3.523 \(\int \frac{x^{16}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=155 \[ \frac{9009 a^2 x}{256 b^8}-\frac{9009 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 b^{17/2}}-\frac{3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac{13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}-\frac{143 x^9}{128 b^4 \left (a+b x^2\right )^2}-\frac{1287 x^7}{256 b^5 \left (a+b x^2\right )}-\frac{3003 a x^3}{256 b^7}-\frac{x^{15}}{10 b \left (a+b x^2\right )^5}+\frac{9009 x^5}{1280 b^6} \]

[Out]

(9009*a^2*x)/(256*b^8) - (3003*a*x^3)/(256*b^7) + (9009*x^5)/(1280*b^6) - x^15/(10*b*(a + b*x^2)^5) - (3*x^13)
/(16*b^2*(a + b*x^2)^4) - (13*x^11)/(32*b^3*(a + b*x^2)^3) - (143*x^9)/(128*b^4*(a + b*x^2)^2) - (1287*x^7)/(2
56*b^5*(a + b*x^2)) - (9009*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*b^(17/2))

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Rubi [A]  time = 0.105612, antiderivative size = 155, 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, Rules used = {28, 288, 302, 205} \[ \frac{9009 a^2 x}{256 b^8}-\frac{9009 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 b^{17/2}}-\frac{3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac{13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}-\frac{143 x^9}{128 b^4 \left (a+b x^2\right )^2}-\frac{1287 x^7}{256 b^5 \left (a+b x^2\right )}-\frac{3003 a x^3}{256 b^7}-\frac{x^{15}}{10 b \left (a+b x^2\right )^5}+\frac{9009 x^5}{1280 b^6} \]

Antiderivative was successfully verified.

[In]

Int[x^16/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(9009*a^2*x)/(256*b^8) - (3003*a*x^3)/(256*b^7) + (9009*x^5)/(1280*b^6) - x^15/(10*b*(a + b*x^2)^5) - (3*x^13)
/(16*b^2*(a + b*x^2)^4) - (13*x^11)/(32*b^3*(a + b*x^2)^3) - (143*x^9)/(128*b^4*(a + b*x^2)^2) - (1287*x^7)/(2
56*b^5*(a + b*x^2)) - (9009*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*b^(17/2))

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^{16}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{x^{16}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac{x^{15}}{10 b \left (a+b x^2\right )^5}+\frac{1}{2} \left (3 b^4\right ) \int \frac{x^{14}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac{x^{15}}{10 b \left (a+b x^2\right )^5}-\frac{3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}+\frac{1}{16} \left (39 b^2\right ) \int \frac{x^{12}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{x^{15}}{10 b \left (a+b x^2\right )^5}-\frac{3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac{13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}+\frac{143}{32} \int \frac{x^{10}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{x^{15}}{10 b \left (a+b x^2\right )^5}-\frac{3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac{13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}-\frac{143 x^9}{128 b^4 \left (a+b x^2\right )^2}+\frac{1287 \int \frac{x^8}{\left (a b+b^2 x^2\right )^2} \, dx}{128 b^2}\\ &=-\frac{x^{15}}{10 b \left (a+b x^2\right )^5}-\frac{3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac{13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}-\frac{143 x^9}{128 b^4 \left (a+b x^2\right )^2}-\frac{1287 x^7}{256 b^5 \left (a+b x^2\right )}+\frac{9009 \int \frac{x^6}{a b+b^2 x^2} \, dx}{256 b^4}\\ &=-\frac{x^{15}}{10 b \left (a+b x^2\right )^5}-\frac{3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac{13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}-\frac{143 x^9}{128 b^4 \left (a+b x^2\right )^2}-\frac{1287 x^7}{256 b^5 \left (a+b x^2\right )}+\frac{9009 \int \left (\frac{a^2}{b^4}-\frac{a x^2}{b^3}+\frac{x^4}{b^2}-\frac{a^3}{b^3 \left (a b+b^2 x^2\right )}\right ) \, dx}{256 b^4}\\ &=\frac{9009 a^2 x}{256 b^8}-\frac{3003 a x^3}{256 b^7}+\frac{9009 x^5}{1280 b^6}-\frac{x^{15}}{10 b \left (a+b x^2\right )^5}-\frac{3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac{13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}-\frac{143 x^9}{128 b^4 \left (a+b x^2\right )^2}-\frac{1287 x^7}{256 b^5 \left (a+b x^2\right )}-\frac{\left (9009 a^3\right ) \int \frac{1}{a b+b^2 x^2} \, dx}{256 b^7}\\ &=\frac{9009 a^2 x}{256 b^8}-\frac{3003 a x^3}{256 b^7}+\frac{9009 x^5}{1280 b^6}-\frac{x^{15}}{10 b \left (a+b x^2\right )^5}-\frac{3 x^{13}}{16 b^2 \left (a+b x^2\right )^4}-\frac{13 x^{11}}{32 b^3 \left (a+b x^2\right )^3}-\frac{143 x^9}{128 b^4 \left (a+b x^2\right )^2}-\frac{1287 x^7}{256 b^5 \left (a+b x^2\right )}-\frac{9009 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 b^{17/2}}\\ \end{align*}

