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

Optimal. Leaf size=84 \[ -\frac{3 b}{2 a^4 \left (a+b x^2\right )}-\frac{b}{2 a^3 \left (a+b x^2\right )^2}-\frac{b}{6 a^2 \left (a+b x^2\right )^3}+\frac{2 b \log \left (a+b x^2\right )}{a^5}-\frac{4 b \log (x)}{a^5}-\frac{1}{2 a^4 x^2} \]

[Out]

-1/(2*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

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Rubi [A]  time = 0.0856393, antiderivative size = 84, 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} \[ -\frac{3 b}{2 a^4 \left (a+b x^2\right )}-\frac{b}{2 a^3 \left (a+b x^2\right )^2}-\frac{b}{6 a^2 \left (a+b x^2\right )^3}+\frac{2 b \log \left (a+b x^2\right )}{a^5}-\frac{4 b \log (x)}{a^5}-\frac{1}{2 a^4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*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 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 )^2} \, dx &=b^4 \int \frac{1}{x^3 \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac{1}{2} b^4 \operatorname{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 \operatorname{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]  time = 0.0642186, size = 70, normalized size = 0.83 \[ -\frac{\frac{a \left (22 a^2 b x^2+3 a^3+30 a b^2 x^4+12 b^3 x^6\right )}{x^2 \left (a+b x^2\right )^3}-12 b \log \left (a+b x^2\right )+24 b \log (x)}{6 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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])/(6*a^5)

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Maple [A]  time = 0.056, size = 77, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{a}^{4}{x}^{2}}}-{\frac{b}{6\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{b}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,b}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}-4\,{\frac{b\ln \left ( x \right ) }{{a}^{5}}}+2\,{\frac{b\ln \left ( b{x}^{2}+a \right ) }{{a}^{5}}} \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)^2,x)

[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

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Maxima [A]  time = 1.01153, size = 134, normalized size = 1.6 \begin{align*} -\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}} \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)^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

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Fricas [B]  time = 1.78155, size = 339, normalized size = 4.04 \begin{align*} -\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 )}} \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)^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)

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Sympy [A]  time = 1.27347, size = 100, normalized size = 1.19 \begin{align*} - \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}} \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)**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

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Giac [A]  time = 1.13287, size = 126, normalized size = 1.5 \begin{align*} -\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}} \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)^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)