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

Optimal. Leaf size=117 \[ \frac{231 a^2 x}{16 b^6}-\frac{231 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 b^{13/2}}-\frac{11 x^9}{24 b^2 \left (a+b x^2\right )^2}-\frac{33 x^7}{16 b^3 \left (a+b x^2\right )}-\frac{77 a x^3}{16 b^5}-\frac{x^{11}}{6 b \left (a+b x^2\right )^3}+\frac{231 x^5}{80 b^4} \]

[Out]

(231*a^2*x)/(16*b^6) - (77*a*x^3)/(16*b^5) + (231*x^5)/(80*b^4) - x^11/(6*b*(a + b*x^2)^3) - (11*x^9)/(24*b^2*
(a + b*x^2)^2) - (33*x^7)/(16*b^3*(a + b*x^2)) - (231*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(16*b^(13/2))

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Rubi [A]  time = 0.072478, antiderivative size = 117, 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, Rules used = {28, 288, 302, 205} \[ \frac{231 a^2 x}{16 b^6}-\frac{231 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 b^{13/2}}-\frac{11 x^9}{24 b^2 \left (a+b x^2\right )^2}-\frac{33 x^7}{16 b^3 \left (a+b x^2\right )}-\frac{77 a x^3}{16 b^5}-\frac{x^{11}}{6 b \left (a+b x^2\right )^3}+\frac{231 x^5}{80 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(231*a^2*x)/(16*b^6) - (77*a*x^3)/(16*b^5) + (231*x^5)/(80*b^4) - x^11/(6*b*(a + b*x^2)^3) - (11*x^9)/(24*b^2*
(a + b*x^2)^2) - (33*x^7)/(16*b^3*(a + b*x^2)) - (231*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(16*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 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^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{x^{12}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{x^{11}}{6 b \left (a+b x^2\right )^3}+\frac{1}{6} \left (11 b^2\right ) \int \frac{x^{10}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{x^{11}}{6 b \left (a+b x^2\right )^3}-\frac{11 x^9}{24 b^2 \left (a+b x^2\right )^2}+\frac{33}{8} \int \frac{x^8}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac{x^{11}}{6 b \left (a+b x^2\right )^3}-\frac{11 x^9}{24 b^2 \left (a+b x^2\right )^2}-\frac{33 x^7}{16 b^3 \left (a+b x^2\right )}+\frac{231 \int \frac{x^6}{a b+b^2 x^2} \, dx}{16 b^2}\\ &=-\frac{x^{11}}{6 b \left (a+b x^2\right )^3}-\frac{11 x^9}{24 b^2 \left (a+b x^2\right )^2}-\frac{33 x^7}{16 b^3 \left (a+b x^2\right )}+\frac{231 \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}{16 b^2}\\ &=\frac{231 a^2 x}{16 b^6}-\frac{77 a x^3}{16 b^5}+\frac{231 x^5}{80 b^4}-\frac{x^{11}}{6 b \left (a+b x^2\right )^3}-\frac{11 x^9}{24 b^2 \left (a+b x^2\right )^2}-\frac{33 x^7}{16 b^3 \left (a+b x^2\right )}-\frac{\left (231 a^3\right ) \int \frac{1}{a b+b^2 x^2} \, dx}{16 b^5}\\ &=\frac{231 a^2 x}{16 b^6}-\frac{77 a x^3}{16 b^5}+\frac{231 x^5}{80 b^4}-\frac{x^{11}}{6 b \left (a+b x^2\right )^3}-\frac{11 x^9}{24 b^2 \left (a+b x^2\right )^2}-\frac{33 x^7}{16 b^3 \left (a+b x^2\right )}-\frac{231 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 b^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.0568608, size = 99, normalized size = 0.85 \[ \frac{1584 a^2 b^3 x^7+7623 a^3 b^2 x^5+9240 a^4 b x^3+3465 a^5 x-176 a b^4 x^9+48 b^5 x^{11}}{240 b^6 \left (a+b x^2\right )^3}-\frac{231 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 b^{13/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3465*a^5*x + 9240*a^4*b*x^3 + 7623*a^3*b^2*x^5 + 1584*a^2*b^3*x^7 - 176*a*b^4*x^9 + 48*b^5*x^11)/(240*b^6*(a
+ b*x^2)^3) - (231*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(16*b^(13/2))

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Maple [A]  time = 0.053, size = 108, normalized size = 0.9 \begin{align*}{\frac{{x}^{5}}{5\,{b}^{4}}}-{\frac{4\,a{x}^{3}}{3\,{b}^{5}}}+10\,{\frac{{a}^{2}x}{{b}^{6}}}+{\frac{89\,{a}^{3}{x}^{5}}{16\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{59\,{a}^{4}{x}^{3}}{6\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{71\,{a}^{5}x}{16\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{231\,{a}^{3}}{16\,{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)^2,x)

[Out]

