3.2448 \(\int \frac{1}{1+\sqrt [5]{x}} \, dx\)

Optimal. Leaf size=45 \[ \frac{5 x^{4/5}}{4}-\frac{5 x^{3/5}}{3}+\frac{5 x^{2/5}}{2}-5 \sqrt [5]{x}+5 \log \left (\sqrt [5]{x}+1\right ) \]

[Out]

-5*x^(1/5) + (5*x^(2/5))/2 - (5*x^(3/5))/3 + (5*x^(4/5))/4 + 5*Log[1 + x^(1/5)]

________________________________________________________________________________________

Rubi [A]  time = 0.0147312, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {190, 43} \[ \frac{5 x^{4/5}}{4}-\frac{5 x^{3/5}}{3}+\frac{5 x^{2/5}}{2}-5 \sqrt [5]{x}+5 \log \left (\sqrt [5]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^(1/5))^(-1),x]

[Out]

-5*x^(1/5) + (5*x^(2/5))/2 - (5*x^(3/5))/3 + (5*x^(4/5))/4 + 5*Log[1 + x^(1/5)]

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[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{1}{1+\sqrt [5]{x}} \, dx &=5 \operatorname{Subst}\left (\int \frac{x^4}{1+x} \, dx,x,\sqrt [5]{x}\right )\\ &=5 \operatorname{Subst}\left (\int \left (-1+x-x^2+x^3+\frac{1}{1+x}\right ) \, dx,x,\sqrt [5]{x}\right )\\ &=-5 \sqrt [5]{x}+\frac{5 x^{2/5}}{2}-\frac{5 x^{3/5}}{3}+\frac{5 x^{4/5}}{4}+5 \log \left (1+\sqrt [5]{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0163861, size = 45, normalized size = 1. \[ \frac{5 x^{4/5}}{4}-\frac{5 x^{3/5}}{3}+\frac{5 x^{2/5}}{2}-5 \sqrt [5]{x}+5 \log \left (\sqrt [5]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^(1/5))^(-1),x]

[Out]

-5*x^(1/5) + (5*x^(2/5))/2 - (5*x^(3/5))/3 + (5*x^(4/5))/4 + 5*Log[1 + x^(1/5)]

________________________________________________________________________________________

Maple [B]  time = 0.044, size = 79, normalized size = 1.8 \begin{align*} \ln \left ( 1+x \right ) +{\frac{5}{2}{x}^{{\frac{2}{5}}}}+4\,\ln \left ( 1+\sqrt [5]{x} \right ) -\ln \left ( -\sqrt{5}\sqrt [5]{x}+2\,{x}^{2/5}-\sqrt [5]{x}+2 \right ) -\ln \left ( \sqrt{5}\sqrt [5]{x}+2\,{x}^{2/5}-\sqrt [5]{x}+2 \right ) +{\frac{5}{4}{x}^{{\frac{4}{5}}}}-5\,\sqrt [5]{x}-{\frac{5}{3}{x}^{{\frac{3}{5}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(1+x)+5/2*x^(2/5)+4*ln(1+x^(1/5))-ln(-5^(1/2)*x^(1/5)+2*x^(2/5)-x^(1/5)+2)-ln(5^(1/2)*x^(1/5)+2*x^(2/5)-x^(1
/5)+2)+5/4*x^(4/5)-5*x^(1/5)-5/3*x^(3/5)

________________________________________________________________________________________

Maxima [A]  time = 1.04067, size = 57, normalized size = 1.27 \begin{align*} \frac{5}{4} \,{\left (x^{\frac{1}{5}} + 1\right )}^{4} - \frac{20}{3} \,{\left (x^{\frac{1}{5}} + 1\right )}^{3} + 15 \,{\left (x^{\frac{1}{5}} + 1\right )}^{2} - 20 \, x^{\frac{1}{5}} + 5 \, \log \left (x^{\frac{1}{5}} + 1\right ) - 20 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/4*(x^(1/5) + 1)^4 - 20/3*(x^(1/5) + 1)^3 + 15*(x^(1/5) + 1)^2 - 20*x^(1/5) + 5*log(x^(1/5) + 1) - 20

________________________________________________________________________________________

Fricas [A]  time = 1.57499, size = 100, normalized size = 2.22 \begin{align*} \frac{5}{4} \, x^{\frac{4}{5}} - \frac{5}{3} \, x^{\frac{3}{5}} + \frac{5}{2} \, x^{\frac{2}{5}} - 5 \, x^{\frac{1}{5}} + 5 \, \log \left (x^{\frac{1}{5}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/4*x^(4/5) - 5/3*x^(3/5) + 5/2*x^(2/5) - 5*x^(1/5) + 5*log(x^(1/5) + 1)

________________________________________________________________________________________

Sympy [A]  time = 4.9917, size = 41, normalized size = 0.91 \begin{align*} \frac{5 x^{\frac{4}{5}}}{4} - \frac{5 x^{\frac{3}{5}}}{3} + \frac{5 x^{\frac{2}{5}}}{2} - 5 \sqrt [5]{x} + 5 \log{\left (\sqrt [5]{x} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x**(4/5)/4 - 5*x**(3/5)/3 + 5*x**(2/5)/2 - 5*x**(1/5) + 5*log(x**(1/5) + 1)

________________________________________________________________________________________

Giac [A]  time = 1.2002, size = 39, normalized size = 0.87 \begin{align*} \frac{5}{4} \, x^{\frac{4}{5}} - \frac{5}{3} \, x^{\frac{3}{5}} + \frac{5}{2} \, x^{\frac{2}{5}} - 5 \, x^{\frac{1}{5}} + 5 \, \log \left (x^{\frac{1}{5}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/4*x^(4/5) - 5/3*x^(3/5) + 5/2*x^(2/5) - 5*x^(1/5) + 5*log(x^(1/5) + 1)