Optimal. Leaf size=117 \[ \frac{a^2 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 b^{4/3}}-\frac{a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{4/3}}+\frac{a x \left (a+b x^3\right )^{2/3}}{9 b}+\frac{1}{6} x^4 \left (a+b x^3\right )^{2/3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.096407, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{a^2 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 b^{4/3}}-\frac{a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{4/3}}+\frac{a x \left (a+b x^3\right )^{2/3}}{9 b}+\frac{1}{6} x^4 \left (a+b x^3\right )^{2/3} \]
Antiderivative was successfully verified.
[In] Int[x^3*(a + b*x^3)^(2/3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 23.1416, size = 156, normalized size = 1.33 \[ \frac{a^{2} \log{\left (- \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{27 b^{\frac{4}{3}}} - \frac{a^{2} \log{\left (\frac{b^{\frac{2}{3}} x^{2}}{\left (a + b x^{3}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{54 b^{\frac{4}{3}}} - \frac{\sqrt{3} a^{2} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + b x^{3}}} + \frac{1}{3}\right ) \right )}}{27 b^{\frac{4}{3}}} + \frac{a x \left (a + b x^{3}\right )^{\frac{2}{3}}}{9 b} + \frac{x^{4} \left (a + b x^{3}\right )^{\frac{2}{3}}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(b*x**3+a)**(2/3),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.187884, size = 143, normalized size = 1.22 \[ \left (a+b x^3\right )^{2/3} \left (\frac{a x}{9 b}+\frac{x^4}{6}\right )-\frac{a^2 \left (\log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )-2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )\right )}{54 b^{4/3}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3*(a + b*x^3)^(2/3),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.041, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(b*x^3+a)^(2/3),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)^(2/3)*x^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.260401, size = 207, normalized size = 1.77 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3} a^{2} \log \left (-\frac{b x -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}}}{x}\right ) - \sqrt{3} a^{2} \log \left (\frac{b x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}} b^{\frac{1}{3}}}{x^{2}}\right ) + 6 \, a^{2} \arctan \left (\frac{\sqrt{3} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}}}{3 \, b x}\right ) + 3 \, \sqrt{3}{\left (3 \, b x^{4} + 2 \, a x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}} b^{\frac{1}{3}}\right )}}{162 \, b^{\frac{4}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(2/3)*x^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 5.22199, size = 39, normalized size = 0.33 \[ \frac{a^{\frac{2}{3}} x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(b*x**3+a)**(2/3),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{2}{3}} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(2/3)*x^3,x, algorithm="giac")
[Out]