3.280 \(\int \frac{8+5 x^{10}}{2 x-x^{11}} \, dx\)

Optimal. Leaf size=17 \[ 4 \log (x)-\frac{9}{10} \log \left (2-x^{10}\right ) \]

[Out]

4*Log[x] - (9*Log[2 - x^10])/10

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Rubi [A]  time = 0.025, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1593, 446, 72} \[ 4 \log (x)-\frac{9}{10} \log \left (2-x^{10}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(8 + 5*x^10)/(2*x - x^11),x]

[Out]

4*Log[x] - (9*Log[2 - x^10])/10

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{8+5 x^{10}}{2 x-x^{11}} \, dx &=\int \frac{8+5 x^{10}}{x \left (2-x^{10}\right )} \, dx\\ &=\frac{1}{10} \operatorname{Subst}\left (\int \frac{8+5 x}{(2-x) x} \, dx,x,x^{10}\right )\\ &=\frac{1}{10} \operatorname{Subst}\left (\int \left (-\frac{9}{-2+x}+\frac{4}{x}\right ) \, dx,x,x^{10}\right )\\ &=4 \log (x)-\frac{9}{10} \log \left (2-x^{10}\right )\\ \end{align*}

Mathematica [A]  time = 0.0051205, size = 17, normalized size = 1. \[ 4 \log (x)-\frac{9}{10} \log \left (2-x^{10}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(8 + 5*x^10)/(2*x - x^11),x]

[Out]

4*Log[x] - (9*Log[2 - x^10])/10

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Maple [A]  time = 0.052, size = 14, normalized size = 0.8 \begin{align*} 4\,\ln \left ( x \right ) -{\frac{9\,\ln \left ({x}^{10}-2 \right ) }{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^10+8)/(-x^11+2*x),x)

[Out]

4*ln(x)-9/10*ln(x^10-2)

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Maxima [A]  time = 1.75488, size = 18, normalized size = 1.06 \begin{align*} -\frac{9}{10} \, \log \left (x^{10} - 2\right ) + 4 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^10+8)/(-x^11+2*x),x, algorithm="maxima")

[Out]

-9/10*log(x^10 - 2) + 4*log(x)

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Fricas [A]  time = 2.06155, size = 43, normalized size = 2.53 \begin{align*} -\frac{9}{10} \, \log \left (x^{10} - 2\right ) + 4 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^10+8)/(-x^11+2*x),x, algorithm="fricas")

[Out]

-9/10*log(x^10 - 2) + 4*log(x)

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Sympy [A]  time = 0.16126, size = 14, normalized size = 0.82 \begin{align*} 4 \log{\left (x \right )} - \frac{9 \log{\left (x^{10} - 2 \right )}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**10+8)/(-x**11+2*x),x)

[Out]

4*log(x) - 9*log(x**10 - 2)/10

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Giac [A]  time = 1.22903, size = 22, normalized size = 1.29 \begin{align*} \frac{2}{5} \, \log \left (x^{10}\right ) - \frac{9}{10} \, \log \left ({\left | x^{10} - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^10+8)/(-x^11+2*x),x, algorithm="giac")

[Out]

2/5*log(x^10) - 9/10*log(abs(x^10 - 2))