3.1460 \(\int \frac{1}{x^9 (a+b x^8)} \, dx\)

Optimal. Leaf size=35 \[ \frac{b \log \left (a+b x^8\right )}{8 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{8 a x^8} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0242778, antiderivative size = 35, 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} \[ \frac{b \log \left (a+b x^8\right )}{8 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(a + b*x^8)),x]

[Out]

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

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

Mathematica [A]  time = 0.0064687, size = 35, normalized size = 1. \[ \frac{b \log \left (a+b x^8\right )}{8 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^9*(a + b*x^8)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 32, normalized size = 0.9 \begin{align*} -{\frac{1}{8\,a{x}^{8}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}}}+{\frac{b\ln \left ( b{x}^{8}+a \right ) }{8\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^9/(b*x^8+a),x)

[Out]

-1/8/a/x^8-b*ln(x)/a^2+1/8*b*ln(b*x^8+a)/a^2

________________________________________________________________________________________

Maxima [A]  time = 0.968044, size = 45, normalized size = 1.29 \begin{align*} \frac{b \log \left (b x^{8} + a\right )}{8 \, a^{2}} - \frac{b \log \left (x^{8}\right )}{8 \, a^{2}} - \frac{1}{8 \, a x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^8+a),x, algorithm="maxima")

[Out]

1/8*b*log(b*x^8 + a)/a^2 - 1/8*b*log(x^8)/a^2 - 1/8/(a*x^8)

________________________________________________________________________________________

Fricas [A]  time = 1.28701, size = 80, normalized size = 2.29 \begin{align*} \frac{b x^{8} \log \left (b x^{8} + a\right ) - 8 \, b x^{8} \log \left (x\right ) - a}{8 \, a^{2} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^8+a),x, algorithm="fricas")

[Out]

1/8*(b*x^8*log(b*x^8 + a) - 8*b*x^8*log(x) - a)/(a^2*x^8)

________________________________________________________________________________________

Sympy [A]  time = 4.82565, size = 31, normalized size = 0.89 \begin{align*} - \frac{1}{8 a x^{8}} - \frac{b \log{\left (x \right )}}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + x^{8} \right )}}{8 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**9/(b*x**8+a),x)

[Out]

-1/(8*a*x**8) - b*log(x)/a**2 + b*log(a/b + x**8)/(8*a**2)

________________________________________________________________________________________

Giac [A]  time = 1.16192, size = 58, normalized size = 1.66 \begin{align*} -\frac{b \log \left (x^{8}\right )}{8 \, a^{2}} + \frac{b \log \left ({\left | b x^{8} + a \right |}\right )}{8 \, a^{2}} + \frac{b x^{8} - a}{8 \, a^{2} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^8+a),x, algorithm="giac")

[Out]

-1/8*b*log(x^8)/a^2 + 1/8*b*log(abs(b*x^8 + a))/a^2 + 1/8*(b*x^8 - a)/(a^2*x^8)