Mathematica [A]  time = 0.0654545, size = 122, normalized size = 0.79 \[ \frac{\frac{\sqrt{b} x \left (16640 a^2 b^5 x^{10}+137995 a^3 b^4 x^8+338910 a^4 b^3 x^6+384384 a^5 b^2 x^4+210210 a^6 b x^2+45045 a^7-1280 a b^6 x^{12}+256 b^7 x^{14}\right )}{\left (a+b x^2\right )^5}-45045 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{1280 b^{17/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^16/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

((Sqrt[b]*x*(45045*a^7 + 210210*a^6*b*x^2 + 384384*a^5*b^2*x^4 + 338910*a^4*b^3*x^6 + 137995*a^3*b^4*x^8 + 166
40*a^2*b^5*x^10 - 1280*a*b^6*x^12 + 256*b^7*x^14))/(a + b*x^2)^5 - 45045*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/
(1280*b^(17/2))

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Maple [A]  time = 0.056, size = 148, normalized size = 1. \begin{align*}{\frac{{x}^{5}}{5\,{b}^{6}}}-2\,{\frac{a{x}^{3}}{{b}^{7}}}+21\,{\frac{{a}^{2}x}{{b}^{8}}}+{\frac{5327\,{a}^{3}{x}^{9}}{256\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{9443\,{a}^{4}{x}^{7}}{128\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{1001\,{a}^{5}{x}^{5}}{10\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{7837\,{a}^{6}{x}^{3}}{128\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{3633\,{a}^{7}x}{256\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{9009\,{a}^{3}}{256\,{b}^{8}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^16/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

1/5*x^5/b^6-2*a*x^3/b^7+21*a^2*x/b^8+5327/256/b^4*a^3/(b*x^2+a)^5*x^9+9443/128/b^5*a^4/(b*x^2+a)^5*x^7+1001/10
/b^6*a^5/(b*x^2+a)^5*x^5+7837/128/b^7*a^6/(b*x^2+a)^5*x^3+3633/256/b^8*a^7/(b*x^2+a)^5*x-9009/256/b^8*a^3/(a*b
)^(1/2)*arctan(b*x/(a*b)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^16/(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 = 1.55159, size = 1049, normalized size = 6.77 \begin{align*} \left [\frac{512 \, b^{7} x^{15} - 2560 \, a b^{6} x^{13} + 33280 \, a^{2} b^{5} x^{11} + 275990 \, a^{3} b^{4} x^{9} + 677820 \, a^{4} b^{3} x^{7} + 768768 \, a^{5} b^{2} x^{5} + 420420 \, a^{6} b x^{3} + 90090 \, a^{7} x + 45045 \,{\left (a^{2} b^{5} x^{10} + 5 \, a^{3} b^{4} x^{8} + 10 \, a^{4} b^{3} x^{6} + 10 \, a^{5} b^{2} x^{4} + 5 \, a^{6} b x^{2} + a^{7}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{2560 \,{\left (b^{13} x^{10} + 5 \, a b^{12} x^{8} + 10 \, a^{2} b^{11} x^{6} + 10 \, a^{3} b^{10} x^{4} + 5 \, a^{4} b^{9} x^{2} + a^{5} b^{8}\right )}}, \frac{256 \, b^{7} x^{15} - 1280 \, a b^{6} x^{13} + 16640 \, a^{2} b^{5} x^{11} + 137995 \, a^{3} b^{4} x^{9} + 338910 \, a^{4} b^{3} x^{7} + 384384 \, a^{5} b^{2} x^{5} + 210210 \, a^{6} b x^{3} + 45045 \, a^{7} x - 45045 \,{\left (a^{2} b^{5} x^{10} + 5 \, a^{3} b^{4} x^{8} + 10 \, a^{4} b^{3} x^{6} + 10 \, a^{5} b^{2} x^{4} + 5 \, a^{6} b x^{2} + a^{7}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right )}{1280 \,{\left (b^{13} x^{10} + 5 \, a b^{12} x^{8} + 10 \, a^{2} b^{11} x^{6} + 10 \, a^{3} b^{10} x^{4} + 5 \, a^{4} b^{9} x^{2} + a^{5} b^{8}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^16/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

