3.152 \(\int \frac{x^6}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=66 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}-\frac{5 a x}{2 b^3}-\frac{x^5}{2 b \left (a+b x^2\right )}+\frac{5 x^3}{6 b^2} \]

[Out]

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

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Rubi [A]  time = 0.07645, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}-\frac{5 a x}{2 b^3}-\frac{x^5}{2 b \left (a+b x^2\right )}+\frac{5 x^3}{6 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + b*x^2)^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(b*x**2+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0773642, size = 60, normalized size = 0.91 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}+\frac{x \left (-\frac{3 a^2}{a+b x^2}-12 a+2 b x^2\right )}{6 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + b*x^2)^2,x]

[Out]

(x*(-12*a + 2*b*x^2 - (3*a^2)/(a + b*x^2)))/(6*b^3) + (5*a^(3/2)*ArcTan[(Sqrt[b]
*x)/Sqrt[a]])/(2*b^(7/2))

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Maple [A]  time = 0.011, size = 57, normalized size = 0.9 \[{\frac{{x}^{3}}{3\,{b}^{2}}}-2\,{\frac{ax}{{b}^{3}}}-{\frac{{a}^{2}x}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{a}^{2}}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b*x^2+a)^2,x)

[Out]

1/3*x^3/b^2-2*a*x/b^3-1/2/b^3*a^2*x/(b*x^2+a)+5/2/b^3*a^2/(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^6/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.206657, size = 1, normalized size = 0.02 \[ \left [\frac{4 \, b^{2} x^{5} - 20 \, a b x^{3} - 30 \, a^{2} x + 15 \,{\left (a b x^{2} + a^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{12 \,{\left (b^{4} x^{2} + a b^{3}\right )}}, \frac{2 \, b^{2} x^{5} - 10 \, a b x^{3} - 15 \, a^{2} x + 15 \,{\left (a b x^{2} + a^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right )}{6 \,{\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

[1/12*(4*b^2*x^5 - 20*a*b*x^3 - 30*a^2*x + 15*(a*b*x^2 + a^2)*sqrt(-a/b)*log((b*
x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^4*x^2 + a*b^3), 1/6*(2*b^2*x^5 - 10
*a*b*x^3 - 15*a^2*x + 15*(a*b*x^2 + a^2)*sqrt(a/b)*arctan(x/sqrt(a/b)))/(b^4*x^2
+ a*b^3)]

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Sympy [A]  time = 1.66467, size = 107, normalized size = 1.62 \[ - \frac{a^{2} x}{2 a b^{3} + 2 b^{4} x^{2}} - \frac{2 a x}{b^{3}} - \frac{5 \sqrt{- \frac{a^{3}}{b^{7}}} \log{\left (x - \frac{b^{3} \sqrt{- \frac{a^{3}}{b^{7}}}}{a} \right )}}{4} + \frac{5 \sqrt{- \frac{a^{3}}{b^{7}}} \log{\left (x + \frac{b^{3} \sqrt{- \frac{a^{3}}{b^{7}}}}{a} \right )}}{4} + \frac{x^{3}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b*x**2+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.20972, size = 82, normalized size = 1.24 \[ \frac{5 \, a^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{3}} - \frac{a^{2} x}{2 \,{\left (b x^{2} + a\right )} b^{3}} + \frac{b^{4} x^{3} - 6 \, a b^{3} x}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

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