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

Optimal. Leaf size=144 \[ \frac{3003 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{15/2}}+\frac{3003}{1280 a^5 x^3 \left (a+b x^2\right )}+\frac{429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac{143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac{13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac{3003 b}{256 a^7 x}-\frac{1001}{256 a^6 x^3}+\frac{1}{10 a x^3 \left (a+b x^2\right )^5} \]

[Out]

-1001/(256*a^6*x^3) + (3003*b)/(256*a^7*x) + 1/(10*a*x^3*(a + b*x^2)^5) + 13/(80*a^2*x^3*(a + b*x^2)^4) + 143/
(480*a^3*x^3*(a + b*x^2)^3) + 429/(640*a^4*x^3*(a + b*x^2)^2) + 3003/(1280*a^5*x^3*(a + b*x^2)) + (3003*b^(3/2
)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(15/2))

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Rubi [A]  time = 0.103098, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {28, 290, 325, 205} \[ \frac{3003 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{15/2}}+\frac{3003}{1280 a^5 x^3 \left (a+b x^2\right )}+\frac{429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac{143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac{13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac{3003 b}{256 a^7 x}-\frac{1001}{256 a^6 x^3}+\frac{1}{10 a x^3 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1001/(256*a^6*x^3) + (3003*b)/(256*a^7*x) + 1/(10*a*x^3*(a + b*x^2)^5) + 13/(80*a^2*x^3*(a + b*x^2)^4) + 143/
(480*a^3*x^3*(a + b*x^2)^3) + 429/(640*a^4*x^3*(a + b*x^2)^2) + 3003/(1280*a^5*x^3*(a + b*x^2)) + (3003*b^(3/2
)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(15/2))

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^4 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{1}{x^4 \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac{1}{10 a x^3 \left (a+b x^2\right )^5}+\frac{\left (13 b^5\right ) \int \frac{1}{x^4 \left (a b+b^2 x^2\right )^5} \, dx}{10 a}\\ &=\frac{1}{10 a x^3 \left (a+b x^2\right )^5}+\frac{13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac{\left (143 b^4\right ) \int \frac{1}{x^4 \left (a b+b^2 x^2\right )^4} \, dx}{80 a^2}\\ &=\frac{1}{10 a x^3 \left (a+b x^2\right )^5}+\frac{13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac{143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac{\left (429 b^3\right ) \int \frac{1}{x^4 \left (a b+b^2 x^2\right )^3} \, dx}{160 a^3}\\ &=\frac{1}{10 a x^3 \left (a+b x^2\right )^5}+\frac{13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac{143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac{429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac{\left (3003 b^2\right ) \int \frac{1}{x^4 \left (a b+b^2 x^2\right )^2} \, dx}{640 a^4}\\ &=\frac{1}{10 a x^3 \left (a+b x^2\right )^5}+\frac{13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac{143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac{429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac{3003}{1280 a^5 x^3 \left (a+b x^2\right )}+\frac{(3003 b) \int \frac{1}{x^4 \left (a b+b^2 x^2\right )} \, dx}{256 a^5}\\ &=-\frac{1001}{256 a^6 x^3}+\frac{1}{10 a x^3 \left (a+b x^2\right )^5}+\frac{13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac{143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac{429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac{3003}{1280 a^5 x^3 \left (a+b x^2\right )}-\frac{\left (3003 b^2\right ) \int \frac{1}{x^2 \left (a b+b^2 x^2\right )} \, dx}{256 a^6}\\ &=-\frac{1001}{256 a^6 x^3}+\frac{3003 b}{256 a^7 x}+\frac{1}{10 a x^3 \left (a+b x^2\right )^5}+\frac{13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac{143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac{429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac{3003}{1280 a^5 x^3 \left (a+b x^2\right )}+\frac{\left (3003 b^3\right ) \int \frac{1}{a b+b^2 x^2} \, dx}{256 a^7}\\ &=-\frac{1001}{256 a^6 x^3}+\frac{3003 b}{256 a^7 x}+\frac{1}{10 a x^3 \left (a+b x^2\right )^5}+\frac{13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac{143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac{429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac{3003}{1280 a^5 x^3 \left (a+b x^2\right )}+\frac{3003 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{15/2}}\\ \end{align*}

