3.15 \(\int t \sqrt [4]{1+t} \, dt\)

Optimal. Leaf size=23 \[ \frac{4}{9} (t+1)^{9/4}-\frac{4}{5} (t+1)^{5/4} \]

[Out]

(-4*(1 + t)^(5/4))/5 + (4*(1 + t)^(9/4))/9

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Rubi [A]  time = 0.0037507, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{4}{9} (t+1)^{9/4}-\frac{4}{5} (t+1)^{5/4} \]

Antiderivative was successfully verified.

[In]

Int[t*(1 + t)^(1/4),t]

[Out]

(-4*(1 + t)^(5/4))/5 + (4*(1 + t)^(9/4))/9

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int t \sqrt [4]{1+t} \, dt &=\int \left (-\sqrt [4]{1+t}+(1+t)^{5/4}\right ) \, dt\\ &=-\frac{4}{5} (1+t)^{5/4}+\frac{4}{9} (1+t)^{9/4}\\ \end{align*}

Mathematica [A]  time = 0.0040473, size = 16, normalized size = 0.7 \[ \frac{4}{45} (t+1)^{5/4} (5 t-4) \]

Antiderivative was successfully verified.

[In]

Integrate[t*(1 + t)^(1/4),t]

[Out]

(4*(1 + t)^(5/4)*(-4 + 5*t))/45

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Maple [A]  time = 0.001, size = 13, normalized size = 0.6 \begin{align*}{\frac{20\,t-16}{45} \left ( 1+t \right ) ^{{\frac{5}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(t*(1+t)^(1/4),t)

[Out]

4/45*(1+t)^(5/4)*(5*t-4)

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Maxima [A]  time = 0.955077, size = 20, normalized size = 0.87 \begin{align*} \frac{4}{9} \,{\left (t + 1\right )}^{\frac{9}{4}} - \frac{4}{5} \,{\left (t + 1\right )}^{\frac{5}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t*(1+t)^(1/4),t, algorithm="maxima")

[Out]

4/9*(t + 1)^(9/4) - 4/5*(t + 1)^(5/4)

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Fricas [A]  time = 0.419432, size = 49, normalized size = 2.13 \begin{align*} \frac{4}{45} \,{\left (5 \, t^{2} + t - 4\right )}{\left (t + 1\right )}^{\frac{1}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t*(1+t)^(1/4),t, algorithm="fricas")

[Out]

4/45*(5*t^2 + t - 4)*(t + 1)^(1/4)

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Sympy [A]  time = 0.882601, size = 34, normalized size = 1.48 \begin{align*} \frac{4 t^{2} \sqrt [4]{t + 1}}{9} + \frac{4 t \sqrt [4]{t + 1}}{45} - \frac{16 \sqrt [4]{t + 1}}{45} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t*(1+t)**(1/4),t)

[Out]

4*t**2*(t + 1)**(1/4)/9 + 4*t*(t + 1)**(1/4)/45 - 16*(t + 1)**(1/4)/45

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Giac [A]  time = 1.06519, size = 20, normalized size = 0.87 \begin{align*} \frac{4}{9} \,{\left (t + 1\right )}^{\frac{9}{4}} - \frac{4}{5} \,{\left (t + 1\right )}^{\frac{5}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t*(1+t)^(1/4),t, algorithm="giac")

[Out]

4/9*(t + 1)^(9/4) - 4/5*(t + 1)^(5/4)