Optimal. Leaf size=21 \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}} \]
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Rubi [A] time = 0.0333973, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}} \]
Antiderivative was successfully verified.
[In] Int[(a^(2/3)*C - a^(1/3)*b^(1/3)*C*x + b^(2/3)*C*x^2)/(a + b*x^3),x]
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Rubi in Sympy [A] time = 16.0681, size = 19, normalized size = 0.9 \[ \frac{C \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{\sqrt [3]{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a**(2/3)*C-a**(1/3)*b**(1/3)*C*x+b**(2/3)*C*x**2)/(b*x**3+a),x)
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Mathematica [A] time = 0.00519268, size = 21, normalized size = 1. \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}} \]
Antiderivative was successfully verified.
[In] Integrate[(a^(2/3)*C - a^(1/3)*b^(1/3)*C*x + b^(2/3)*C*x^2)/(a + b*x^3),x]
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Maple [B] time = 0.007, size = 218, normalized size = 10.4 \[{\frac{C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{C}{6\,b}{a}^{{\frac{2}{3}}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C\sqrt{3}}{3\,b}{a}^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C}{3}\sqrt [3]{a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{C}{6}\sqrt [3]{a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{C\sqrt{3}}{3}\sqrt [3]{a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3}{\frac{1}{\sqrt [3]{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a^(2/3)*C-a^(1/3)*b^(1/3)*C*x+b^(2/3)*C*x^2)/(b*x^3+a),x)
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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 - C*a^(1/3)*b^(1/3)*x + C*a^(2/3))/(b*x^3 + a),x, algorithm="maxima")
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Fricas [A] time = 0.256511, size = 23, normalized size = 1.1 \[ \frac{C \log \left (b x + a^{\frac{1}{3}} b^{\frac{2}{3}}\right )}{b^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*b^(2/3)*x^2 - C*a^(1/3)*b^(1/3)*x + C*a^(2/3))/(b*x^3 + a),x, algorithm="fricas")
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Sympy [A] time = 0.623161, size = 20, normalized size = 0.95 \[ \frac{C \log{\left (\sqrt [3]{a} b^{\frac{2}{3}} + b x \right )}}{\sqrt [3]{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a**(2/3)*C-a**(1/3)*b**(1/3)*C*x+b**(2/3)*C*x**2)/(b*x**3+a),x)
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GIAC/XCAS [A] time = 0.223282, size = 22, normalized size = 1.05 \[ \frac{C{\rm ln}\left ({\left | b^{\frac{1}{3}} x + a^{\frac{1}{3}} \right |}\right )}{b^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*b^(2/3)*x^2 - C*a^(1/3)*b^(1/3)*x + C*a^(2/3))/(b*x^3 + a),x, algorithm="giac")
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