Integrand size = 15, antiderivative size = 108 \[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=2 \sqrt {x}+\frac {3 \arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {3 \arctan \left (1+\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {3 \log \left (1-\sqrt {2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt {2}} \]
-3/2*arctan(-1+x^(1/6)*2^(1/2))*2^(1/2)-3/2*arctan(1+x^(1/6)*2^(1/2))*2^(1 /2)-3/4*ln(1+x^(1/3)-x^(1/6)*2^(1/2))*2^(1/2)+3/4*ln(1+x^(1/3)+x^(1/6)*2^( 1/2))*2^(1/2)+2*x^(1/2)
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=2 \sqrt {x}-\frac {3 \arctan \left (\frac {-1+\sqrt [3]{x}}{\sqrt {2} \sqrt [6]{x}}\right )}{\sqrt {2}}+\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [6]{x}}{1+\sqrt [3]{x}}\right )}{\sqrt {2}} \]
2*Sqrt[x] - (3*ArcTan[(-1 + x^(1/3))/(Sqrt[2]*x^(1/6))])/Sqrt[2] + (3*ArcT anh[(Sqrt[2]*x^(1/6))/(1 + x^(1/3))])/Sqrt[2]
Time = 0.31 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.15, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {10, 864, 262, 266, 826, 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.
| \(\displaystyle \int \frac {\sqrt {x}}{x+\sqrt [3]{x}} \, dx\) |
| \(\Big \downarrow \) 10 |
| \(\displaystyle \int \frac {\sqrt [6]{x}}{x^{2/3}+1}dx\) |
| \(\Big \downarrow \) 864 |
| \(\displaystyle 3 \int \frac {x^{5/6}}{x^{2/3}+1}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle 3 \left (\frac {2 \sqrt {x}}{3}-\int \frac {\sqrt [6]{x}}{x^{2/3}+1}d\sqrt [3]{x}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle 3 \left (\frac {2 \sqrt {x}}{3}-2 \int \frac {x^{2/3}}{x^{4/3}+1}d\sqrt [6]{x}\right )\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle 3 \left (\frac {2 \sqrt {x}}{3}-2 \left (\frac {1}{2} \int \frac {x^{2/3}+1}{x^{4/3}+1}d\sqrt [6]{x}-\frac {1}{2} \int \frac {1-x^{2/3}}{x^{4/3}+1}d\sqrt [6]{x}\right )\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 3 \left (\frac {2 \sqrt {x}}{3}-2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^{2/3}-\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}+\frac {1}{2} \int \frac {1}{x^{2/3}+\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}\right )-\frac {1}{2} \int \frac {1-x^{2/3}}{x^{4/3}+1}d\sqrt [6]{x}\right )\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 3 \left (\frac {2 \sqrt {x}}{3}-2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-x^{2/3}-1}d\left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x^{2/3}-1}d\left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-x^{2/3}}{x^{4/3}+1}d\sqrt [6]{x}\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 3 \left (\frac {2 \sqrt {x}}{3}-2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-x^{2/3}}{x^{4/3}+1}d\sqrt [6]{x}\right )\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 3 \left (\frac {2 \sqrt {x}}{3}-2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt [6]{x}}{x^{2/3}-\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [6]{x}+1\right )}{x^{2/3}+\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \left (\frac {2 \sqrt {x}}{3}-2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt [6]{x}}{x^{2/3}-\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [6]{x}+1\right )}{x^{2/3}+\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \left (\frac {2 \sqrt {x}}{3}-2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt [6]{x}}{x^{2/3}-\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt [6]{x}+1}{x^{2/3}+\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 3 \left (\frac {2 \sqrt {x}}{3}-2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (x^{2/3}-\sqrt {2} \sqrt [6]{x}+1\right )}{2 \sqrt {2}}-\frac {\log \left (x^{2/3}+\sqrt {2} \sqrt [6]{x}+1\right )}{2 \sqrt {2}}\right )\right )\right )\) |
3*((2*Sqrt[x])/3 - 2*((-(ArcTan[1 - Sqrt[2]*x^(1/6)]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*x^(1/6)]/Sqrt[2])/2 + (Log[1 - Sqrt[2]*x^(1/6) + x^(2/3)]/(2*Sqrt [2]) - Log[1 + Sqrt[2]*x^(1/6) + x^(2/3)]/(2*Sqrt[2]))/2))
3.6.77.3.1 Defintions of rubi rules used
Int[(u_.)*((e_.)*(x_))^(m_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x _Symbol] :> Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*(a + b*x^(s - r))^p, x], x] /; FreeQ[{a, b, e, m, r, s}, x] && IntegerQ[p] && (IntegerQ[p*r] || GtQ [e, 0]) && PosQ[s - r]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((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]
