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

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

[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 = 0.125541, 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 \[ \frac{2}{5} t \sqrt{t^3+4}-\frac{8\ 2^{2/3} \sqrt{2+\sqrt{3}} \left (t+2^{2/3}\right ) \sqrt{\frac{t^2-2^{2/3} t+2 \sqrt [3]{2}}{\left (t+2^{2/3} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{t+2^{2/3} \left (1-\sqrt{3}\right )}{t+2^{2/3} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [4]{3} \sqrt{\frac{t+2^{2/3}}{\left (t+2^{2/3} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{t^3+4}} \]

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])

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Rubi in Sympy [A]  time = 2.24772, size = 162, normalized size = 0.94 \[ \frac{2 t \sqrt{t^{3} + 4}}{5} - \frac{8 \cdot 3^{\frac{3}{4}} \sqrt{\frac{2^{\frac{2}{3}} t^{2} - 2 \sqrt [3]{2} t + 4}{\left (\sqrt [3]{2} t + 2 + 2 \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (2 t + 2 \cdot 2^{\frac{2}{3}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{2} t - 2 \sqrt{3} + 2}{\sqrt [3]{2} t + 2 + 2 \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{15 \sqrt{\frac{2 \sqrt [3]{2} t + 4}{\left (\sqrt [3]{2} t + 2 + 2 \sqrt{3}\right )^{2}}} \sqrt{t^{3} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(t**3/(t**3+4)**(1/2),t)

[Out]

2*t*sqrt(t**3 + 4)/5 - 8*3**(3/4)*sqrt((2**(2/3)*t**2 - 2*2**(1/3)*t + 4)/(2**(1
/3)*t + 2 + 2*sqrt(3))**2)*sqrt(sqrt(3) + 2)*(2*t + 2*2**(2/3))*elliptic_f(asin(
(2**(1/3)*t - 2*sqrt(3) + 2)/(2**(1/3)*t + 2 + 2*sqrt(3))), -7 - 4*sqrt(3))/(15*
sqrt((2*2**(1/3)*t + 4)/(2**(1/3)*t + 2 + 2*sqrt(3))**2)*sqrt(t**3 + 4))

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Mathematica [C]  time = 0.266495, size = 122, normalized size = 0.71 \[ \frac{6 t \left (t^3+4\right )-8 \sqrt [6]{-2} 3^{3/4} \sqrt{-\sqrt [6]{-1} \left (\sqrt [3]{2} t+2 (-1)^{2/3}\right )} \sqrt{(-2)^{2/3} t^2+2 \sqrt [3]{-2} t+4} F\left (\sin ^{-1}\left (\frac{\sqrt{\left (-i+\sqrt{3}\right ) \left (\sqrt [3]{2} t+2\right )}}{2 \sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{15 \sqrt{t^3+4}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[t^3/Sqrt[4 + t^3],t]

[Out]

(6*t*(4 + t^3) - 8*(-2)^(1/6)*3^(3/4)*Sqrt[-((-1)^(1/6)*(2*(-1)^(2/3) + 2^(1/3)*
t))]*Sqrt[4 + 2*(-2)^(1/3)*t + (-2)^(2/3)*t^2]*EllipticF[ArcSin[Sqrt[(-I + Sqrt[
3])*(2 + 2^(1/3)*t)]/(2*3^(1/4))], (-1)^(1/3)])/(15*Sqrt[4 + t^3])

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Maple [A]  time = 0.658, size = 168, normalized size = 1. \[{\frac{2\,t}{5}\sqrt{{t}^{3}+4}}+{{\frac{8\,i}{15}}\sqrt{3}{2}^{{\frac{2}{3}}}\sqrt{i \left ( t-{\frac{{2}^{{\frac{2}{3}}}}{2}}-{\frac{i}{2}}\sqrt{3}{2}^{{\frac{2}{3}}} \right ) \sqrt{3}\sqrt [3]{2}}\sqrt{{\frac{{2}^{{\frac{2}{3}}}+t}{{\frac{3\,{2}^{2/3}}{2}}+{\frac{i}{2}}\sqrt{3}{2}^{{\frac{2}{3}}}}}}\sqrt{-i \left ( t-{\frac{{2}^{{\frac{2}{3}}}}{2}}+{\frac{i}{2}}\sqrt{3}{2}^{{\frac{2}{3}}} \right ) \sqrt{3}\sqrt [3]{2}}{\it EllipticF} \left ({\frac{\sqrt{6}}{6}\sqrt{i \left ( t-{\frac{{2}^{{\frac{2}{3}}}}{2}}-{\frac{i}{2}}\sqrt{3}{2}^{{\frac{2}{3}}} \right ) \sqrt{3}\sqrt [3]{2}}},\sqrt{{\frac{i\sqrt{3}{2}^{{\frac{2}{3}}}}{{\frac{3\,{2}^{2/3}}{2}}+{\frac{i}{2}}\sqrt{3}{2}^{{\frac{2}{3}}}}}} \right ){\frac{1}{\sqrt{{t}^{3}+4}}}} \]

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. \[ \int \frac{t^{3}}{\sqrt{t^{3} + 4}}\,{d t} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(t^3/sqrt(t^3 + 4),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. \[{\rm integral}\left (\frac{t^{3}}{\sqrt{t^{3} + 4}}, t\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(t^3/sqrt(t^3 + 4),t, algorithm="fricas")

[Out]

integral(t^3/sqrt(t^3 + 4), t)

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Sympy [A]  time = 0.899525, size = 31, normalized size = 0.18 \[ \frac{t^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{t^{3} e^{i \pi }}{4}} \right )}}{6 \Gamma \left (\frac{7}{3}\right )} \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{t^{3}}{\sqrt{t^{3} + 4}}\,{d t} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(t^3/sqrt(t^3 + 4),t, algorithm="giac")

[Out]

integrate(t^3/sqrt(t^3 + 4), t)