3.522 \(\int \frac{1}{x^5 (a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=140 \[ \frac{15 b^2}{2 a^7 \left (a+b x^2\right )}+\frac{5 b^2}{2 a^6 \left (a+b x^2\right )^2}+\frac{b^2}{a^5 \left (a+b x^2\right )^3}+\frac{3 b^2}{8 a^4 \left (a+b x^2\right )^4}+\frac{b^2}{10 a^3 \left (a+b x^2\right )^5}-\frac{21 b^2 \log \left (a+b x^2\right )}{2 a^8}+\frac{21 b^2 \log (x)}{a^8}+\frac{3 b}{a^7 x^2}-\frac{1}{4 a^6 x^4} \]

[Out]

-1/(4*a^6*x^4) + (3*b)/(a^7*x^2) + b^2/(10*a^3*(a + b*x^2)^5) + (3*b^2)/(8*a^4*(a + b*x^2)^4) + b^2/(a^5*(a +
b*x^2)^3) + (5*b^2)/(2*a^6*(a + b*x^2)^2) + (15*b^2)/(2*a^7*(a + b*x^2)) + (21*b^2*Log[x])/a^8 - (21*b^2*Log[a
+ b*x^2])/(2*a^8)

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Rubi [A]  time = 0.144679, antiderivative size = 140, 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} \[ \frac{15 b^2}{2 a^7 \left (a+b x^2\right )}+\frac{5 b^2}{2 a^6 \left (a+b x^2\right )^2}+\frac{b^2}{a^5 \left (a+b x^2\right )^3}+\frac{3 b^2}{8 a^4 \left (a+b x^2\right )^4}+\frac{b^2}{10 a^3 \left (a+b x^2\right )^5}-\frac{21 b^2 \log \left (a+b x^2\right )}{2 a^8}+\frac{21 b^2 \log (x)}{a^8}+\frac{3 b}{a^7 x^2}-\frac{1}{4 a^6 x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

-1/(4*a^6*x^4) + (3*b)/(a^7*x^2) + b^2/(10*a^3*(a + b*x^2)^5) + (3*b^2)/(8*a^4*(a + b*x^2)^4) + b^2/(a^5*(a +
b*x^2)^3) + (5*b^2)/(2*a^6*(a + b*x^2)^2) + (15*b^2)/(2*a^7*(a + b*x^2)) + (21*b^2*Log[x])/a^8 - (21*b^2*Log[a
+ b*x^2])/(2*a^8)

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 )^3} \, dx &=b^6 \int \frac{1}{x^5 \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac{1}{2} b^6 \operatorname{Subst}\left (\int \frac{1}{x^3 \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^3}-\frac{6}{a^7 b^5 x^2}+\frac{21}{a^8 b^4 x}-\frac{1}{a^3 b^3 (a+b x)^6}-\frac{3}{a^4 b^3 (a+b x)^5}-\frac{6}{a^5 b^3 (a+b x)^4}-\frac{10}{a^6 b^3 (a+b x)^3}-\frac{15}{a^7 b^3 (a+b x)^2}-\frac{21}{a^8 b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{4 a^6 x^4}+\frac{3 b}{a^7 x^2}+\frac{b^2}{10 a^3 \left (a+b x^2\right )^5}+\frac{3 b^2}{8 a^4 \left (a+b x^2\right )^4}+\frac{b^2}{a^5 \left (a+b x^2\right )^3}+\frac{5 b^2}{2 a^6 \left (a+b x^2\right )^2}+\frac{15 b^2}{2 a^7 \left (a+b x^2\right )}+\frac{21 b^2 \log (x)}{a^8}-\frac{21 b^2 \log \left (a+b x^2\right )}{2 a^8}\\ \end{align*}

