\(\int \frac {1}{1+\sqrt [5]{x}} \, dx\) [2448]

Optimal result

Integrand size = 9, antiderivative size = 45 \[ \int \frac {1}{1+\sqrt [5]{x}} \, dx=-5 \sqrt [5]{x}+\frac {5 x^{2/5}}{2}-\frac {5 x^{3/5}}{3}+\frac {5 x^{4/5}}{4}+5 \log \left (1+\sqrt [5]{x}\right ) \]

[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))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {196, 45} \[ \int \frac {1}{1+\sqrt [5]{x}} \, dx=\frac {5 x^{4/5}}{4}-\frac {5 x^{3/5}}{3}+\frac {5 x^{2/5}}{2}-5 \sqrt [5]{x}+5 \log \left (\sqrt [5]{x}+1\right ) \]

[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 45

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 196

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*} \text {integral}& = 5 \text {Subst}\left (\int \frac {x^4}{1+x} \, dx,x,\sqrt [5]{x}\right ) \\ & = 5 \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {1}{1+\sqrt [5]{x}} \, dx=\frac {5}{12} \left (-12+6 \sqrt [5]{x}-4 x^{2/5}+3 x^{3/5}\right ) \sqrt [5]{x}+5 \log \left (1+\sqrt [5]{x}\right ) \]

[In]

Integrate[(1 + x^(1/5))^(-1),x]

[Out]

(5*(-12 + 6*x^(1/5) - 4*x^(2/5) + 3*x^(3/5))*x^(1/5))/12 + 5*Log[1 + x^(1/5)]

Maple [A] (verified)

Time = 3.78 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.67

method	result	size
derivativedivides	\(-5 x^{\frac {1}{5}}+\frac {5 x^{\frac {2}{5}}}{2}-\frac {5 x^{\frac {3}{5}}}{3}+\frac {5 x^{\frac {4}{5}}}{4}+5 \ln \left (1+x^{\frac {1}{5}}\right )\)	\(30\)
meijerg	\(-\frac {x^{\frac {1}{5}} \left (-15 x^{\frac {3}{5}}+20 x^{\frac {2}{5}}-30 x^{\frac {1}{5}}+60\right )}{12}+5 \ln \left (1+x^{\frac {1}{5}}\right )\)	\(32\)
trager	\(-5 x^{\frac {1}{5}}+\frac {5 x^{\frac {2}{5}}}{2}-\frac {5 x^{\frac {3}{5}}}{3}+\frac {5 x^{\frac {4}{5}}}{4}+\ln \left (-5 x^{\frac {4}{5}}-10 x^{\frac {3}{5}}-10 x^{\frac {2}{5}}-5 x^{\frac {1}{5}}-x -1\right )\)	\(48\)
default	\(-5 x^{\frac {1}{5}}-\frac {5 x^{\frac {3}{5}}}{3}+\frac {5 x^{\frac {2}{5}}}{2}+\frac {5 x^{\frac {4}{5}}}{4}+4 \ln \left (1+x^{\frac {1}{5}}\right )+\frac {\ln \left (-\sqrt {5}\, x^{\frac {1}{5}}+2 x^{\frac {2}{5}}-x^{\frac {1}{5}}+2\right ) \sqrt {5}}{2}-\frac {\ln \left (\sqrt {5}\, x^{\frac {1}{5}}+2 x^{\frac {2}{5}}-x^{\frac {1}{5}}+2\right ) \sqrt {5}}{2}-\frac {\ln \left (\sqrt {5}\, x^{\frac {1}{5}}+2 x^{\frac {2}{5}}-x^{\frac {1}{5}}+2\right )}{2}+\ln \left (1+x \right )-\frac {\left (\sqrt {5}+1\right ) \ln \left (-\sqrt {5}\, x^{\frac {1}{5}}+2 x^{\frac {2}{5}}-x^{\frac {1}{5}}+2\right )}{4}+\frac {\left (\sqrt {5}-1\right ) \ln \left (\sqrt {5}\, x^{\frac {1}{5}}+2 x^{\frac {2}{5}}-x^{\frac {1}{5}}+2\right )}{4}-\frac {\ln \left (-\sqrt {5}\, x^{\frac {1}{5}}+2 x^{\frac {2}{5}}-x^{\frac {1}{5}}+2\right )}{2}+\frac {2 \left (4-\frac {\left (-\sqrt {5}-1\right )^{2}}{4}\right ) \arctan \left (\frac {-\sqrt {5}+4 x^{\frac {1}{5}}-1}{\sqrt {10-2 \sqrt {5}}}\right )}{\sqrt {10-2 \sqrt {5}}}-\frac {2 \left (-4-\frac {\left (-\sqrt {5}+1\right ) \left (\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {\sqrt {5}+4 x^{\frac {1}{5}}-1}{\sqrt {10+2 \sqrt {5}}}\right )}{\sqrt {10+2 \sqrt {5}}}-\frac {2 \left (-\sqrt {5}+1-\frac {\left (\sqrt {5}+1\right ) \left (-\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {-\sqrt {5}+4 x^{\frac {1}{5}}-1}{\sqrt {10-2 \sqrt {5}}}\right )}{\sqrt {10-2 \sqrt {5}}}+\frac {2 \left (-\sqrt {5}-1-\frac {\left (\sqrt {5}-1\right )^{2}}{4}\right ) \arctan \left (\frac {\sqrt {5}+4 x^{\frac {1}{5}}-1}{\sqrt {10+2 \sqrt {5}}}\right )}{\sqrt {10+2 \sqrt {5}}}+\frac {\left (-\sqrt {5}-1\right ) \ln \left (-\sqrt {5}\, x^{\frac {1}{5}}+2 x^{\frac {2}{5}}-x^{\frac {1}{5}}+2\right )}{4}-\frac {\left (-\sqrt {5}+1\right ) \ln \left (\sqrt {5}\, x^{\frac {1}{5}}+2 x^{\frac {2}{5}}-x^{\frac {1}{5}}+2\right )}{4}\)	\(442\)

