3.144 \(\int \frac{1}{x^9 (a+b x^2)} \, dx\)

Optimal. Leaf size=75 \[ \frac{b^3}{2 a^4 x^2}-\frac{b^2}{4 a^3 x^4}-\frac{b^4 \log \left (a+b x^2\right )}{2 a^5}+\frac{b^4 \log (x)}{a^5}+\frac{b}{6 a^2 x^6}-\frac{1}{8 a x^8} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0408999, antiderivative size = 75, normalized size of antiderivative = 1., 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} \[ \frac{b^3}{2 a^4 x^2}-\frac{b^2}{4 a^3 x^4}-\frac{b^4 \log \left (a+b x^2\right )}{2 a^5}+\frac{b^4 \log (x)}{a^5}+\frac{b}{6 a^2 x^6}-\frac{1}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(a + b*x^2)),x]

[Out]

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

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^9 \left (a+b x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^5 (a+b x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a x^5}-\frac{b}{a^2 x^4}+\frac{b^2}{a^3 x^3}-\frac{b^3}{a^4 x^2}+\frac{b^4}{a^5 x}-\frac{b^5}{a^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{8 a x^8}+\frac{b}{6 a^2 x^6}-\frac{b^2}{4 a^3 x^4}+\frac{b^3}{2 a^4 x^2}+\frac{b^4 \log (x)}{a^5}-\frac{b^4 \log \left (a+b x^2\right )}{2 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0065807, size = 75, normalized size = 1. \[ \frac{b^3}{2 a^4 x^2}-\frac{b^2}{4 a^3 x^4}-\frac{b^4 \log \left (a+b x^2\right )}{2 a^5}+\frac{b^4 \log (x)}{a^5}+\frac{b}{6 a^2 x^6}-\frac{1}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^9*(a + b*x^2)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 66, normalized size = 0.9 \begin{align*} -{\frac{1}{8\,a{x}^{8}}}+{\frac{b}{6\,{a}^{2}{x}^{6}}}-{\frac{{b}^{2}}{4\,{a}^{3}{x}^{4}}}+{\frac{{b}^{3}}{2\,{a}^{4}{x}^{2}}}+{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{5}}}-{\frac{{b}^{4}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^9/(b*x^2+a),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 2.47016, size = 93, normalized size = 1.24 \begin{align*} -\frac{b^{4} \log \left (b x^{2} + a\right )}{2 \, a^{5}} + \frac{b^{4} \log \left (x^{2}\right )}{2 \, a^{5}} + \frac{12 \, b^{3} x^{6} - 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} - 3 \, a^{3}}{24 \, a^{4} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^2+a),x, algorithm="maxima")

[Out]

-1/2*b^4*log(b*x^2 + a)/a^5 + 1/2*b^4*log(x^2)/a^5 + 1/24*(12*b^3*x^6 - 6*a*b^2*x^4 + 4*a^2*b*x^2 - 3*a^3)/(a^
4*x^8)

________________________________________________________________________________________

Fricas [A]  time = 1.25234, size = 159, normalized size = 2.12 \begin{align*} -\frac{12 \, b^{4} x^{8} \log \left (b x^{2} + a\right ) - 24 \, b^{4} x^{8} \log \left (x\right ) - 12 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, a^{5} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^2+a),x, algorithm="fricas")

[Out]

-1/24*(12*b^4*x^8*log(b*x^2 + a) - 24*b^4*x^8*log(x) - 12*a*b^3*x^6 + 6*a^2*b^2*x^4 - 4*a^3*b*x^2 + 3*a^4)/(a^
5*x^8)

________________________________________________________________________________________

Sympy [A]  time = 0.683919, size = 68, normalized size = 0.91 \begin{align*} \frac{- 3 a^{3} + 4 a^{2} b x^{2} - 6 a b^{2} x^{4} + 12 b^{3} x^{6}}{24 a^{4} x^{8}} + \frac{b^{4} \log{\left (x \right )}}{a^{5}} - \frac{b^{4} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**9/(b*x**2+a),x)

[Out]

(-3*a**3 + 4*a**2*b*x**2 - 6*a*b**2*x**4 + 12*b**3*x**6)/(24*a**4*x**8) + b**4*log(x)/a**5 - b**4*log(a/b + x*
*2)/(2*a**5)

________________________________________________________________________________________

Giac [A]  time = 2.74214, size = 109, normalized size = 1.45 \begin{align*} \frac{b^{4} \log \left (x^{2}\right )}{2 \, a^{5}} - \frac{b^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{5}} - \frac{25 \, b^{4} x^{8} - 12 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, a^{5} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^2+a),x, algorithm="giac")

[Out]

1/2*b^4*log(x^2)/a^5 - 1/2*b^4*log(abs(b*x^2 + a))/a^5 - 1/24*(25*b^4*x^8 - 12*a*b^3*x^6 + 6*a^2*b^2*x^4 - 4*a
^3*b*x^2 + 3*a^4)/(a^5*x^8)