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

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

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 144, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.06, size = 113, normalized size = 0.78 \[ \frac {\frac {\sqrt {a} \left (-1280 a^6+16640 a^5 b x^2+137995 a^4 b^2 x^4+338910 a^3 b^3 x^6+384384 a^2 b^4 x^8+210210 a b^5 x^{10}+45045 b^6 x^{12}\right )}{x^3 \left (a+b x^2\right )^5}+45045 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3840 a^{15/2}} \]

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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fricas [A]  time = 1.02, size = 436, normalized size = 3.03 \[ \left [\frac {90090 \, b^{6} x^{12} + 420420 \, a b^{5} x^{10} + 768768 \, a^{2} b^{4} x^{8} + 677820 \, a^{3} b^{3} x^{6} + 275990 \, a^{4} b^{2} x^{4} + 33280 \, a^{5} b x^{2} - 2560 \, a^{6} + 45045 \, {\left (b^{6} x^{13} + 5 \, a b^{5} x^{11} + 10 \, a^{2} b^{4} x^{9} + 10 \, a^{3} b^{3} x^{7} + 5 \, a^{4} b^{2} x^{5} + a^{5} b x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{7680 \, {\left (a^{7} b^{5} x^{13} + 5 \, a^{8} b^{4} x^{11} + 10 \, a^{9} b^{3} x^{9} + 10 \, a^{10} b^{2} x^{7} + 5 \, a^{11} b x^{5} + a^{12} x^{3}\right )}}, \frac {45045 \, b^{6} x^{12} + 210210 \, a b^{5} x^{10} + 384384 \, a^{2} b^{4} x^{8} + 338910 \, a^{3} b^{3} x^{6} + 137995 \, a^{4} b^{2} x^{4} + 16640 \, a^{5} b x^{2} - 1280 \, a^{6} + 45045 \, {\left (b^{6} x^{13} + 5 \, a b^{5} x^{11} + 10 \, a^{2} b^{4} x^{9} + 10 \, a^{3} b^{3} x^{7} + 5 \, a^{4} b^{2} x^{5} + a^{5} b x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{3840 \, {\left (a^{7} b^{5} x^{13} + 5 \, a^{8} b^{4} x^{11} + 10 \, a^{9} b^{3} x^{9} + 10 \, a^{10} b^{2} x^{7} + 5 \, a^{11} b x^{5} + a^{12} x^{3}\right )}}\right ] \]

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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giac [A]  time = 0.16, size = 104, normalized size = 0.72 \[ \frac {3003 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{7}} + \frac {18 \, b x^{2} - a}{3 \, a^{7} x^{3}} + \frac {22005 \, b^{6} x^{9} + 96290 \, a b^{5} x^{7} + 160384 \, a^{2} b^{4} x^{5} + 121310 \, a^{3} b^{3} x^{3} + 35595 \, a^{4} b^{2} x}{3840 \, {\left (b x^{2} + a\right )}^{5} a^{7}} \]

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)

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maple [A]  time = 0.02, size = 139, normalized size = 0.97 \[ \frac {1467 b^{6} x^{9}}{256 \left (b \,x^{2}+a \right )^{5} a^{7}}+\frac {9629 b^{5} x^{7}}{384 \left (b \,x^{2}+a \right )^{5} a^{6}}+\frac {1253 b^{4} x^{5}}{30 \left (b \,x^{2}+a \right )^{5} a^{5}}+\frac {12131 b^{3} x^{3}}{384 \left (b \,x^{2}+a \right )^{5} a^{4}}+\frac {2373 b^{2} x}{256 \left (b \,x^{2}+a \right )^{5} a^{3}}+\frac {3003 b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \sqrt {a b}\, a^{7}}+\frac {6 b}{a^{7} x}-\frac {1}{3 a^{6} x^{3}} \]

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(1/(a*b)^(1/2)*b*x)

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maxima [A]  time = 3.10, size = 152, normalized size = 1.06 \[ \frac {45045 \, b^{6} x^{12} + 210210 \, a b^{5} x^{10} + 384384 \, a^{2} b^{4} x^{8} + 338910 \, a^{3} b^{3} x^{6} + 137995 \, a^{4} b^{2} x^{4} + 16640 \, a^{5} b x^{2} - 1280 \, a^{6}}{3840 \, {\left (a^{7} b^{5} x^{13} + 5 \, a^{8} b^{4} x^{11} + 10 \, a^{9} b^{3} x^{9} + 10 \, a^{10} b^{2} x^{7} + 5 \, a^{11} b x^{5} + a^{12} x^{3}\right )}} + \frac {3003 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{7}} \]

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]

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 + 16
640*a^5*b*x^2 - 1280*a^6)/(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) + 3003/256*b^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^7)

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mupad [B]  time = 4.62, size = 146, normalized size = 1.01 \[ \frac {\frac {13\,b\,x^2}{3\,a^2}-\frac {1}{3\,a}+\frac {27599\,b^2\,x^4}{768\,a^3}+\frac {11297\,b^3\,x^6}{128\,a^4}+\frac {1001\,b^4\,x^8}{10\,a^5}+\frac {7007\,b^5\,x^{10}}{128\,a^6}+\frac {3003\,b^6\,x^{12}}{256\,a^7}}{a^5\,x^3+5\,a^4\,b\,x^5+10\,a^3\,b^2\,x^7+10\,a^2\,b^3\,x^9+5\,a\,b^4\,x^{11}+b^5\,x^{13}}+\frac {3003\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,a^{15/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((13*b*x^2)/(3*a^2) - 1/(3*a) + (27599*b^2*x^4)/(768*a^3) + (11297*b^3*x^6)/(128*a^4) + (1001*b^4*x^8)/(10*a^5
) + (7007*b^5*x^10)/(128*a^6) + (3003*b^6*x^12)/(256*a^7))/(a^5*x^3 + b^5*x^13 + 5*a^4*b*x^5 + 5*a*b^4*x^11 +
10*a^3*b^2*x^7 + 10*a^2*b^3*x^9) + (3003*b^(3/2)*atan((b^(1/2)*x)/a^(1/2)))/(256*a^(15/2))

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sympy [A]  time = 0.89, size = 209, normalized size = 1.45 \[ - \frac {3003 \sqrt {- \frac {b^{3}}{a^{15}}} \log {\left (- \frac {a^{8} \sqrt {- \frac {b^{3}}{a^{15}}}}{b^{2}} + x \right )}}{512} + \frac {3003 \sqrt {- \frac {b^{3}}{a^{15}}} \log {\left (\frac {a^{8} \sqrt {- \frac {b^{3}}{a^{15}}}}{b^{2}} + x \right )}}{512} + \frac {- 1280 a^{6} + 16640 a^{5} b x^{2} + 137995 a^{4} b^{2} x^{4} + 338910 a^{3} b^{3} x^{6} + 384384 a^{2} b^{4} x^{8} + 210210 a b^{5} x^{10} + 45045 b^{6} x^{12}}{3840 a^{12} x^{3} + 19200 a^{11} b x^{5} + 38400 a^{10} b^{2} x^{7} + 38400 a^{9} b^{3} x^{9} + 19200 a^{8} b^{4} x^{11} + 3840 a^{7} b^{5} x^{13}} \]

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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