3.525 \(\int \frac{x^{12}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=131 \[ -\frac{11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac{33 x^7}{160 b^3 \left (a+b x^2\right )^3}-\frac{231 x^5}{640 b^4 \left (a+b x^2\right )^2}-\frac{231 x^3}{256 b^5 \left (a+b x^2\right )}-\frac{693 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 b^{13/2}}-\frac{x^{11}}{10 b \left (a+b x^2\right )^5}+\frac{693 x}{256 b^6} \]

[Out]

(693*x)/(256*b^6) - x^11/(10*b*(a + b*x^2)^5) - (11*x^9)/(80*b^2*(a + b*x^2)^4) - (33*x^7)/(160*b^3*(a + b*x^2
)^3) - (231*x^5)/(640*b^4*(a + b*x^2)^2) - (231*x^3)/(256*b^5*(a + b*x^2)) - (693*Sqrt[a]*ArcTan[(Sqrt[b]*x)/S
qrt[a]])/(256*b^(13/2))

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Rubi [A]  time = 0.0777952, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {28, 288, 321, 205} \[ -\frac{11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac{33 x^7}{160 b^3 \left (a+b x^2\right )^3}-\frac{231 x^5}{640 b^4 \left (a+b x^2\right )^2}-\frac{231 x^3}{256 b^5 \left (a+b x^2\right )}-\frac{693 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 b^{13/2}}-\frac{x^{11}}{10 b \left (a+b x^2\right )^5}+\frac{693 x}{256 b^6} \]

Antiderivative was successfully verified.

[In]

Int[x^12/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(693*x)/(256*b^6) - x^11/(10*b*(a + b*x^2)^5) - (11*x^9)/(80*b^2*(a + b*x^2)^4) - (33*x^7)/(160*b^3*(a + b*x^2
)^3) - (231*x^5)/(640*b^4*(a + b*x^2)^2) - (231*x^3)/(256*b^5*(a + b*x^2)) - (693*Sqrt[a]*ArcTan[(Sqrt[b]*x)/S
qrt[a]])/(256*b^(13/2))

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 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{x^{12}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac{x^{11}}{10 b \left (a+b x^2\right )^5}+\frac{1}{10} \left (11 b^4\right ) \int \frac{x^{10}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac{x^{11}}{10 b \left (a+b x^2\right )^5}-\frac{11 x^9}{80 b^2 \left (a+b x^2\right )^4}+\frac{1}{80} \left (99 b^2\right ) \int \frac{x^8}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{x^{11}}{10 b \left (a+b x^2\right )^5}-\frac{11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac{33 x^7}{160 b^3 \left (a+b x^2\right )^3}+\frac{231}{160} \int \frac{x^6}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{x^{11}}{10 b \left (a+b x^2\right )^5}-\frac{11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac{33 x^7}{160 b^3 \left (a+b x^2\right )^3}-\frac{231 x^5}{640 b^4 \left (a+b x^2\right )^2}+\frac{231 \int \frac{x^4}{\left (a b+b^2 x^2\right )^2} \, dx}{128 b^2}\\ &=-\frac{x^{11}}{10 b \left (a+b x^2\right )^5}-\frac{11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac{33 x^7}{160 b^3 \left (a+b x^2\right )^3}-\frac{231 x^5}{640 b^4 \left (a+b x^2\right )^2}-\frac{231 x^3}{256 b^5 \left (a+b x^2\right )}+\frac{693 \int \frac{x^2}{a b+b^2 x^2} \, dx}{256 b^4}\\ &=\frac{693 x}{256 b^6}-\frac{x^{11}}{10 b \left (a+b x^2\right )^5}-\frac{11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac{33 x^7}{160 b^3 \left (a+b x^2\right )^3}-\frac{231 x^5}{640 b^4 \left (a+b x^2\right )^2}-\frac{231 x^3}{256 b^5 \left (a+b x^2\right )}-\frac{(693 a) \int \frac{1}{a b+b^2 x^2} \, dx}{256 b^5}\\ &=\frac{693 x}{256 b^6}-\frac{x^{11}}{10 b \left (a+b x^2\right )^5}-\frac{11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac{33 x^7}{160 b^3 \left (a+b x^2\right )^3}-\frac{231 x^5}{640 b^4 \left (a+b x^2\right )^2}-\frac{231 x^3}{256 b^5 \left (a+b x^2\right )}-\frac{693 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 b^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.0509986, size = 100, normalized size = 0.76 \[ \frac{\frac{\sqrt{b} x \left (26070 a^2 b^3 x^6+29568 a^3 b^2 x^4+16170 a^4 b x^2+3465 a^5+10615 a b^4 x^8+1280 b^5 x^{10}\right )}{\left (a+b x^2\right )^5}-3465 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{1280 b^{13/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^12/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

