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

Optimal. Leaf size=66 \[ -\frac {3 b^2 \log \left (a+b x^2\right )}{2 a^4}+\frac {3 b^2 \log (x)}{a^4}+\frac {b^2}{2 a^3 \left (a+b x^2\right )}+\frac {b}{a^3 x^2}-\frac {1}{4 a^2 x^4} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 44} \[ \frac {b^2}{2 a^3 \left (a+b x^2\right )}-\frac {3 b^2 \log \left (a+b x^2\right )}{2 a^4}+\frac {3 b^2 \log (x)}{a^4}+\frac {b}{a^3 x^2}-\frac {1}{4 a^2 x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x^2)^2),x]

[Out]

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

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])

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]]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 57, normalized size = 0.86 \[ \frac {-6 b^2 \log \left (a+b x^2\right )+a \left (\frac {2 b^2}{a+b x^2}-\frac {a}{x^4}+\frac {4 b}{x^2}\right )+12 b^2 \log (x)}{4 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x^2)^2),x]

[Out]

(a*(-(a/x^4) + (4*b)/x^2 + (2*b^2)/(a + b*x^2)) + 12*b^2*Log[x] - 6*b^2*Log[a + b*x^2])/(4*a^4)

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fricas [A]  time = 0.87, size = 90, normalized size = 1.36 \[ \frac {6 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - a^{3} - 6 \, {\left (b^{3} x^{6} + a b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 12 \, {\left (b^{3} x^{6} + a b^{2} x^{4}\right )} \log \relax (x)}{4 \, {\left (a^{4} b x^{6} + a^{5} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(6*a*b^2*x^4 + 3*a^2*b*x^2 - a^3 - 6*(b^3*x^6 + a*b^2*x^4)*log(b*x^2 + a) + 12*(b^3*x^6 + a*b^2*x^4)*log(x
))/(a^4*b*x^6 + a^5*x^4)

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giac [A]  time = 0.63, size = 86, normalized size = 1.30 \[ \frac {3 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac {3 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4}} + \frac {3 \, b^{3} x^{2} + 4 \, a b^{2}}{2 \, {\left (b x^{2} + a\right )} a^{4}} - \frac {9 \, b^{2} x^{4} - 4 \, a b x^{2} + a^{2}}{4 \, a^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*b^2*log(x^2)/a^4 - 3/2*b^2*log(abs(b*x^2 + a))/a^4 + 1/2*(3*b^3*x^2 + 4*a*b^2)/((b*x^2 + a)*a^4) - 1/4*(9*
b^2*x^4 - 4*a*b*x^2 + a^2)/(a^4*x^4)

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maple [A]  time = 0.01, size = 61, normalized size = 0.92 \[ \frac {b^{2}}{2 \left (b \,x^{2}+a \right ) a^{3}}+\frac {3 b^{2} \ln \relax (x )}{a^{4}}-\frac {3 b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{4}}+\frac {b}{a^{3} x^{2}}-\frac {1}{4 a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.32, size = 70, normalized size = 1.06 \[ \frac {6 \, b^{2} x^{4} + 3 \, a b x^{2} - a^{2}}{4 \, {\left (a^{3} b x^{6} + a^{4} x^{4}\right )}} - \frac {3 \, b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac {3 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(6*b^2*x^4 + 3*a*b*x^2 - a^2)/(a^3*b*x^6 + a^4*x^4) - 3/2*b^2*log(b*x^2 + a)/a^4 + 3/2*b^2*log(x^2)/a^4

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mupad [B]  time = 4.80, size = 67, normalized size = 1.02 \[ \frac {\frac {3\,b\,x^2}{4\,a^2}-\frac {1}{4\,a}+\frac {3\,b^2\,x^4}{2\,a^3}}{b\,x^6+a\,x^4}-\frac {3\,b^2\,\ln \left (b\,x^2+a\right )}{2\,a^4}+\frac {3\,b^2\,\ln \relax (x)}{a^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + b*x^2)^2),x)

[Out]

((3*b*x^2)/(4*a^2) - 1/(4*a) + (3*b^2*x^4)/(2*a^3))/(a*x^4 + b*x^6) - (3*b^2*log(a + b*x^2))/(2*a^4) + (3*b^2*
log(x))/a^4

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sympy [A]  time = 0.44, size = 68, normalized size = 1.03 \[ \frac {- a^{2} + 3 a b x^{2} + 6 b^{2} x^{4}}{4 a^{4} x^{4} + 4 a^{3} b x^{6}} + \frac {3 b^{2} \log {\relax (x )}}{a^{4}} - \frac {3 b^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a**2 + 3*a*b*x**2 + 6*b**2*x**4)/(4*a**4*x**4 + 4*a**3*b*x**6) + 3*b**2*log(x)/a**4 - 3*b**2*log(a/b + x**2)
/(2*a**4)

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