Mathematica [A]  time = 0.059476, size = 113, normalized size = 0.78 \[ \frac{\frac{\sqrt{a} \left (384384 a^2 b^4 x^8+338910 a^3 b^3 x^6+137995 a^4 b^2 x^4+16640 a^5 b x^2-1280 a^6+210210 a b^5 x^{10}+45045 b^6 x^{12}\right )}{x^3 \left (a+b x^2\right )^5}+45045 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3840 a^{15/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[a]*(-1280*a^6 + 16640*a^5*b*x^2 + 137995*a^4*b^2*x^4 + 338910*a^3*b^3*x^6 + 384384*a^2*b^4*x^8 + 210210
*a*b^5*x^10 + 45045*b^6*x^12))/(x^3*(a + b*x^2)^5) + 45045*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3840*a^(15/2)
)

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Maple [A]  time = 0.062, size = 139, normalized size = 1. \begin{align*} -{\frac{1}{3\,{a}^{6}{x}^{3}}}+6\,{\frac{b}{{a}^{7}x}}+{\frac{1467\,{b}^{6}{x}^{9}}{256\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{9629\,{b}^{5}{x}^{7}}{384\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{1253\,{b}^{4}{x}^{5}}{30\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{12131\,{b}^{3}{x}^{3}}{384\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{2373\,{b}^{2}x}{256\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{3003\,{b}^{2}}{256\,{a}^{7}}\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^4/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-1/3/a^6/x^3+6*b/a^7/x+1467/256/a^7*b^6/(b*x^2+a)^5*x^9+9629/384/a^6*b^5/(b*x^2+a)^5*x^7+1253/30/a^5*b^4/(b*x^
2+a)^5*x^5+12131/384/a^4*b^3/(b*x^2+a)^5*x^3+2373/256/a^3*b^2/(b*x^2+a)^5*x+3003/256/a^7*b^2/(a*b)^(1/2)*arcta
n(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^4/(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 = 1.51335, size = 1006, normalized size = 6.99 \begin{align*} \left [\frac{90090 \, b^{6} x^{12} + 420420 \, a b^{5} x^{10} + 768768 \, a^{2} b^{4} x^{8} + 677820 \, a^{3} b^{3} x^{6} + 275990 \, a^{4} b^{2} x^{4} + 33280 \, a^{5} b x^{2} - 2560 \, a^{6} + 45045 \,{\left (b^{6} x^{13} + 5 \, a b^{5} x^{11} + 10 \, a^{2} b^{4} x^{9} + 10 \, a^{3} b^{3} x^{7} + 5 \, a^{4} b^{2} x^{5} + a^{5} b x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{7680 \,{\left (a^{7} b^{5} x^{13} + 5 \, a^{8} b^{4} x^{11} + 10 \, a^{9} b^{3} x^{9} + 10 \, a^{10} b^{2} x^{7} + 5 \, a^{11} b x^{5} + a^{12} x^{3}\right )}}, \frac{45045 \, b^{6} x^{12} + 210210 \, a b^{5} x^{10} + 384384 \, a^{2} b^{4} x^{8} + 338910 \, a^{3} b^{3} x^{6} + 137995 \, a^{4} b^{2} x^{4} + 16640 \, a^{5} b x^{2} - 1280 \, a^{6} + 45045 \,{\left (b^{6} x^{13} + 5 \, a b^{5} x^{11} + 10 \, a^{2} b^{4} x^{9} + 10 \, a^{3} b^{3} x^{7} + 5 \, a^{4} b^{2} x^{5} + a^{5} b x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{3840 \,{\left (a^{7} b^{5} x^{13} + 5 \, a^{8} b^{4} x^{11} + 10 \, a^{9} b^{3} x^{9} + 10 \, a^{10} b^{2} x^{7} + 5 \, a^{11} b x^{5} + a^{12} x^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/7680*(90090*b^6*x^12 + 420420*a*b^5*x^10 + 768768*a^2*b^4*x^8 + 677820*a^3*b^3*x^6 + 275990*a^4*b^2*x^4 + 3
3280*a^5*b*x^2 - 2560*a^6 + 45045*(b^6*x^13 + 5*a*b^5*x^11 + 10*a^2*b^4*x^9 + 10*a^3*b^3*x^7 + 5*a^4*b^2*x^5 +
a^5*b*x^3)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^7*b^5*x^13 + 5*a^8*b^4*x^11 + 10*a^
9*b^3*x^9 + 10*a^10*b^2*x^7 + 5*a^11*b*x^5 + a^12*x^3), 1/3840*(45045*b^6*x^12 + 210210*a*b^5*x^10 + 384384*a^
2*b^4*x^8 + 338910*a^3*b^3*x^6 + 137995*a^4*b^2*x^4 + 16640*a^5*b*x^2 - 1280*a^6 + 45045*(b^6*x^13 + 5*a*b^5*x
^11 + 10*a^2*b^4*x^9 + 10*a^3*b^3*x^7 + 5*a^4*b^2*x^5 + a^5*b*x^3)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^7*b^5*x^1
3 + 5*a^8*b^4*x^11 + 10*a^9*b^3*x^9 + 10*a^10*b^2*x^7 + 5*a^11*b*x^5 + a^12*x^3)]

