3.1295 \(\int \frac{x^{14}}{1+x^5} \, dx\)

Optimal. Leaf size=25 \[ \frac{x^{10}}{10}-\frac{x^5}{5}+\frac{1}{5} \log \left (x^5+1\right ) \]

[Out]

-x^5/5 + x^10/10 + Log[1 + x^5]/5

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Rubi [A]  time = 0.0118298, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {266, 43} \[ \frac{x^{10}}{10}-\frac{x^5}{5}+\frac{1}{5} \log \left (x^5+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^14/(1 + x^5),x]

[Out]

-x^5/5 + x^10/10 + Log[1 + x^5]/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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^{14}}{1+x^5} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{x^2}{1+x} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (-1+x+\frac{1}{1+x}\right ) \, dx,x,x^5\right )\\ &=-\frac{x^5}{5}+\frac{x^{10}}{10}+\frac{1}{5} \log \left (1+x^5\right )\\ \end{align*}

Mathematica [A]  time = 0.0045356, size = 22, normalized size = 0.88 \[ \frac{1}{10} \left (x^{10}-2 x^5+2 \log \left (x^5+1\right )-3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^14/(1 + x^5),x]

[Out]

(-3 - 2*x^5 + x^10 + 2*Log[1 + x^5])/10

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Maple [A]  time = 0.003, size = 20, normalized size = 0.8 \begin{align*} -{\frac{{x}^{5}}{5}}+{\frac{{x}^{10}}{10}}+{\frac{\ln \left ({x}^{5}+1 \right ) }{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14/(x^5+1),x)

[Out]

-1/5*x^5+1/10*x^10+1/5*ln(x^5+1)

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Maxima [A]  time = 0.983718, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{10} \, x^{10} - \frac{1}{5} \, x^{5} + \frac{1}{5} \, \log \left (x^{5} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(x^5+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.7088, size = 54, normalized size = 2.16 \begin{align*} \frac{1}{10} \, x^{10} - \frac{1}{5} \, x^{5} + \frac{1}{5} \, \log \left (x^{5} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(x^5+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.103791, size = 17, normalized size = 0.68 \begin{align*} \frac{x^{10}}{10} - \frac{x^{5}}{5} + \frac{\log{\left (x^{5} + 1 \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**14/(x**5+1),x)

[Out]

x**10/10 - x**5/5 + log(x**5 + 1)/5

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Giac [A]  time = 1.22031, size = 27, normalized size = 1.08 \begin{align*} \frac{1}{10} \, x^{10} - \frac{1}{5} \, x^{5} + \frac{1}{5} \, \log \left ({\left | x^{5} + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(x^5+1),x, algorithm="giac")

[Out]

1/10*x^10 - 1/5*x^5 + 1/5*log(abs(x^5 + 1))