3.534 \(\int \frac{1}{x^6 (a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=157 \[ -\frac{9009 b^2}{256 a^8 x}-\frac{9009 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{17/2}}+\frac{3003 b}{256 a^7 x^3}+\frac{1287}{256 a^5 x^5 \left (a+b x^2\right )}+\frac{143}{128 a^4 x^5 \left (a+b x^2\right )^2}+\frac{13}{32 a^3 x^5 \left (a+b x^2\right )^3}+\frac{3}{16 a^2 x^5 \left (a+b x^2\right )^4}-\frac{9009}{1280 a^6 x^5}+\frac{1}{10 a x^5 \left (a+b x^2\right )^5} \]

[Out]

-9009/(1280*a^6*x^5) + (3003*b)/(256*a^7*x^3) - (9009*b^2)/(256*a^8*x) + 1/(10*a*x^5*(a + b*x^2)^5) + 3/(16*a^
2*x^5*(a + b*x^2)^4) + 13/(32*a^3*x^5*(a + b*x^2)^3) + 143/(128*a^4*x^5*(a + b*x^2)^2) + 1287/(256*a^5*x^5*(a
+ b*x^2)) - (9009*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(17/2))

________________________________________________________________________________________

Rubi [A]  time = 0.122192, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {28, 290, 325, 205} \[ -\frac{9009 b^2}{256 a^8 x}-\frac{9009 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{17/2}}+\frac{3003 b}{256 a^7 x^3}+\frac{1287}{256 a^5 x^5 \left (a+b x^2\right )}+\frac{143}{128 a^4 x^5 \left (a+b x^2\right )^2}+\frac{13}{32 a^3 x^5 \left (a+b x^2\right )^3}+\frac{3}{16 a^2 x^5 \left (a+b x^2\right )^4}-\frac{9009}{1280 a^6 x^5}+\frac{1}{10 a x^5 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

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

Mathematica [A]  time = 0.0628365, size = 123, normalized size = 0.78 \[ -\frac{384384 a^2 b^5 x^{10}+338910 a^3 b^4 x^8+137995 a^4 b^3 x^6+16640 a^5 b^2 x^4-1280 a^6 b x^2+256 a^7+210210 a b^6 x^{12}+45045 b^7 x^{14}}{1280 a^8 x^5 \left (a+b x^2\right )^5}-\frac{9009 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{17/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

-(256*a^7 - 1280*a^6*b*x^2 + 16640*a^5*b^2*x^4 + 137995*a^4*b^3*x^6 + 338910*a^3*b^4*x^8 + 384384*a^2*b^5*x^10
+ 210210*a*b^6*x^12 + 45045*b^7*x^14)/(1280*a^8*x^5*(a + b*x^2)^5) - (9009*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]
])/(256*a^(17/2))

________________________________________________________________________________________

Maple [A]  time = 0.061, size = 150, normalized size = 1. \begin{align*} -{\frac{1}{5\,{a}^{6}{x}^{5}}}-21\,{\frac{{b}^{2}}{{a}^{8}x}}+2\,{\frac{b}{{a}^{7}{x}^{3}}}-{\frac{3633\,{b}^{7}{x}^{9}}{256\,{a}^{8} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{7837\,{b}^{6}{x}^{7}}{128\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{1001\,{b}^{5}{x}^{5}}{10\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{9443\,{b}^{4}{x}^{3}}{128\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{5327\,{b}^{3}x}{256\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{9009\,{b}^{3}}{256\,{a}^{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(1/x^6/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

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

________________________________________________________________________________________

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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.53888, size = 1073, normalized size = 6.83 \begin{align*} \left [-\frac{90090 \, b^{7} x^{14} + 420420 \, a b^{6} x^{12} + 768768 \, a^{2} b^{5} x^{10} + 677820 \, a^{3} b^{4} x^{8} + 275990 \, a^{4} b^{3} x^{6} + 33280 \, a^{5} b^{2} x^{4} - 2560 \, a^{6} b x^{2} + 512 \, a^{7} - 45045 \,{\left (b^{7} x^{15} + 5 \, a b^{6} x^{13} + 10 \, a^{2} b^{5} x^{11} + 10 \, a^{3} b^{4} x^{9} + 5 \, a^{4} b^{3} x^{7} + a^{5} 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 )}{2560 \,{\left (a^{8} b^{5} x^{15} + 5 \, a^{9} b^{4} x^{13} + 10 \, a^{10} b^{3} x^{11} + 10 \, a^{11} b^{2} x^{9} + 5 \, a^{12} b x^{7} + a^{13} x^{5}\right )}}, -\frac{45045 \, b^{7} x^{14} + 210210 \, a b^{6} x^{12} + 384384 \, a^{2} b^{5} x^{10} + 338910 \, a^{3} b^{4} x^{8} + 137995 \, a^{4} b^{3} x^{6} + 16640 \, a^{5} b^{2} x^{4} - 1280 \, a^{6} b x^{2} + 256 \, a^{7} + 45045 \,{\left (b^{7} x^{15} + 5 \, a b^{6} x^{13} + 10 \, a^{2} b^{5} x^{11} + 10 \, a^{3} b^{4} x^{9} + 5 \, a^{4} b^{3} x^{7} + a^{5} b^{2} x^{5}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{1280 \,{\left (a^{8} b^{5} x^{15} + 5 \, a^{9} b^{4} x^{13} + 10 \, a^{10} b^{3} x^{11} + 10 \, a^{11} b^{2} x^{9} + 5 \, a^{12} b x^{7} + a^{13} 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)^3,x, algorithm="fricas")