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Sympy [A]  time = 9.09644, size = 209, normalized size = 1.45 \begin{align*} - \frac{3003 \sqrt{- \frac{b^{3}}{a^{15}}} \log{\left (- \frac{a^{8} \sqrt{- \frac{b^{3}}{a^{15}}}}{b^{2}} + x \right )}}{512} + \frac{3003 \sqrt{- \frac{b^{3}}{a^{15}}} \log{\left (\frac{a^{8} \sqrt{- \frac{b^{3}}{a^{15}}}}{b^{2}} + x \right )}}{512} + \frac{- 1280 a^{6} + 16640 a^{5} b x^{2} + 137995 a^{4} b^{2} x^{4} + 338910 a^{3} b^{3} x^{6} + 384384 a^{2} b^{4} x^{8} + 210210 a b^{5} x^{10} + 45045 b^{6} x^{12}}{3840 a^{12} x^{3} + 19200 a^{11} b x^{5} + 38400 a^{10} b^{2} x^{7} + 38400 a^{9} b^{3} x^{9} + 19200 a^{8} b^{4} x^{11} + 3840 a^{7} b^{5} x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3003*sqrt(-b**3/a**15)*log(-a**8*sqrt(-b**3/a**15)/b**2 + x)/512 + 3003*sqrt(-b**3/a**15)*log(a**8*sqrt(-b**3
/a**15)/b**2 + x)/512 + (-1280*a**6 + 16640*a**5*b*x**2 + 137995*a**4*b**2*x**4 + 338910*a**3*b**3*x**6 + 3843
84*a**2*b**4*x**8 + 210210*a*b**5*x**10 + 45045*b**6*x**12)/(3840*a**12*x**3 + 19200*a**11*b*x**5 + 38400*a**1
0*b**2*x**7 + 38400*a**9*b**3*x**9 + 19200*a**8*b**4*x**11 + 3840*a**7*b**5*x**13)

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Giac [A]  time = 1.14116, size = 140, normalized size = 0.97 \begin{align*} \frac{3003 \, b^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} a^{7}} + \frac{18 \, b x^{2} - a}{3 \, a^{7} x^{3}} + \frac{22005 \, b^{6} x^{9} + 96290 \, a b^{5} x^{7} + 160384 \, a^{2} b^{4} x^{5} + 121310 \, a^{3} b^{3} x^{3} + 35595 \, a^{4} b^{2} x}{3840 \,{\left (b x^{2} + a\right )}^{5} a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3003/256*b^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^7) + 1/3*(18*b*x^2 - a)/(a^7*x^3) + 1/3840*(22005*b^6*x^9 + 96
290*a*b^5*x^7 + 160384*a^2*b^4*x^5 + 121310*a^3*b^3*x^3 + 35595*a^4*b^2*x)/((b*x^2 + a)^5*a^7)