Optimal. Leaf size=71 \[ \frac{C \log \left (\sqrt [3]{\frac{a}{b}}+x\right )}{b}-\frac{2 \left (\frac{a}{b}\right )^{2/3} \left (C \sqrt [3]{\frac{a}{b}}+B\right ) \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} a} \]
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Rubi [A] time = 0.158123, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{C \log \left (\sqrt [3]{\frac{a}{b}}+x\right )}{b}-\frac{2 \left (\frac{a}{b}\right )^{2/3} \left (C \sqrt [3]{\frac{a}{b}}+B\right ) \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} a} \]
Antiderivative was successfully verified.
[In] Int[((a/b)^(1/3)*B + 2*(a/b)^(2/3)*C + B*x + C*x^2)/(a + b*x^3),x]
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Rubi in Sympy [A] time = 17.3761, size = 54, normalized size = 0.76 \[ \frac{C \log{\left (x + \sqrt [3]{\frac{a}{b}} \right )}}{b} - \frac{2 \sqrt{3} \left (\frac{B}{\sqrt [3]{\frac{a}{b}}} + C\right ) \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 x}{3 \sqrt [3]{\frac{a}{b}}} + \frac{1}{3}\right ) \right )}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((a/b)**(1/3)*B+2*(a/b)**(2/3)*C+B*x+C*x**2)/(b*x**3+a),x)
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Mathematica [B] time = 0.634627, size = 247, normalized size = 3.48 \[ \frac{\sqrt [3]{b} \left (a^{2/3} B-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (2 C \sqrt [3]{\frac{a}{b}}+B\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} \left (\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (2 C \sqrt [3]{\frac{a}{b}}+B\right )-a^{2/3} B\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (2 C \sqrt [3]{\frac{a}{b}}+B\right )+\sqrt [3]{a} B\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )+2 a C \log \left (a+b x^3\right )}{6 a b} \]
Antiderivative was successfully verified.
[In] Integrate[((a/b)^(1/3)*B + 2*(a/b)^(2/3)*C + B*x + C*x^2)/(a + b*x^3),x]
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Maple [A] time = 0.007, size = 121, normalized size = 1.7 \[{\frac{2\,C}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{C}{3\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) }+{\frac{2\,C\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) }+{\frac{2\,B\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3\,b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((a/b)^(1/3)*B+2*(a/b)^(2/3)*C+B*x+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*x^2 + B*x + 2*C*(a/b)^(2/3) + B*(a/b)^(1/3))/(b*x^3 + a),x, algorithm="maxima")
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Fricas [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*x^2 + B*x + 2*C*(a/b)^(2/3) + B*(a/b)^(1/3))/(b*x^3 + a),x, algorithm="fricas")
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: PolynomialDivisionFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((a/b)**(1/3)*B+2*(a/b)**(2/3)*C+B*x+C*x**2)/(b*x**3+a),x)
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GIAC/XCAS [A] time = 0.248145, size = 351, normalized size = 4.94 \[ -\frac{{\left (C b^{2} \left (-\frac{a}{b}\right )^{\frac{2}{3}} + B b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (a b^{2}\right )^{\frac{1}{3}} B b + 2 \, \left (a b^{2}\right )^{\frac{2}{3}} C\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{2}} + \frac{\sqrt{3}{\left ({\left (9 \, \left (-a^{2} b^{4}\right )^{\frac{1}{3}} a b^{2} - 27^{\frac{5}{6}} \left (-a^{2} b^{4}\right )^{\frac{5}{6}}\right )} B + 18 \,{\left (\sqrt{3} a^{2} b^{3} i + a^{2} b^{3}\right )} C\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{54 \, a^{2} b^{4}} + \frac{{\left ({\left (27 \, \left (-a^{2} b^{4}\right )^{\frac{1}{3}} a b^{2} + 27^{\frac{5}{6}} \left (-a^{2} b^{4}\right )^{\frac{5}{6}}\right )} B - 18 \,{\left (\sqrt{3} a^{2} b^{3} i - 3 \, a^{2} b^{3}\right )} C\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{108 \, a^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + 2*C*(a/b)^(2/3) + B*(a/b)^(1/3))/(b*x^3 + a),x, algorithm="giac")
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