Optimal. Leaf size=75 \[ -\frac{2 \left (\frac{a}{b}\right )^{2/3} \left (B-C \sqrt [3]{\frac{a}{b}}\right ) \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{\frac{a}{b}}}+1}{\sqrt{3}}\right )}{\sqrt{3} a}-\frac{C \log \left (\sqrt [3]{\frac{a}{b}}-x\right )}{b} \]
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Rubi [A] time = 0.174261, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 \left (\frac{a}{b}\right )^{2/3} \left (B-C \sqrt [3]{\frac{a}{b}}\right ) \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{\frac{a}{b}}}+1}{\sqrt{3}}\right )}{\sqrt{3} a}-\frac{C \log \left (\sqrt [3]{\frac{a}{b}}-x\right )}{b} \]
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 = 20.804, size = 54, normalized size = 0.72 \[ - \frac{C \log{\left (x - \sqrt [3]{\frac{a}{b}} \right )}}{b} + \frac{2 \sqrt{3} \left (- \frac{B}{\sqrt [3]{\frac{a}{b}}} + C\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3 \sqrt [3]{\frac{a}{b}}} + \frac{1}{3}\right ) \right )}}{3 b} \]
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 = 0.608016, size = 244, normalized size = 3.25 \[ \frac{\sqrt [3]{b} \left (a^{2/3} B+\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (2 C \sqrt [3]{\frac{a}{b}}-B\right )\right ) \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt [3]{b} \left (a^{2/3} B+\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (2 C \sqrt [3]{\frac{a}{b}}-B\right )\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )-2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (B-2 C \sqrt [3]{\frac{a}{b}}\right )+\sqrt [3]{a} B\right ) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )-2 a C \log \left (a-b x^3\right )}{6 a b} \]
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 = 0.008, size = 124, normalized size = 1.7 \[ -{\frac{2\,C}{3\,b}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{C}{3\,b}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) }+{\frac{2\,C\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) } \right ) }-{\frac{2\,B\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{C\ln \left ( b{x}^{3}-a \right ) }{3\,b}} \]
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 = 0.223387, size = 169, normalized size = 2.25 \[ \frac{2 \, \sqrt{3}{\left (C a b - \left (a b^{2}\right )^{\frac{2}{3}} B\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{2}} - \frac{{\left (C b^{2} \left (\frac{a}{b}\right )^{\frac{2}{3}} + B b^{2} \left (\frac{a}{b}\right )^{\frac{1}{3}} - \left (a b^{2}\right )^{\frac{1}{3}} B b + 2 \, \left (a b^{2}\right )^{\frac{2}{3}} C\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{2}} \]
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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