3.41 \(\int \frac{t^3}{\sqrt{4+t^3}} \, dt\)

Optimal. Leaf size=172 \[ \frac{2}{5} t \sqrt{t^3+4}-\frac{8\ 2^{2/3} \sqrt{2+\sqrt{3}} \left (t+2^{2/3}\right ) \sqrt{\frac{t^2-2^{2/3} t+2 \sqrt [3]{2}}{\left (t+2^{2/3} \left (1+\sqrt{3}\right )\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{t+2^{2/3} \left (1-\sqrt{3}\right )}{t+2^{2/3} \left (1+\sqrt{3}\right )}\right ),-7-4 \sqrt{3}\right )}{5 \sqrt [4]{3} \sqrt{\frac{t+2^{2/3}}{\left (t+2^{2/3} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{t^3+4}} \]

[Out]

(2*t*Sqrt[4 + t^3])/5 - (8*2^(2/3)*Sqrt[2 + Sqrt[3]]*(2^(2/3) + t)*Sqrt[(2*2^(1/3) - 2^(2/3)*t + t^2)/(2^(2/3)
*(1 + Sqrt[3]) + t)^2]*EllipticF[ArcSin[(2^(2/3)*(1 - Sqrt[3]) + t)/(2^(2/3)*(1 + Sqrt[3]) + t)], -7 - 4*Sqrt[
3]])/(5*3^(1/4)*Sqrt[(2^(2/3) + t)/(2^(2/3)*(1 + Sqrt[3]) + t)^2]*Sqrt[4 + t^3])

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Rubi [A]  time = 0.0288655, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {321, 218} \[ \frac{2}{5} t \sqrt{t^3+4}-\frac{8\ 2^{2/3} \sqrt{2+\sqrt{3}} \left (t+2^{2/3}\right ) \sqrt{\frac{t^2-2^{2/3} t+2 \sqrt [3]{2}}{\left (t+2^{2/3} \left (1+\sqrt{3}\right )\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{t+2^{2/3} \left (1-\sqrt{3}\right )}{t+2^{2/3} \left (1+\sqrt{3}\right )}\right ),-7-4 \sqrt{3}\right )}{5 \sqrt [4]{3} \sqrt{\frac{t+2^{2/3}}{\left (t+2^{2/3} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{t^3+4}} \]

Antiderivative was successfully verified.

[In]

Int[t^3/Sqrt[4 + t^3],t]

[Out]

(2*t*Sqrt[4 + t^3])/5 - (8*2^(2/3)*Sqrt[2 + Sqrt[3]]*(2^(2/3) + t)*Sqrt[(2*2^(1/3) - 2^(2/3)*t + t^2)/(2^(2/3)
*(1 + Sqrt[3]) + t)^2]*EllipticF[ArcSin[(2^(2/3)*(1 - Sqrt[3]) + t)/(2^(2/3)*(1 + Sqrt[3]) + t)], -7 - 4*Sqrt[
3]])/(5*3^(1/4)*Sqrt[(2^(2/3) + t)/(2^(2/3)*(1 + Sqrt[3]) + t)^2]*Sqrt[4 + t^3])

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rubi steps

\begin{align*} \int \frac{t^3}{\sqrt{4+t^3}} \, dt &=\frac{2}{5} t \sqrt{4+t^3}-\frac{8}{5} \int \frac{1}{\sqrt{4+t^3}} \, dt\\ &=\frac{2}{5} t \sqrt{4+t^3}-\frac{8\ 2^{2/3} \sqrt{2+\sqrt{3}} \left (2^{2/3}+t\right ) \sqrt{\frac{2 \sqrt [3]{2}-2^{2/3} t+t^2}{\left (2^{2/3} \left (1+\sqrt{3}\right )+t\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1-\sqrt{3}\right )+t}{2^{2/3} \left (1+\sqrt{3}\right )+t}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [4]{3} \sqrt{\frac{2^{2/3}+t}{\left (2^{2/3} \left (1+\sqrt{3}\right )+t\right )^2}} \sqrt{4+t^3}}\\ \end{align*}