[Out]

[-1/2560*(90090*b^7*x^14 + 420420*a*b^6*x^12 + 768768*a^2*b^5*x^10 + 677820*a^3*b^4*x^8 + 275990*a^4*b^3*x^6 +
33280*a^5*b^2*x^4 - 2560*a^6*b*x^2 + 512*a^7 - 45045*(b^7*x^15 + 5*a*b^6*x^13 + 10*a^2*b^5*x^11 + 10*a^3*b^4*
x^9 + 5*a^4*b^3*x^7 + a^5*b^2*x^5)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^8*b^5*x^15 +
5*a^9*b^4*x^13 + 10*a^10*b^3*x^11 + 10*a^11*b^2*x^9 + 5*a^12*b*x^7 + a^13*x^5), -1/1280*(45045*b^7*x^14 + 210
210*a*b^6*x^12 + 384384*a^2*b^5*x^10 + 338910*a^3*b^4*x^8 + 137995*a^4*b^3*x^6 + 16640*a^5*b^2*x^4 - 1280*a^6*
b*x^2 + 256*a^7 + 45045*(b^7*x^15 + 5*a*b^6*x^13 + 10*a^2*b^5*x^11 + 10*a^3*b^4*x^9 + 5*a^4*b^3*x^7 + a^5*b^2*
x^5)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^8*b^5*x^15 + 5*a^9*b^4*x^13 + 10*a^10*b^3*x^11 + 10*a^11*b^2*x^9 + 5*a^
12*b*x^7 + a^13*x^5)]

________________________________________________________________________________________

Sympy [A]  time = 19.3699, size = 221, normalized size = 1.41 \begin{align*} \frac{9009 \sqrt{- \frac{b^{5}}{a^{17}}} \log{\left (- \frac{a^{9} \sqrt{- \frac{b^{5}}{a^{17}}}}{b^{3}} + x \right )}}{512} - \frac{9009 \sqrt{- \frac{b^{5}}{a^{17}}} \log{\left (\frac{a^{9} \sqrt{- \frac{b^{5}}{a^{17}}}}{b^{3}} + x \right )}}{512} - \frac{256 a^{7} - 1280 a^{6} b x^{2} + 16640 a^{5} b^{2} x^{4} + 137995 a^{4} b^{3} x^{6} + 338910 a^{3} b^{4} x^{8} + 384384 a^{2} b^{5} x^{10} + 210210 a b^{6} x^{12} + 45045 b^{7} x^{14}}{1280 a^{13} x^{5} + 6400 a^{12} b x^{7} + 12800 a^{11} b^{2} x^{9} + 12800 a^{10} b^{3} x^{11} + 6400 a^{9} b^{4} x^{13} + 1280 a^{8} b^{5} x^{15}} \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)**3,x)

[Out]

9009*sqrt(-b**5/a**17)*log(-a**9*sqrt(-b**5/a**17)/b**3 + x)/512 - 9009*sqrt(-b**5/a**17)*log(a**9*sqrt(-b**5/
a**17)/b**3 + x)/512 - (256*a**7 - 1280*a**6*b*x**2 + 16640*a**5*b**2*x**4 + 137995*a**4*b**3*x**6 + 338910*a*
*3*b**4*x**8 + 384384*a**2*b**5*x**10 + 210210*a*b**6*x**12 + 45045*b**7*x**14)/(1280*a**13*x**5 + 6400*a**12*
b*x**7 + 12800*a**11*b**2*x**9 + 12800*a**10*b**3*x**11 + 6400*a**9*b**4*x**13 + 1280*a**8*b**5*x**15)

________________________________________________________________________________________

Giac [A]  time = 1.11959, size = 155, normalized size = 0.99 \begin{align*} -\frac{9009 \, b^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} a^{8}} - \frac{45045 \, b^{7} x^{14} + 210210 \, a b^{6} x^{12} + 384384 \, a^{2} b^{5} x^{10} + 338910 \, a^{3} b^{4} x^{8} + 137995 \, a^{4} b^{3} x^{6} + 16640 \, a^{5} b^{2} x^{4} - 1280 \, a^{6} b x^{2} + 256 \, a^{7}}{1280 \,{\left (b x^{3} + a x\right )}^{5} a^{8}} \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)^3,x, algorithm="giac")

[Out]

-9009/256*b^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^8) - 1/1280*(45045*b^7*x^14 + 210210*a*b^6*x^12 + 384384*a^2*
b^5*x^10 + 338910*a^3*b^4*x^8 + 137995*a^4*b^3*x^6 + 16640*a^5*b^2*x^4 - 1280*a^6*b*x^2 + 256*a^7)/((b*x^3 + a
*x)^5*a^8)