1/5*x^5/b^4-4/3*a*x^3/b^5+10*a^2*x/b^6+89/16/b^4*a^3/(b*x^2+a)^3*x^5+59/6/b^5*a^4/(b*x^2+a)^3*x^3+71/16/b^6*a^
5/(b*x^2+a)^3*x-231/16/b^6*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^12/(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 = 1.73812, size = 706, normalized size = 6.03 \begin{align*} \left [\frac{96 \, b^{5} x^{11} - 352 \, a b^{4} x^{9} + 3168 \, a^{2} b^{3} x^{7} + 15246 \, a^{3} b^{2} x^{5} + 18480 \, a^{4} b x^{3} + 6930 \, a^{5} x + 3465 \,{\left (a^{2} b^{3} x^{6} + 3 \, a^{3} b^{2} x^{4} + 3 \, 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 )}{480 \,{\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}}, \frac{48 \, b^{5} x^{11} - 176 \, a b^{4} x^{9} + 1584 \, a^{2} b^{3} x^{7} + 7623 \, a^{3} b^{2} x^{5} + 9240 \, a^{4} b x^{3} + 3465 \, a^{5} x - 3465 \,{\left (a^{2} b^{3} x^{6} + 3 \, a^{3} b^{2} x^{4} + 3 \, a^{4} b x^{2} + a^{5}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right )}{240 \,{\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} 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)^2,x, algorithm="fricas")

[Out]

[1/480*(96*b^5*x^11 - 352*a*b^4*x^9 + 3168*a^2*b^3*x^7 + 15246*a^3*b^2*x^5 + 18480*a^4*b*x^3 + 6930*a^5*x + 34
65*(a^2*b^3*x^6 + 3*a^3*b^2*x^4 + 3*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^9*x^6 + 3*a*b^8*x^4 + 3*a^2*b^7*x^2 + a^3*b^6), 1/240*(48*b^5*x^11 - 176*a*b^4*x^9 + 1584*a^2*b^3*x^7 +
7623*a^3*b^2*x^5 + 9240*a^4*b*x^3 + 3465*a^5*x - 3465*(a^2*b^3*x^6 + 3*a^3*b^2*x^4 + 3*a^4*b*x^2 + a^5)*sqrt(a
/b)*arctan(b*x*sqrt(a/b)/a))/(b^9*x^6 + 3*a*b^8*x^4 + 3*a^2*b^7*x^2 + a^3*b^6)]

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Sympy [A]  time = 0.853864, size = 172, normalized size = 1.47 \begin{align*} \frac{10 a^{2} x}{b^{6}} - \frac{4 a x^{3}}{3 b^{5}} + \frac{231 \sqrt{- \frac{a^{5}}{b^{13}}} \log{\left (x - \frac{b^{6} \sqrt{- \frac{a^{5}}{b^{13}}}}{a^{2}} \right )}}{32} - \frac{231 \sqrt{- \frac{a^{5}}{b^{13}}} \log{\left (x + \frac{b^{6} \sqrt{- \frac{a^{5}}{b^{13}}}}{a^{2}} \right )}}{32} + \frac{213 a^{5} x + 472 a^{4} b x^{3} + 267 a^{3} b^{2} x^{5}}{48 a^{3} b^{6} + 144 a^{2} b^{7} x^{2} + 144 a b^{8} x^{4} + 48 b^{9} x^{6}} + \frac{x^{5}}{5 b^{4}} \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)**2,x)

[Out]

10*a**2*x/b**6 - 4*a*x**3/(3*b**5) + 231*sqrt(-a**5/b**13)*log(x - b**6*sqrt(-a**5/b**13)/a**2)/32 - 231*sqrt(
-a**5/b**13)*log(x + b**6*sqrt(-a**5/b**13)/a**2)/32 + (213*a**5*x + 472*a**4*b*x**3 + 267*a**3*b**2*x**5)/(48
*a**3*b**6 + 144*a**2*b**7*x**2 + 144*a*b**8*x**4 + 48*b**9*x**6) + x**5/(5*b**4)

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Giac [A]  time = 1.12034, size = 130, normalized size = 1.11 \begin{align*} -\frac{231 \, a^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{16 \, \sqrt{a b} b^{6}} + \frac{267 \, a^{3} b^{2} x^{5} + 472 \, a^{4} b x^{3} + 213 \, a^{5} x}{48 \,{\left (b x^{2} + a\right )}^{3} b^{6}} + \frac{3 \, b^{16} x^{5} - 20 \, a b^{15} x^{3} + 150 \, a^{2} b^{14} x}{15 \, b^{20}} \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)^2,x, algorithm="giac")

[Out]

-231/16*a^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/48*(267*a^3*b^2*x^5 + 472*a^4*b*x^3 + 213*a^5*x)/((b*x^2
+ a)^3*b^6) + 1/15*(3*b^16*x^5 - 20*a*b^15*x^3 + 150*a^2*b^14*x)/b^20