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]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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]]))
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] :> 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]
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]
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]
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]
Time = 0.95 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61
| method | result | size |
| derivativedivides | \(2 \sqrt {x}-\frac {3 \sqrt {2}\, \left (\ln \left (\frac {1+x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {2}}{1+x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {2}}\right )+2 \arctan \left (1+x^{\frac {1}{6}} \sqrt {2}\right )+2 \arctan \left (-1+x^{\frac {1}{6}} \sqrt {2}\right )\right )}{4}\) | \(66\) |
| default | \(2 \sqrt {x}-\frac {3 \sqrt {2}\, \left (\ln \left (\frac {1+x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {2}}{1+x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {2}}\right )+2 \arctan \left (1+x^{\frac {1}{6}} \sqrt {2}\right )+2 \arctan \left (-1+x^{\frac {1}{6}} \sqrt {2}\right )\right )}{4}\) | \(66\) |
| meijerg | \(2 \sqrt {x}-\frac {3 \sqrt {x}\, \left (\frac {\sqrt {2}\, \ln \left (1+x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {2}\right )}{2 \sqrt {x}}+\frac {\sqrt {2}\, \arctan \left (\frac {x^{\frac {1}{6}} \sqrt {2}}{2-x^{\frac {1}{6}} \sqrt {2}}\right )}{\sqrt {x}}-\frac {\sqrt {2}\, \ln \left (1+x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {2}\right )}{2 \sqrt {x}}+\frac {\sqrt {2}\, \arctan \left (\frac {x^{\frac {1}{6}} \sqrt {2}}{2+x^{\frac {1}{6}} \sqrt {2}}\right )}{\sqrt {x}}\right )}{2}\) | \(112\) |
2*x^(1/2)-3/4*2^(1/2)*(ln((1+x^(1/3)-x^(1/6)*2^(1/2))/(1+x^(1/3)+x^(1/6)*2 ^(1/2)))+2*arctan(1+x^(1/6)*2^(1/2))+2*arctan(-1+x^(1/6)*2^(1/2)))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=-\left (\frac {3}{4} i - \frac {3}{4}\right ) \, \sqrt {2} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, x^{\frac {1}{6}}\right ) + \left (\frac {3}{4} i + \frac {3}{4}\right ) \, \sqrt {2} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, x^{\frac {1}{6}}\right ) - \left (\frac {3}{4} i + \frac {3}{4}\right ) \, \sqrt {2} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, x^{\frac {1}{6}}\right ) + \left (\frac {3}{4} i - \frac {3}{4}\right ) \, \sqrt {2} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, x^{\frac {1}{6}}\right ) + 2 \, \sqrt {x} \]
-(3/4*I - 3/4)*sqrt(2)*log((I + 1)*sqrt(2) + 2*x^(1/6)) + (3/4*I + 3/4)*sq rt(2)*log(-(I - 1)*sqrt(2) + 2*x^(1/6)) - (3/4*I + 3/4)*sqrt(2)*log((I - 1 )*sqrt(2) + 2*x^(1/6)) + (3/4*I - 3/4)*sqrt(2)*log(-(I + 1)*sqrt(2) + 2*x^ (1/6)) + 2*sqrt(x)
\[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=\int \frac {\sqrt {x}}{\sqrt [3]{x} + x}\, dx \]
Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=-\frac {3}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, x^{\frac {1}{6}}\right )}\right ) - \frac {3}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, x^{\frac {1}{6}}\right )}\right ) + \frac {3}{4} \, \sqrt {2} \log \left (\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) - \frac {3}{4} \, \sqrt {2} \log \left (-\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) + 2 \, \sqrt {x} \]
-3/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*x^(1/6))) - 3/2*sqrt(2)*arcta n(-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) + 2*sqrt(x)
Time = 0.35 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=-\frac {3}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, x^{\frac {1}{6}}\right )}\right ) - \frac {3}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, x^{\frac {1}{6}}\right )}\right ) + \frac {3}{4} \, \sqrt {2} \log \left (\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) - \frac {3}{4} \, \sqrt {2} \log \left (-\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) + 2 \, \sqrt {x} \]
-3/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*x^(1/6))) - 3/2*sqrt(2)*arcta n(-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) + 2*sqrt(x)
Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=2\,\sqrt {x}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x^{1/6}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {3}{2}+\frac {3}{2}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x^{1/6}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {3}{2}-\frac {3}{2}{}\mathrm {i}\right ) \]