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

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.02, size = 45, normalized size = 1.00 \[ \frac {5 x^{4/5}}{4}-\frac {5 x^{3/5}}{3}+\frac {5 x^{2/5}}{2}-5 \sqrt [5]{x}+5 \log \left (\sqrt [5]{x}+1\right ) \]

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)]

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fricas [A]  time = 1.61, size = 29, normalized size = 0.64 \[ \frac {5}{4} \, x^{\frac {4}{5}} - \frac {5}{3} \, x^{\frac {3}{5}} + \frac {5}{2} \, x^{\frac {2}{5}} - 5 \, x^{\frac {1}{5}} + 5 \, \log \left (x^{\frac {1}{5}} + 1\right ) \]

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)

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giac [A]  time = 0.19, size = 29, normalized size = 0.64 \[ \frac {5}{4} \, x^{\frac {4}{5}} - \frac {5}{3} \, x^{\frac {3}{5}} + \frac {5}{2} \, x^{\frac {2}{5}} - 5 \, x^{\frac {1}{5}} + 5 \, \log \left (x^{\frac {1}{5}} + 1\right ) \]

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)

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maple [B]  time = 0.06, size = 79, normalized size = 1.76 \[ 4 \ln \left (x^{\frac {1}{5}}+1\right )+\ln \left (x +1\right )-\ln \left (2 x^{\frac {2}{5}}-\sqrt {5}\, x^{\frac {1}{5}}-x^{\frac {1}{5}}+2\right )-\ln \left (2 x^{\frac {2}{5}}+\sqrt {5}\, x^{\frac {1}{5}}-x^{\frac {1}{5}}+2\right )+\frac {5 x^{\frac {4}{5}}}{4}-\frac {5 x^{\frac {3}{5}}}{3}+\frac {5 x^{\frac {2}{5}}}{2}-5 x^{\frac {1}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.50, size = 42, normalized size = 0.93 \[ \frac {5}{4} \, {\left (x^{\frac {1}{5}} + 1\right )}^{4} - \frac {20}{3} \, {\left (x^{\frac {1}{5}} + 1\right )}^{3} + 15 \, {\left (x^{\frac {1}{5}} + 1\right )}^{2} - 20 \, x^{\frac {1}{5}} + 5 \, \log \left (x^{\frac {1}{5}} + 1\right ) - 20 \]

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

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mupad [B]  time = 0.02, size = 29, normalized size = 0.64 \[ 5\,\ln \left (x^{1/5}+1\right )-5\,x^{1/5}+\frac {5\,x^{2/5}}{2}-\frac {5\,x^{3/5}}{3}+\frac {5\,x^{4/5}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/5) + 1),x)

[Out]

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

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sympy [A]  time = 4.08, size = 41, normalized size = 0.91 \[ \frac {5 x^{\frac {4}{5}}}{4} - \frac {5 x^{\frac {3}{5}}}{3} + \frac {5 x^{\frac {2}{5}}}{2} - 5 \sqrt [5]{x} + 5 \log {\left (\sqrt [5]{x} + 1 \right )} \]

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)

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