Integrand size = 15, antiderivative size = 58 \[ \int \frac {\sqrt [3]{x}}{1+\sqrt {x}} \, dx=-3 \sqrt [3]{x}+\frac {6 x^{5/6}}{5}-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [6]{x}}{\sqrt {3}}\right )-3 \log \left (1+\sqrt [6]{x}\right )+\log \left (1+\sqrt {x}\right ) \] Output:
-3*x^(1/3)+6/5*x^(5/6)-2*3^(1/2)*arctan(1/3*(1-2*x^(1/6))*3^(1/2))-3*ln(1+ x^(1/6))+ln(1+x^(1/2))
Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt [3]{x}}{1+\sqrt {x}} \, dx=-3 \sqrt [3]{x}+\frac {6 x^{5/6}}{5}-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [6]{x}}{\sqrt {3}}\right )-2 \log \left (1+\sqrt [6]{x}\right )+\log \left (1-\sqrt [6]{x}+\sqrt [3]{x}\right ) \] Input:
Integrate[x^(1/3)/(1 + Sqrt[x]),x]
Output:
-3*x^(1/3) + (6*x^(5/6))/5 - 2*Sqrt[3]*ArcTan[(1 - 2*x^(1/6))/Sqrt[3]] - 2 *Log[1 + x^(1/6)] + Log[1 - x^(1/6) + x^(1/3)]
Time = 0.32 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {864, 60, 60, 68, 16, 1083, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt [3]{x}}{\sqrt {x}+1} \, dx\) |
\(\Big \downarrow \) 864 |
\(\displaystyle 2 \int \frac {x^{5/6}}{\sqrt {x}+1}d\sqrt {x}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle 2 \left (\frac {3 x^{5/6}}{5}-\int \frac {\sqrt [3]{x}}{\sqrt {x}+1}d\sqrt {x}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle 2 \left (\int \frac {1}{\left (\sqrt {x}+1\right ) \sqrt [6]{x}}d\sqrt {x}+\frac {3 x^{5/6}}{5}-\frac {3 \sqrt [3]{x}}{2}\right )\) |
\(\Big \downarrow \) 68 |
\(\displaystyle 2 \left (-\frac {3}{2} \int \frac {1}{\sqrt [6]{x}+1}d\sqrt [6]{x}+\frac {3}{2} \int \frac {1}{x-\sqrt [6]{x}+1}d\sqrt [6]{x}+\frac {3 x^{5/6}}{5}-\frac {3 \sqrt [3]{x}}{2}+\frac {1}{2} \log \left (\sqrt {x}+1\right )\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle 2 \left (\frac {3}{2} \int \frac {1}{x-\sqrt [6]{x}+1}d\sqrt [6]{x}+\frac {3 x^{5/6}}{5}-\frac {3 \sqrt [3]{x}}{2}-\frac {3}{2} \log \left (\sqrt [6]{x}+1\right )+\frac {1}{2} \log \left (\sqrt {x}+1\right )\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle 2 \left (-3 \int \frac {1}{-x-3}d\left (2 \sqrt [6]{x}-1\right )+\frac {3 x^{5/6}}{5}-\frac {3 \sqrt [3]{x}}{2}-\frac {3}{2} \log \left (\sqrt [6]{x}+1\right )+\frac {1}{2} \log \left (\sqrt {x}+1\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle 2 \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{x}-1}{\sqrt {3}}\right )+\frac {3 x^{5/6}}{5}-\frac {3 \sqrt [3]{x}}{2}-\frac {3}{2} \log \left (\sqrt [6]{x}+1\right )+\frac {1}{2} \log \left (\sqrt {x}+1\right )\right )\) |
Input:
Int[x^(1/3)/(1 + Sqrt[x]),x]
Output:
2*((-3*x^(1/3))/2 + (3*x^(5/6))/5 + Sqrt[3]*ArcTan[(-1 + 2*x^(1/6))/Sqrt[3 ]] - (3*Log[1 + x^(1/6)])/2 + Log[1 + Sqrt[x]]/2)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[-(b*c - a*d)/b, 3]}, Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[n]}, Simp[k Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x ^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Time = 0.46 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {6 x^{\frac {5}{6}}}{5}-3 x^{\frac {1}{3}}-2 \ln \left (1+x^{\frac {1}{6}}\right )+\ln \left (x^{\frac {1}{3}}-x^{\frac {1}{6}}+1\right )+2 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{6}}-1\right ) \sqrt {3}}{3}\right )\) | \(49\) |
default | \(\frac {6 x^{\frac {5}{6}}}{5}-3 x^{\frac {1}{3}}-2 \ln \left (1+x^{\frac {1}{6}}\right )+\ln \left (x^{\frac {1}{3}}-x^{\frac {1}{6}}+1\right )+2 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{6}}-1\right ) \sqrt {3}}{3}\right )\) | \(49\) |
meijerg | \(-\frac {3 x^{\frac {1}{3}} \left (-8 \sqrt {x}+20\right )}{20}+2 x^{\frac {1}{3}} \left (-\frac {\ln \left (1+x^{\frac {1}{6}}\right )}{x^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {1}{3}}-x^{\frac {1}{6}}+1\right )}{2 x^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x^{\frac {1}{6}}}{2-x^{\frac {1}{6}}}\right )}{x^{\frac {1}{3}}}\right )\) | \(72\) |
Input:
int(x^(1/3)/(1+x^(1/2)),x,method=_RETURNVERBOSE)
Output:
6/5*x^(5/6)-3*x^(1/3)-2*ln(1+x^(1/6))+ln(x^(1/3)-x^(1/6)+1)+2*3^(1/2)*arct an(1/3*(2*x^(1/6)-1)*3^(1/2))
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt [3]{x}}{1+\sqrt {x}} \, dx=2 \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - 3 \, x^{\frac {1}{3}} + \log \left (x^{\frac {1}{3}} - x^{\frac {1}{6}} + 1\right ) - 2 \, \log \left (x^{\frac {1}{6}} + 1\right ) \] Input:
integrate(x^(1/3)/(1+x^(1/2)),x, algorithm="fricas")
Output:
2*sqrt(3)*arctan(2/3*sqrt(3)*x^(1/6) - 1/3*sqrt(3)) + 6/5*x^(5/6) - 3*x^(1 /3) + log(x^(1/3) - x^(1/6) + 1) - 2*log(x^(1/6) + 1)
Result contains complex when optimal does not.
