\(\int \frac {1}{x (a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [520]

Optimal result

Integrand size = 24, antiderivative size = 102 \[ \int \frac {1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {1}{10 a \left (a+b x^2\right )^5}+\frac {1}{8 a^2 \left (a+b x^2\right )^4}+\frac {1}{6 a^3 \left (a+b x^2\right )^3}+\frac {1}{4 a^4 \left (a+b x^2\right )^2}+\frac {1}{2 a^5 \left (a+b x^2\right )}+\frac {\log (x)}{a^6}-\frac {\log \left (a+b x^2\right )}{2 a^6} \]

[Out]

1/10/a/(b*x^2+a)^5+1/8/a^2/(b*x^2+a)^4+1/6/a^3/(b*x^2+a)^3+1/4/a^4/(b*x^2+a)^2+1/2/a^5/(b*x^2+a)+ln(x)/a^6-1/2
*ln(b*x^2+a)/a^6

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 272, 46} \[ \int \frac {1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\log \left (a+b x^2\right )}{2 a^6}+\frac {\log (x)}{a^6}+\frac {1}{2 a^5 \left (a+b x^2\right )}+\frac {1}{4 a^4 \left (a+b x^2\right )^2}+\frac {1}{6 a^3 \left (a+b x^2\right )^3}+\frac {1}{8 a^2 \left (a+b x^2\right )^4}+\frac {1}{10 a \left (a+b x^2\right )^5} \]

[In]

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

[Out]

1/(10*a*(a + b*x^2)^5) + 1/(8*a^2*(a + b*x^2)^4) + 1/(6*a^3*(a + b*x^2)^3) + 1/(4*a^4*(a + b*x^2)^2) + 1/(2*a^
5*(a + b*x^2)) + Log[x]/a^6 - Log[a + b*x^2]/(2*a^6)

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 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = b^6 \int \frac {1}{x \left (a b+b^2 x^2\right )^6} \, dx \\ & = \frac {1}{2} b^6 \text {Subst}\left (\int \frac {1}{x \left (a b+b^2 x\right )^6} \, dx,x,x^2\right ) \\ & = \frac {1}{2} b^6 \text {Subst}\left (\int \left (\frac {1}{a^6 b^6 x}-\frac {1}{a b^5 (a+b x)^6}-\frac {1}{a^2 b^5 (a+b x)^5}-\frac {1}{a^3 b^5 (a+b x)^4}-\frac {1}{a^4 b^5 (a+b x)^3}-\frac {1}{a^5 b^5 (a+b x)^2}-\frac {1}{a^6 b^5 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{10 a \left (a+b x^2\right )^5}+\frac {1}{8 a^2 \left (a+b x^2\right )^4}+\frac {1}{6 a^3 \left (a+b x^2\right )^3}+\frac {1}{4 a^4 \left (a+b x^2\right )^2}+\frac {1}{2 a^5 \left (a+b x^2\right )}+\frac {\log (x)}{a^6}-\frac {\log \left (a+b x^2\right )}{2 a^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {a \left (137 a^4+385 a^3 b x^2+470 a^2 b^2 x^4+270 a b^3 x^6+60 b^4 x^8\right )}{\left (a+b x^2\right )^5}+120 \log (x)-60 \log \left (a+b x^2\right )}{120 a^6} \]

[In]

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

[Out]

((a*(137*a^4 + 385*a^3*b*x^2 + 470*a^2*b^2*x^4 + 270*a*b^3*x^6 + 60*b^4*x^8))/(a + b*x^2)^5 + 120*Log[x] - 60*
Log[a + b*x^2])/(120*a^6)

