Optimal. Leaf size=61 \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{2 C \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{b}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0833869, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{2 C \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
[In] Int[(2*a^(2/3)*C + b^(2/3)*C*x^2)/(a + b*x^3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.1518, size = 63, normalized size = 1.03 \[ \frac{C \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{\sqrt [3]{b}} - \frac{2 \sqrt{3} C \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*a**(2/3)*C+b**(2/3)*C*x**2)/(b*x**3+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0416317, size = 95, normalized size = 1.56 \[ \frac{C \left (-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\log \left (a+b x^3\right )+2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{3 \sqrt [3]{b}} \]
Antiderivative was successfully verified.
[In] Integrate[(2*a^(2/3)*C + b^(2/3)*C*x^2)/(a + b*x^3),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.005, size = 117, normalized size = 1.9 \[{\frac{2\,C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,C\sqrt{3}}{3\,b}{a}^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3}{\frac{1}{\sqrt [3]{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*a^(2/3)*C+b^(2/3)*C*x^2)/(b*x^3+a),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((C*b^(2/3)*x^2 + 2*C*a^(2/3))/(b*x^3 + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.244809, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{\frac{1}{3}} C b \sqrt{-\frac{1}{b^{\frac{2}{3}}}} \log \left (\frac{2 \, a^{\frac{1}{3}} b x^{2} - 2 \, a^{\frac{2}{3}} b^{\frac{2}{3}} x + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a^{\frac{2}{3}} b x - a b^{\frac{2}{3}}\right )} \sqrt{-\frac{1}{b^{\frac{2}{3}}}} - a b^{\frac{1}{3}}}{a^{\frac{1}{3}} b x^{2} - a^{\frac{2}{3}} b^{\frac{2}{3}} x + a b^{\frac{1}{3}}}\right ) + C b^{\frac{2}{3}} \log \left (a^{\frac{2}{3}} b x + a b^{\frac{2}{3}}\right )}{b}, -\frac{2 \, \sqrt{\frac{1}{3}} C b^{\frac{2}{3}} \arctan \left (-\frac{\sqrt{\frac{1}{3}}{\left (2 \, a^{\frac{2}{3}} b x - a b^{\frac{2}{3}}\right )}}{a b^{\frac{2}{3}}}\right ) - C b^{\frac{2}{3}} \log \left (a^{\frac{2}{3}} b x + a b^{\frac{2}{3}}\right )}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*b^(2/3)*x^2 + 2*C*a^(2/3))/(b*x^3 + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.964292, size = 70, normalized size = 1.15 \[ \operatorname{RootSum}{\left (3 t^{3} b^{\frac{5}{3}} - 3 t^{2} C b^{\frac{4}{3}} + t C^{2} b - C^{3} b^{\frac{2}{3}}, \left ( t \mapsto t \log{\left (x + \frac{3 t \sqrt [3]{a} \sqrt [3]{b} - C \sqrt [3]{a}}{2 C \sqrt [3]{b}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*a**(2/3)*C+b**(2/3)*C*x**2)/(b*x**3+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*b^(2/3)*x^2 + 2*C*a^(2/3))/(b*x^3 + a),x, algorithm="giac")
[Out]