3.524 \(\int \frac{x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=142 \[ \frac{3003 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 b^{15/2}}-\frac{3003 a x}{256 b^7}-\frac{3003 x^5}{1280 b^5 \left (a+b x^2\right )}-\frac{429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac{143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac{13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac{x^{13}}{10 b \left (a+b x^2\right )^5}+\frac{1001 x^3}{256 b^6} \]

[Out]

(-3003*a*x)/(256*b^7) + (1001*x^3)/(256*b^6) - x^13/(10*b*(a + b*x^2)^5) - (13*x
^11)/(80*b^2*(a + b*x^2)^4) - (143*x^9)/(480*b^3*(a + b*x^2)^3) - (429*x^7)/(640
*b^4*(a + b*x^2)^2) - (3003*x^5)/(1280*b^5*(a + b*x^2)) + (3003*a^(3/2)*ArcTan[(
Sqrt[b]*x)/Sqrt[a]])/(256*b^(15/2))

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Rubi [A]  time = 0.242471, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{3003 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 b^{15/2}}-\frac{3003 a x}{256 b^7}-\frac{3003 x^5}{1280 b^5 \left (a+b x^2\right )}-\frac{429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac{143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac{13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac{x^{13}}{10 b \left (a+b x^2\right )^5}+\frac{1001 x^3}{256 b^6} \]

Antiderivative was successfully verified.

[In]  Int[x^14/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(-3003*a*x)/(256*b^7) + (1001*x^3)/(256*b^6) - x^13/(10*b*(a + b*x^2)^5) - (13*x
^11)/(80*b^2*(a + b*x^2)^4) - (143*x^9)/(480*b^3*(a + b*x^2)^3) - (429*x^7)/(640
*b^4*(a + b*x^2)^2) - (3003*x^5)/(1280*b^5*(a + b*x^2)) + (3003*a^(3/2)*ArcTan[(
Sqrt[b]*x)/Sqrt[a]])/(256*b^(15/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3003 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 b^{\frac{15}{2}}} - \frac{x^{13}}{10 b \left (a + b x^{2}\right )^{5}} - \frac{13 x^{11}}{80 b^{2} \left (a + b x^{2}\right )^{4}} - \frac{143 x^{9}}{480 b^{3} \left (a + b x^{2}\right )^{3}} - \frac{429 x^{7}}{640 b^{4} \left (a + b x^{2}\right )^{2}} - \frac{3003 x^{5}}{1280 b^{5} \left (a + b x^{2}\right )} + \frac{1001 x^{3}}{256 b^{6}} - \frac{3003 \int a\, dx}{256 b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

3003*a**(3/2)*atan(sqrt(b)*x/sqrt(a))/(256*b**(15/2)) - x**13/(10*b*(a + b*x**2)
**5) - 13*x**11/(80*b**2*(a + b*x**2)**4) - 143*x**9/(480*b**3*(a + b*x**2)**3)
- 429*x**7/(640*b**4*(a + b*x**2)**2) - 3003*x**5/(1280*b**5*(a + b*x**2)) + 100
1*x**3/(256*b**6) - 3003*Integral(a, x)/(256*b**7)

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Mathematica [A]  time = 0.104396, size = 111, normalized size = 0.78 \[ \frac{45045 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\frac{\sqrt{b} x \left (-45045 a^6-210210 a^5 b x^2-384384 a^4 b^2 x^4-338910 a^3 b^3 x^6-137995 a^2 b^4 x^8-16640 a b^5 x^{10}+1280 b^6 x^{12}\right )}{\left (a+b x^2\right )^5}}{3840 b^{15/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^14/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

((Sqrt[b]*x*(-45045*a^6 - 210210*a^5*b*x^2 - 384384*a^4*b^2*x^4 - 338910*a^3*b^3
*x^6 - 137995*a^2*b^4*x^8 - 16640*a*b^5*x^10 + 1280*b^6*x^12))/(a + b*x^2)^5 + 4
5045*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3840*b^(15/2))

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Maple [A]  time = 0.019, size = 137, normalized size = 1. \[{\frac{{x}^{3}}{3\,{b}^{6}}}-6\,{\frac{ax}{{b}^{7}}}-{\frac{2373\,{a}^{2}{x}^{9}}{256\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{12131\,{a}^{3}{x}^{7}}{384\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{1253\,{a}^{4}{x}^{5}}{30\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{9629\,{a}^{5}{x}^{3}}{384\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{1467\,{a}^{6}x}{256\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{3003\,{a}^{2}}{256\,{b}^{7}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

