Optimal. Leaf size=145 \[ -\frac{a \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{2/3}}-\frac{a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{2/3}}+\frac{a \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{18 b^{2/3}}+\frac{1}{3} x^2 \sqrt [3]{a+b x^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.15952, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{a \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{2/3}}-\frac{a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{2/3}}+\frac{a \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{18 b^{2/3}}+\frac{1}{3} x^2 \sqrt [3]{a+b x^3} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x^3)^(1/3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 21.2556, size = 134, normalized size = 0.92 \[ - \frac{a \log{\left (- \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{9 b^{\frac{2}{3}}} + \frac{a \log{\left (\frac{b^{\frac{2}{3}} x^{2}}{\left (a + b x^{3}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{18 b^{\frac{2}{3}}} - \frac{\sqrt{3} a \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + b x^{3}}} + \frac{1}{3}\right ) \right )}}{9 b^{\frac{2}{3}}} + \frac{x^{2} \sqrt [3]{a + b x^{3}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x**3+a)**(1/3),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0450405, size = 63, normalized size = 0.43 \[ \frac{x^2 \left (a \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )+2 \left (a+b x^3\right )\right )}{6 \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x^3)^(1/3),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.033, size = 0, normalized size = 0. \[ \int x\sqrt [3]{b{x}^{3}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x^3+a)^(1/3),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(1/3)*x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.235518, size = 223, normalized size = 1.54 \[ \frac{\sqrt{3}{\left (6 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{1}{3}} x^{2} + 2 \, \sqrt{3} a \log \left (\frac{b x +{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{1}{3}}}{x}\right ) - \sqrt{3} a \log \left (\frac{b^{2} x^{2} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{1}{3}} b x +{\left (b x^{3} + a\right )}^{\frac{2}{3}} \left (-b^{2}\right )^{\frac{2}{3}}}{x^{2}}\right ) + 6 \, a \arctan \left (-\frac{\sqrt{3} b x - 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{1}{3}}}{3 \, b x}\right )\right )}}{54 \, \left (-b^{2}\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(1/3)*x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 4.08574, size = 39, normalized size = 0.27 \[ \frac{\sqrt [3]{a} x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{5}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x**3+a)**(1/3),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{1}{3}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(1/3)*x,x, algorithm="giac")
[Out]