Optimal. Leaf size=39 \[ -\frac{2 B \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}} \]
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Rubi [A] time = 0.0609289, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ -\frac{2 B \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}} \]
Antiderivative was successfully verified.
[In] Int[(a^(1/3)*b^(1/3)*B + b^(2/3)*B*x)/(a + b*x^3),x]
[Out]
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Rubi in Sympy [A] time = 12.1477, size = 44, normalized size = 1.13 \[ - \frac{2 \sqrt{3} B \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a**(1/3)*b**(1/3)*B+b**(2/3)*B*x)/(b*x**3+a),x)
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Mathematica [A] time = 0.0368662, size = 35, normalized size = 0.9 \[ -\frac{2 B \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a}} \]
Antiderivative was successfully verified.
[In] Integrate[(a^(1/3)*b^(1/3)*B + b^(2/3)*B*x)/(a + b*x^3),x]
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Maple [B] time = 0.005, size = 195, normalized size = 5. \[{\frac{B}{3}\sqrt [3]{a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{6}\sqrt [3]{a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B\sqrt{3}}{3}\sqrt [3]{a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{3}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{b}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{6}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{b}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{b}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a^(1/3)*b^(1/3)*B+b^(2/3)*B*x)/(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((B*b^(2/3)*x + B*a^(1/3)*b^(1/3))/(b*x^3 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247734, size = 1, normalized size = 0.03 \[ \left [\sqrt{\frac{1}{3}} B \sqrt{-\frac{1}{a^{\frac{2}{3}}}} \log \left (\frac{2 \, a^{\frac{1}{3}} b x^{2} - 2 \, a^{\frac{2}{3}} b^{\frac{2}{3}} x + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b^{\frac{2}{3}} x - a^{\frac{4}{3}} b^{\frac{1}{3}}\right )} \sqrt{-\frac{1}{a^{\frac{2}{3}}}} - a b^{\frac{1}{3}}}{a^{\frac{1}{3}} b x^{2} - a^{\frac{2}{3}} b^{\frac{2}{3}} x + a b^{\frac{1}{3}}}\right ), -\frac{2 \, \sqrt{\frac{1}{3}} B \arctan \left (-\frac{\sqrt{\frac{1}{3}}{\left (2 \, a^{\frac{2}{3}} b x - a b^{\frac{2}{3}}\right )}}{a b^{\frac{2}{3}}}\right )}{a^{\frac{1}{3}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*b^(2/3)*x + B*a^(1/3)*b^(1/3))/(b*x^3 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.821074, size = 88, normalized size = 2.26 \[ \frac{B \left (- \frac{\sqrt{3} i \log{\left (x + \frac{- B \sqrt [3]{a} - \sqrt{3} i B \sqrt [3]{a}}{2 B \sqrt [3]{b}} \right )}}{3} + \frac{\sqrt{3} i \log{\left (x + \frac{- B \sqrt [3]{a} + \sqrt{3} i B \sqrt [3]{a}}{2 B \sqrt [3]{b}} \right )}}{3}\right )}{\sqrt [3]{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a**(1/3)*b**(1/3)*B+b**(2/3)*B*x)/(b*x**3+a),x)
[Out]
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GIAC/XCAS [A] time = 0.225475, size = 65, normalized size = 1.67 \[ \frac{2 \, \sqrt{3} B b^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, b^{\frac{2}{3}} x - a^{\frac{1}{3}} b^{\frac{1}{3}}\right )}}{3 \, \sqrt{a^{\frac{2}{3}} b^{\frac{2}{3}}}}\right )}{3 \, \sqrt{a^{\frac{2}{3}} b^{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*b^(2/3)*x + B*a^(1/3)*b^(1/3))/(b*x^3 + a),x, algorithm="giac")
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