3.1289 \(\int \frac{1}{x^6 (2 b+b x^5)} \, dx\)

Optimal. Leaf size=33 \[ -\frac{1}{10 b x^5}+\frac{\log \left (x^5+2\right )}{20 b}-\frac{\log (x)}{4 b} \]

[Out]

-1/(10*b*x^5) - Log[x]/(4*b) + Log[2 + x^5]/(20*b)

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Rubi [A]  time = 0.0181239, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 44} \[ -\frac{1}{10 b x^5}+\frac{\log \left (x^5+2\right )}{20 b}-\frac{\log (x)}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(2*b + b*x^5)),x]

[Out]

-1/(10*b*x^5) - Log[x]/(4*b) + Log[2 + x^5]/(20*b)

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^6 \left (2 b+b x^5\right )} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x^2 (2 b+b x)} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (\frac{1}{2 b x^2}-\frac{1}{4 b x}+\frac{1}{4 b (2+x)}\right ) \, dx,x,x^5\right )\\ &=-\frac{1}{10 b x^5}-\frac{\log (x)}{4 b}+\frac{\log \left (2+x^5\right )}{20 b}\\ \end{align*}

Mathematica [A]  time = 0.004133, size = 33, normalized size = 1. \[ -\frac{1}{10 b x^5}+\frac{\log \left (x^5+2\right )}{20 b}-\frac{\log (x)}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(2*b + b*x^5)),x]

[Out]

-1/(10*b*x^5) - Log[x]/(4*b) + Log[2 + x^5]/(20*b)

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Maple [A]  time = 0.006, size = 28, normalized size = 0.9 \begin{align*} -{\frac{1}{10\,b{x}^{5}}}-{\frac{\ln \left ( x \right ) }{4\,b}}+{\frac{\ln \left ({x}^{5}+2 \right ) }{20\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(b*x^5+2*b),x)

[Out]

-1/10/b/x^5-1/4*ln(x)/b+1/20*ln(x^5+2)/b

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Maxima [A]  time = 1.02313, size = 39, normalized size = 1.18 \begin{align*} \frac{\log \left (x^{5} + 2\right )}{20 \, b} - \frac{\log \left (x^{5}\right )}{20 \, b} - \frac{1}{10 \, b x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^5+2*b),x, algorithm="maxima")

[Out]

1/20*log(x^5 + 2)/b - 1/20*log(x^5)/b - 1/10/(b*x^5)

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Fricas [A]  time = 1.71003, size = 70, normalized size = 2.12 \begin{align*} \frac{x^{5} \log \left (x^{5} + 2\right ) - 5 \, x^{5} \log \left (x\right ) - 2}{20 \, b x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^5+2*b),x, algorithm="fricas")

[Out]

1/20*(x^5*log(x^5 + 2) - 5*x^5*log(x) - 2)/(b*x^5)

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Sympy [A]  time = 0.801041, size = 24, normalized size = 0.73 \begin{align*} - \frac{\log{\left (x \right )}}{4 b} + \frac{\log{\left (x^{5} + 2 \right )}}{20 b} - \frac{1}{10 b x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(b*x**5+2*b),x)

[Out]

-log(x)/(4*b) + log(x**5 + 2)/(20*b) - 1/(10*b*x**5)

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Giac [A]  time = 1.18481, size = 46, normalized size = 1.39 \begin{align*} \frac{\log \left ({\left | x^{5} + 2 \right |}\right )}{20 \, b} - \frac{\log \left ({\left | x \right |}\right )}{4 \, b} + \frac{x^{5} - 2}{20 \, b x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^5+2*b),x, algorithm="giac")

[Out]

1/20*log(abs(x^5 + 2))/b - 1/4*log(abs(x))/b + 1/20*(x^5 - 2)/(b*x^5)