Integrand size = 11, antiderivative size = 32 \[ \int \frac {x}{(a+b x)^{4/3}} \, dx=\frac {3 a}{b^2 \sqrt [3]{a+b x}}+\frac {3 (a+b x)^{2/3}}{2 b^2} \]
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Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x}{(a+b x)^{4/3}} \, dx=\frac {3 a}{b^2 \sqrt [3]{a+b x}}+\frac {3 (a+b x)^{2/3}}{2 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a}{b (a+b x)^{4/3}}+\frac {1}{b \sqrt [3]{a+b x}}\right ) \, dx \\ & = \frac {3 a}{b^2 \sqrt [3]{a+b x}}+\frac {3 (a+b x)^{2/3}}{2 b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {x}{(a+b x)^{4/3}} \, dx=\frac {3 (3 a+b x)}{2 b^2 \sqrt [3]{a+b x}} \]
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Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62
method | result | size |
gosper | \(\frac {\frac {3 b x}{2}+\frac {9 a}{2}}{\left (b x +a \right )^{\frac {1}{3}} b^{2}}\) | \(20\) |
trager | \(\frac {\frac {3 b x}{2}+\frac {9 a}{2}}{\left (b x +a \right )^{\frac {1}{3}} b^{2}}\) | \(20\) |
pseudoelliptic | \(\frac {3 b x +9 a}{2 \left (b x +a \right )^{\frac {1}{3}} b^{2}}\) | \(21\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2}+\frac {3 a}{\left (b x +a \right )^{\frac {1}{3}}}}{b^{2}}\) | \(25\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2}+\frac {3 a}{\left (b x +a \right )^{\frac {1}{3}}}}{b^{2}}\) | \(25\) |
risch | \(\frac {3 a}{b^{2} \left (b x +a \right )^{\frac {1}{3}}}+\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2 b^{2}}\) | \(27\) |
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none
Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {x}{(a+b x)^{4/3}} \, dx=\frac {3 \, {\left (b x + 3 \, a\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{2 \, {\left (b^{3} x + a b^{2}\right )}} \]
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Time = 0.35 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {x}{(a+b x)^{4/3}} \, dx=\begin {cases} \frac {9 a}{2 b^{2} \sqrt [3]{a + b x}} + \frac {3 x}{2 b \sqrt [3]{a + b x}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {4}{3}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {x}{(a+b x)^{4/3}} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {2}{3}}}{2 \, b^{2}} + \frac {3 \, a}{{\left (b x + a\right )}^{\frac {1}{3}} b^{2}} \]
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Time = 0.45 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {x}{(a+b x)^{4/3}} \, dx=\frac {3 \, {\left (\frac {{\left (b x + a\right )}^{\frac {2}{3}}}{b} + \frac {2 \, a}{{\left (b x + a\right )}^{\frac {1}{3}} b}\right )}}{2 \, b} \]
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Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62 \[ \int \frac {x}{(a+b x)^{4/3}} \, dx=\frac {9\,a+3\,b\,x}{2\,b^2\,{\left (a+b\,x\right )}^{1/3}} \]
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