Mathematica [A]  time = 0.0592197, size = 107, normalized size = 0.76 \[ \frac{\frac{a \left (3290 a^2 b^4 x^8+2695 a^3 b^3 x^6+959 a^4 b^2 x^4+70 a^5 b x^2-10 a^6+1890 a b^5 x^{10}+420 b^6 x^{12}\right )}{x^4 \left (a+b x^2\right )^5}-420 b^2 \log \left (a+b x^2\right )+840 b^2 \log (x)}{40 a^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

((a*(-10*a^6 + 70*a^5*b*x^2 + 959*a^4*b^2*x^4 + 2695*a^3*b^3*x^6 + 3290*a^2*b^4*x^8 + 1890*a*b^5*x^10 + 420*b^
6*x^12))/(x^4*(a + b*x^2)^5) + 840*b^2*Log[x] - 420*b^2*Log[a + b*x^2])/(40*a^8)

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Maple [A]  time = 0.058, size = 129, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{x}^{4}{a}^{6}}}+3\,{\frac{b}{{a}^{7}{x}^{2}}}+{\frac{{b}^{2}}{10\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{3\,{b}^{2}}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{{b}^{2}}{{a}^{5} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{5\,{b}^{2}}{2\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{15\,{b}^{2}}{2\,{a}^{7} \left ( b{x}^{2}+a \right ) }}+21\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{8}}}-{\frac{21\,{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{8}}} \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)^3,x)

[Out]

-1/4/x^4/a^6+3*b/a^7/x^2+1/10*b^2/a^3/(b*x^2+a)^5+3/8*b^2/a^4/(b*x^2+a)^4+b^2/a^5/(b*x^2+a)^3+5/2*b^2/a^6/(b*x
^2+a)^2+15/2*b^2/a^7/(b*x^2+a)+21*b^2*ln(x)/a^8-21/2*b^2*ln(b*x^2+a)/a^8

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Maxima [A]  time = 1.04972, size = 213, normalized size = 1.52 \begin{align*} \frac{420 \, b^{6} x^{12} + 1890 \, a b^{5} x^{10} + 3290 \, a^{2} b^{4} x^{8} + 2695 \, a^{3} b^{3} x^{6} + 959 \, a^{4} b^{2} x^{4} + 70 \, a^{5} b x^{2} - 10 \, a^{6}}{40 \,{\left (a^{7} b^{5} x^{14} + 5 \, a^{8} b^{4} x^{12} + 10 \, a^{9} b^{3} x^{10} + 10 \, a^{10} b^{2} x^{8} + 5 \, a^{11} b x^{6} + a^{12} x^{4}\right )}} - \frac{21 \, b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{8}} + \frac{21 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{8}} \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)^3,x, algorithm="maxima")

[Out]

1/40*(420*b^6*x^12 + 1890*a*b^5*x^10 + 3290*a^2*b^4*x^8 + 2695*a^3*b^3*x^6 + 959*a^4*b^2*x^4 + 70*a^5*b*x^2 -
10*a^6)/(a^7*b^5*x^14 + 5*a^8*b^4*x^12 + 10*a^9*b^3*x^10 + 10*a^10*b^2*x^8 + 5*a^11*b*x^6 + a^12*x^4) - 21/2*b
^2*log(b*x^2 + a)/a^8 + 21/2*b^2*log(x^2)/a^8

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Fricas [B]  time = 1.54522, size = 586, normalized size = 4.19 \begin{align*} \frac{420 \, a b^{6} x^{12} + 1890 \, a^{2} b^{5} x^{10} + 3290 \, a^{3} b^{4} x^{8} + 2695 \, a^{4} b^{3} x^{6} + 959 \, a^{5} b^{2} x^{4} + 70 \, a^{6} b x^{2} - 10 \, a^{7} - 420 \,{\left (b^{7} x^{14} + 5 \, a b^{6} x^{12} + 10 \, a^{2} b^{5} x^{10} + 10 \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{6} + a^{5} b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 840 \,{\left (b^{7} x^{14} + 5 \, a b^{6} x^{12} + 10 \, a^{2} b^{5} x^{10} + 10 \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{6} + a^{5} b^{2} x^{4}\right )} \log \left (x\right )}{40 \,{\left (a^{8} b^{5} x^{14} + 5 \, a^{9} b^{4} x^{12} + 10 \, a^{10} b^{3} x^{10} + 10 \, a^{11} b^{2} x^{8} + 5 \, a^{12} b x^{6} + a^{13} 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)^3,x, algorithm="fricas")

