3.503 \(\int \frac{x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\)

Optimal. Leaf size=104 \[ \frac{105 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 b^{11/2}}-\frac{105 a x}{16 b^5}-\frac{21 x^5}{16 b^3 \left (a+b x^2\right )}-\frac{3 x^7}{8 b^2 \left (a+b x^2\right )^2}-\frac{x^9}{6 b \left (a+b x^2\right )^3}+\frac{35 x^3}{16 b^4} \]

[Out]

(-105*a*x)/(16*b^5) + (35*x^3)/(16*b^4) - x^9/(6*b*(a + b*x^2)^3) - (3*x^7)/(8*b
^2*(a + b*x^2)^2) - (21*x^5)/(16*b^3*(a + b*x^2)) + (105*a^(3/2)*ArcTan[(Sqrt[b]
*x)/Sqrt[a]])/(16*b^(11/2))

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Rubi [A]  time = 0.158399, antiderivative size = 104, 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 \[ \frac{105 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 b^{11/2}}-\frac{105 a x}{16 b^5}-\frac{21 x^5}{16 b^3 \left (a+b x^2\right )}-\frac{3 x^7}{8 b^2 \left (a+b x^2\right )^2}-\frac{x^9}{6 b \left (a+b x^2\right )^3}+\frac{35 x^3}{16 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^10/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

(-105*a*x)/(16*b^5) + (35*x^3)/(16*b^4) - x^9/(6*b*(a + b*x^2)^3) - (3*x^7)/(8*b
^2*(a + b*x^2)^2) - (21*x^5)/(16*b^3*(a + b*x^2)) + (105*a^(3/2)*ArcTan[(Sqrt[b]
*x)/Sqrt[a]])/(16*b^(11/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{105 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{11}{2}}} - \frac{x^{9}}{6 b \left (a + b x^{2}\right )^{3}} - \frac{3 x^{7}}{8 b^{2} \left (a + b x^{2}\right )^{2}} - \frac{21 x^{5}}{16 b^{3} \left (a + b x^{2}\right )} + \frac{35 x^{3}}{16 b^{4}} - \frac{105 \int a\, dx}{16 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

105*a**(3/2)*atan(sqrt(b)*x/sqrt(a))/(16*b**(11/2)) - x**9/(6*b*(a + b*x**2)**3)
- 3*x**7/(8*b**2*(a + b*x**2)**2) - 21*x**5/(16*b**3*(a + b*x**2)) + 35*x**3/(1
6*b**4) - 105*Integral(a, x)/(16*b**5)

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Mathematica [A]  time = 0.0832548, size = 89, normalized size = 0.86 \[ \frac{315 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\frac{\sqrt{b} x \left (-315 a^4-840 a^3 b x^2-693 a^2 b^2 x^4-144 a b^3 x^6+16 b^4 x^8\right )}{\left (a+b x^2\right )^3}}{48 b^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^10/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

((Sqrt[b]*x*(-315*a^4 - 840*a^3*b*x^2 - 693*a^2*b^2*x^4 - 144*a*b^3*x^6 + 16*b^4
*x^8))/(a + b*x^2)^3 + 315*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(48*b^(11/2))

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Maple [A]  time = 0.015, size = 97, normalized size = 0.9 \[{\frac{{x}^{3}}{3\,{b}^{4}}}-4\,{\frac{ax}{{b}^{5}}}-{\frac{55\,{a}^{2}{x}^{5}}{16\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{35\,{a}^{3}{x}^{3}}{6\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{41\,{a}^{4}x}{16\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{105\,{a}^{2}}{16\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10/(b^2*x^4+2*a*b*x^2+a^2)^2,x)

[Out]

1/3*x^3/b^4-4*a*x/b^5-55/16/b^3*a^2/(b*x^2+a)^3*x^5-35/6/b^4*a^3/(b*x^2+a)^3*x^3
-41/16/b^5*a^4/(b*x^2+a)^3*x+105/16/b^5*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^10/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270931, size = 1, normalized size = 0.01 \[ \left [\frac{32 \, b^{4} x^{9} - 288 \, a b^{3} x^{7} - 1386 \, a^{2} b^{2} x^{5} - 1680 \, a^{3} b x^{3} - 630 \, a^{4} x + 315 \,{\left (a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + 3 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{96 \,{\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}}, \frac{16 \, b^{4} x^{9} - 144 \, a b^{3} x^{7} - 693 \, a^{2} b^{2} x^{5} - 840 \, a^{3} b x^{3} - 315 \, a^{4} x + 315 \,{\left (a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + 3 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right )}{48 \,{\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="fricas")

[Out]

[1/96*(32*b^4*x^9 - 288*a*b^3*x^7 - 1386*a^2*b^2*x^5 - 1680*a^3*b*x^3 - 630*a^4*
x + 315*(a*b^3*x^6 + 3*a^2*b^2*x^4 + 3*a^3*b*x^2 + a^4)*sqrt(-a/b)*log((b*x^2 +
2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 + a^3
*b^5), 1/48*(16*b^4*x^9 - 144*a*b^3*x^7 - 693*a^2*b^2*x^5 - 840*a^3*b*x^3 - 315*
a^4*x + 315*(a*b^3*x^6 + 3*a^2*b^2*x^4 + 3*a^3*b*x^2 + a^4)*sqrt(a/b)*arctan(x/s
qrt(a/b)))/(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 + a^3*b^5)]

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Sympy [A]  time = 2.78368, size = 155, normalized size = 1.49 \[ - \frac{4 a x}{b^{5}} - \frac{105 \sqrt{- \frac{a^{3}}{b^{11}}} \log{\left (x - \frac{b^{5} \sqrt{- \frac{a^{3}}{b^{11}}}}{a} \right )}}{32} + \frac{105 \sqrt{- \frac{a^{3}}{b^{11}}} \log{\left (x + \frac{b^{5} \sqrt{- \frac{a^{3}}{b^{11}}}}{a} \right )}}{32} - \frac{123 a^{4} x + 280 a^{3} b x^{3} + 165 a^{2} b^{2} x^{5}}{48 a^{3} b^{5} + 144 a^{2} b^{6} x^{2} + 144 a b^{7} x^{4} + 48 b^{8} x^{6}} + \frac{x^{3}}{3 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

-4*a*x/b**5 - 105*sqrt(-a**3/b**11)*log(x - b**5*sqrt(-a**3/b**11)/a)/32 + 105*s
qrt(-a**3/b**11)*log(x + b**5*sqrt(-a**3/b**11)/a)/32 - (123*a**4*x + 280*a**3*b
*x**3 + 165*a**2*b**2*x**5)/(48*a**3*b**5 + 144*a**2*b**6*x**2 + 144*a*b**7*x**4
+ 48*b**8*x**6) + x**3/(3*b**4)

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GIAC/XCAS [A]  time = 0.269506, size = 113, normalized size = 1.09 \[ \frac{105 \, a^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{16 \, \sqrt{a b} b^{5}} - \frac{165 \, a^{2} b^{2} x^{5} + 280 \, a^{3} b x^{3} + 123 \, a^{4} x}{48 \,{\left (b x^{2} + a\right )}^{3} b^{5}} + \frac{b^{8} x^{3} - 12 \, a b^{7} x}{3 \, b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="giac")

[Out]

105/16*a^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) - 1/48*(165*a^2*b^2*x^5 + 280*a
^3*b*x^3 + 123*a^4*x)/((b*x^2 + a)^3*b^5) + 1/3*(b^8*x^3 - 12*a*b^7*x)/b^12