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

Optimal. Leaf size=41 \[ -6 \sqrt [6]{t}+2 \sqrt {t}-\frac {6 t^{5/6}}{5}+\frac {6 t^{7/6}}{7}+6 \tan ^{-1}\left (\sqrt [6]{t}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {348, 52, 65, 209} \begin {gather*} \frac {6 t^{7/6}}{7}-\frac {6 t^{5/6}}{5}+2 \sqrt {t}-6 \sqrt [6]{t}+6 \tan ^{-1}\left (\sqrt [6]{t}\right ) \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt &=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 \tan ^{-1}\left (\sqrt [6]{t}\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 42, normalized size = 1.02 \begin {gather*} \frac {2}{35} \left (-105 \sqrt [6]{t}+35 \sqrt {t}-21 t^{5/6}+15 t^{7/6}\right )+6 \tan ^{-1}\left (\sqrt [6]{t}\right ) \end {gather*}

Antiderivative was successfully verified.

[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)]

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Mathics [A]
time = 3.60, size = 27, normalized size = 0.66 \begin {gather*} -6 t^{\frac {1}{6}}+2 \sqrt {t}-\frac {6 t^{\frac {5}{6}}}{5}+\frac {6 t^{\frac {7}{6}}}{7}+6 \text {ArcTan}\left [t^{\frac {1}{6}}\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[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)]

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Maple [A]
time = 0.05, size = 28, normalized size = 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	\(-\frac {2 t^{\frac {1}{6}} \left (-45 t +63 t^{\frac {2}{3}}-105 t^{\frac {1}{3}}+315\right )}{105}+6 \arctan \left (t^{\frac {1}{6}}\right )\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]
time = 0.36, size = 27, normalized size = 0.66 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [A]
time = 0.33, size = 25, normalized size = 0.61 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [A]
time = 1.67, size = 37, normalized size = 0.90 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]
time = 0.00, size = 46, normalized size = 1.12 \begin {gather*} 6 \left (\frac {1}{7} t^{\frac {1}{6}} t-\frac {1}{5} t^{\frac {1}{3}} \sqrt {t}+\frac {\sqrt {t}}{3}-t^{\frac {1}{6}}+\arctan \left (t^{\frac {1}{6}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.04, size = 27, normalized size = 0.66 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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