Mathematica [C]  time = 0.00516, size = 34, normalized size = 0.2 \[ \frac{2}{5} t \left (\sqrt{t^3+4}-2 \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{1}{2},\frac{4}{3},-\frac{t^3}{4}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[t^3/Sqrt[4 + t^3],t]

[Out]

(2*t*(Sqrt[4 + t^3] - 2*Hypergeometric2F1[1/3, 1/2, 4/3, -t^3/4]))/5

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Maple [A]  time = 0.098, size = 168, normalized size = 1. \begin{align*}{\frac{2\,t}{5}\sqrt{{t}^{3}+4}}+{{\frac{8\,i}{15}}\sqrt{3}{2}^{{\frac{2}{3}}}\sqrt{i \left ( t-{\frac{{2}^{{\frac{2}{3}}}}{2}}-{\frac{i}{2}}\sqrt{3}{2}^{{\frac{2}{3}}} \right ) \sqrt{3}\sqrt [3]{2}}\sqrt{{\frac{{2}^{{\frac{2}{3}}}+t}{{\frac{3\,{2}^{2/3}}{2}}+{\frac{i}{2}}\sqrt{3}{2}^{{\frac{2}{3}}}}}}\sqrt{-i \left ( t-{\frac{{2}^{{\frac{2}{3}}}}{2}}+{\frac{i}{2}}\sqrt{3}{2}^{{\frac{2}{3}}} \right ) \sqrt{3}\sqrt [3]{2}}{\it EllipticF} \left ({\frac{\sqrt{6}}{6}\sqrt{i \left ( t-{\frac{{2}^{{\frac{2}{3}}}}{2}}-{\frac{i}{2}}\sqrt{3}{2}^{{\frac{2}{3}}} \right ) \sqrt{3}\sqrt [3]{2}}},\sqrt{{\frac{i\sqrt{3}{2}^{{\frac{2}{3}}}}{{\frac{3\,{2}^{2/3}}{2}}+{\frac{i}{2}}\sqrt{3}{2}^{{\frac{2}{3}}}}}} \right ){\frac{1}{\sqrt{{t}^{3}+4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(t^3/(t^3+4)^(1/2),t)

[Out]

2/5*t*(t^3+4)^(1/2)+8/15*I*3^(1/2)*2^(2/3)*(I*(t-1/2*2^(2/3)-1/2*I*3^(1/2)*2^(2/3))*3^(1/2)*2^(1/3))^(1/2)*((2
^(2/3)+t)/(3/2*2^(2/3)+1/2*I*3^(1/2)*2^(2/3)))^(1/2)*(-I*(t-1/2*2^(2/3)+1/2*I*3^(1/2)*2^(2/3))*3^(1/2)*2^(1/3)
)^(1/2)/(t^3+4)^(1/2)*EllipticF(1/6*6^(1/2)*(I*(t-1/2*2^(2/3)-1/2*I*3^(1/2)*2^(2/3))*3^(1/2)*2^(1/3))^(1/2),(I
*3^(1/2)*2^(2/3)/(3/2*2^(2/3)+1/2*I*3^(1/2)*2^(2/3)))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{t^{3}}{\sqrt{t^{3} + 4}}\,{d t} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^3/(t^3+4)^(1/2),t, algorithm="maxima")

[Out]

integrate(t^3/sqrt(t^3 + 4), t)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{t^{3}}{\sqrt{t^{3} + 4}}, t\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^3/(t^3+4)^(1/2),t, algorithm="fricas")

[Out]

integral(t^3/sqrt(t^3 + 4), t)

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Sympy [A]  time = 0.61008, size = 31, normalized size = 0.18 \begin{align*} \frac{t^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{t^{3} e^{i \pi }}{4}} \right )}}{6 \Gamma \left (\frac{7}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t**3/(t**3+4)**(1/2),t)

[Out]

t**4*gamma(4/3)*hyper((1/2, 4/3), (7/3,), t**3*exp_polar(I*pi)/4)/(6*gamma(7/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{t^{3}}{\sqrt{t^{3} + 4}}\,{d t} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^3/(t^3+4)^(1/2),t, algorithm="giac")

[Out]

integrate(t^3/sqrt(t^3 + 4), t)