[1/2560*(512*b^7*x^15 - 2560*a*b^6*x^13 + 33280*a^2*b^5*x^11 + 275990*a^3*b^4*x^9 + 677820*a^4*b^3*x^7 + 76876
8*a^5*b^2*x^5 + 420420*a^6*b*x^3 + 90090*a^7*x + 45045*(a^2*b^5*x^10 + 5*a^3*b^4*x^8 + 10*a^4*b^3*x^6 + 10*a^5
*b^2*x^4 + 5*a^6*b*x^2 + a^7)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^13*x^10 + 5*a*b^1
2*x^8 + 10*a^2*b^11*x^6 + 10*a^3*b^10*x^4 + 5*a^4*b^9*x^2 + a^5*b^8), 1/1280*(256*b^7*x^15 - 1280*a*b^6*x^13 +
16640*a^2*b^5*x^11 + 137995*a^3*b^4*x^9 + 338910*a^4*b^3*x^7 + 384384*a^5*b^2*x^5 + 210210*a^6*b*x^3 + 45045*
a^7*x - 45045*(a^2*b^5*x^10 + 5*a^3*b^4*x^8 + 10*a^4*b^3*x^6 + 10*a^5*b^2*x^4 + 5*a^6*b*x^2 + a^7)*sqrt(a/b)*a
rctan(b*x*sqrt(a/b)/a))/(b^13*x^10 + 5*a*b^12*x^8 + 10*a^2*b^11*x^6 + 10*a^3*b^10*x^4 + 5*a^4*b^9*x^2 + a^5*b^
8)]

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Sympy [A]  time = 1.53132, size = 218, normalized size = 1.41 \begin{align*} \frac{21 a^{2} x}{b^{8}} - \frac{2 a x^{3}}{b^{7}} + \frac{9009 \sqrt{- \frac{a^{5}}{b^{17}}} \log{\left (x - \frac{b^{8} \sqrt{- \frac{a^{5}}{b^{17}}}}{a^{2}} \right )}}{512} - \frac{9009 \sqrt{- \frac{a^{5}}{b^{17}}} \log{\left (x + \frac{b^{8} \sqrt{- \frac{a^{5}}{b^{17}}}}{a^{2}} \right )}}{512} + \frac{18165 a^{7} x + 78370 a^{6} b x^{3} + 128128 a^{5} b^{2} x^{5} + 94430 a^{4} b^{3} x^{7} + 26635 a^{3} b^{4} x^{9}}{1280 a^{5} b^{8} + 6400 a^{4} b^{9} x^{2} + 12800 a^{3} b^{10} x^{4} + 12800 a^{2} b^{11} x^{6} + 6400 a b^{12} x^{8} + 1280 b^{13} x^{10}} + \frac{x^{5}}{5 b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**16/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

21*a**2*x/b**8 - 2*a*x**3/b**7 + 9009*sqrt(-a**5/b**17)*log(x - b**8*sqrt(-a**5/b**17)/a**2)/512 - 9009*sqrt(-
a**5/b**17)*log(x + b**8*sqrt(-a**5/b**17)/a**2)/512 + (18165*a**7*x + 78370*a**6*b*x**3 + 128128*a**5*b**2*x*
*5 + 94430*a**4*b**3*x**7 + 26635*a**3*b**4*x**9)/(1280*a**5*b**8 + 6400*a**4*b**9*x**2 + 12800*a**3*b**10*x**
4 + 12800*a**2*b**11*x**6 + 6400*a*b**12*x**8 + 1280*b**13*x**10) + x**5/(5*b**6)

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Giac [A]  time = 1.13854, size = 158, normalized size = 1.02 \begin{align*} -\frac{9009 \, a^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} b^{8}} + \frac{26635 \, a^{3} b^{4} x^{9} + 94430 \, a^{4} b^{3} x^{7} + 128128 \, a^{5} b^{2} x^{5} + 78370 \, a^{6} b x^{3} + 18165 \, a^{7} x}{1280 \,{\left (b x^{2} + a\right )}^{5} b^{8}} + \frac{b^{24} x^{5} - 10 \, a b^{23} x^{3} + 105 \, a^{2} b^{22} x}{5 \, b^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^16/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

-9009/256*a^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^8) + 1/1280*(26635*a^3*b^4*x^9 + 94430*a^4*b^3*x^7 + 128128*a
^5*b^2*x^5 + 78370*a^6*b*x^3 + 18165*a^7*x)/((b*x^2 + a)^5*b^8) + 1/5*(b^24*x^5 - 10*a*b^23*x^3 + 105*a^2*b^22
*x)/b^30