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]

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

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Rubi [A]  time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.70 \[ \frac {4}{45} (t+1)^{5/4} (5 t-4) \]

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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fricas [A]  time = 0.39, size = 15, normalized size = 0.65 \[ \frac {4}{45} \, {\left (5 \, t^{2} + t - 4\right )} {\left (t + 1\right )}^{\frac {1}{4}} \]

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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giac [A]  time = 0.01, size = 15, normalized size = 0.65 \[ \frac {4}{9} \, {\left (t + 1\right )}^{\frac {9}{4}} - \frac {4}{5} \, {\left (t + 1\right )}^{\frac {5}{4}} \]

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)

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maple [A]  time = 0.00, size = 13, normalized size = 0.57 \[ \frac {4 \left (t +1\right )^{\frac {5}{4}} \left (5 t -4\right )}{45} \]

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 = 0.50, size = 15, normalized size = 0.65 \[ \frac {4}{9} \, {\left (t + 1\right )}^{\frac {9}{4}} - \frac {4}{5} \, {\left (t + 1\right )}^{\frac {5}{4}} \]

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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mupad [B]  time = 0.03, size = 12, normalized size = 0.52 \[ \frac {4\,\left (5\,t-4\right )\,{\left (t+1\right )}^{5/4}}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(t*(t + 1)^(1/4),t)

[Out]

(4*(5*t - 4)*(t + 1)^(5/4))/45

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sympy [A]  time = 1.01, size = 34, normalized size = 1.48 \[ \frac {4 t^{2} \sqrt [4]{t + 1}}{9} + \frac {4 t \sqrt [4]{t + 1}}{45} - \frac {16 \sqrt [4]{t + 1}}{45} \]

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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