3.485 \(\int \frac{x^8}{a^2+2 a b x^2+b^2 x^4} \, dx\)

Optimal. Leaf size=79 \[ -\frac{7 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{9/2}}+\frac{7 a^2 x}{2 b^4}-\frac{7 a x^3}{6 b^3}-\frac{x^7}{2 b \left (a+b x^2\right )}+\frac{7 x^5}{10 b^2} \]

[Out]

(7*a^2*x)/(2*b^4) - (7*a*x^3)/(6*b^3) + (7*x^5)/(10*b^2) - x^7/(2*b*(a + b*x^2))
- (7*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(9/2))

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Rubi [A]  time = 0.110292, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{7 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{9/2}}+\frac{7 a^2 x}{2 b^4}-\frac{7 a x^3}{6 b^3}-\frac{x^7}{2 b \left (a+b x^2\right )}+\frac{7 x^5}{10 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(7*a^2*x)/(2*b^4) - (7*a*x^3)/(6*b^3) + (7*x^5)/(10*b^2) - x^7/(2*b*(a + b*x^2))
- (7*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(9/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{7 a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{9}{2}}} - \frac{7 a x^{3}}{6 b^{3}} - \frac{x^{7}}{2 b \left (a + b x^{2}\right )} + \frac{7 x^{5}}{10 b^{2}} + \frac{7 \int a^{2}\, dx}{2 b^{4}} \]

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),x)

[Out]

-7*a**(5/2)*atan(sqrt(b)*x/sqrt(a))/(2*b**(9/2)) - 7*a*x**3/(6*b**3) - x**7/(2*b
*(a + b*x**2)) + 7*x**5/(10*b**2) + 7*Integral(a**2, x)/(2*b**4)

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Mathematica [A]  time = 0.0849795, size = 71, normalized size = 0.9 \[ \frac{x \left (\frac{15 a^3}{a+b x^2}+90 a^2-20 a b x^2+6 b^2 x^4\right )}{30 b^4}-\frac{7 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(90*a^2 - 20*a*b*x^2 + 6*b^2*x^4 + (15*a^3)/(a + b*x^2)))/(30*b^4) - (7*a^(5/
2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(9/2))

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Maple [A]  time = 0.012, size = 68, normalized size = 0.9 \[{\frac{{x}^{5}}{5\,{b}^{2}}}-{\frac{2\,a{x}^{3}}{3\,{b}^{3}}}+3\,{\frac{{a}^{2}x}{{b}^{4}}}+{\frac{{a}^{3}x}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{7\,{a}^{3}}{2\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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),x)

[Out]

1/5*x^5/b^2-2/3*a*x^3/b^3+3*a^2*x/b^4+1/2/b^4*a^3*x/(b*x^2+a)-7/2/b^4*a^3/(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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263148, size = 1, normalized size = 0.01 \[ \left [\frac{12 \, b^{3} x^{7} - 28 \, a b^{2} x^{5} + 140 \, a^{2} b x^{3} + 210 \, a^{3} x + 105 \,{\left (a^{2} b x^{2} + a^{3}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{60 \,{\left (b^{5} x^{2} + a b^{4}\right )}}, \frac{6 \, b^{3} x^{7} - 14 \, a b^{2} x^{5} + 70 \, a^{2} b x^{3} + 105 \, a^{3} x - 105 \,{\left (a^{2} b x^{2} + a^{3}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right )}{30 \,{\left (b^{5} x^{2} + a b^{4}\right )}}\right ] \]

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),x, algorithm="fricas")

[Out]

[1/60*(12*b^3*x^7 - 28*a*b^2*x^5 + 140*a^2*b*x^3 + 210*a^3*x + 105*(a^2*b*x^2 +
a^3)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^5*x^2 + a*b^
4), 1/30*(6*b^3*x^7 - 14*a*b^2*x^5 + 70*a^2*b*x^3 + 105*a^3*x - 105*(a^2*b*x^2 +
a^3)*sqrt(a/b)*arctan(x/sqrt(a/b)))/(b^5*x^2 + a*b^4)]

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Sympy [A]  time = 1.63131, size = 124, normalized size = 1.57 \[ \frac{a^{3} x}{2 a b^{4} + 2 b^{5} x^{2}} + \frac{3 a^{2} x}{b^{4}} - \frac{2 a x^{3}}{3 b^{3}} + \frac{7 \sqrt{- \frac{a^{5}}{b^{9}}} \log{\left (x - \frac{b^{4} \sqrt{- \frac{a^{5}}{b^{9}}}}{a^{2}} \right )}}{4} - \frac{7 \sqrt{- \frac{a^{5}}{b^{9}}} \log{\left (x + \frac{b^{4} \sqrt{- \frac{a^{5}}{b^{9}}}}{a^{2}} \right )}}{4} + \frac{x^{5}}{5 b^{2}} \]

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),x)

[Out]

a**3*x/(2*a*b**4 + 2*b**5*x**2) + 3*a**2*x/b**4 - 2*a*x**3/(3*b**3) + 7*sqrt(-a*
*5/b**9)*log(x - b**4*sqrt(-a**5/b**9)/a**2)/4 - 7*sqrt(-a**5/b**9)*log(x + b**4
*sqrt(-a**5/b**9)/a**2)/4 + x**5/(5*b**2)

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GIAC/XCAS [A]  time = 0.270136, size = 99, normalized size = 1.25 \[ -\frac{7 \, a^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{4}} + \frac{a^{3} x}{2 \,{\left (b x^{2} + a\right )} b^{4}} + \frac{3 \, b^{8} x^{5} - 10 \, a b^{7} x^{3} + 45 \, a^{2} b^{6} x}{15 \, b^{10}} \]

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),x, algorithm="giac")

[Out]

-7/2*a^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/2*a^3*x/((b*x^2 + a)*b^4) + 1
/15*(3*b^8*x^5 - 10*a*b^7*x^3 + 45*a^2*b^6*x)/b^10