Optimal. Leaf size=109 \[ \frac{3 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3}}-\frac{\sqrt [3]{b} \log (a+b x)}{2 a^{4/3}}+\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{4/3}}-\frac{3}{a \sqrt [3]{x}} \]
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Rubi [A] time = 0.0879252, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{3 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3}}-\frac{\sqrt [3]{b} \log (a+b x)}{2 a^{4/3}}+\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{4/3}}-\frac{3}{a \sqrt [3]{x}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(4/3)*(a + b*x)),x]
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Rubi in Sympy [A] time = 11.1336, size = 104, normalized size = 0.95 \[ - \frac{3}{a \sqrt [3]{x}} + \frac{3 \sqrt [3]{b} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{2 a^{\frac{4}{3}}} - \frac{\sqrt [3]{b} \log{\left (a + b x \right )}}{2 a^{\frac{4}{3}}} + \frac{\sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{a^{\frac{4}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(4/3)/(b*x+a),x)
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Mathematica [A] time = 0.0671427, size = 127, normalized size = 1.17 \[ \frac{-\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )+2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )-\frac{6 \sqrt [3]{a}}{\sqrt [3]{x}}}{2 a^{4/3}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(4/3)*(a + b*x)),x]
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Maple [A] time = 0.012, size = 104, normalized size = 1. \[ -3\,{\frac{1}{a\sqrt [3]{x}}}+{\frac{1}{a}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{2\,a}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}}{a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(4/3)/(b*x+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(1/((b*x + a)*x^(4/3)),x, algorithm="maxima")
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Fricas [A] time = 0.232659, size = 169, normalized size = 1.55 \[ -\frac{2 \, \sqrt{3} x^{\frac{1}{3}} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3}{\left (a \left (\frac{b}{a}\right )^{\frac{2}{3}} - 2 \, b x^{\frac{1}{3}}\right )}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + x^{\frac{1}{3}} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (-a x^{\frac{1}{3}} \left (\frac{b}{a}\right )^{\frac{2}{3}} + b x^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 2 \, x^{\frac{1}{3}} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (a \left (\frac{b}{a}\right )^{\frac{2}{3}} + b x^{\frac{1}{3}}\right ) + 6}{2 \, a x^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*x^(4/3)),x, algorithm="fricas")
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Sympy [A] time = 3.91452, size = 182, normalized size = 1.67 \[ \frac{\Gamma \left (- \frac{1}{3}\right )}{a \sqrt [3]{x} \Gamma \left (\frac{2}{3}\right )} - \frac{\sqrt [3]{b} e^{\frac{10 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right )} - \frac{\sqrt [3]{b} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right )} - \frac{\sqrt [3]{b} e^{\frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(4/3)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.221063, size = 169, normalized size = 1.55 \[ \frac{b \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a^{2}} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{a^{2} b} - \frac{3}{a x^{\frac{1}{3}}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{2 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*x^(4/3)),x, algorithm="giac")
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