3.501 \(\int \frac{1}{x^5 (a^2+2 a b x^2+b^2 x^4)^2} \, dx\)

Optimal. Leaf size=101 \[ \frac{3 b^2}{a^5 \left (a+b x^2\right )}+\frac{3 b^2}{4 a^4 \left (a+b x^2\right )^2}+\frac{b^2}{6 a^3 \left (a+b x^2\right )^3}-\frac{5 b^2 \log \left (a+b x^2\right )}{a^6}+\frac{10 b^2 \log (x)}{a^6}+\frac{2 b}{a^5 x^2}-\frac{1}{4 a^4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.0981371, antiderivative size = 101, 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} \[ \frac{3 b^2}{a^5 \left (a+b x^2\right )}+\frac{3 b^2}{4 a^4 \left (a+b x^2\right )^2}+\frac{b^2}{6 a^3 \left (a+b x^2\right )^3}-\frac{5 b^2 \log \left (a+b x^2\right )}{a^6}+\frac{10 b^2 \log (x)}{a^6}+\frac{2 b}{a^5 x^2}-\frac{1}{4 a^4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]

[Out]

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

Mathematica [A]  time = 0.054207, size = 85, normalized size = 0.84 \[ \frac{\frac{a \left (110 a^2 b^2 x^4+15 a^3 b x^2-3 a^4+150 a b^3 x^6+60 b^4 x^8\right )}{x^4 \left (a+b x^2\right )^3}-60 b^2 \log \left (a+b x^2\right )+120 b^2 \log (x)}{12 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]

[Out]

((a*(-3*a^4 + 15*a^3*b*x^2 + 110*a^2*b^2*x^4 + 150*a*b^3*x^6 + 60*b^4*x^8))/(x^4*(a + b*x^2)^3) + 120*b^2*Log[
x] - 60*b^2*Log[a + b*x^2])/(12*a^6)

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Maple [A]  time = 0.055, size = 96, normalized size = 1. \begin{align*} -{\frac{1}{4\,{a}^{4}{x}^{4}}}+2\,{\frac{b}{{a}^{5}{x}^{2}}}+{\frac{{b}^{2}}{6\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{3\,{b}^{2}}{4\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+3\,{\frac{{b}^{2}}{{a}^{5} \left ( b{x}^{2}+a \right ) }}+10\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{6}}}-5\,{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b^2*x^4+2*a*b*x^2+a^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.02538, size = 154, normalized size = 1.52 \begin{align*} \frac{60 \, b^{4} x^{8} + 150 \, a b^{3} x^{6} + 110 \, a^{2} b^{2} x^{4} + 15 \, a^{3} b x^{2} - 3 \, a^{4}}{12 \,{\left (a^{5} b^{3} x^{10} + 3 \, a^{6} b^{2} x^{8} + 3 \, a^{7} b x^{6} + a^{8} x^{4}\right )}} - \frac{5 \, b^{2} \log \left (b x^{2} + a\right )}{a^{6}} + \frac{5 \, b^{2} \log \left (x^{2}\right )}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")

[Out]

1/12*(60*b^4*x^8 + 150*a*b^3*x^6 + 110*a^2*b^2*x^4 + 15*a^3*b*x^2 - 3*a^4)/(a^5*b^3*x^10 + 3*a^6*b^2*x^8 + 3*a
^7*b*x^6 + a^8*x^4) - 5*b^2*log(b*x^2 + a)/a^6 + 5*b^2*log(x^2)/a^6

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Fricas [A]  time = 1.8134, size = 375, normalized size = 3.71 \begin{align*} \frac{60 \, a b^{4} x^{8} + 150 \, a^{2} b^{3} x^{6} + 110 \, a^{3} b^{2} x^{4} + 15 \, a^{4} b x^{2} - 3 \, a^{5} - 60 \,{\left (b^{5} x^{10} + 3 \, a b^{4} x^{8} + 3 \, a^{2} b^{3} x^{6} + a^{3} b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 120 \,{\left (b^{5} x^{10} + 3 \, a b^{4} x^{8} + 3 \, a^{2} b^{3} x^{6} + a^{3} b^{2} x^{4}\right )} \log \left (x\right )}{12 \,{\left (a^{6} b^{3} x^{10} + 3 \, a^{7} b^{2} x^{8} + 3 \, a^{8} b x^{6} + a^{9} x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")

[Out]

1/12*(60*a*b^4*x^8 + 150*a^2*b^3*x^6 + 110*a^3*b^2*x^4 + 15*a^4*b*x^2 - 3*a^5 - 60*(b^5*x^10 + 3*a*b^4*x^8 + 3
*a^2*b^3*x^6 + a^3*b^2*x^4)*log(b*x^2 + a) + 120*(b^5*x^10 + 3*a*b^4*x^8 + 3*a^2*b^3*x^6 + a^3*b^2*x^4)*log(x)
)/(a^6*b^3*x^10 + 3*a^7*b^2*x^8 + 3*a^8*b*x^6 + a^9*x^4)

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Sympy [A]  time = 2.25184, size = 116, normalized size = 1.15 \begin{align*} \frac{- 3 a^{4} + 15 a^{3} b x^{2} + 110 a^{2} b^{2} x^{4} + 150 a b^{3} x^{6} + 60 b^{4} x^{8}}{12 a^{8} x^{4} + 36 a^{7} b x^{6} + 36 a^{6} b^{2} x^{8} + 12 a^{5} b^{3} x^{10}} + \frac{10 b^{2} \log{\left (x \right )}}{a^{6}} - \frac{5 b^{2} \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

(-3*a**4 + 15*a**3*b*x**2 + 110*a**2*b**2*x**4 + 150*a*b**3*x**6 + 60*b**4*x**8)/(12*a**8*x**4 + 36*a**7*b*x**
6 + 36*a**6*b**2*x**8 + 12*a**5*b**3*x**10) + 10*b**2*log(x)/a**6 - 5*b**2*log(a/b + x**2)/a**6

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Giac [A]  time = 1.15233, size = 146, normalized size = 1.45 \begin{align*} \frac{5 \, b^{2} \log \left (x^{2}\right )}{a^{6}} - \frac{5 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{6}} + \frac{110 \, b^{5} x^{6} + 366 \, a b^{4} x^{4} + 411 \, a^{2} b^{3} x^{2} + 157 \, a^{3} b^{2}}{12 \,{\left (b x^{2} + a\right )}^{3} a^{6}} - \frac{30 \, b^{2} x^{4} - 8 \, a b x^{2} + a^{2}}{4 \, a^{6} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")

[Out]

5*b^2*log(x^2)/a^6 - 5*b^2*log(abs(b*x^2 + a))/a^6 + 1/12*(110*b^5*x^6 + 366*a*b^4*x^4 + 411*a^2*b^3*x^2 + 157
*a^3*b^2)/((b*x^2 + a)^3*a^6) - 1/4*(30*b^2*x^4 - 8*a*b*x^2 + a^2)/(a^6*x^4)