\(\int \frac {\sqrt {x}}{1+x^{2/3}} \, dx\) [280]

Optimal result

Integrand size = 15, antiderivative size = 88 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=-6 \sqrt [6]{x}+\frac {6 x^{5/6}}{5}-\frac {3 \arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}+\frac {3 \arctan \left (1+\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}+\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [6]{x}}{1+\sqrt [3]{x}}\right )}{\sqrt {2}} \] Output:

-6*x^(1/6)+6/5*x^(5/6)+3/2*arctan(-1+2^(1/2)*x^(1/6))*2^(1/2)+3/2*arctan(1 
+2^(1/2)*x^(1/6))*2^(1/2)+3/2*arctanh(2^(1/2)*x^(1/6)/(1+x^(1/3)))*2^(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=\frac {6}{5} \left (-5+x^{2/3}\right ) \sqrt [6]{x}+\frac {3 \arctan \left (\frac {-1+\sqrt [3]{x}}{\sqrt {2} \sqrt [6]{x}}\right )}{\sqrt {2}}+\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [6]{x}}{1+\sqrt [3]{x}}\right )}{\sqrt {2}} \] Input:

Integrate[Sqrt[x]/(1 + x^(2/3)),x]
 

Output:

(6*(-5 + x^(2/3))*x^(1/6))/5 + (3*ArcTan[(-1 + x^(1/3))/(Sqrt[2]*x^(1/6))] 
)/Sqrt[2] + (3*ArcTanh[(Sqrt[2]*x^(1/6))/(1 + x^(1/3))])/Sqrt[2]
 

Rubi [A] (warning: unable to verify)

Time = 0.52 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.49, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {864, 262, 262, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {\sqrt {x}}{x^{2/3}+1} \, dx\)

\(\Big \downarrow \) 864

\(\displaystyle 3 \int \frac {x^{7/6}}{x^{2/3}+1}d\sqrt [3]{x}\)

\(\Big \downarrow \) 262

\(\displaystyle 3 \left (\frac {2 x^{5/6}}{5}-\int \frac {\sqrt {x}}{x^{2/3}+1}d\sqrt [3]{x}\right )\)

\(\Big \downarrow \) 262

\(\displaystyle 3 \left (\int \frac {1}{\left (x^{2/3}+1\right ) \sqrt [6]{x}}d\sqrt [3]{x}+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\)

\(\Big \downarrow \) 266

\(\displaystyle 3 \left (2 \int \frac {1}{x^{4/3}+1}d\sqrt [6]{x}+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\)

\(\Big \downarrow \) 755

\(\displaystyle 3 \left (2 \left (\frac {1}{2} \int \frac {1-x^{2/3}}{x^{4/3}+1}d\sqrt [6]{x}+\frac {1}{2} \int \frac {x^{2/3}+1}{x^{4/3}+1}d\sqrt [6]{x}\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle 3 \left (2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^{2/3}-\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}+\frac {1}{2} \int \frac {1}{x^{2/3}+\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}\right )+\frac {1}{2} \int \frac {1-x^{2/3}}{x^{4/3}+1}d\sqrt [6]{x}\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle 3 \left (2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-x^{2/3}-1}d\left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x^{2/3}-1}d\left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-x^{2/3}}{x^{4/3}+1}d\sqrt [6]{x}\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle 3 \left (2 \left (\frac {1}{2} \int \frac {1-x^{2/3}}{x^{4/3}+1}d\sqrt [6]{x}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle 3 \left (2 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt [6]{x}}{x^{2/3}-\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [6]{x}+1\right )}{x^{2/3}+\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 3 \left (2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt [6]{x}}{x^{2/3}-\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [6]{x}+1\right )}{x^{2/3}+\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 3 \left (2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt [6]{x}}{x^{2/3}-\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt [6]{x}+1}{x^{2/3}+\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle 3 \left (2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (x^{2/3}+\sqrt {2} \sqrt [6]{x}+1\right )}{2 \sqrt {2}}-\frac {\log \left (x^{2/3}-\sqrt {2} \sqrt [6]{x}+1\right )}{2 \sqrt {2}}\right )\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\)

Input:

Int[Sqrt[x]/(1 + x^(2/3)),x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 864