1/3*x^3/b^6-6*a*x/b^7-2373/256/b^3*a^2/(b*x^2+a)^5*x^9-12131/384/b^4*a^3/(b*x^2+
a)^5*x^7-1253/30/b^5*a^4/(b*x^2+a)^5*x^5-9629/384/b^6*a^5/(b*x^2+a)^5*x^3-1467/2
56/b^7*a^6/(b*x^2+a)^5*x+3003/256/b^7*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^14/(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 = 0.269229, size = 1, normalized size = 0.01 \[ \left [\frac{2560 \, b^{6} x^{13} - 33280 \, a b^{5} x^{11} - 275990 \, a^{2} b^{4} x^{9} - 677820 \, a^{3} b^{3} x^{7} - 768768 \, a^{4} b^{2} x^{5} - 420420 \, a^{5} b x^{3} - 90090 \, a^{6} x + 45045 \,{\left (a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 10 \, a^{4} b^{2} x^{4} + 5 \, a^{5} b x^{2} + a^{6}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{7680 \,{\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}}, \frac{1280 \, b^{6} x^{13} - 16640 \, a b^{5} x^{11} - 137995 \, a^{2} b^{4} x^{9} - 338910 \, a^{3} b^{3} x^{7} - 384384 \, a^{4} b^{2} x^{5} - 210210 \, a^{5} b x^{3} - 45045 \, a^{6} x + 45045 \,{\left (a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 10 \, a^{4} b^{2} x^{4} + 5 \, a^{5} b x^{2} + a^{6}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right )}{3840 \,{\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="fricas")

[Out]

[1/7680*(2560*b^6*x^13 - 33280*a*b^5*x^11 - 275990*a^2*b^4*x^9 - 677820*a^3*b^3*
x^7 - 768768*a^4*b^2*x^5 - 420420*a^5*b*x^3 - 90090*a^6*x + 45045*(a*b^5*x^10 +
5*a^2*b^4*x^8 + 10*a^3*b^3*x^6 + 10*a^4*b^2*x^4 + 5*a^5*b*x^2 + a^6)*sqrt(-a/b)*
log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^12*x^10 + 5*a*b^11*x^8 + 10*
a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5*b^7), 1/3840*(1280*b^6*x^13
- 16640*a*b^5*x^11 - 137995*a^2*b^4*x^9 - 338910*a^3*b^3*x^7 - 384384*a^4*b^2*x^
5 - 210210*a^5*b*x^3 - 45045*a^6*x + 45045*(a*b^5*x^10 + 5*a^2*b^4*x^8 + 10*a^3*
b^3*x^6 + 10*a^4*b^2*x^4 + 5*a^5*b*x^2 + a^6)*sqrt(a/b)*arctan(x/sqrt(a/b)))/(b^
12*x^10 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5*
b^7)]

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Sympy [A]  time = 5.28372, size = 202, normalized size = 1.42 \[ - \frac{6 a x}{b^{7}} - \frac{3003 \sqrt{- \frac{a^{3}}{b^{15}}} \log{\left (x - \frac{b^{7} \sqrt{- \frac{a^{3}}{b^{15}}}}{a} \right )}}{512} + \frac{3003 \sqrt{- \frac{a^{3}}{b^{15}}} \log{\left (x + \frac{b^{7} \sqrt{- \frac{a^{3}}{b^{15}}}}{a} \right )}}{512} - \frac{22005 a^{6} x + 96290 a^{5} b x^{3} + 160384 a^{4} b^{2} x^{5} + 121310 a^{3} b^{3} x^{7} + 35595 a^{2} b^{4} x^{9}}{3840 a^{5} b^{7} + 19200 a^{4} b^{8} x^{2} + 38400 a^{3} b^{9} x^{4} + 38400 a^{2} b^{10} x^{6} + 19200 a b^{11} x^{8} + 3840 b^{12} x^{10}} + \frac{x^{3}}{3 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

-6*a*x/b**7 - 3003*sqrt(-a**3/b**15)*log(x - b**7*sqrt(-a**3/b**15)/a)/512 + 300
3*sqrt(-a**3/b**15)*log(x + b**7*sqrt(-a**3/b**15)/a)/512 - (22005*a**6*x + 9629
0*a**5*b*x**3 + 160384*a**4*b**2*x**5 + 121310*a**3*b**3*x**7 + 35595*a**2*b**4*
x**9)/(3840*a**5*b**7 + 19200*a**4*b**8*x**2 + 38400*a**3*b**9*x**4 + 38400*a**2
*b**10*x**6 + 19200*a*b**11*x**8 + 3840*b**12*x**10) + x**3/(3*b**6)

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GIAC/XCAS [A]  time = 0.271406, size = 143, normalized size = 1.01 \[ \frac{3003 \, a^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} b^{7}} - \frac{35595 \, a^{2} b^{4} x^{9} + 121310 \, a^{3} b^{3} x^{7} + 160384 \, a^{4} b^{2} x^{5} + 96290 \, a^{5} b x^{3} + 22005 \, a^{6} x}{3840 \,{\left (b x^{2} + a\right )}^{5} b^{7}} + \frac{b^{12} x^{3} - 18 \, a b^{11} x}{3 \, b^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="giac")

[Out]

3003/256*a^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^7) - 1/3840*(35595*a^2*b^4*x^9 +
121310*a^3*b^3*x^7 + 160384*a^4*b^2*x^5 + 96290*a^5*b*x^3 + 22005*a^6*x)/((b*x^
2 + a)^5*b^7) + 1/3*(b^12*x^3 - 18*a*b^11*x)/b^18