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

Optimal. Leaf size=125 \[ \frac{7 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{9/2} b^{3/2}}+\frac{7 x}{256 a^4 b \left (a+b x^2\right )}+\frac{7 x}{384 a^3 b \left (a+b x^2\right )^2}+\frac{7 x}{480 a^2 b \left (a+b x^2\right )^3}+\frac{x}{80 a b \left (a+b x^2\right )^4}-\frac{x}{10 b \left (a+b x^2\right )^5} \]

[Out]

-x/(10*b*(a + b*x^2)^5) + x/(80*a*b*(a + b*x^2)^4) + (7*x)/(480*a^2*b*(a + b*x^2
)^3) + (7*x)/(384*a^3*b*(a + b*x^2)^2) + (7*x)/(256*a^4*b*(a + b*x^2)) + (7*ArcT
an[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(9/2)*b^(3/2))

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Rubi [A]  time = 0.167364, antiderivative size = 125, 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 \[ \frac{7 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{9/2} b^{3/2}}+\frac{7 x}{256 a^4 b \left (a+b x^2\right )}+\frac{7 x}{384 a^3 b \left (a+b x^2\right )^2}+\frac{7 x}{480 a^2 b \left (a+b x^2\right )^3}+\frac{x}{80 a b \left (a+b x^2\right )^4}-\frac{x}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

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

[Out]

-x/(10*b*(a + b*x^2)^5) + x/(80*a*b*(a + b*x^2)^4) + (7*x)/(480*a^2*b*(a + b*x^2
)^3) + (7*x)/(384*a^3*b*(a + b*x^2)^2) + (7*x)/(256*a^4*b*(a + b*x^2)) + (7*ArcT
an[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(9/2)*b^(3/2))

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Rubi in Sympy [A]  time = 31.4254, size = 109, normalized size = 0.87 \[ - \frac{x}{10 b \left (a + b x^{2}\right )^{5}} + \frac{x}{80 a b \left (a + b x^{2}\right )^{4}} + \frac{7 x}{480 a^{2} b \left (a + b x^{2}\right )^{3}} + \frac{7 x}{384 a^{3} b \left (a + b x^{2}\right )^{2}} + \frac{7 x}{256 a^{4} b \left (a + b x^{2}\right )} + \frac{7 \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 a^{\frac{9}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(10*b*(a + b*x**2)**5) + x/(80*a*b*(a + b*x**2)**4) + 7*x/(480*a**2*b*(a + b*
x**2)**3) + 7*x/(384*a**3*b*(a + b*x**2)**2) + 7*x/(256*a**4*b*(a + b*x**2)) + 7
*atan(sqrt(b)*x/sqrt(a))/(256*a**(9/2)*b**(3/2))

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Mathematica [A]  time = 0.0983973, size = 91, normalized size = 0.73 \[ \frac{7 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{9/2} b^{3/2}}+\frac{-105 a^4 x+790 a^3 b x^3+896 a^2 b^2 x^5+490 a b^3 x^7+105 b^4 x^9}{3840 a^4 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

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

[Out]

(-105*a^4*x + 790*a^3*b*x^3 + 896*a^2*b^2*x^5 + 490*a*b^3*x^7 + 105*b^4*x^9)/(38
40*a^4*b*(a + b*x^2)^5) + (7*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(9/2)*b^(3/2))

