Optimal. Leaf size=70 \[ \frac{C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}-\frac{2 C \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{-b}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.12748, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}-\frac{2 C \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{-b}} \]
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 = 19.8648, size = 70, normalized size = 1. \[ \frac{C \log{\left (\sqrt [3]{a} - x \sqrt [3]{- b} \right )}}{\sqrt [3]{- b}} - \frac{2 \sqrt{3} C \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 x \sqrt [3]{- b}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{- b}} \]
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 = 0.057948, size = 116, normalized size = 1.66 \[ -\frac{C \left (-b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+(-b)^{2/3} \log \left (a+b x^3\right )\right )}{3 b} \]
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 = 0.009, size = 122, normalized size = 1.7 \[ -{\frac{2\,C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,C\sqrt{3}}{3\,b}{a}^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3\,b} \left ( -b \right ) ^{{\frac{2}{3}}}} \]
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 = 0.249605, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{\frac{1}{3}} C b \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a^{\frac{1}{3}} b x^{2} - 2 \, a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} x - 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a^{\frac{2}{3}} b x - a \left (-b\right )^{\frac{2}{3}}\right )} \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} + a \left (-b\right )^{\frac{1}{3}}}{a^{\frac{1}{3}} b x^{2} - a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} x - a \left (-b\right )^{\frac{1}{3}}}\right ) - C \left (-b\right )^{\frac{2}{3}} \log \left (a^{\frac{2}{3}} b x + a \left (-b\right )^{\frac{2}{3}}\right )}{b}, -\frac{2 \, \sqrt{\frac{1}{3}} C b \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, a^{\frac{2}{3}} b x - a \left (-b\right )^{\frac{2}{3}}\right )}}{a b \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}}}\right ) + C \left (-b\right )^{\frac{2}{3}} \log \left (a^{\frac{2}{3}} b x + a \left (-b\right )^{\frac{2}{3}}\right )}{b}\right ] \]
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 = 1.06539, size = 73, normalized size = 1.04 \[ - \operatorname{RootSum}{\left (3 t^{3} b^{2} - 3 t^{2} C b \left (- b\right )^{\frac{2}{3}} + t C^{2} \left (- b\right )^{\frac{4}{3}} - C^{3} b, \left ( t \mapsto t \log{\left (\frac{3 t \sqrt [3]{a}}{2 C} - \frac{\sqrt [3]{a} \left (- b\right )^{\frac{2}{3}}}{2 b} + x \right )} \right )\right )} \]
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]