}}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,a}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

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

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Maxima [A]  time = 0.732851, size = 163, normalized size = 1.5 \[ \frac{300 \, a b^{4} x^{8} + 900 \, a^{2} b^{3} x^{6} + 1100 \, a^{3} b^{2} x^{4} + 625 \, a^{4} b x^{2} + 137 \, a^{5}}{120 \,{\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} + \frac{\log \left (b x^{2} + a\right )}{2 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="maxima")

[Out]

1/120*(300*a*b^4*x^8 + 900*a^2*b^3*x^6 + 1100*a^3*b^2*x^4 + 625*a^4*b*x^2 + 137*
a^5)/(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*x^2
+ a^5*b^6) + 1/2*log(b*x^2 + a)/b^6

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Fricas [A]  time = 0.267725, size = 227, normalized size = 2.08 \[ \frac{300 \, a b^{4} x^{8} + 900 \, a^{2} b^{3} x^{6} + 1100 \, a^{3} b^{2} x^{4} + 625 \, a^{4} b x^{2} + 137 \, a^{5} + 60 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{120 \,{\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="fricas")

[Out]

1/120*(300*a*b^4*x^8 + 900*a^2*b^3*x^6 + 1100*a^3*b^2*x^4 + 625*a^4*b*x^2 + 137*
a^5 + 60*(b^5*x^10 + 5*a*b^4*x^8 + 10*a^2*b^3*x^6 + 10*a^3*b^2*x^4 + 5*a^4*b*x^2
+ a^5)*log(b*x^2 + a))/(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b^9*x^6 + 10*a^3*b^8*
x^4 + 5*a^4*b^7*x^2 + a^5*b^6)

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Sympy [A]  time = 4.62984, size = 124, normalized size = 1.14 \[ \frac{137 a^{5} + 625 a^{4} b x^{2} + 1100 a^{3} b^{2} x^{4} + 900 a^{2} b^{3} x^{6} + 300 a b^{4} x^{8}}{120 a^{5} b^{6} + 600 a^{4} b^{7} x^{2} + 1200 a^{3} b^{8} x^{4} + 1200 a^{2} b^{9} x^{6} + 600 a b^{10} x^{8} + 120 b^{11} x^{10}} + \frac{\log{\left (a + b x^{2} \right )}}{2 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

(137*a**5 + 625*a**4*b*x**2 + 1100*a**3*b**2*x**4 + 900*a**2*b**3*x**6 + 300*a*b
**4*x**8)/(120*a**5*b**6 + 600*a**4*b**7*x**2 + 1200*a**3*b**8*x**4 + 1200*a**2*
b**9*x**6 + 600*a*b**10*x**8 + 120*b**11*x**10) + log(a + b*x**2)/(2*b**6)

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GIAC/XCAS [A]  time = 0.272478, size = 101, normalized size = 0.93 \[ \frac{{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} - \frac{137 \, b^{4} x^{10} + 385 \, a b^{3} x^{8} + 470 \, a^{2} b^{2} x^{6} + 270 \, a^{3} b x^{4} + 60 \, a^{4} x^{2}}{120 \,{\left (b x^{2} + a\right )}^{5} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="giac")

[Out]

1/2*ln(abs(b*x^2 + a))/b^6 - 1/120*(137*b^4*x^10 + 385*a*b^3*x^8 + 470*a^2*b^2*x
^6 + 270*a^3*b*x^4 + 60*a^4*x^2)/((b*x^2 + a)^5*b^5)