3.1296 \(\int \frac{x^9}{1+x^5} \, dx\)

Optimal. Leaf size=18 \[ \frac{x^5}{5}-\frac{1}{5} \log \left (x^5+1\right ) \]

[Out]

x^5/5 - Log[1 + x^5]/5

________________________________________________________________________________________

Rubi [A]  time = 0.0080608, antiderivative size = 18, 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} \[ \frac{x^5}{5}-\frac{1}{5} \log \left (x^5+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^9/(1 + x^5),x]

[Out]

x^5/5 - 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^9}{1+x^5} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{x}{1+x} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (1+\frac{1}{-1-x}\right ) \, dx,x,x^5\right )\\ &=\frac{x^5}{5}-\frac{1}{5} \log \left (1+x^5\right )\\ \end{align*}

Mathematica [A]  time = 0.0022557, size = 18, normalized size = 1. \[ \frac{x^5}{5}-\frac{1}{5} \log \left (x^5+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(1 + x^5),x]

[Out]

x^5/5 - Log[1 + x^5]/5

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 15, normalized size = 0.8 \begin{align*}{\frac{{x}^{5}}{5}}-{\frac{\ln \left ({x}^{5}+1 \right ) }{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(x^5+1),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.04527, size = 19, normalized size = 1.06 \begin{align*} \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^9/(x^5+1),x, algorithm="maxima")

[Out]

1/5*x^5 - 1/5*log(x^5 + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.58839, size = 38, normalized size = 2.11 \begin{align*} \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^9/(x^5+1),x, algorithm="fricas")

[Out]

1/5*x^5 - 1/5*log(x^5 + 1)

________________________________________________________________________________________

Sympy [A]  time = 0.105383, size = 12, normalized size = 0.67 \begin{align*} \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**9/(x**5+1),x)

[Out]

x**5/5 - log(x**5 + 1)/5

________________________________________________________________________________________

Giac [A]  time = 1.1438, size = 20, normalized size = 1.11 \begin{align*} \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^9/(x^5+1),x, algorithm="giac")

[Out]

1/5*x^5 - 1/5*log(abs(x^5 + 1))