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

Optimal. Leaf size=102 \[ \frac{1}{2 a^5 \left (a+b x^2\right )}+\frac{1}{4 a^4 \left (a+b x^2\right )^2}+\frac{1}{6 a^3 \left (a+b x^2\right )^3}+\frac{1}{8 a^2 \left (a+b x^2\right )^4}-\frac{\log \left (a+b x^2\right )}{2 a^6}+\frac{\log (x)}{a^6}+\frac{1}{10 a \left (a+b x^2\right )^5} \]

[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)

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Rubi [A]  time = 0.105126, antiderivative size = 102, 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} \[ \frac{1}{2 a^5 \left (a+b x^2\right )}+\frac{1}{4 a^4 \left (a+b x^2\right )^2}+\frac{1}{6 a^3 \left (a+b x^2\right )^3}+\frac{1}{8 a^2 \left (a+b x^2\right )^4}-\frac{\log \left (a+b x^2\right )}{2 a^6}+\frac{\log (x)}{a^6}+\frac{1}{10 a \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[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 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 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{1}{x \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac{1}{2} b^6 \operatorname{Subst}\left (\int \frac{1}{x \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}-\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]  time = 0.0490998, size = 76, normalized size = 0.75 \[ \frac{\frac{a \left (470 a^2 b^2 x^4+385 a^3 b x^2+137 a^4+270 a b^3 x^6+60 b^4 x^8\right )}{\left (a+b x^2\right )^5}-60 \log \left (a+b x^2\right )+120 \log (x)}{120 a^6} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.056, size = 91, normalized size = 0.9 \begin{align*}{\frac{1}{10\,a \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{1}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{1}{6\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{1}{4\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{1}{2\,{a}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{6}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]  time = 1.3507, size = 170, normalized size = 1.67 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.50483, size = 489, normalized size = 4.79 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 2.73205, size = 128, normalized size = 1.25 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.14841, size = 124, normalized size = 1.22 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)