3.1858 \(\int \frac{x^4}{\left (a+\frac{b}{x^2}\right )^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0995307, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{9/2}}+\frac{7 b^2 x}{2 a^4}-\frac{7 b x^3}{6 a^3}+\frac{7 x^5}{10 a^2}-\frac{x^7}{2 a \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b/x^2)^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{7 b^{2} \int \frac{1}{a^{3}}\, dx}{2 a} - \frac{x^{7}}{2 a \left (a x^{2} + b\right )} + \frac{7 x^{5}}{10 a^{2}} - \frac{7 b x^{3}}{6 a^{3}} - \frac{7 b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.11083, size = 71, normalized size = 0.9 \[ \frac{x \left (6 a^2 x^4+\frac{15 b^3}{a x^2+b}-20 a b x^2+90 b^2\right )}{30 a^4}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b/x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.007, size = 68, normalized size = 0.9 \[{\frac{{x}^{5}}{5\,{a}^{2}}}-{\frac{2\,b{x}^{3}}{3\,{a}^{3}}}+3\,{\frac{{b}^{2}x}{{a}^{4}}}+{\frac{{b}^{3}x}{2\,{a}^{4} \left ( a{x}^{2}+b \right ) }}-{\frac{7\,{b}^{3}}{2\,{a}^{4}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(a+b/x^2)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235178, size = 1, normalized size = 0.01 \[ \left [\frac{12 \, a^{3} x^{7} - 28 \, a^{2} b x^{5} + 140 \, a b^{2} x^{3} + 210 \, b^{3} x + 105 \,{\left (a b^{2} x^{2} + b^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right )}{60 \,{\left (a^{5} x^{2} + a^{4} b\right )}}, \frac{6 \, a^{3} x^{7} - 14 \, a^{2} b x^{5} + 70 \, a b^{2} x^{3} + 105 \, b^{3} x - 105 \,{\left (a b^{2} x^{2} + b^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{x}{\sqrt{\frac{b}{a}}}\right )}{30 \,{\left (a^{5} x^{2} + a^{4} b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.79283, size = 124, normalized size = 1.57 \[ \frac{b^{3} x}{2 a^{5} x^{2} + 2 a^{4} b} + \frac{7 \sqrt{- \frac{b^{5}}{a^{9}}} \log{\left (- \frac{a^{4} \sqrt{- \frac{b^{5}}{a^{9}}}}{b^{2}} + x \right )}}{4} - \frac{7 \sqrt{- \frac{b^{5}}{a^{9}}} \log{\left (\frac{a^{4} \sqrt{- \frac{b^{5}}{a^{9}}}}{b^{2}} + x \right )}}{4} + \frac{x^{5}}{5 a^{2}} - \frac{2 b x^{3}}{3 a^{3}} + \frac{3 b^{2} x}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(a+b/x**2)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.219047, size = 99, normalized size = 1.25 \[ -\frac{7 \, b^{3} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{4}} + \frac{b^{3} x}{2 \,{\left (a x^{2} + b\right )} a^{4}} + \frac{3 \, a^{8} x^{5} - 10 \, a^{7} b x^{3} + 45 \, a^{6} b^{2} x}{15 \, a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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