Optimal. Leaf size=123 \[ \frac{3 a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{7/3}}-\frac{a^{4/3} \log (a+b x)}{2 b^{7/3}}-\frac{\sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{7/3}}-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{4/3}}{4 b} \]
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Rubi [A] time = 0.0563835, antiderivative size = 123, 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, Rules used = {50, 58, 617, 204, 31} \[ \frac{3 a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{7/3}}-\frac{a^{4/3} \log (a+b x)}{2 b^{7/3}}-\frac{\sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{7/3}}-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{4/3}}{4 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 58
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^{4/3}}{a+b x} \, dx &=\frac{3 x^{4/3}}{4 b}-\frac{a \int \frac{\sqrt [3]{x}}{a+b x} \, dx}{b}\\ &=-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{4/3}}{4 b}+\frac{a^2 \int \frac{1}{x^{2/3} (a+b x)} \, dx}{b^2}\\ &=-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{4/3}}{4 b}-\frac{a^{4/3} \log (a+b x)}{2 b^{7/3}}+\frac{\left (3 a^{5/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{a^{2/3}}{b^{2/3}}-\frac{\sqrt [3]{a} x}{\sqrt [3]{b}}+x^2} \, dx,x,\sqrt [3]{x}\right )}{2 b^{8/3}}+\frac{\left (3 a^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{2 b^{7/3}}\\ &=-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{4/3}}{4 b}+\frac{3 a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{7/3}}-\frac{a^{4/3} \log (a+b x)}{2 b^{7/3}}+\frac{\left (3 a^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{b^{7/3}}\\ &=-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{4/3}}{4 b}-\frac{\sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{7/3}}+\frac{3 a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{7/3}}-\frac{a^{4/3} \log (a+b x)}{2 b^{7/3}}\\ \end{align*}
Mathematica [A] time = 0.0578726, size = 140, normalized size = 1.14 \[ \frac{-2 a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )+4 a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )-4 \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )-12 a \sqrt [3]{b} \sqrt [3]{x}+3 b^{4/3} x^{4/3}}{4 b^{7/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 121, normalized size = 1. \begin{align*}{\frac{3}{4\,b}{x}^{{\frac{4}{3}}}}-3\,{\frac{a\sqrt [3]{x}}{{b}^{2}}}+{\frac{{a}^{2}}{{b}^{3}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}}{2\,{b}^{3}}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{{\frac{a}{b}}}\sqrt [3]{x}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{a}^{2}\sqrt{3}}{{b}^{3}}\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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56, size = 312, normalized size = 2.54 \begin{align*} \frac{4 \, \sqrt{3} a \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x^{\frac{1}{3}} \left (\frac{a}{b}\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right ) - 2 \, a \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 ) + 4 \, a \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 3 \,{\left (b x - 4 \, a\right )} x^{\frac{1}{3}}}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 43.7102, size = 240, normalized size = 1.95 \begin{align*} \begin{cases} \tilde{\infty } x^{\frac{4}{3}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{3 x^{\frac{7}{3}}}{7 a} & \text{for}\: b = 0 \\\frac{3 x^{\frac{4}{3}}}{4 b} & \text{for}\: a = 0 \\- \frac{\sqrt [3]{-1} a^{\frac{4}{3}} \log{\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac{1}{b}} + \sqrt [3]{x} \right )}}{b^{3} \left (\frac{1}{b}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{-1} a^{\frac{4}{3}} \log{\left (4 \left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \left (\frac{1}{b}\right )^{\frac{2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{\frac{1}{b}} + 4 x^{\frac{2}{3}} \right )}}{2 b^{3} \left (\frac{1}{b}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{-1} \sqrt{3} a^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{2 \left (-1\right )^{\frac{2}{3}} \sqrt{3} \sqrt [3]{x}}{3 \sqrt [3]{a} \sqrt [3]{\frac{1}{b}}} \right )}}{b^{3} \left (\frac{1}{b}\right )^{\frac{2}{3}}} - \frac{3 a \sqrt [3]{x}}{b^{2}} + \frac{3 x^{\frac{4}{3}}}{4 b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0752, size = 184, normalized size = 1.5 \begin{align*} -\frac{a \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{b^{2}} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} a \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^{3}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} a \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 \, b^{3}} + \frac{3 \,{\left (b^{3} x^{\frac{4}{3}} - 4 \, a b^{2} x^{\frac{1}{3}}\right )}}{4 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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