\(\int \frac {1+\sqrt {x}}{(1+\sqrt [3]{x}) \sqrt {x}} \, dx\) [917]

Optimal result

Integrand size = 22, antiderivative size = 42 \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=6 \sqrt [6]{x}-3 \sqrt [3]{x}+\frac {3 x^{2/3}}{2}-6 \arctan \left (\sqrt [6]{x}\right )+3 \log \left (1+\sqrt [3]{x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6820, 1607, 1816, 649, 209, 266} \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=-6 \arctan \left (\sqrt [6]{x}\right )+\frac {3 x^{2/3}}{2}-3 \sqrt [3]{x}+6 \sqrt [6]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]

[In]

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

[Out]

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1816

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1+\frac {1}{\sqrt {x}}}{1+\sqrt [3]{x}} \, dx \\ & = 6 \text {Subst}\left (\int \frac {x^2+x^5}{1+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \text {Subst}\left (\int \frac {x^2 \left (1+x^3\right )}{1+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \text {Subst}\left (\int \left (1-x+x^3-\frac {1-x}{1+x^2}\right ) \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}-3 \sqrt [3]{x}+\frac {3 x^{2/3}}{2}-6 \text {Subst}\left (\int \frac {1-x}{1+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}-3 \sqrt [3]{x}+\frac {3 x^{2/3}}{2}-6 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [6]{x}\right )+6 \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}-3 \sqrt [3]{x}+\frac {3 x^{2/3}}{2}-6 \tan ^{-1}\left (\sqrt [6]{x}\right )+3 \log \left (1+\sqrt [3]{x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=\frac {3}{2} \left (4-2 \sqrt [6]{x}+\sqrt {x}\right ) \sqrt [6]{x}-6 \arctan \left (\sqrt [6]{x}\right )+3 \log \left (1+\sqrt [3]{x}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(6 x^{\frac {1}{6}}-3 x^{\frac {1}{3}}+\frac {3 x^{\frac {2}{3}}}{2}-6 \arctan \left (x^{\frac {1}{6}}\right )+3 \ln \left (1+x^{\frac {1}{3}}\right )\)	\(31\)
meijerg	\(6 x^{\frac {1}{6}}-6 \arctan \left (x^{\frac {1}{6}}\right )-\frac {x^{\frac {1}{3}} \left (-3 x^{\frac {1}{3}}+6\right )}{2}+3 \ln \left (1+x^{\frac {1}{3}}\right )\)	\(33\)
default	\(\ln \left (x +1\right )+\frac {3 x^{\frac {2}{3}}}{2}-\ln \left (x^{\frac {2}{3}}-x^{\frac {1}{3}}+1\right )+2 \ln \left (1+x^{\frac {1}{3}}\right )-3 x^{\frac {1}{3}}+6 x^{\frac {1}{6}}-6 \arctan \left (x^{\frac {1}{6}}\right )\)	\(48\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=\frac {3}{2} \, x^{\frac {2}{3}} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \arctan \left (x^{\frac {1}{6}}\right ) + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 4.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.93 \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=6 \sqrt [6]{x} + \frac {3 x^{\frac {2}{3}}}{2} - 3 \sqrt [3]{x} + 3 \log {\left (\sqrt [3]{x} + 1 \right )} - 6 \operatorname {atan}{\left (\sqrt [6]{x} \right )} \]

[In]

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

[Out]

6*x**(1/6) + 3*x**(2/3)/2 - 3*x**(1/3) + 3*log(x**(1/3) + 1) - 6*atan(x**(1/6))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=\frac {3}{2} \, x^{\frac {2}{3}} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \arctan \left (x^{\frac {1}{6}}\right ) + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=\frac {3}{2} \, x^{\frac {2}{3}} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \arctan \left (x^{\frac {1}{6}}\right ) + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=\frac {3\,x^{2/3}}{2}+3\,\ln \left (\left (-6+x^{1/6}\,6{}\mathrm {i}\right )\,\left (6+x^{1/6}\,6{}\mathrm {i}\right )\right )-3\,x^{1/3}-6\,\mathrm {atan}\left (x^{1/6}\right )+6\,x^{1/6} \]

[In]

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

[Out]

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