[Out]

1/40*(420*a*b^6*x^12 + 1890*a^2*b^5*x^10 + 3290*a^3*b^4*x^8 + 2695*a^4*b^3*x^6 + 959*a^5*b^2*x^4 + 70*a^6*b*x^
2 - 10*a^7 - 420*(b^7*x^14 + 5*a*b^6*x^12 + 10*a^2*b^5*x^10 + 10*a^3*b^4*x^8 + 5*a^4*b^3*x^6 + a^5*b^2*x^4)*lo
g(b*x^2 + a) + 840*(b^7*x^14 + 5*a*b^6*x^12 + 10*a^2*b^5*x^10 + 10*a^3*b^4*x^8 + 5*a^4*b^3*x^6 + a^5*b^2*x^4)*
log(x))/(a^8*b^5*x^14 + 5*a^9*b^4*x^12 + 10*a^10*b^3*x^10 + 10*a^11*b^2*x^8 + 5*a^12*b*x^6 + a^13*x^4)

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Sympy [A]  time = 13.4509, size = 165, normalized size = 1.18 \begin{align*} \frac{- 10 a^{6} + 70 a^{5} b x^{2} + 959 a^{4} b^{2} x^{4} + 2695 a^{3} b^{3} x^{6} + 3290 a^{2} b^{4} x^{8} + 1890 a b^{5} x^{10} + 420 b^{6} x^{12}}{40 a^{12} x^{4} + 200 a^{11} b x^{6} + 400 a^{10} b^{2} x^{8} + 400 a^{9} b^{3} x^{10} + 200 a^{8} b^{4} x^{12} + 40 a^{7} b^{5} x^{14}} + \frac{21 b^{2} \log{\left (x \right )}}{a^{8}} - \frac{21 b^{2} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{8}} \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)**3,x)

[Out]

(-10*a**6 + 70*a**5*b*x**2 + 959*a**4*b**2*x**4 + 2695*a**3*b**3*x**6 + 3290*a**2*b**4*x**8 + 1890*a*b**5*x**1
0 + 420*b**6*x**12)/(40*a**12*x**4 + 200*a**11*b*x**6 + 400*a**10*b**2*x**8 + 400*a**9*b**3*x**10 + 200*a**8*b
**4*x**12 + 40*a**7*b**5*x**14) + 21*b**2*log(x)/a**8 - 21*b**2*log(a/b + x**2)/(2*a**8)

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Giac [A]  time = 1.15962, size = 176, normalized size = 1.26 \begin{align*} \frac{21 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{8}} - \frac{21 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{8}} - \frac{63 \, b^{2} x^{4} - 12 \, a b x^{2} + a^{2}}{4 \, a^{8} x^{4}} + \frac{959 \, b^{7} x^{10} + 5095 \, a b^{6} x^{8} + 10890 \, a^{2} b^{5} x^{6} + 11730 \, a^{3} b^{4} x^{4} + 6390 \, a^{4} b^{3} x^{2} + 1418 \, a^{5} b^{2}}{40 \,{\left (b x^{2} + a\right )}^{5} a^{8}} \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)^3,x, algorithm="giac")

[Out]

21/2*b^2*log(x^2)/a^8 - 21/2*b^2*log(abs(b*x^2 + a))/a^8 - 1/4*(63*b^2*x^4 - 12*a*b*x^2 + a^2)/(a^8*x^4) + 1/4
0*(959*b^7*x^10 + 5095*a*b^6*x^8 + 10890*a^2*b^5*x^6 + 11730*a^3*b^4*x^4 + 6390*a^4*b^3*x^2 + 1418*a^5*b^2)/((
b*x^2 + a)^5*a^8)