Optimal. Leaf size=105 \[ \frac{3}{2} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{1}{2} a^{4/3} \log (x)+3 a \sqrt [3]{a+b x}+\frac{3}{4} (a+b x)^{4/3} \]
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Rubi [A] time = 0.103591, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{3}{2} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{1}{2} a^{4/3} \log (x)+3 a \sqrt [3]{a+b x}+\frac{3}{4} (a+b x)^{4/3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(4/3)/x,x]
[Out]
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Rubi in Sympy [A] time = 9.63422, size = 97, normalized size = 0.92 \[ - \frac{a^{\frac{4}{3}} \log{\left (x \right )}}{2} + \frac{3 a^{\frac{4}{3}} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x} \right )}}{2} - \sqrt{3} a^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x}}{3}\right )}{\sqrt [3]{a}} \right )} + 3 a \sqrt [3]{a + b x} + \frac{3 \left (a + b x\right )^{\frac{4}{3}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(4/3)/x,x)
[Out]
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Mathematica [C] time = 0.0373007, size = 74, normalized size = 0.7 \[ \left (\frac{15 a}{4}+\frac{3 b x}{4}\right ) \sqrt [3]{a+b x}-\frac{3 a^2 \left (\frac{a+b x}{b x}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x}\right )}{2 (a+b x)^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(4/3)/x,x]
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Maple [A] time = 0.007, size = 95, normalized size = 0.9 \[{\frac{3}{4} \left ( bx+a \right ) ^{{\frac{4}{3}}}}+3\,a\sqrt [3]{bx+a}+{a}^{{\frac{4}{3}}}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ) -{\frac{1}{2}{a}^{{\frac{4}{3}}}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{bx+a}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ) }-{a}^{{\frac{4}{3}}}\sqrt{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(4/3)/x,x)
[Out]
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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*x + a)^(4/3)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236972, size = 126, normalized size = 1.2 \[ -\sqrt{3} a^{\frac{4}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right ) - \frac{1}{2} \, a^{\frac{4}{3}} \log \left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) + a^{\frac{4}{3}} \log \left ({\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) + \frac{3}{4} \,{\left (b x + 5 \, a\right )}{\left (b x + a\right )}^{\frac{1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(4/3)/x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.56517, size = 209, normalized size = 1.99 \[ \frac{7 a^{\frac{4}{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{7}{3}\right )}{3 \Gamma \left (\frac{10}{3}\right )} + \frac{7 a^{\frac{4}{3}} e^{\frac{4 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} e^{\frac{2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{7}{3}\right )}{3 \Gamma \left (\frac{10}{3}\right )} + \frac{7 a^{\frac{4}{3}} e^{\frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} e^{\frac{4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{7}{3}\right )}{3 \Gamma \left (\frac{10}{3}\right )} + \frac{7 a \sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} \Gamma \left (\frac{7}{3}\right )}{\Gamma \left (\frac{10}{3}\right )} + \frac{7 b^{\frac{4}{3}} \left (\frac{a}{b} + x\right )^{\frac{4}{3}} \Gamma \left (\frac{7}{3}\right )}{4 \Gamma \left (\frac{10}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(4/3)/x,x)
[Out]
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GIAC/XCAS [A] time = 0.510789, size = 131, normalized size = 1.25 \[ -\sqrt{3} a^{\frac{4}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right ) - \frac{1}{2} \, a^{\frac{4}{3}}{\rm ln}\left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) + a^{\frac{4}{3}}{\rm ln}\left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right ) + \frac{3}{4} \,{\left (b x + a\right )}^{\frac{4}{3}} + 3 \,{\left (b x + a\right )}^{\frac{1}{3}} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(4/3)/x,x, algorithm="giac")
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