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

Optimal. Leaf size=116 \[ -\frac{5 b}{2 a^6 \left (a+b x^2\right )}-\frac{b}{a^5 \left (a+b x^2\right )^2}-\frac{b}{2 a^4 \left (a+b x^2\right )^3}-\frac{b}{4 a^3 \left (a+b x^2\right )^4}-\frac{b}{10 a^2 \left (a+b x^2\right )^5}+\frac{3 b \log \left (a+b x^2\right )}{a^7}-\frac{6 b \log (x)}{a^7}-\frac{1}{2 a^6 x^2} \]

[Out]

-1/(2*a^6*x^2) - b/(10*a^2*(a + b*x^2)^5) - b/(4*a^3*(a + b*x^2)^4) - b/(2*a^4*(a + b*x^2)^3) - b/(a^5*(a + b*
x^2)^2) - (5*b)/(2*a^6*(a + b*x^2)) - (6*b*Log[x])/a^7 + (3*b*Log[a + b*x^2])/a^7

________________________________________________________________________________________

Rubi [A]  time = 0.125673, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 266, 44} \[ -\frac{5 b}{2 a^6 \left (a+b x^2\right )}-\frac{b}{a^5 \left (a+b x^2\right )^2}-\frac{b}{2 a^4 \left (a+b x^2\right )^3}-\frac{b}{4 a^3 \left (a+b x^2\right )^4}-\frac{b}{10 a^2 \left (a+b x^2\right )^5}+\frac{3 b \log \left (a+b x^2\right )}{a^7}-\frac{6 b \log (x)}{a^7}-\frac{1}{2 a^6 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*a^6*x^2) - b/(10*a^2*(a + b*x^2)^5) - b/(4*a^3*(a + b*x^2)^4) - b/(2*a^4*(a + b*x^2)^3) - b/(a^5*(a + b*
x^2)^2) - (5*b)/(2*a^6*(a + b*x^2)) - (6*b*Log[x])/a^7 + (3*b*Log[a + b*x^2])/a^7

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 266

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]]

Rule 44

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] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{1}{x^3 \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac{1}{2} b^6 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a b+b^2 x\right )^6} \, dx,x,x^2\right )\\ &=\frac{1}{2} b^6 \operatorname{Subst}\left (\int \left (\frac{1}{a^6 b^6 x^2}-\frac{6}{a^7 b^5 x}+\frac{1}{a^2 b^4 (a+b x)^6}+\frac{2}{a^3 b^4 (a+b x)^5}+\frac{3}{a^4 b^4 (a+b x)^4}+\frac{4}{a^5 b^4 (a+b x)^3}+\frac{5}{a^6 b^4 (a+b x)^2}+\frac{6}{a^7 b^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 a^6 x^2}-\frac{b}{10 a^2 \left (a+b x^2\right )^5}-\frac{b}{4 a^3 \left (a+b x^2\right )^4}-\frac{b}{2 a^4 \left (a+b x^2\right )^3}-\frac{b}{a^5 \left (a+b x^2\right )^2}-\frac{5 b}{2 a^6 \left (a+b x^2\right )}-\frac{6 b \log (x)}{a^7}+\frac{3 b \log \left (a+b x^2\right )}{a^7}\\ \end{align*}

Mathematica [A]  time = 0.0805515, size = 92, normalized size = 0.79 \[ -\frac{\frac{a \left (470 a^2 b^3 x^6+385 a^3 b^2 x^4+137 a^4 b x^2+10 a^5+270 a b^4 x^8+60 b^5 x^{10}\right )}{x^2 \left (a+b x^2\right )^5}-60 b \log \left (a+b x^2\right )+120 b \log (x)}{20 a^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(10*a^5 + 137*a^4*b*x^2 + 385*a^3*b^2*x^4 + 470*a^2*b^3*x^6 + 270*a*b^4*x^8 + 60*b^5*x^10))/(x^2*(a + b*x
^2)^5) + 120*b*Log[x] - 60*b*Log[a + b*x^2])/(20*a^7)

________________________________________________________________________________________

Maple [A]  time = 0.057, size = 107, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}{a}^{6}}}-{\frac{b}{10\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{b}{4\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{4}}}-{\frac{b}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{b}{{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,b}{2\,{a}^{6} \left ( b{x}^{2}+a \right ) }}-6\,{\frac{b\ln \left ( x \right ) }{{a}^{7}}}+3\,{\frac{b\ln \left ( b{x}^{2}+a \right ) }{{a}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/x^2/a^6-1/10*b/a^2/(b*x^2+a)^5-1/4*b/a^3/(b*x^2+a)^4-1/2*b/a^4/(b*x^2+a)^3-b/a^5/(b*x^2+a)^2-5/2*b/a^6/(b
*x^2+a)-6*b*ln(x)/a^7+3*b*ln(b*x^2+a)/a^7

