3.511 \(\int \frac{1}{x^6 (a^2+2 a b x^2+b^2 x^4)^2} \, dx\)

Optimal. Leaf size=119 \[ -\frac{231 b^2}{16 a^6 x}-\frac{231 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 a^{13/2}}+\frac{77 b}{16 a^5 x^3}+\frac{33}{16 a^3 x^5 \left (a+b x^2\right )}+\frac{11}{24 a^2 x^5 \left (a+b x^2\right )^2}-\frac{231}{80 a^4 x^5}+\frac{1}{6 a x^5 \left (a+b x^2\right )^3} \]

[Out]

-231/(80*a^4*x^5) + (77*b)/(16*a^5*x^3) - (231*b^2)/(16*a^6*x) + 1/(6*a*x^5*(a + b*x^2)^3) + 11/(24*a^2*x^5*(a
+ b*x^2)^2) + 33/(16*a^3*x^5*(a + b*x^2)) - (231*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(16*a^(13/2))

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Rubi [A]  time = 0.0785271, antiderivative size = 119, 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, 290, 325, 205} \[ -\frac{231 b^2}{16 a^6 x}-\frac{231 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 a^{13/2}}+\frac{77 b}{16 a^5 x^3}+\frac{33}{16 a^3 x^5 \left (a+b x^2\right )}+\frac{11}{24 a^2 x^5 \left (a+b x^2\right )^2}-\frac{231}{80 a^4 x^5}+\frac{1}{6 a x^5 \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]

[Out]

-231/(80*a^4*x^5) + (77*b)/(16*a^5*x^3) - (231*b^2)/(16*a^6*x) + 1/(6*a*x^5*(a + b*x^2)^3) + 11/(24*a^2*x^5*(a
+ b*x^2)^2) + 33/(16*a^3*x^5*(a + b*x^2)) - (231*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(16*a^(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 290

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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{1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{1}{x^6 \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac{1}{6 a x^5 \left (a+b x^2\right )^3}+\frac{\left (11 b^3\right ) \int \frac{1}{x^6 \left (a b+b^2 x^2\right )^3} \, dx}{6 a}\\ &=\frac{1}{6 a x^5 \left (a+b x^2\right )^3}+\frac{11}{24 a^2 x^5 \left (a+b x^2\right )^2}+\frac{\left (33 b^2\right ) \int \frac{1}{x^6 \left (a b+b^2 x^2\right )^2} \, dx}{8 a^2}\\ &=\frac{1}{6 a x^5 \left (a+b x^2\right )^3}+\frac{11}{24 a^2 x^5 \left (a+b x^2\right )^2}+\frac{33}{16 a^3 x^5 \left (a+b x^2\right )}+\frac{(231 b) \int \frac{1}{x^6 \left (a b+b^2 x^2\right )} \, dx}{16 a^3}\\ &=-\frac{231}{80 a^4 x^5}+\frac{1}{6 a x^5 \left (a+b x^2\right )^3}+\frac{11}{24 a^2 x^5 \left (a+b x^2\right )^2}+\frac{33}{16 a^3 x^5 \left (a+b x^2\right )}-\frac{\left (231 b^2\right ) \int \frac{1}{x^4 \left (a b+b^2 x^2\right )} \, dx}{16 a^4}\\ &=-\frac{231}{80 a^4 x^5}+\frac{77 b}{16 a^5 x^3}+\frac{1}{6 a x^5 \left (a+b x^2\right )^3}+\frac{11}{24 a^2 x^5 \left (a+b x^2\right )^2}+\frac{33}{16 a^3 x^5 \left (a+b x^2\right )}+\frac{\left (231 b^3\right ) \int \frac{1}{x^2 \left (a b+b^2 x^2\right )} \, dx}{16 a^5}\\ &=-\frac{231}{80 a^4 x^5}+\frac{77 b}{16 a^5 x^3}-\frac{231 b^2}{16 a^6 x}+\frac{1}{6 a x^5 \left (a+b x^2\right )^3}+\frac{11}{24 a^2 x^5 \left (a+b x^2\right )^2}+\frac{33}{16 a^3 x^5 \left (a+b x^2\right )}-\frac{\left (231 b^4\right ) \int \frac{1}{a b+b^2 x^2} \, dx}{16 a^6}\\ &=-\frac{231}{80 a^4 x^5}+\frac{77 b}{16 a^5 x^3}-\frac{231 b^2}{16 a^6 x}+\frac{1}{6 a x^5 \left (a+b x^2\right )^3}+\frac{11}{24 a^2 x^5 \left (a+b x^2\right )^2}+\frac{33}{16 a^3 x^5 \left (a+b x^2\right )}-\frac{231 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 a^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.051214, size = 101, normalized size = 0.85 \[ -\frac{7623 a^2 b^3 x^6+1584 a^3 b^2 x^4-176 a^4 b x^2+48 a^5+9240 a b^4 x^8+3465 b^5 x^{10}}{240 a^6 x^5 \left (a+b x^2\right )^3}-\frac{231 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 a^{13/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]

[Out]

-(48*a^5 - 176*a^4*b*x^2 + 1584*a^3*b^2*x^4 + 7623*a^2*b^3*x^6 + 9240*a*b^4*x^8 + 3465*b^5*x^10)/(240*a^6*x^5*
(a + b*x^2)^3) - (231*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(16*a^(13/2))

