\(\int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt\) [319]

Optimal result

Integrand size = 15, antiderivative size = 41 \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=-6 \sqrt [6]{t}+2 \sqrt {t}-\frac {6 t^{5/6}}{5}+\frac {6 t^{7/6}}{7}+6 \arctan \left (\sqrt [6]{t}\right ) \]

[Out]

-6*t^(1/6)-6/5*t^(5/6)+6/7*t^(7/6)+6*arctan(t^(1/6))+2*t^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {348, 52, 65, 209} \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=6 \arctan \left (\sqrt [6]{t}\right )+\frac {6 t^{7/6}}{7}-\frac {6 t^{5/6}}{5}+2 \sqrt {t}-6 \sqrt [6]{t} \]

[In]

Int[Sqrt[t]/(1 + t^(1/3)),t]

[Out]

-6*t^(1/6) + 2*Sqrt[t] - (6*t^(5/6))/5 + (6*t^(7/6))/7 + 6*ArcTan[t^(1/6)]

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 348

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[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]

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {t^{7/2}}{1+t} \, dt,t,\sqrt [3]{t}\right ) \\ & = \frac {6 t^{7/6}}{7}-3 \text {Subst}\left (\int \frac {t^{5/2}}{1+t} \, dt,t,\sqrt [3]{t}\right ) \\ & = -\frac {6 t^{5/6}}{5}+\frac {6 t^{7/6}}{7}+3 \text {Subst}\left (\int \frac {t^{3/2}}{1+t} \, dt,t,\sqrt [3]{t}\right ) \\ & = 2 \sqrt {t}-\frac {6 t^{5/6}}{5}+\frac {6 t^{7/6}}{7}-3 \text {Subst}\left (\int \frac {\sqrt {t}}{1+t} \, dt,t,\sqrt [3]{t}\right ) \\ & = -6 \sqrt [6]{t}+2 \sqrt {t}-\frac {6 t^{5/6}}{5}+\frac {6 t^{7/6}}{7}+3 \text {Subst}\left (\int \frac {1}{\sqrt {t} (1+t)} \, dt,t,\sqrt [3]{t}\right ) \\ & = -6 \sqrt [6]{t}+2 \sqrt {t}-\frac {6 t^{5/6}}{5}+\frac {6 t^{7/6}}{7}+6 \text {Subst}\left (\int \frac {1}{1+t^2} \, dt,t,\sqrt [6]{t}\right ) \\ & = -6 \sqrt [6]{t}+2 \sqrt {t}-\frac {6 t^{5/6}}{5}+\frac {6 t^{7/6}}{7}+6 \arctan \left (\sqrt [6]{t}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=\frac {2}{35} \left (-105 \sqrt [6]{t}+35 \sqrt {t}-21 t^{5/6}+15 t^{7/6}\right )+6 \arctan \left (\sqrt [6]{t}\right ) \]

[In]

Integrate[Sqrt[t]/(1 + t^(1/3)),t]

[Out]

(2*(-105*t^(1/6) + 35*Sqrt[t] - 21*t^(5/6) + 15*t^(7/6)))/35 + 6*ArcTan[t^(1/6)]

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.68

method	result	size
derivativedivides	\(-6 t^{\frac {1}{6}}-\frac {6 t^{\frac {5}{6}}}{5}+\frac {6 t^{\frac {7}{6}}}{7}+6 \arctan \left (t^{\frac {1}{6}}\right )+2 \sqrt {t}\)	\(28\)
default	\(-6 t^{\frac {1}{6}}-\frac {6 t^{\frac {5}{6}}}{5}+\frac {6 t^{\frac {7}{6}}}{7}+6 \arctan \left (t^{\frac {1}{6}}\right )+2 \sqrt {t}\)	\(28\)
meijerg	\(-\frac {2 t^{\frac {1}{6}} \left (-45 t +63 t^{\frac {2}{3}}-105 t^{\frac {1}{3}}+315\right )}{105}+6 \arctan \left (t^{\frac {1}{6}}\right )\)	\(28\)

[In]

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

[Out]

-6*t^(1/6)-6/5*t^(5/6)+6/7*t^(7/6)+6*arctan(t^(1/6))+2*t^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=\frac {6}{7} \, {\left (t - 7\right )} t^{\frac {1}{6}} - \frac {6}{5} \, t^{\frac {5}{6}} + 2 \, \sqrt {t} + 6 \, \arctan \left (t^{\frac {1}{6}}\right ) \]

[In]

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

[Out]

6/7*(t - 7)*t^(1/6) - 6/5*t^(5/6) + 2*sqrt(t) + 6*arctan(t^(1/6))

Sympy [A] (verification not implemented)

Time = 1.70 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=\frac {6 t^{\frac {7}{6}}}{7} - \frac {6 t^{\frac {5}{6}}}{5} - 6 \sqrt [6]{t} + 2 \sqrt {t} + 6 \operatorname {atan}{\left (\sqrt [6]{t} \right )} \]

[In]

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

[Out]

6*t**(7/6)/7 - 6*t**(5/6)/5 - 6*t**(1/6) + 2*sqrt(t) + 6*atan(t**(1/6))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=\frac {6}{7} \, t^{\frac {7}{6}} - \frac {6}{5} \, t^{\frac {5}{6}} + 2 \, \sqrt {t} - 6 \, t^{\frac {1}{6}} + 6 \, \arctan \left (t^{\frac {1}{6}}\right ) \]

[In]

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

[Out]

6/7*t^(7/6) - 6/5*t^(5/6) + 2*sqrt(t) - 6*t^(1/6) + 6*arctan(t^(1/6))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=\frac {6}{7} \, t^{\frac {7}{6}} - \frac {6}{5} \, t^{\frac {5}{6}} + 2 \, \sqrt {t} - 6 \, t^{\frac {1}{6}} + 6 \, \arctan \left (t^{\frac {1}{6}}\right ) \]

[In]

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

[Out]

6/7*t^(7/6) - 6/5*t^(5/6) + 2*sqrt(t) - 6*t^(1/6) + 6*arctan(t^(1/6))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=6\,\mathrm {atan}\left (t^{1/6}\right )+2\,\sqrt {t}-6\,t^{1/6}-\frac {6\,t^{5/6}}{5}+\frac {6\,t^{7/6}}{7} \]

[In]

int(t^(1/2)/(t^(1/3) + 1),t)

[Out]

6*atan(t^(1/6)) + 2*t^(1/2) - 6*t^(1/6) - (6*t^(5/6))/5 + (6*t^(7/6))/7