Integrand size = 13, antiderivative size = 123 \[ \int \frac {x^{4/3}}{a+b x} \, dx=-\frac {3 a \sqrt [3]{x}}{b^2}+\frac {3 x^{4/3}}{4 b}-\frac {\sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{7/3}}+\frac {3 a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{7/3}}-\frac {a^{4/3} \log (a+b x)}{2 b^{7/3}} \]
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Time = 0.04 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {52, 60, 631, 210, 31} \[ \int \frac {x^{4/3}}{a+b x} \, dx=-\frac {\sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{7/3}}+\frac {3 a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{7/3}}-\frac {a^{4/3} \log (a+b x)}{2 b^{7/3}}-\frac {3 a \sqrt [3]{x}}{b^2}+\frac {3 x^{4/3}}{4 b} \]
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Rule 31
Rule 52
Rule 60
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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*}
Time = 0.07 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.14 \[ \int \frac {x^{4/3}}{a+b x} \, dx=\frac {-12 a \sqrt [3]{b} \sqrt [3]{x}+3 b^{4/3} x^{4/3}-4 \sqrt {3} a^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+4 a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )-2 a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{4 b^{7/3}} \]
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Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(-\frac {3 \left (-\frac {b \,x^{\frac {4}{3}}}{4}+a \,x^{\frac {1}{3}}\right )}{b^{2}}+\frac {3 \left (\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) a^{2}}{b^{2}}\) | \(123\) |
default | \(-\frac {3 \left (-\frac {b \,x^{\frac {4}{3}}}{4}+a \,x^{\frac {1}{3}}\right )}{b^{2}}+\frac {3 \left (\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) a^{2}}{b^{2}}\) | \(123\) |
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Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int \frac {x^{4/3}}{a+b x} \, dx=\frac {4 \, \sqrt {3} a \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - 2 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 4 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (b x - 4 \, a\right )} x^{\frac {1}{3}}}{4 \, b^{2}} \]
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Time = 26.34 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.41 \[ \int \frac {x^{4/3}}{a+b x} \, dx=\begin {cases} \tilde {\infty } x^{\frac {4}{3}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {7}{3}}}{7 a} & \text {for}\: b = 0 \\\frac {3 x^{\frac {4}{3}}}{4 b} & \text {for}\: a = 0 \\- \frac {3 a \sqrt [3]{x}}{b^{2}} - \frac {a \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{b^{2}} + \frac {a \sqrt [3]{- \frac {a}{b}} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{2 b^{2}} + \frac {\sqrt {3} a \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{b^{2}} + \frac {3 x^{\frac {4}{3}}}{4 b} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.04 \[ \int \frac {x^{4/3}}{a+b x} \, dx=\frac {\sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {a^{2} \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {a^{2} \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {3 \, {\left (b x^{\frac {4}{3}} - 4 \, a x^{\frac {1}{3}}\right )}}{4 \, b^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.11 \[ \int \frac {x^{4/3}}{a+b x} \, dx=-\frac {a \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{b^{2}} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{3}} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} a \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, b^{3}} + \frac {3 \, {\left (b^{3} x^{\frac {4}{3}} - 4 \, a b^{2} x^{\frac {1}{3}}\right )}}{4 \, b^{4}} \]
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Time = 0.07 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.02 \[ \int \frac {x^{4/3}}{a+b x} \, dx=\frac {3\,x^{4/3}}{4\,b}-\frac {3\,a\,x^{1/3}}{b^2}+\frac {a^{4/3}\,\ln \left (\frac {9\,a^{7/3}}{b^{1/3}}+9\,a^2\,x^{1/3}\right )}{b^{7/3}}+\frac {a^{4/3}\,\ln \left (9\,a^2\,x^{1/3}+\frac {9\,a^{7/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{1/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{7/3}}-\frac {a^{4/3}\,\ln \left (9\,a^2\,x^{1/3}-\frac {9\,a^{7/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{1/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{7/3}} \]
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