\(\int \frac {1}{x^3 (a^2+2 a b x^2+b^2 x^4)^2} \, dx\) [500]

Optimal result

Integrand size = 24, antiderivative size = 84 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {1}{2 a^4 x^2}-\frac {b}{6 a^2 \left (a+b x^2\right )^3}-\frac {b}{2 a^3 \left (a+b x^2\right )^2}-\frac {3 b}{2 a^4 \left (a+b x^2\right )}-\frac {4 b \log (x)}{a^5}+\frac {2 b \log \left (a+b x^2\right )}{a^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 84, 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} \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {2 b \log \left (a+b x^2\right )}{a^5}-\frac {4 b \log (x)}{a^5}-\frac {3 b}{2 a^4 \left (a+b x^2\right )}-\frac {1}{2 a^4 x^2}-\frac {b}{2 a^3 \left (a+b x^2\right )^2}-\frac {b}{6 a^2 \left (a+b x^2\right )^3} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {\frac {a \left (3 a^3+22 a^2 b x^2+30 a b^2 x^4+12 b^3 x^6\right )}{x^2 \left (a+b x^2\right )^3}+24 b \log (x)-12 b \log \left (a+b x^2\right )}{6 a^5} \]

[In]

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

[Out]

-1/6*((a*(3*a^3 + 22*a^2*b*x^2 + 30*a*b^2*x^4 + 12*b^3*x^6))/(x^2*(a + b*x^2)^3) + 24*b*Log[x] - 12*b*Log[a +
b*x^2])/a^5

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.90

method	result	size
norman	\(\frac {-\frac {1}{2 a}+\frac {6 b^{2} x^{4}}{a^{3}}+\frac {9 b^{3} x^{6}}{a^{4}}+\frac {11 b^{4} x^{8}}{3 a^{5}}}{x^{2} \left (b \,x^{2}+a \right )^{3}}-\frac {4 b \ln \left (x \right )}{a^{5}}+\frac {2 b \ln \left (b \,x^{2}+a \right )}{a^{5}}\)	\(76\)
default	\(-\frac {1}{2 a^{4} x^{2}}-\frac {4 b \ln \left (x \right )}{a^{5}}+\frac {b^{2} \left (\frac {4 \ln \left (b \,x^{2}+a \right )}{b}-\frac {a^{2}}{b \left (b \,x^{2}+a \right )^{2}}-\frac {a^{3}}{3 b \left (b \,x^{2}+a \right )^{3}}-\frac {3 a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{5}}\)	\(89\)
risch	\(\frac {-\frac {2 b^{3} x^{6}}{a^{4}}-\frac {5 b^{2} x^{4}}{a^{3}}-\frac {11 b \,x^{2}}{3 a^{2}}-\frac {1}{2 a}}{x^{2} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right ) \left (b \,x^{2}+a \right )}-\frac {4 b \ln \left (x \right )}{a^{5}}+\frac {2 b \ln \left (-b \,x^{2}-a \right )}{a^{5}}\)	\(97\)
parallelrisch	\(-\frac {24 b^{4} \ln \left (x \right ) x^{8}-12 \ln \left (b \,x^{2}+a \right ) x^{8} b^{4}-22 b^{4} x^{8}+72 a \,b^{3} \ln \left (x \right ) x^{6}-36 \ln \left (b \,x^{2}+a \right ) x^{6} a \,b^{3}-54 a \,b^{3} x^{6}+72 a^{2} b^{2} \ln \left (x \right ) x^{4}-36 \ln \left (b \,x^{2}+a \right ) x^{4} a^{2} b^{2}-36 a^{2} b^{2} x^{4}+24 a^{3} b \ln \left (x \right ) x^{2}-12 \ln \left (b \,x^{2}+a \right ) x^{2} a^{3} b +3 a^{4}}{6 a^{5} x^{2} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right ) \left (b \,x^{2}+a \right )}\)	\(186\)

[In]

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

[Out]

(-1/2/a+6*b^2/a^3*x^4+9*b^3/a^4*x^6+11/3*b^4/a^5*x^8)/x^2/(b*x^2+a)^3-4*b*ln(x)/a^5+2*b*ln(b*x^2+a)/a^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (76) = 152\).