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.83

method	result	size
norman	\(\frac {-\frac {5 b \,x^{2}}{2 a^{2}}-\frac {15 b^{2} x^{4}}{2 a^{3}}-\frac {55 b^{3} x^{6}}{6 a^{4}}-\frac {125 b^{4} x^{8}}{24 a^{5}}-\frac {137 b^{5} x^{10}}{120 a^{6}}}{\left (b \,x^{2}+a \right )^{5}}+\frac {\ln \left (x \right )}{a^{6}}-\frac {\ln \left (b \,x^{2}+a \right )}{2 a^{6}}\)	\(85\)
risch	\(\frac {\frac {b^{4} x^{8}}{2 a^{5}}+\frac {9 b^{3} x^{6}}{4 a^{4}}+\frac {47 b^{2} x^{4}}{12 a^{3}}+\frac {77 b \,x^{2}}{24 a^{2}}+\frac {137}{120 a}}{\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}+\frac {\ln \left (x \right )}{a^{6}}-\frac {\ln \left (b \,x^{2}+a \right )}{2 a^{6}}\)	\(99\)
default	\(\frac {\ln \left (x \right )}{a^{6}}-\frac {b \left (-\frac {a^{4}}{4 b \left (b \,x^{2}+a \right )^{4}}-\frac {a^{5}}{5 b \left (b \,x^{2}+a \right )^{5}}+\frac {\ln \left (b \,x^{2}+a \right )}{b}-\frac {a^{3}}{3 b \left (b \,x^{2}+a \right )^{3}}-\frac {a^{2}}{2 b \left (b \,x^{2}+a \right )^{2}}-\frac {a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{6}}\)	\(110\)
parallelrisch	\(\frac {120 \ln \left (x \right ) x^{10} b^{5}-60 \ln \left (b \,x^{2}+a \right ) x^{10} b^{5}-137 x^{10} b^{5}+600 \ln \left (x \right ) x^{8} a \,b^{4}-300 \ln \left (b \,x^{2}+a \right ) x^{8} a \,b^{4}-625 a \,x^{8} b^{4}+1200 \ln \left (x \right ) x^{6} a^{2} b^{3}-600 \ln \left (b \,x^{2}+a \right ) x^{6} a^{2} b^{3}-1100 a^{2} x^{6} b^{3}+1200 \ln \left (x \right ) x^{4} a^{3} b^{2}-600 \ln \left (b \,x^{2}+a \right ) x^{4} a^{3} b^{2}-900 a^{3} x^{4} b^{2}+600 \ln \left (x \right ) x^{2} a^{4} b -300 \ln \left (b \,x^{2}+a \right ) x^{2} a^{4} b -300 x^{2} a^{4} b +120 \ln \left (x \right ) a^{5}-60 \ln \left (b \,x^{2}+a \right ) a^{5}}{120 a^{6} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} \left (b \,x^{2}+a \right )}\)	\(250\)

[In]

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

[Out]

(-5/2*b/a^2*x^2-15/2*b^2/a^3*x^4-55/6*b^3/a^4*x^6-125/24*b^4/a^5*x^8-137/120*b^5/a^6*x^10)/(b*x^2+a)^5+ln(x)/a
^6-1/2*ln(b*x^2+a)/a^6

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (90) = 180\).

Time = 0.25 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.18 \[ \int \frac {1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {60 \, a b^{4} x^{8} + 270 \, a^{2} b^{3} x^{6} + 470 \, a^{3} b^{2} x^{4} + 385 \, a^{4} b x^{2} + 137 \, a^{5} - 60 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right ) + 120 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \log \left (x\right )}{120 \, {\left (a^{6} b^{5} x^{10} + 5 \, a^{7} b^{4} x^{8} + 10 \, a^{8} b^{3} x^{6} + 10 \, a^{9} b^{2} x^{4} + 5 \, a^{10} b x^{2} + a^{11}\right )}} \]

[In]

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

[Out]

1/120*(60*a*b^4*x^8 + 270*a^2*b^3*x^6 + 470*a^3*b^2*x^4 + 385*a^4*b*x^2 + 137*a^5 - 60*(b^5*x^10 + 5*a*b^4*x^8
+ 10*a^2*b^3*x^6 + 10*a^3*b^2*x^4 + 5*a^4*b*x^2 + a^5)*log(b*x^2 + a) + 120*(b^5*x^10 + 5*a*b^4*x^8 + 10*a^2*
b^3*x^6 + 10*a^3*b^2*x^4 + 5*a^4*b*x^2 + a^5)*log(x))/(a^6*b^5*x^10 + 5*a^7*b^4*x^8 + 10*a^8*b^3*x^6 + 10*a^9*
b^2*x^4 + 5*a^10*b*x^2 + a^11)

