Integrand size = 11, antiderivative size = 35 \[ \int \frac {(a+b x)^3}{x} \, dx=3 a^2 b x+\frac {3}{2} a b^2 x^2+\frac {b^3 x^3}{3}+a^3 \log (x) \]
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Time = 0.01 (sec) , antiderivative size = 35, 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} \[ \int \frac {(a+b x)^3}{x} \, dx=a^3 \log (x)+3 a^2 b x+\frac {3}{2} a b^2 x^2+\frac {b^3 x^3}{3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (3 a^2 b+\frac {a^3}{x}+3 a b^2 x+b^3 x^2\right ) \, dx \\ & = 3 a^2 b x+\frac {3}{2} a b^2 x^2+\frac {b^3 x^3}{3}+a^3 \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^3}{x} \, dx=3 a^2 b x+\frac {3}{2} a b^2 x^2+\frac {b^3 x^3}{3}+a^3 \log (x) \]
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Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91
method | result | size |
default | \(3 a^{2} b x +\frac {3 a \,b^{2} x^{2}}{2}+\frac {b^{3} x^{3}}{3}+a^{3} \ln \left (x \right )\) | \(32\) |
norman | \(3 a^{2} b x +\frac {3 a \,b^{2} x^{2}}{2}+\frac {b^{3} x^{3}}{3}+a^{3} \ln \left (x \right )\) | \(32\) |
risch | \(3 a^{2} b x +\frac {3 a \,b^{2} x^{2}}{2}+\frac {b^{3} x^{3}}{3}+a^{3} \ln \left (x \right )\) | \(32\) |
parallelrisch | \(3 a^{2} b x +\frac {3 a \,b^{2} x^{2}}{2}+\frac {b^{3} x^{3}}{3}+a^{3} \ln \left (x \right )\) | \(32\) |
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none
Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^3}{x} \, dx=\frac {1}{3} \, b^{3} x^{3} + \frac {3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left (x\right ) \]
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Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^3}{x} \, dx=a^{3} \log {\left (x \right )} + 3 a^{2} b x + \frac {3 a b^{2} x^{2}}{2} + \frac {b^{3} x^{3}}{3} \]
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none
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^3}{x} \, dx=\frac {1}{3} \, b^{3} x^{3} + \frac {3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left (x\right ) \]
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none
Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^3}{x} \, dx=\frac {1}{3} \, b^{3} x^{3} + \frac {3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left ({\left | x \right |}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^3}{x} \, dx=a^3\,\ln \left (x\right )+\frac {b^3\,x^3}{3}+\frac {3\,a\,b^2\,x^2}{2}+3\,a^2\,b\,x \]
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