((Sqrt[b]*x*(3465*a^5 + 16170*a^4*b*x^2 + 29568*a^3*b^2*x^4 + 26070*a^2*b^3*x^6 + 10615*a*b^4*x^8 + 1280*b^5*x
^10))/(a + b*x^2)^5 - 3465*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(1280*b^(13/2))

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Maple [A]  time = 0.056, size = 123, normalized size = 0.9 \begin{align*}{\frac{x}{{b}^{6}}}+{\frac{843\,a{x}^{9}}{256\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{1327\,{a}^{2}{x}^{7}}{128\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{131\,{a}^{3}{x}^{5}}{10\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{977\,{a}^{4}{x}^{3}}{128\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{437\,{a}^{5}x}{256\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{693\,a}{256\,{b}^{6}}\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^12/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

x/b^6+843/256/b^2*a/(b*x^2+a)^5*x^9+1327/128/b^3*a^2/(b*x^2+a)^5*x^7+131/10/b^4*a^3/(b*x^2+a)^5*x^5+977/128/b^
5*a^4/(b*x^2+a)^5*x^3+437/256/b^6*a^5/(b*x^2+a)^5*x-693/256/b^6*a/(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^12/(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 = 1.50938, size = 903, normalized size = 6.89 \begin{align*} \left [\frac{2560 \, b^{5} x^{11} + 21230 \, a b^{4} x^{9} + 52140 \, a^{2} b^{3} x^{7} + 59136 \, a^{3} b^{2} x^{5} + 32340 \, a^{4} b x^{3} + 6930 \, a^{5} x + 3465 \,{\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 )} \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^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}}, \frac{1280 \, b^{5} x^{11} + 10615 \, a b^{4} x^{9} + 26070 \, a^{2} b^{3} x^{7} + 29568 \, a^{3} b^{2} x^{5} + 16170 \, a^{4} b x^{3} + 3465 \, a^{5} x - 3465 \,{\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 )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right )}{1280 \,{\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

[1/2560*(2560*b^5*x^11 + 21230*a*b^4*x^9 + 52140*a^2*b^3*x^7 + 59136*a^3*b^2*x^5 + 32340*a^4*b*x^3 + 6930*a^5*
x + 3465*(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)*sqrt(-a/b)*log((b*x^2
- 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*
x^2 + a^5*b^6), 1/1280*(1280*b^5*x^11 + 10615*a*b^4*x^9 + 26070*a^2*b^3*x^7 + 29568*a^3*b^2*x^5 + 16170*a^4*b*
x^3 + 3465*a^5*x - 3465*(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)*sqrt(a/
b)*arctan(b*x*sqrt(a/b)/a))/(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*x^2 + a^5*
b^6)]

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Sympy [A]  time = 1.38736, size = 178, normalized size = 1.36 \begin{align*} \frac{693 \sqrt{- \frac{a}{b^{13}}} \log{\left (- b^{6} \sqrt{- \frac{a}{b^{13}}} + x \right )}}{512} - \frac{693 \sqrt{- \frac{a}{b^{13}}} \log{\left (b^{6} \sqrt{- \frac{a}{b^{13}}} + x \right )}}{512} + \frac{2185 a^{5} x + 9770 a^{4} b x^{3} + 16768 a^{3} b^{2} x^{5} + 13270 a^{2} b^{3} x^{7} + 4215 a b^{4} x^{9}}{1280 a^{5} b^{6} + 6400 a^{4} b^{7} x^{2} + 12800 a^{3} b^{8} x^{4} + 12800 a^{2} b^{9} x^{6} + 6400 a b^{10} x^{8} + 1280 b^{11} x^{10}} + \frac{x}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**12/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

693*sqrt(-a/b**13)*log(-b**6*sqrt(-a/b**13) + x)/512 - 693*sqrt(-a/b**13)*log(b**6*sqrt(-a/b**13) + x)/512 + (
2185*a**5*x + 9770*a**4*b*x**3 + 16768*a**3*b**2*x**5 + 13270*a**2*b**3*x**7 + 4215*a*b**4*x**9)/(1280*a**5*b*
*6 + 6400*a**4*b**7*x**2 + 12800*a**3*b**8*x**4 + 12800*a**2*b**9*x**6 + 6400*a*b**10*x**8 + 1280*b**11*x**10)
+ x/b**6

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Giac [A]  time = 1.14657, size = 117, normalized size = 0.89 \begin{align*} -\frac{693 \, a \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} b^{6}} + \frac{x}{b^{6}} + \frac{4215 \, a b^{4} x^{9} + 13270 \, a^{2} b^{3} x^{7} + 16768 \, a^{3} b^{2} x^{5} + 9770 \, a^{4} b x^{3} + 2185 \, a^{5} x}{1280 \,{\left (b x^{2} + a\right )}^{5} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

-693/256*a*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + x/b^6 + 1/1280*(4215*a*b^4*x^9 + 13270*a^2*b^3*x^7 + 16768*
a^3*b^2*x^5 + 9770*a^4*b*x^3 + 2185*a^5*x)/((b*x^2 + a)^5*b^6)