3.527 \(\int \frac{x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=122 \[ \frac{7 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{3/2} b^{9/2}}+\frac{7 x}{256 a b^4 \left (a+b x^2\right )}-\frac{7 x}{128 b^4 \left (a+b x^2\right )^2}-\frac{7 x^3}{96 b^3 \left (a+b x^2\right )^3}-\frac{7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac{x^7}{10 b \left (a+b x^2\right )^5} \]

[Out]

-x^7/(10*b*(a + b*x^2)^5) - (7*x^5)/(80*b^2*(a + b*x^2)^4) - (7*x^3)/(96*b^3*(a
+ b*x^2)^3) - (7*x)/(128*b^4*(a + b*x^2)^2) + (7*x)/(256*a*b^4*(a + b*x^2)) + (7
*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(3/2)*b^(9/2))

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Rubi [A]  time = 0.182228, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{7 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{3/2} b^{9/2}}+\frac{7 x}{256 a b^4 \left (a+b x^2\right )}-\frac{7 x}{128 b^4 \left (a+b x^2\right )^2}-\frac{7 x^3}{96 b^3 \left (a+b x^2\right )^3}-\frac{7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac{x^7}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-x^7/(10*b*(a + b*x^2)^5) - (7*x^5)/(80*b^2*(a + b*x^2)^4) - (7*x^3)/(96*b^3*(a
+ b*x^2)^3) - (7*x)/(128*b^4*(a + b*x^2)^2) + (7*x)/(256*a*b^4*(a + b*x^2)) + (7
*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(3/2)*b^(9/2))

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Rubi in Sympy [A]  time = 35.0716, size = 112, normalized size = 0.92 \[ - \frac{x^{7}}{10 b \left (a + b x^{2}\right )^{5}} - \frac{7 x^{5}}{80 b^{2} \left (a + b x^{2}\right )^{4}} - \frac{7 x^{3}}{96 b^{3} \left (a + b x^{2}\right )^{3}} - \frac{7 x}{128 b^{4} \left (a + b x^{2}\right )^{2}} + \frac{7 x}{256 a b^{4} \left (a + b x^{2}\right )} + \frac{7 \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 a^{\frac{3}{2}} b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

-x**7/(10*b*(a + b*x**2)**5) - 7*x**5/(80*b**2*(a + b*x**2)**4) - 7*x**3/(96*b**
3*(a + b*x**2)**3) - 7*x/(128*b**4*(a + b*x**2)**2) + 7*x/(256*a*b**4*(a + b*x**
2)) + 7*atan(sqrt(b)*x/sqrt(a))/(256*a**(3/2)*b**(9/2))

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Mathematica [A]  time = 0.105656, size = 91, normalized size = 0.75 \[ \frac{7 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{3/2} b^{9/2}}-\frac{x \left (105 a^4+490 a^3 b x^2+896 a^2 b^2 x^4+790 a b^3 x^6-105 b^4 x^8\right )}{3840 a b^4 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(x*(105*a^4 + 490*a^3*b*x^2 + 896*a^2*b^2*x^4 + 790*a*b^3*x^6 - 105*b^4*x^8))/(
3840*a*b^4*(a + b*x^2)^5) + (7*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(3/2)*b^(9/2)
)

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Maple [A]  time = 0.015, size = 80, normalized size = 0.7 \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{5}} \left ({\frac{7\,{x}^{9}}{256\,a}}-{\frac{79\,{x}^{7}}{384\,b}}-{\frac{7\,a{x}^{5}}{30\,{b}^{2}}}-{\frac{49\,{a}^{2}{x}^{3}}{384\,{b}^{3}}}-{\frac{7\,{a}^{3}x}{256\,{b}^{4}}} \right ) }+{\frac{7}{256\,a{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

(7/256/a*x^9-79/384/b*x^7-7/30*a/b^2*x^5-49/384*a^2/b^3*x^3-7/256*a^3/b^4*x)/(b*
x^2+a)^5+7/256/a/b^4/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270399, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (105 \, b^{4} x^{9} - 790 \, a b^{3} x^{7} - 896 \, a^{2} b^{2} x^{5} - 490 \, a^{3} b x^{3} - 105 \, a^{4} x\right )} \sqrt{-a b}}{7680 \,{\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \sqrt{-a b}}, \frac{105 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (105 \, b^{4} x^{9} - 790 \, a b^{3} x^{7} - 896 \, a^{2} b^{2} x^{5} - 490 \, a^{3} b x^{3} - 105 \, a^{4} x\right )} \sqrt{a b}}{3840 \,{\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="fricas")

[Out]

[1/7680*(105*(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((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) + 2*(105*b^4*x^9
- 790*a*b^3*x^7 - 896*a^2*b^2*x^5 - 490*a^3*b*x^3 - 105*a^4*x)*sqrt(-a*b))/((a*
b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6
*b^4)*sqrt(-a*b)), 1/3840*(105*(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)*arctan(sqrt(a*b)*x/a) + (105*b^4*x^9 - 790*a*b^3*x
^7 - 896*a^2*b^2*x^5 - 490*a^3*b*x^3 - 105*a^4*x)*sqrt(a*b))/((a*b^9*x^10 + 5*a^
2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^4)*sqrt(a*b)
)]

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Sympy [A]  time = 4.49052, size = 194, normalized size = 1.59 \[ - \frac{7 \sqrt{- \frac{1}{a^{3} b^{9}}} \log{\left (- a^{2} b^{4} \sqrt{- \frac{1}{a^{3} b^{9}}} + x \right )}}{512} + \frac{7 \sqrt{- \frac{1}{a^{3} b^{9}}} \log{\left (a^{2} b^{4} \sqrt{- \frac{1}{a^{3} b^{9}}} + x \right )}}{512} + \frac{- 105 a^{4} x - 490 a^{3} b x^{3} - 896 a^{2} b^{2} x^{5} - 790 a b^{3} x^{7} + 105 b^{4} x^{9}}{3840 a^{6} b^{4} + 19200 a^{5} b^{5} x^{2} + 38400 a^{4} b^{6} x^{4} + 38400 a^{3} b^{7} x^{6} + 19200 a^{2} b^{8} x^{8} + 3840 a b^{9} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

-7*sqrt(-1/(a**3*b**9))*log(-a**2*b**4*sqrt(-1/(a**3*b**9)) + x)/512 + 7*sqrt(-1
/(a**3*b**9))*log(a**2*b**4*sqrt(-1/(a**3*b**9)) + x)/512 + (-105*a**4*x - 490*a
**3*b*x**3 - 896*a**2*b**2*x**5 - 790*a*b**3*x**7 + 105*b**4*x**9)/(3840*a**6*b*
*4 + 19200*a**5*b**5*x**2 + 38400*a**4*b**6*x**4 + 38400*a**3*b**7*x**6 + 19200*
a**2*b**8*x**8 + 3840*a*b**9*x**10)

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GIAC/XCAS [A]  time = 0.271863, size = 113, normalized size = 0.93 \[ \frac{7 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} a b^{4}} + \frac{105 \, b^{4} x^{9} - 790 \, a b^{3} x^{7} - 896 \, a^{2} b^{2} x^{5} - 490 \, a^{3} b x^{3} - 105 \, a^{4} x}{3840 \,{\left (b x^{2} + a\right )}^{5} a b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="giac")

[Out]

7/256*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^4) + 1/3840*(105*b^4*x^9 - 790*a*b^3*
x^7 - 896*a^2*b^2*x^5 - 490*a^3*b*x^3 - 105*a^4*x)/((b*x^2 + a)^5*a*b^4)