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]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {6 x^{\frac {5}{6}}}{5}-6 x^{\frac {1}{6}}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {x^{\frac {1}{3}}+\sqrt {2}\, x^{\frac {1}{6}}+1}{x^{\frac {1}{3}}-\sqrt {2}\, x^{\frac {1}{6}}+1}\right )+2 \arctan \left (1+\sqrt {2}\, x^{\frac {1}{6}}\right )+2 \arctan \left (-1+\sqrt {2}\, x^{\frac {1}{6}}\right )\right )}{4}\)	\(71\)
default	\(\frac {6 x^{\frac {5}{6}}}{5}-6 x^{\frac {1}{6}}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {x^{\frac {1}{3}}+\sqrt {2}\, x^{\frac {1}{6}}+1}{x^{\frac {1}{3}}-\sqrt {2}\, x^{\frac {1}{6}}+1}\right )+2 \arctan \left (1+\sqrt {2}\, x^{\frac {1}{6}}\right )+2 \arctan \left (-1+\sqrt {2}\, x^{\frac {1}{6}}\right )\right )}{4}\)	\(71\)
meijerg	\(-\frac {2 x^{\frac {1}{6}} \left (-9 x^{\frac {2}{3}}+45\right )}{15}+\frac {3 x^{\frac {1}{6}} \left (-\frac {\sqrt {2}\, \ln \left (x^{\frac {1}{3}}-\sqrt {2}\, x^{\frac {1}{6}}+1\right )}{2 x^{\frac {1}{6}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{6}}}{2-\sqrt {2}\, x^{\frac {1}{6}}}\right )}{x^{\frac {1}{6}}}+\frac {\sqrt {2}\, \ln \left (x^{\frac {1}{3}}+\sqrt {2}\, x^{\frac {1}{6}}+1\right )}{2 x^{\frac {1}{6}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{6}}}{2+\sqrt {2}\, x^{\frac {1}{6}}}\right )}{x^{\frac {1}{6}}}\right )}{2}\)	\(119\)

Input:

int(x^(1/2)/(1+x^(2/3)),x,method=_RETURNVERBOSE)
 

Output:

6/5*x^(5/6)-6*x^(1/6)+3/4*2^(1/2)*(ln((x^(1/3)+2^(1/2)*x^(1/6)+1)/(x^(1/3) 
-2^(1/2)*x^(1/6)+1))+2*arctan(1+2^(1/2)*x^(1/6))+2*arctan(-1+2^(1/2)*x^(1/ 
6)))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=\frac {3}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x^{\frac {1}{6}} + 1\right ) + \frac {3}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x^{\frac {1}{6}} - 1\right ) + \frac {3}{4} \, \sqrt {2} \log \left (\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) - \frac {3}{4} \, \sqrt {2} \log \left (-\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - 6 \, x^{\frac {1}{6}} \] Input:

integrate(x^(1/2)/(1+x^(2/3)),x, algorithm="fricas")
 

Output:

3/2*sqrt(2)*arctan(sqrt(2)*x^(1/6) + 1) + 3/2*sqrt(2)*arctan(sqrt(2)*x^(1/ 
6) - 1) + 3/4*sqrt(2)*log(sqrt(2)*x^(1/6) + x^(1/3) + 1) - 3/4*sqrt(2)*log 
(-sqrt(2)*x^(1/6) + x^(1/3) + 1) + 6/5*x^(5/6) - 6*x^(1/6)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.86 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.12 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=\frac {27 x^{\frac {5}{6}} \Gamma \left (\frac {9}{4}\right )}{10 \Gamma \left (\frac {13}{4}\right )} - \frac {27 \sqrt [6]{x} \Gamma \left (\frac {9}{4}\right )}{2 \Gamma \left (\frac {13}{4}\right )} - \frac {27 e^{- \frac {i \pi }{4}} \log {\left (- \sqrt [6]{x} e^{\frac {i \pi }{4}} + 1 \right )} \Gamma \left (\frac {9}{4}\right )}{8 \Gamma \left (\frac {13}{4}\right )} + \frac {27 i e^{- \frac {i \pi }{4}} \log {\left (- \sqrt [6]{x} e^{\frac {3 i \pi }{4}} + 1 \right )} \Gamma \left (\frac {9}{4}\right )}{8 \Gamma \left (\frac {13}{4}\right )} + \frac {27 e^{- \frac {i \pi }{4}} \log {\left (- \sqrt [6]{x} e^{\frac {5 i \pi }{4}} + 1 \right )} \Gamma \left (\frac {9}{4}\right )}{8 \Gamma \left (\frac {13}{4}\right )} - \frac {27 i e^{- \frac {i \pi }{4}} \log {\left (- \sqrt [6]{x} e^{\frac {7 i \pi }{4}} + 1 \right )} \Gamma \left (\frac {9}{4}\right )}{8 \Gamma \left (\frac {13}{4}\right )} \] Input:

integrate(x**(1/2)/(1+x**(2/3)),x)
 

Output:

