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

Optimal. Leaf size=133 \[ \frac{231}{256 a^5 x \left (a+b x^2\right )}+\frac{231}{640 a^4 x \left (a+b x^2\right )^2}+\frac{33}{160 a^3 x \left (a+b x^2\right )^3}+\frac{11}{80 a^2 x \left (a+b x^2\right )^4}-\frac{693 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{13/2}}-\frac{693}{256 a^6 x}+\frac{1}{10 a x \left (a+b x^2\right )^5} \]

[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))

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Rubi [A]  time = 0.0893844, antiderivative size = 133, 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} \[ \frac{231}{256 a^5 x \left (a+b x^2\right )}+\frac{231}{640 a^4 x \left (a+b x^2\right )^2}+\frac{33}{160 a^3 x \left (a+b x^2\right )^3}+\frac{11}{80 a^2 x \left (a+b x^2\right )^4}-\frac{693 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{13/2}}-\frac{693}{256 a^6 x}+\frac{1}{10 a x \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[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 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^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=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]  time = 0.0549293, size = 101, normalized size = 0.76 \[ -\frac{29568 a^2 b^3 x^6+26070 a^3 b^2 x^4+10615 a^4 b x^2+1280 a^5+16170 a b^4 x^8+3465 b^5 x^{10}}{1280 a^6 x \left (a+b x^2\right )^5}-\frac{693 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{13/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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*(a + b*x^2)^5) - (693*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(13/2))

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Maple [A]  time = 0.057, size = 126, normalized size = 1. \begin{align*} -{\frac{1}{{a}^{6}x}}-{\frac{437\,{b}^{5}{x}^{9}}{256\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{977\,{b}^{4}{x}^{7}}{128\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{131\,{b}^{3}{x}^{5}}{10\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{1327\,{b}^{2}{x}^{3}}{128\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{843\,bx}{256\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{693\,b}{256\,{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^2/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-1/a^6/x-437/256/a^6*b^5/(b*x^2+a)^5*x^9-977/128/a^5*b^4/(b*x^2+a)^5*x^7-131/10/a^4*b^3/(b*x^2+a)^5*x^5-1327/1
28/a^3*b^2/(b*x^2+a)^5*x^3-843/256/a^2*b/(b*x^2+a)^5*x-693/256/a^6*b/(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^2/(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 = 1.53474, size = 906, normalized size = 6.81 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [A]  time = 4.07907, size = 185, normalized size = 1.39 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.13629, size = 122, normalized size = 0.92 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)