Time = 0.25 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.94 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {12 \, a b^{3} x^{6} + 30 \, a^{2} b^{2} x^{4} + 22 \, a^{3} b x^{2} + 3 \, a^{4} - 12 \, {\left (b^{4} x^{8} + 3 \, a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + a^{3} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 24 \, {\left (b^{4} x^{8} + 3 \, a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + a^{3} b x^{2}\right )} \log \left (x\right )}{6 \, {\left (a^{5} b^{3} x^{8} + 3 \, a^{6} b^{2} x^{6} + 3 \, a^{7} b x^{4} + a^{8} x^{2}\right )}} \]

[In]

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

[Out]

-1/6*(12*a*b^3*x^6 + 30*a^2*b^2*x^4 + 22*a^3*b*x^2 + 3*a^4 - 12*(b^4*x^8 + 3*a*b^3*x^6 + 3*a^2*b^2*x^4 + a^3*b
*x^2)*log(b*x^2 + a) + 24*(b^4*x^8 + 3*a*b^3*x^6 + 3*a^2*b^2*x^4 + a^3*b*x^2)*log(x))/(a^5*b^3*x^8 + 3*a^6*b^2
*x^6 + 3*a^7*b*x^4 + a^8*x^2)

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {- 3 a^{3} - 22 a^{2} b x^{2} - 30 a b^{2} x^{4} - 12 b^{3} x^{6}}{6 a^{7} x^{2} + 18 a^{6} b x^{4} + 18 a^{5} b^{2} x^{6} + 6 a^{4} b^{3} x^{8}} - \frac {4 b \log {\left (x \right )}}{a^{5}} + \frac {2 b \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{5}} \]

[In]

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

[Out]

(-3*a**3 - 22*a**2*b*x**2 - 30*a*b**2*x**4 - 12*b**3*x**6)/(6*a**7*x**2 + 18*a**6*b*x**4 + 18*a**5*b**2*x**6 +
6*a**4*b**3*x**8) - 4*b*log(x)/a**5 + 2*b*log(a/b + x**2)/a**5

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {12 \, b^{3} x^{6} + 30 \, a b^{2} x^{4} + 22 \, a^{2} b x^{2} + 3 \, a^{3}}{6 \, {\left (a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{6} + 3 \, a^{6} b x^{4} + a^{7} x^{2}\right )}} + \frac {2 \, b \log \left (b x^{2} + a\right )}{a^{5}} - \frac {2 \, b \log \left (x^{2}\right )}{a^{5}} \]

[In]

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

[Out]

-1/6*(12*b^3*x^6 + 30*a*b^2*x^4 + 22*a^2*b*x^2 + 3*a^3)/(a^4*b^3*x^8 + 3*a^5*b^2*x^6 + 3*a^6*b*x^4 + a^7*x^2)
+ 2*b*log(b*x^2 + a)/a^5 - 2*b*log(x^2)/a^5

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {2 \, b \log \left (x^{2}\right )}{a^{5}} + \frac {2 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{a^{5}} + \frac {4 \, b x^{2} - a}{2 \, a^{5} x^{2}} - \frac {22 \, b^{4} x^{6} + 75 \, a b^{3} x^{4} + 87 \, a^{2} b^{2} x^{2} + 35 \, a^{3} b}{6 \, {\left (b x^{2} + a\right )}^{3} a^{5}} \]

[In]

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

[Out]

-2*b*log(x^2)/a^5 + 2*b*log(abs(b*x^2 + a))/a^5 + 1/2*(4*b*x^2 - a)/(a^5*x^2) - 1/6*(22*b^4*x^6 + 75*a*b^3*x^4
+ 87*a^2*b^2*x^2 + 35*a^3*b)/((b*x^2 + a)^3*a^5)

Mupad [B] (verification not implemented)

Time = 13.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {2\,b\,\ln \left (b\,x^2+a\right )}{a^5}-\frac {\frac {1}{2\,a}+\frac {11\,b\,x^2}{3\,a^2}+\frac {5\,b^2\,x^4}{a^3}+\frac {2\,b^3\,x^6}{a^4}}{a^3\,x^2+3\,a^2\,b\,x^4+3\,a\,b^2\,x^6+b^3\,x^8}-\frac {4\,b\,\ln \left (x\right )}{a^5} \]

[In]

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

[Out]

(2*b*log(a + b*x^2))/a^5 - (1/(2*a) + (11*b*x^2)/(3*a^2) + (5*b^2*x^4)/a^3 + (2*b^3*x^6)/a^4)/(a^3*x^2 + b^3*x
^8 + 3*a^2*b*x^4 + 3*a*b^2*x^6) - (4*b*log(x))/a^5