\(\int \frac {1}{x^2 (a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [532]

Optimal result

Integrand size = 24, antiderivative size = 133 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {693}{256 a^6 x}+\frac {1}{10 a x \left (a+b x^2\right )^5}+\frac {11}{80 a^2 x \left (a+b x^2\right )^4}+\frac {33}{160 a^3 x \left (a+b x^2\right )^3}+\frac {231}{640 a^4 x \left (a+b x^2\right )^2}+\frac {231}{256 a^5 x \left (a+b x^2\right )}-\frac {693 \sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{13/2}} \]

[Out]

-693/256/a^6/x+1/10/a/x/(b*x^2+a)^5+11/80/a^2/x/(b*x^2+a)^4+33/160/a^3/x/(b*x^2+a)^3+231/640/a^4/x/(b*x^2+a)^2
+231/256/a^5/x/(b*x^2+a)-693/256*arctan(x*b^(1/2)/a^(1/2))*b^(1/2)/a^(13/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 296, 331, 211} \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {693 \sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{13/2}}-\frac {693}{256 a^6 x}+\frac {231}{256 a^5 x \left (a+b x^2\right )}+\frac {231}{640 a^4 x \left (a+b x^2\right )^2}+\frac {33}{160 a^3 x \left (a+b x^2\right )^3}+\frac {11}{80 a^2 x \left (a+b x^2\right )^4}+\frac {1}{10 a x \left (a+b x^2\right )^5} \]

[In]

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

[Out]

-693/(256*a^6*x) + 1/(10*a*x*(a + b*x^2)^5) + 11/(80*a^2*x*(a + b*x^2)^4) + 33/(160*a^3*x*(a + b*x^2)^3) + 231
/(640*a^4*x*(a + b*x^2)^2) + 231/(256*a^5*x*(a + b*x^2)) - (693*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(1
3/2))

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 211

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

Rule 296

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] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 331

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]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {1280 a^5+10615 a^4 b x^2+26070 a^3 b^2 x^4+29568 a^2 b^3 x^6+16170 a b^4 x^8+3465 b^5 x^{10}}{1280 a^6 x \left (a+b x^2\right )^5}-\frac {693 \sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{13/2}} \]

[In]

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

[Out]

-1/1280*(1280*a^5 + 10615*a^4*b*x^2 + 26070*a^3*b^2*x^4 + 29568*a^2*b^3*x^6 + 16170*a*b^4*x^8 + 3465*b^5*x^10)
/(a^6*x*(a + b*x^2)^5) - (693*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(13/2))

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.65

method	result	size
default	\(-\frac {1}{a^{6} x}-\frac {b \left (\frac {\frac {437}{256} b^{4} x^{9}+\frac {977}{128} a \,b^{3} x^{7}+\frac {131}{10} a^{2} b^{2} x^{5}+\frac {1327}{128} a^{3} b \,x^{3}+\frac {843}{256} a^{4} x}{\left (b \,x^{2}+a \right )^{5}}+\frac {693 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \sqrt {a b}}\right )}{a^{6}}\)	\(87\)
risch	\(\frac {-\frac {693 b^{5} x^{10}}{256 a^{6}}-\frac {1617 b^{4} x^{8}}{128 a^{5}}-\frac {231 b^{3} x^{6}}{10 a^{4}}-\frac {2607 b^{2} x^{4}}{128 a^{3}}-\frac {2123 b \,x^{2}}{256 a^{2}}-\frac {1}{a}}{x \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} \left (b \,x^{2}+a \right )}+\frac {693 \sqrt {-a b}\, \ln \left (-b x +\sqrt {-a b}\right )}{512 a^{7}}-\frac {693 \sqrt {-a b}\, \ln \left (-b x -\sqrt {-a b}\right )}{512 a^{7}}\)	\(142\)

[In]

int(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

-1/a^6/x-b/a^6*((437/256*b^4*x^9+977/128*a*b^3*x^7+131/10*a^2*b^2*x^5+1327/128*a^3*b*x^3+843/256*a^4*x)/(b*x^2
+a)^5+693/256/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 400, normalized size of antiderivative = 3.01 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\left [-\frac {6930 \, b^{5} x^{10} + 32340 \, a b^{4} x^{8} + 59136 \, a^{2} b^{3} x^{6} + 52140 \, a^{3} b^{2} x^{4} + 21230 \, a^{4} b x^{2} + 2560 \, a^{5} - 3465 \, {\left (b^{5} x^{11} + 5 \, a b^{4} x^{9} + 10 \, a^{2} b^{3} x^{7} + 10 \, a^{3} b^{2} x^{5} + 5 \, a^{4} b x^{3} + a^{5} x\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{2560 \, {\left (a^{6} b^{5} x^{11} + 5 \, a^{7} b^{4} x^{9} + 10 \, a^{8} b^{3} x^{7} + 10 \, a^{9} b^{2} x^{5} + 5 \, a^{10} b x^{3} + a^{11} x\right )}}, -\frac {3465 \, b^{5} x^{10} + 16170 \, a b^{4} x^{8} + 29568 \, a^{2} b^{3} x^{6} + 26070 \, a^{3} b^{2} x^{4} + 10615 \, a^{4} b x^{2} + 1280 \, a^{5} + 3465 \, {\left (b^{5} x^{11} + 5 \, a b^{4} x^{9} + 10 \, a^{2} b^{3} x^{7} + 10 \, a^{3} b^{2} x^{5} + 5 \, a^{4} b x^{3} + a^{5} x\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{1280 \, {\left (a^{6} b^{5} x^{11} + 5 \, a^{7} b^{4} x^{9} + 10 \, a^{8} b^{3} x^{7} + 10 \, a^{9} b^{2} x^{5} + 5 \, a^{10} b x^{3} + a^{11} x\right )}}\right ] \]