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Maple [A]  time = 0.056, size = 110, normalized size = 0.9 \begin{align*} -{\frac{1}{5\,{a}^{4}{x}^{5}}}-10\,{\frac{{b}^{2}}{{a}^{6}x}}+{\frac{4\,b}{3\,{a}^{5}{x}^{3}}}-{\frac{71\,{b}^{5}{x}^{5}}{16\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{59\,{b}^{4}{x}^{3}}{6\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{89\,{b}^{3}x}{16\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{231\,{b}^{3}}{16\,{a}^{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(1/x^6/(b^2*x^4+2*a*b*x^2+a^2)^2,x)

[Out]

-1/5/a^4/x^5-10*b^2/a^6/x+4/3*b/a^5/x^3-71/16/a^6*b^5/(b*x^2+a)^3*x^5-59/6/a^5*b^4/(b*x^2+a)^3*x^3-89/16/a^4*b
^3/(b*x^2+a)^3*x-231/16/a^6*b^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(1/x^6/(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 = 1.7744, size = 725, normalized size = 6.09 \begin{align*} \left [-\frac{6930 \, b^{5} x^{10} + 18480 \, a b^{4} x^{8} + 15246 \, a^{2} b^{3} x^{6} + 3168 \, a^{3} b^{2} x^{4} - 352 \, a^{4} b x^{2} + 96 \, a^{5} - 3465 \,{\left (b^{5} x^{11} + 3 \, a b^{4} x^{9} + 3 \, a^{2} b^{3} x^{7} + a^{3} b^{2} x^{5}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{480 \,{\left (a^{6} b^{3} x^{11} + 3 \, a^{7} b^{2} x^{9} + 3 \, a^{8} b x^{7} + a^{9} x^{5}\right )}}, -\frac{3465 \, b^{5} x^{10} + 9240 \, a b^{4} x^{8} + 7623 \, a^{2} b^{3} x^{6} + 1584 \, a^{3} b^{2} x^{4} - 176 \, a^{4} b x^{2} + 48 \, a^{5} + 3465 \,{\left (b^{5} x^{11} + 3 \, a b^{4} x^{9} + 3 \, a^{2} b^{3} x^{7} + a^{3} b^{2} x^{5}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{240 \,{\left (a^{6} b^{3} x^{11} + 3 \, a^{7} b^{2} x^{9} + 3 \, a^{8} b x^{7} + a^{9} x^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")

[Out]

[-1/480*(6930*b^5*x^10 + 18480*a*b^4*x^8 + 15246*a^2*b^3*x^6 + 3168*a^3*b^2*x^4 - 352*a^4*b*x^2 + 96*a^5 - 346
5*(b^5*x^11 + 3*a*b^4*x^9 + 3*a^2*b^3*x^7 + a^3*b^2*x^5)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2
+ a)))/(a^6*b^3*x^11 + 3*a^7*b^2*x^9 + 3*a^8*b*x^7 + a^9*x^5), -1/240*(3465*b^5*x^10 + 9240*a*b^4*x^8 + 7623*a
^2*b^3*x^6 + 1584*a^3*b^2*x^4 - 176*a^4*b*x^2 + 48*a^5 + 3465*(b^5*x^11 + 3*a*b^4*x^9 + 3*a^2*b^3*x^7 + a^3*b^
2*x^5)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^6*b^3*x^11 + 3*a^7*b^2*x^9 + 3*a^8*b*x^7 + a^9*x^5)]

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Sympy [A]  time = 3.05541, size = 173, normalized size = 1.45 \begin{align*} \frac{231 \sqrt{- \frac{b^{5}}{a^{13}}} \log{\left (- \frac{a^{7} \sqrt{- \frac{b^{5}}{a^{13}}}}{b^{3}} + x \right )}}{32} - \frac{231 \sqrt{- \frac{b^{5}}{a^{13}}} \log{\left (\frac{a^{7} \sqrt{- \frac{b^{5}}{a^{13}}}}{b^{3}} + x \right )}}{32} - \frac{48 a^{5} - 176 a^{4} b x^{2} + 1584 a^{3} b^{2} x^{4} + 7623 a^{2} b^{3} x^{6} + 9240 a b^{4} x^{8} + 3465 b^{5} x^{10}}{240 a^{9} x^{5} + 720 a^{8} b x^{7} + 720 a^{7} b^{2} x^{9} + 240 a^{6} b^{3} x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

231*sqrt(-b**5/a**13)*log(-a**7*sqrt(-b**5/a**13)/b**3 + x)/32 - 231*sqrt(-b**5/a**13)*log(a**7*sqrt(-b**5/a**
13)/b**3 + x)/32 - (48*a**5 - 176*a**4*b*x**2 + 1584*a**3*b**2*x**4 + 7623*a**2*b**3*x**6 + 9240*a*b**4*x**8 +
3465*b**5*x**10)/(240*a**9*x**5 + 720*a**8*b*x**7 + 720*a**7*b**2*x**9 + 240*a**6*b**3*x**11)

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Giac [A]  time = 1.15355, size = 126, normalized size = 1.06 \begin{align*} -\frac{231 \, b^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{16 \, \sqrt{a b} a^{6}} - \frac{213 \, b^{5} x^{5} + 472 \, a b^{4} x^{3} + 267 \, a^{2} b^{3} x}{48 \,{\left (b x^{2} + a\right )}^{3} a^{6}} - \frac{150 \, b^{2} x^{4} - 20 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{6} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")

[Out]

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