________________________________________________________________________________________

Maxima [A]  time = 1.02722, size = 193, normalized size = 1.66 \begin{align*} -\frac{60 \, b^{5} x^{10} + 270 \, a b^{4} x^{8} + 470 \, a^{2} b^{3} x^{6} + 385 \, a^{3} b^{2} x^{4} + 137 \, a^{4} b x^{2} + 10 \, a^{5}}{20 \,{\left (a^{6} b^{5} x^{12} + 5 \, a^{7} b^{4} x^{10} + 10 \, a^{8} b^{3} x^{8} + 10 \, a^{9} b^{2} x^{6} + 5 \, a^{10} b x^{4} + a^{11} x^{2}\right )}} + \frac{3 \, b \log \left (b x^{2} + a\right )}{a^{7}} - \frac{3 \, b \log \left (x^{2}\right )}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(60*b^5*x^10 + 270*a*b^4*x^8 + 470*a^2*b^3*x^6 + 385*a^3*b^2*x^4 + 137*a^4*b*x^2 + 10*a^5)/(a^6*b^5*x^12
+ 5*a^7*b^4*x^10 + 10*a^8*b^3*x^8 + 10*a^9*b^2*x^6 + 5*a^10*b*x^4 + a^11*x^2) + 3*b*log(b*x^2 + a)/a^7 - 3*b*
log(x^2)/a^7

________________________________________________________________________________________

Fricas [B]  time = 1.47088, size = 545, normalized size = 4.7 \begin{align*} -\frac{60 \, a b^{5} x^{10} + 270 \, a^{2} b^{4} x^{8} + 470 \, a^{3} b^{3} x^{6} + 385 \, a^{4} b^{2} x^{4} + 137 \, a^{5} b x^{2} + 10 \, a^{6} - 60 \,{\left (b^{6} x^{12} + 5 \, a b^{5} x^{10} + 10 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 5 \, a^{4} b^{2} x^{4} + a^{5} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 120 \,{\left (b^{6} x^{12} + 5 \, a b^{5} x^{10} + 10 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 5 \, a^{4} b^{2} x^{4} + a^{5} b x^{2}\right )} \log \left (x\right )}{20 \,{\left (a^{7} b^{5} x^{12} + 5 \, a^{8} b^{4} x^{10} + 10 \, a^{9} b^{3} x^{8} + 10 \, a^{10} b^{2} x^{6} + 5 \, a^{11} b x^{4} + a^{12} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(60*a*b^5*x^10 + 270*a^2*b^4*x^8 + 470*a^3*b^3*x^6 + 385*a^4*b^2*x^4 + 137*a^5*b*x^2 + 10*a^6 - 60*(b^6*
x^12 + 5*a*b^5*x^10 + 10*a^2*b^4*x^8 + 10*a^3*b^3*x^6 + 5*a^4*b^2*x^4 + a^5*b*x^2)*log(b*x^2 + a) + 120*(b^6*x
^12 + 5*a*b^5*x^10 + 10*a^2*b^4*x^8 + 10*a^3*b^3*x^6 + 5*a^4*b^2*x^4 + a^5*b*x^2)*log(x))/(a^7*b^5*x^12 + 5*a^
8*b^4*x^10 + 10*a^9*b^3*x^8 + 10*a^10*b^2*x^6 + 5*a^11*b*x^4 + a^12*x^2)

________________________________________________________________________________________

Sympy [A]  time = 6.15536, size = 148, normalized size = 1.28 \begin{align*} - \frac{10 a^{5} + 137 a^{4} b x^{2} + 385 a^{3} b^{2} x^{4} + 470 a^{2} b^{3} x^{6} + 270 a b^{4} x^{8} + 60 b^{5} x^{10}}{20 a^{11} x^{2} + 100 a^{10} b x^{4} + 200 a^{9} b^{2} x^{6} + 200 a^{8} b^{3} x^{8} + 100 a^{7} b^{4} x^{10} + 20 a^{6} b^{5} x^{12}} - \frac{6 b \log{\left (x \right )}}{a^{7}} + \frac{3 b \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(10*a**5 + 137*a**4*b*x**2 + 385*a**3*b**2*x**4 + 470*a**2*b**3*x**6 + 270*a*b**4*x**8 + 60*b**5*x**10)/(20*a
**11*x**2 + 100*a**10*b*x**4 + 200*a**9*b**2*x**6 + 200*a**8*b**3*x**8 + 100*a**7*b**4*x**10 + 20*a**6*b**5*x*
*12) - 6*b*log(x)/a**7 + 3*b*log(a/b + x**2)/a**7

________________________________________________________________________________________

Giac [A]  time = 1.16342, size = 155, normalized size = 1.34 \begin{align*} -\frac{3 \, b \log \left (x^{2}\right )}{a^{7}} + \frac{3 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{a^{7}} + \frac{6 \, b x^{2} - a}{2 \, a^{7} x^{2}} - \frac{137 \, b^{6} x^{10} + 735 \, a b^{5} x^{8} + 1590 \, a^{2} b^{4} x^{6} + 1740 \, a^{3} b^{3} x^{4} + 970 \, a^{4} b^{2} x^{2} + 224 \, a^{5} b}{20 \,{\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^3/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

-3*b*log(x^2)/a^7 + 3*b*log(abs(b*x^2 + a))/a^7 + 1/2*(6*b*x^2 - a)/(a^7*x^2) - 1/20*(137*b^6*x^10 + 735*a*b^5
*x^8 + 1590*a^2*b^4*x^6 + 1740*a^3*b^3*x^4 + 970*a^4*b^2*x^2 + 224*a^5*b)/((b*x^2 + a)^5*a^7)