Optimal. Leaf size=109 \[ -\frac{3 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{4/3}}+\frac{\sqrt [3]{a} \log (a+b x)}{2 b^{4/3}}+\frac{\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{4/3}}+\frac{3 \sqrt [3]{x}}{b} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0880264, 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 \[ -\frac{3 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{4/3}}+\frac{\sqrt [3]{a} \log (a+b x)}{2 b^{4/3}}+\frac{\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{4/3}}+\frac{3 \sqrt [3]{x}}{b} \]
Antiderivative was successfully verified.
[In] Int[x^(1/3)/(a + b*x),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 11.231, size = 104, normalized size = 0.95 \[ - \frac{3 \sqrt [3]{a} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{2 b^{\frac{4}{3}}} + \frac{\sqrt [3]{a} \log{\left (a + b x \right )}}{2 b^{\frac{4}{3}}} + \frac{\sqrt{3} \sqrt [3]{a} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{b^{\frac{4}{3}}} + \frac{3 \sqrt [3]{x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/3)/(b*x+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0276174, size = 126, normalized size = 1.16 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )+6 \sqrt [3]{b} \sqrt [3]{x}}{2 b^{4/3}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(1/3)/(a + b*x),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 108, normalized size = 1. \[ 3\,{\frac{\sqrt [3]{x}}{b}}-{\frac{a}{{b}^{2}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a}{2\,{b}^{2}}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a\sqrt{3}}{{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/3)/(b*x+a),x)
[Out]
_______________________________________________________________________________________
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(x^(1/3)/(b*x + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.229134, size = 150, normalized size = 1.38 \[ -\frac{2 \, \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 6 \, x^{\frac{1}{3}}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(1/3)/(b*x + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 3.57049, size = 180, normalized size = 1.65 \[ \frac{4 \sqrt [3]{a} e^{\frac{5 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{3 b^{\frac{4}{3}} \Gamma \left (\frac{7}{3}\right )} - \frac{4 \sqrt [3]{a} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{3 b^{\frac{4}{3}} \Gamma \left (\frac{7}{3}\right )} + \frac{4 \sqrt [3]{a} e^{\frac{i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{3 b^{\frac{4}{3}} \Gamma \left (\frac{7}{3}\right )} + \frac{4 \sqrt [3]{x} \Gamma \left (\frac{4}{3}\right )}{b \Gamma \left (\frac{7}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/3)/(b*x+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.224201, size = 161, normalized size = 1.48 \[ \frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{b} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{b^{2}} + \frac{3 \, x^{\frac{1}{3}}}{b} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(1/3)/(b*x + a),x, algorithm="giac")
[Out]