[In]

int(1/(1+x^(1/5)),x,method=_RETURNVERBOSE)

[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))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64 \[ \int \frac {1}{1+\sqrt [5]{x}} \, dx=\frac {5}{4} \, x^{\frac {4}{5}} - \frac {5}{3} \, x^{\frac {3}{5}} + \frac {5}{2} \, x^{\frac {2}{5}} - 5 \, x^{\frac {1}{5}} + 5 \, \log \left (x^{\frac {1}{5}} + 1\right ) \]

[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] (verification not implemented)

Time = 22.45 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {1}{1+\sqrt [5]{x}} \, dx=\frac {5 x^{\frac {4}{5}}}{4} - \frac {5 x^{\frac {3}{5}}}{3} + \frac {5 x^{\frac {2}{5}}}{2} - 5 \sqrt [5]{x} + 5 \log {\left (\sqrt [5]{x} + 1 \right )} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int \frac {1}{1+\sqrt [5]{x}} \, dx=\frac {5}{4} \, {\left (x^{\frac {1}{5}} + 1\right )}^{4} - \frac {20}{3} \, {\left (x^{\frac {1}{5}} + 1\right )}^{3} + 15 \, {\left (x^{\frac {1}{5}} + 1\right )}^{2} - 20 \, x^{\frac {1}{5}} + 5 \, \log \left (x^{\frac {1}{5}} + 1\right ) - 20 \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64 \[ \int \frac {1}{1+\sqrt [5]{x}} \, dx=\frac {5}{4} \, x^{\frac {4}{5}} - \frac {5}{3} \, x^{\frac {3}{5}} + \frac {5}{2} \, x^{\frac {2}{5}} - 5 \, x^{\frac {1}{5}} + 5 \, \log \left (x^{\frac {1}{5}} + 1\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64 \[ \int \frac {1}{1+\sqrt [5]{x}} \, dx=5\,\ln \left (x^{1/5}+1\right )-5\,x^{1/5}+\frac {5\,x^{2/5}}{2}-\frac {5\,x^{3/5}}{3}+\frac {5\,x^{4/5}}{4} \]

[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