[In]

integrate(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.41 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {693 \sqrt {- \frac {b}{a^{13}}} \log {\left (- \frac {a^{7} \sqrt {- \frac {b}{a^{13}}}}{b} + x \right )}}{512} - \frac {693 \sqrt {- \frac {b}{a^{13}}} \log {\left (\frac {a^{7} \sqrt {- \frac {b}{a^{13}}}}{b} + x \right )}}{512} + \frac {- 1280 a^{5} - 10615 a^{4} b x^{2} - 26070 a^{3} b^{2} x^{4} - 29568 a^{2} b^{3} x^{6} - 16170 a b^{4} x^{8} - 3465 b^{5} x^{10}}{1280 a^{11} x + 6400 a^{10} b x^{3} + 12800 a^{9} b^{2} x^{5} + 12800 a^{8} b^{3} x^{7} + 6400 a^{7} b^{4} x^{9} + 1280 a^{6} b^{5} x^{11}} \]

[In]

integrate(1/x**2/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

693*sqrt(-b/a**13)*log(-a**7*sqrt(-b/a**13)/b + x)/512 - 693*sqrt(-b/a**13)*log(a**7*sqrt(-b/a**13)/b + x)/512
+ (-1280*a**5 - 10615*a**4*b*x**2 - 26070*a**3*b**2*x**4 - 29568*a**2*b**3*x**6 - 16170*a*b**4*x**8 - 3465*b*
*5*x**10)/(1280*a**11*x + 6400*a**10*b*x**3 + 12800*a**9*b**2*x**5 + 12800*a**8*b**3*x**7 + 6400*a**7*b**4*x**
9 + 1280*a**6*b**5*x**11)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {3465 \, b^{5} x^{10} + 16170 \, a b^{4} x^{8} + 29568 \, a^{2} b^{3} x^{6} + 26070 \, a^{3} b^{2} x^{4} + 10615 \, a^{4} b x^{2} + 1280 \, a^{5}}{1280 \, {\left (a^{6} b^{5} x^{11} + 5 \, a^{7} b^{4} x^{9} + 10 \, a^{8} b^{3} x^{7} + 10 \, a^{9} b^{2} x^{5} + 5 \, a^{10} b x^{3} + a^{11} x\right )}} - \frac {693 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{6}} \]

[In]

integrate(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

-1/1280*(3465*b^5*x^10 + 16170*a*b^4*x^8 + 29568*a^2*b^3*x^6 + 26070*a^3*b^2*x^4 + 10615*a^4*b*x^2 + 1280*a^5)
/(a^6*b^5*x^11 + 5*a^7*b^4*x^9 + 10*a^8*b^3*x^7 + 10*a^9*b^2*x^5 + 5*a^10*b*x^3 + a^11*x) - 693/256*b*arctan(b
*x/sqrt(a*b))/(sqrt(a*b)*a^6)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {693 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{6}} - \frac {1}{a^{6} x} - \frac {2185 \, b^{5} x^{9} + 9770 \, a b^{4} x^{7} + 16768 \, a^{2} b^{3} x^{5} + 13270 \, a^{3} b^{2} x^{3} + 4215 \, a^{4} b x}{1280 \, {\left (b x^{2} + a\right )}^{5} a^{6}} \]

[In]

integrate(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\frac {1}{a}+\frac {2123\,b\,x^2}{256\,a^2}+\frac {2607\,b^2\,x^4}{128\,a^3}+\frac {231\,b^3\,x^6}{10\,a^4}+\frac {1617\,b^4\,x^8}{128\,a^5}+\frac {693\,b^5\,x^{10}}{256\,a^6}}{a^5\,x+5\,a^4\,b\,x^3+10\,a^3\,b^2\,x^5+10\,a^2\,b^3\,x^7+5\,a\,b^4\,x^9+b^5\,x^{11}}-\frac {693\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,a^{13/2}} \]

[In]

int(1/(x^2*(a^2 + b^2*x^4 + 2*a*b*x^2)^3),x)

[Out]

- (1/a + (2123*b*x^2)/(256*a^2) + (2607*b^2*x^4)/(128*a^3) + (231*b^3*x^6)/(10*a^4) + (1617*b^4*x^8)/(128*a^5)
+ (693*b^5*x^10)/(256*a^6))/(a^5*x + b^5*x^11 + 5*a^4*b*x^3 + 5*a*b^4*x^9 + 10*a^3*b^2*x^5 + 10*a^2*b^3*x^7)
- (693*b^(1/2)*atan((b^(1/2)*x)/a^(1/2)))/(256*a^(13/2))