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.25 \[ \int \frac {1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {137 a^{4} + 385 a^{3} b x^{2} + 470 a^{2} b^{2} x^{4} + 270 a b^{3} x^{6} + 60 b^{4} x^{8}}{120 a^{10} + 600 a^{9} b x^{2} + 1200 a^{8} b^{2} x^{4} + 1200 a^{7} b^{3} x^{6} + 600 a^{6} b^{4} x^{8} + 120 a^{5} b^{5} x^{10}} + \frac {\log {\left (x \right )}}{a^{6}} - \frac {\log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{6}} \]

[In]

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

[Out]

(137*a**4 + 385*a**3*b*x**2 + 470*a**2*b**2*x**4 + 270*a*b**3*x**6 + 60*b**4*x**8)/(120*a**10 + 600*a**9*b*x**
2 + 1200*a**8*b**2*x**4 + 1200*a**7*b**3*x**6 + 600*a**6*b**4*x**8 + 120*a**5*b**5*x**10) + log(x)/a**6 - log(
a/b + x**2)/(2*a**6)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {60 \, b^{4} x^{8} + 270 \, a b^{3} x^{6} + 470 \, a^{2} b^{2} x^{4} + 385 \, a^{3} b x^{2} + 137 \, a^{4}}{120 \, {\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )}} - \frac {\log \left (b x^{2} + a\right )}{2 \, a^{6}} + \frac {\log \left (x^{2}\right )}{2 \, a^{6}} \]

[In]

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

[Out]

1/120*(60*b^4*x^8 + 270*a*b^3*x^6 + 470*a^2*b^2*x^4 + 385*a^3*b*x^2 + 137*a^4)/(a^5*b^5*x^10 + 5*a^6*b^4*x^8 +
10*a^7*b^3*x^6 + 10*a^8*b^2*x^4 + 5*a^9*b*x^2 + a^10) - 1/2*log(b*x^2 + a)/a^6 + 1/2*log(x^2)/a^6

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\log \left (x^{2}\right )}{2 \, a^{6}} - \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{6}} + \frac {137 \, b^{5} x^{10} + 745 \, a b^{4} x^{8} + 1640 \, a^{2} b^{3} x^{6} + 1840 \, a^{3} b^{2} x^{4} + 1070 \, a^{4} b x^{2} + 274 \, a^{5}}{120 \, {\left (b x^{2} + a\right )}^{5} a^{6}} \]

[In]

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

[Out]

1/2*log(x^2)/a^6 - 1/2*log(abs(b*x^2 + a))/a^6 + 1/120*(137*b^5*x^10 + 745*a*b^4*x^8 + 1640*a^2*b^3*x^6 + 1840
*a^3*b^2*x^4 + 1070*a^4*b*x^2 + 274*a^5)/((b*x^2 + a)^5*a^6)

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\ln \left (x\right )}{a^6}-\frac {\ln \left (b\,x^2+a\right )}{2\,a^6}+\frac {\frac {137}{120\,a}+\frac {77\,b\,x^2}{24\,a^2}+\frac {47\,b^2\,x^4}{12\,a^3}+\frac {9\,b^3\,x^6}{4\,a^4}+\frac {b^4\,x^8}{2\,a^5}}{a^5+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^6+5\,a\,b^4\,x^8+b^5\,x^{10}} \]

[In]

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

[Out]

log(x)/a^6 - log(a + b*x^2)/(2*a^6) + (137/(120*a) + (77*b*x^2)/(24*a^2) + (47*b^2*x^4)/(12*a^3) + (9*b^3*x^6)
/(4*a^4) + (b^4*x^8)/(2*a^5))/(a^5 + b^5*x^10 + 5*a^4*b*x^2 + 5*a*b^4*x^8 + 10*a^3*b^2*x^4 + 10*a^2*b^3*x^6)