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

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Rubi [A]  time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \[ 4 \log (x)-\frac {9}{10} \log \left (2-x^{10}\right ) \]

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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fricas [A]  time = 1.27, size = 13, normalized size = 0.76 \[ -\frac {9}{10} \, \log \left (x^{10} - 2\right ) + 4 \, \log \relax (x) \]

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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giac [A]  time = 0.19, size = 16, normalized size = 0.94 \[ \frac {2}{5} \, \log \left (x^{10}\right ) - \frac {9}{10} \, \log \left ({\left | x^{10} - 2 \right |}\right ) \]

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))

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maple [A]  time = 0.05, size = 14, normalized size = 0.82 \[ 4 \ln \relax (x )-\frac {9 \ln \left (x^{10}-2\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.94, size = 13, normalized size = 0.76 \[ -\frac {9}{10} \, \log \left (x^{10} - 2\right ) + 4 \, \log \relax (x) \]

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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mupad [B]  time = 0.14, size = 13, normalized size = 0.76 \[ 4\,\ln \relax (x)-\frac {9\,\ln \left (x^{10}-2\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^10 + 8)/(2*x - x^11),x)

[Out]

4*log(x) - (9*log(x^10 - 2))/10

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sympy [A]  time = 0.15, size = 14, normalized size = 0.82 \[ 4 \log {\relax (x )} - \frac {9 \log {\left (x^{10} - 2 \right )}}{10} \]

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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