Optimal. Leaf size=33 \[ \frac{x \left (a+b x^3\right )^{4/3} \, _2F_1\left (1,\frac{5}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{a} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0250243, antiderivative size = 46, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\sqrt [3]{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^3)^(1/3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 3.37183, size = 39, normalized size = 1.18 \[ \frac{x \sqrt [3]{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{\sqrt [3]{1 + \frac{b x^{3}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a)**(1/3),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.400137, size = 196, normalized size = 5.94 \[ \frac{3 \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt [3]{a+b x^3} F_1\left (\frac{4}{3};-\frac{1}{3},-\frac{1}{3};\frac{7}{3};-\frac{i \left (\sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a}\right )}{\sqrt{3} \sqrt [3]{a}},\frac{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right )}{4 \sqrt [3]{2} \sqrt [3]{b} \sqrt [3]{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt [3]{\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x^3)^(1/3),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.045, size = 0, normalized size = 0. \[ \int \sqrt [3]{b{x}^{3}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a)^(1/3),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(1/3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac{1}{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(1/3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 2.2394, size = 37, normalized size = 1.12 \[ \frac{\sqrt [3]{a} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**3+a)**(1/3),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(1/3),x, algorithm="giac")
[Out]