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Maple [A]  time = 0.015, size = 80, normalized size = 0.6 \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{5}} \left ({\frac{7\,{b}^{3}{x}^{9}}{256\,{a}^{4}}}+{\frac{49\,{b}^{2}{x}^{7}}{384\,{a}^{3}}}+{\frac{7\,b{x}^{5}}{30\,{a}^{2}}}+{\frac{79\,{x}^{3}}{384\,a}}-{\frac{7\,x}{256\,b}} \right ) }+{\frac{7}{256\,{a}^{4}b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(7/256*b^3/a^4*x^9+49/384*b^2/a^3*x^7+7/30*b/a^2*x^5+79/384/a*x^3-7/256/b*x)/(b*
x^2+a)^5+7/256/a^4/b/(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^2/(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 = 0.266787, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (105 \, b^{4} x^{9} + 490 \, a b^{3} x^{7} + 896 \, a^{2} b^{2} x^{5} + 790 \, a^{3} b x^{3} - 105 \, a^{4} x\right )} \sqrt{-a b}}{7680 \,{\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \sqrt{-a b}}, \frac{105 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (105 \, b^{4} x^{9} + 490 \, a b^{3} x^{7} + 896 \, a^{2} b^{2} x^{5} + 790 \, a^{3} b x^{3} - 105 \, a^{4} x\right )} \sqrt{a b}}{3840 \,{\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(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
+ 490*a*b^3*x^7 + 896*a^2*b^2*x^5 + 790*a^3*b*x^3 - 105*a^4*x)*sqrt(-a*b))/((a^
4*b^6*x^10 + 5*a^5*b^5*x^8 + 10*a^6*b^4*x^6 + 10*a^7*b^3*x^4 + 5*a^8*b^2*x^2 + a
^9*b)*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 + 490*a*b^3*x
^7 + 896*a^2*b^2*x^5 + 790*a^3*b*x^3 - 105*a^4*x)*sqrt(a*b))/((a^4*b^6*x^10 + 5*
a^5*b^5*x^8 + 10*a^6*b^4*x^6 + 10*a^7*b^3*x^4 + 5*a^8*b^2*x^2 + a^9*b)*sqrt(a*b)
)]

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Sympy [A]  time = 4.06985, size = 190, normalized size = 1.52 \[ - \frac{7 \sqrt{- \frac{1}{a^{9} b^{3}}} \log{\left (- a^{5} b \sqrt{- \frac{1}{a^{9} b^{3}}} + x \right )}}{512} + \frac{7 \sqrt{- \frac{1}{a^{9} b^{3}}} \log{\left (a^{5} b \sqrt{- \frac{1}{a^{9} b^{3}}} + x \right )}}{512} + \frac{- 105 a^{4} x + 790 a^{3} b x^{3} + 896 a^{2} b^{2} x^{5} + 490 a b^{3} x^{7} + 105 b^{4} x^{9}}{3840 a^{9} b + 19200 a^{8} b^{2} x^{2} + 38400 a^{7} b^{3} x^{4} + 38400 a^{6} b^{4} x^{6} + 19200 a^{5} b^{5} x^{8} + 3840 a^{4} b^{6} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*sqrt(-1/(a**9*b**3))*log(-a**5*b*sqrt(-1/(a**9*b**3)) + x)/512 + 7*sqrt(-1/(a
**9*b**3))*log(a**5*b*sqrt(-1/(a**9*b**3)) + x)/512 + (-105*a**4*x + 790*a**3*b*
x**3 + 896*a**2*b**2*x**5 + 490*a*b**3*x**7 + 105*b**4*x**9)/(3840*a**9*b + 1920
0*a**8*b**2*x**2 + 38400*a**7*b**3*x**4 + 38400*a**6*b**4*x**6 + 19200*a**5*b**5
*x**8 + 3840*a**4*b**6*x**10)

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GIAC/XCAS [A]  time = 0.270129, size = 113, normalized size = 0.9 \[ \frac{7 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} a^{4} b} + \frac{105 \, b^{4} x^{9} + 490 \, a b^{3} x^{7} + 896 \, a^{2} b^{2} x^{5} + 790 \, a^{3} b x^{3} - 105 \, a^{4} x}{3840 \,{\left (b x^{2} + a\right )}^{5} a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(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^4*b) + 1/3840*(105*b^4*x^9 + 490*a*b^3*
x^7 + 896*a^2*b^2*x^5 + 790*a^3*b*x^3 - 105*a^4*x)/((b*x^2 + a)^5*a^4*b)