Integrand size = 10, antiderivative size = 67 \[ \int (a+b x)^3 \log (x) \, dx=-a^3 x-\frac {3}{4} a^2 b x^2-\frac {1}{3} a b^2 x^3-\frac {b^3 x^4}{16}-\frac {a^4 \log (x)}{4 b}+\frac {(a+b x)^4 \log (x)}{4 b} \]
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Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {32, 2350, 12, 45} \[ \int (a+b x)^3 \log (x) \, dx=-\frac {a^4 \log (x)}{4 b}-a^3 x-\frac {3}{4} a^2 b x^2-\frac {1}{3} a b^2 x^3+\frac {\log (x) (a+b x)^4}{4 b}-\frac {b^3 x^4}{16} \]
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Rule 12
Rule 32
Rule 45
Rule 2350
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^4 \log (x)}{4 b}-\int \frac {(a+b x)^4}{4 b x} \, dx \\ & = \frac {(a+b x)^4 \log (x)}{4 b}-\frac {\int \frac {(a+b x)^4}{x} \, dx}{4 b} \\ & = \frac {(a+b x)^4 \log (x)}{4 b}-\frac {\int \left (4 a^3 b+\frac {a^4}{x}+6 a^2 b^2 x+4 a b^3 x^2+b^4 x^3\right ) \, dx}{4 b} \\ & = -a^3 x-\frac {3}{4} a^2 b x^2-\frac {1}{3} a b^2 x^3-\frac {b^3 x^4}{16}-\frac {a^4 \log (x)}{4 b}+\frac {(a+b x)^4 \log (x)}{4 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.21 \[ \int (a+b x)^3 \log (x) \, dx=-a^3 x-\frac {3}{4} a^2 b x^2-\frac {1}{3} a b^2 x^3-\frac {b^3 x^4}{16}+a^3 x \log (x)+\frac {3}{2} a^2 b x^2 \log (x)+a b^2 x^3 \log (x)+\frac {1}{4} b^3 x^4 \log (x) \]
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Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.87
method | result | size |
risch | \(-a^{3} x -\frac {3 a^{2} b \,x^{2}}{4}-\frac {a \,b^{2} x^{3}}{3}-\frac {b^{3} x^{4}}{16}-\frac {a^{4} \ln \left (x \right )}{4 b}+\frac {\left (b x +a \right )^{4} \ln \left (x \right )}{4 b}\) | \(58\) |
default | \(b^{3} \left (-\frac {x^{4}}{16}+\frac {x^{4} \ln \left (x \right )}{4}\right )+3 b^{2} a \left (-\frac {x^{3}}{9}+\frac {x^{3} \ln \left (x \right )}{3}\right )+3 a^{2} b \left (-\frac {x^{2}}{4}+\frac {x^{2} \ln \left (x \right )}{2}\right )+a^{3} \left (-x +x \ln \left (x \right )\right )\) | \(69\) |
norman | \(a^{3} x \ln \left (x \right )+a \,b^{2} x^{3} \ln \left (x \right )-a^{3} x -\frac {b^{3} x^{4}}{16}-\frac {a \,b^{2} x^{3}}{3}-\frac {3 a^{2} b \,x^{2}}{4}+\frac {b^{3} x^{4} \ln \left (x \right )}{4}+\frac {3 a^{2} b \,x^{2} \ln \left (x \right )}{2}\) | \(72\) |
parallelrisch | \(a^{3} x \ln \left (x \right )+a \,b^{2} x^{3} \ln \left (x \right )-a^{3} x -\frac {b^{3} x^{4}}{16}-\frac {a \,b^{2} x^{3}}{3}-\frac {3 a^{2} b \,x^{2}}{4}+\frac {b^{3} x^{4} \ln \left (x \right )}{4}+\frac {3 a^{2} b \,x^{2} \ln \left (x \right )}{2}\) | \(72\) |
parts | \(\frac {b^{3} x^{4} \ln \left (x \right )}{4}+a \,b^{2} x^{3} \ln \left (x \right )+\frac {3 a^{2} b \,x^{2} \ln \left (x \right )}{2}+a^{3} x \ln \left (x \right )+\frac {a^{4} \ln \left (x \right )}{4 b}-\frac {\frac {b^{4} x^{4}}{4}+\frac {4 a \,b^{3} x^{3}}{3}+3 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4} \ln \left (x \right )}{4 b}\) | \(97\) |
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Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.03 \[ \int (a+b x)^3 \log (x) \, dx=-\frac {1}{16} \, b^{3} x^{4} - \frac {1}{3} \, a b^{2} x^{3} - \frac {3}{4} \, a^{2} b x^{2} - a^{3} x + \frac {1}{4} \, {\left (b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x\right )} \log \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06 \[ \int (a+b x)^3 \log (x) \, dx=- a^{3} x - \frac {3 a^{2} b x^{2}}{4} - \frac {a b^{2} x^{3}}{3} - \frac {b^{3} x^{4}}{16} + \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac {b^{3} x^{4}}{4}\right ) \log {\left (x \right )} \]
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Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.03 \[ \int (a+b x)^3 \log (x) \, dx=-\frac {1}{16} \, b^{3} x^{4} - \frac {1}{3} \, a b^{2} x^{3} - \frac {3}{4} \, a^{2} b x^{2} - a^{3} x + \frac {1}{4} \, {\left (b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x\right )} \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06 \[ \int (a+b x)^3 \log (x) \, dx=\frac {1}{4} \, b^{3} x^{4} \log \left (x\right ) - \frac {1}{16} \, b^{3} x^{4} + a b^{2} x^{3} \log \left (x\right ) - \frac {1}{3} \, a b^{2} x^{3} + \frac {3}{2} \, a^{2} b x^{2} \log \left (x\right ) - \frac {3}{4} \, a^{2} b x^{2} + a^{3} x \log \left (x\right ) - a^{3} x \]
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Time = 0.44 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06 \[ \int (a+b x)^3 \log (x) \, dx=a^3\,x\,\ln \left (x\right )-\frac {b^3\,x^4}{16}-\frac {3\,a^2\,b\,x^2}{4}-\frac {a\,b^2\,x^3}{3}-a^3\,x+\frac {b^3\,x^4\,\ln \left (x\right )}{4}+\frac {3\,a^2\,b\,x^2\,\ln \left (x\right )}{2}+a\,b^2\,x^3\,\ln \left (x\right ) \]
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