Time = 0.83 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.38 \[ \int \frac {\sqrt [3]{x}}{1+\sqrt {x}} \, dx=\frac {16 x^{\frac {5}{6}} \Gamma \left (\frac {8}{3}\right )}{5 \Gamma \left (\frac {11}{3}\right )} - \frac {8 \sqrt [3]{x} \Gamma \left (\frac {8}{3}\right )}{\Gamma \left (\frac {11}{3}\right )} - \frac {16 e^{- \frac {2 i \pi }{3}} \log {\left (- \sqrt [6]{x} e^{\frac {i \pi }{3}} + 1 \right )} \Gamma \left (\frac {8}{3}\right )}{3 \Gamma \left (\frac {11}{3}\right )} - \frac {16 \log {\left (- \sqrt [6]{x} e^{i \pi } + 1 \right )} \Gamma \left (\frac {8}{3}\right )}{3 \Gamma \left (\frac {11}{3}\right )} - \frac {16 e^{\frac {2 i \pi }{3}} \log {\left (- \sqrt [6]{x} e^{\frac {5 i \pi }{3}} + 1 \right )} \Gamma \left (\frac {8}{3}\right )}{3 \Gamma \left (\frac {11}{3}\right )} \] Input:
integrate(x**(1/3)/(1+x**(1/2)),x)
Output:
16*x**(5/6)*gamma(8/3)/(5*gamma(11/3)) - 8*x**(1/3)*gamma(8/3)/gamma(11/3) - 16*exp(-2*I*pi/3)*log(-x**(1/6)*exp_polar(I*pi/3) + 1)*gamma(8/3)/(3*ga mma(11/3)) - 16*log(-x**(1/6)*exp_polar(I*pi) + 1)*gamma(8/3)/(3*gamma(11/ 3)) - 16*exp(2*I*pi/3)*log(-x**(1/6)*exp_polar(5*I*pi/3) + 1)*gamma(8/3)/( 3*gamma(11/3))
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt [3]{x}}{1+\sqrt {x}} \, dx=2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{6}} - 1\right )}\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - 3 \, x^{\frac {1}{3}} + \log \left (x^{\frac {1}{3}} - x^{\frac {1}{6}} + 1\right ) - 2 \, \log \left (x^{\frac {1}{6}} + 1\right ) \] Input:
integrate(x^(1/3)/(1+x^(1/2)),x, algorithm="maxima")
Output:
2*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/6) - 1)) + 6/5*x^(5/6) - 3*x^(1/3) + log(x^(1/3) - x^(1/6) + 1) - 2*log(x^(1/6) + 1)
Time = 0.13 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt [3]{x}}{1+\sqrt {x}} \, dx=2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{6}} - 1\right )}\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - 3 \, x^{\frac {1}{3}} + \log \left (x^{\frac {1}{3}} - x^{\frac {1}{6}} + 1\right ) - 2 \, \log \left (x^{\frac {1}{6}} + 1\right ) \] Input:
integrate(x^(1/3)/(1+x^(1/2)),x, algorithm="giac")
Output:
2*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/6) - 1)) + 6/5*x^(5/6) - 3*x^(1/3) + log(x^(1/3) - x^(1/6) + 1) - 2*log(x^(1/6) + 1)
Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt [3]{x}}{1+\sqrt {x}} \, dx=\frac {6\,x^{5/6}}{5}-\ln \left (9\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2+36\,x^{1/6}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )+\ln \left (9\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2+36\,x^{1/6}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )-3\,x^{1/3}-2\,\ln \left (36\,x^{1/6}+36\right ) \] Input:
int(x^(1/3)/(x^(1/2) + 1),x)
Output:
log(9*(3^(1/2)*1i + 1)^2 + 36*x^(1/6))*(3^(1/2)*1i + 1) - log(9*(3^(1/2)*1 i - 1)^2 + 36*x^(1/6))*(3^(1/2)*1i - 1) - 2*log(36*x^(1/6) + 36) - 3*x^(1/ 3) + (6*x^(5/6))/5
Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt [3]{x}}{1+\sqrt {x}} \, dx=2 \sqrt {3}\, \mathit {atan} \left (\frac {2 x^{\frac {1}{6}}-1}{\sqrt {3}}\right )+\frac {6 x^{\frac {5}{6}}}{5}-3 x^{\frac {1}{3}}-2 \,\mathrm {log}\left (x^{\frac {1}{6}}+1\right )+\mathrm {log}\left (-x^{\frac {1}{6}}+x^{\frac {1}{3}}+1\right ) \] Input:
int(x^(1/3)/(1+x^(1/2)),x)
Output:
(10*sqrt(3)*atan((2*x**(1/6) - 1)/sqrt(3)) + 6*x**(5/6) - 15*x**(1/3) - 10 *log(x**(1/6) + 1) + 5*log( - x**(1/6) + x**(1/3) + 1))/5