27*x**(5/6)*gamma(9/4)/(10*gamma(13/4)) - 27*x**(1/6)*gamma(9/4)/(2*gamma( 
13/4)) - 27*exp(-I*pi/4)*log(-x**(1/6)*exp_polar(I*pi/4) + 1)*gamma(9/4)/( 
8*gamma(13/4)) + 27*I*exp(-I*pi/4)*log(-x**(1/6)*exp_polar(3*I*pi/4) + 1)* 
gamma(9/4)/(8*gamma(13/4)) + 27*exp(-I*pi/4)*log(-x**(1/6)*exp_polar(5*I*p 
i/4) + 1)*gamma(9/4)/(8*gamma(13/4)) - 27*I*exp(-I*pi/4)*log(-x**(1/6)*exp 
_polar(7*I*pi/4) + 1)*gamma(9/4)/(8*gamma(13/4))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=\frac {3}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, x^{\frac {1}{6}}\right )}\right ) + \frac {3}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, x^{\frac {1}{6}}\right )}\right ) + \frac {3}{4} \, \sqrt {2} \log \left (\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) - \frac {3}{4} \, \sqrt {2} \log \left (-\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - 6 \, x^{\frac {1}{6}} \] Input:

integrate(x^(1/2)/(1+x^(2/3)),x, algorithm="maxima")
 

Output:

3/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*x^(1/6))) + 3/2*sqrt(2)*arctan 
(-1/2*sqrt(2)*(sqrt(2) - 2*x^(1/6))) + 3/4*sqrt(2)*log(sqrt(2)*x^(1/6) + x 
^(1/3) + 1) - 3/4*sqrt(2)*log(-sqrt(2)*x^(1/6) + x^(1/3) + 1) + 6/5*x^(5/6 
) - 6*x^(1/6)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=\frac {3}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, x^{\frac {1}{6}}\right )}\right ) + \frac {3}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, x^{\frac {1}{6}}\right )}\right ) + \frac {3}{4} \, \sqrt {2} \log \left (\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) - \frac {3}{4} \, \sqrt {2} \log \left (-\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - 6 \, x^{\frac {1}{6}} \] Input:

integrate(x^(1/2)/(1+x^(2/3)),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

3/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*x^(1/6))) + 3/2*sqrt(2)*arctan 
(-1/2*sqrt(2)*(sqrt(2) - 2*x^(1/6))) + 3/4*sqrt(2)*log(sqrt(2)*x^(1/6) + x 
^(1/3) + 1) - 3/4*sqrt(2)*log(-sqrt(2)*x^(1/6) + x^(1/3) + 1) + 6/5*x^(5/6 
) - 6*x^(1/6)
 

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=\frac {6\,x^{5/6}}{5}-6\,x^{1/6}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x^{1/6}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{2}+\frac {3}{2}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x^{1/6}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{2}-\frac {3}{2}{}\mathrm {i}\right ) \] Input:

int(x^(1/2)/(x^(2/3) + 1),x)
 

Output:

2^(1/2)*atan(2^(1/2)*x^(1/6)*(1/2 - 1i/2))*(3/2 + 3i/2) + 2^(1/2)*atan(2^( 
1/2)*x^(1/6)*(1/2 + 1i/2))*(3/2 - 3i/2) - 6*x^(1/6) + (6*x^(5/6))/5
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=\frac {3 \sqrt {2}\, \mathit {atan} \left (\frac {2 x^{\frac {1}{6}}-\sqrt {2}}{\sqrt {2}}\right )}{2}+\frac {3 \sqrt {2}\, \mathit {atan} \left (\frac {2 x^{\frac {1}{6}}+\sqrt {2}}{\sqrt {2}}\right )}{2}+\frac {6 x^{\frac {5}{6}}}{5}-6 x^{\frac {1}{6}}-\frac {3 \sqrt {2}\, \mathrm {log}\left (-x^{\frac {1}{6}} \sqrt {2}+x^{\frac {1}{3}}+1\right )}{4}+\frac {3 \sqrt {2}\, \mathrm {log}\left (x^{\frac {1}{6}} \sqrt {2}+x^{\frac {1}{3}}+1\right )}{4} \] Input:

int(x^(1/2)/(1+x^(2/3)),x)
 

Output:

(3*(10*sqrt(2)*atan((2*x**(1/6) - sqrt(2))/sqrt(2)) + 10*sqrt(2)*atan((2*x 
**(1/6) + sqrt(2))/sqrt(2)) + 8*x**(5/6) - 40*x**(1/6) - 5*sqrt(2)*log( - 
x**(1/6)*sqrt(2) + x**(1/3) + 1) + 5*sqrt(2)*log(x**(1/6)*sqrt(2) + x**(1/ 
3) + 1)))/20