Integrand size = 18, antiderivative size = 55 \[ \int x (a+b x) (a c-b c x)^3 \, dx=\frac {1}{2} a^4 c^3 x^2-\frac {2}{3} a^3 b c^3 x^3+\frac {2}{5} a b^3 c^3 x^5-\frac {1}{6} b^4 c^3 x^6 \]
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Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {76} \[ \int x (a+b x) (a c-b c x)^3 \, dx=\frac {1}{2} a^4 c^3 x^2-\frac {2}{3} a^3 b c^3 x^3+\frac {2}{5} a b^3 c^3 x^5-\frac {1}{6} b^4 c^3 x^6 \]
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Rule 76
Rubi steps \begin{align*} \text {integral}& = \int \left (a^4 c^3 x-2 a^3 b c^3 x^2+2 a b^3 c^3 x^4-b^4 c^3 x^5\right ) \, dx \\ & = \frac {1}{2} a^4 c^3 x^2-\frac {2}{3} a^3 b c^3 x^3+\frac {2}{5} a b^3 c^3 x^5-\frac {1}{6} b^4 c^3 x^6 \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int x (a+b x) (a c-b c x)^3 \, dx=c^3 \left (\frac {a^4 x^2}{2}-\frac {2}{3} a^3 b x^3+\frac {2}{5} a b^3 x^5-\frac {b^4 x^6}{6}\right ) \]
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Time = 0.36 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71
method | result | size |
gosper | \(\frac {x^{2} \left (-5 b^{4} x^{4}+12 a \,b^{3} x^{3}-20 a^{3} b x +15 a^{4}\right ) c^{3}}{30}\) | \(39\) |
default | \(\frac {1}{2} a^{4} c^{3} x^{2}-\frac {2}{3} a^{3} b \,c^{3} x^{3}+\frac {2}{5} a \,b^{3} c^{3} x^{5}-\frac {1}{6} b^{4} c^{3} x^{6}\) | \(48\) |
norman | \(\frac {1}{2} a^{4} c^{3} x^{2}-\frac {2}{3} a^{3} b \,c^{3} x^{3}+\frac {2}{5} a \,b^{3} c^{3} x^{5}-\frac {1}{6} b^{4} c^{3} x^{6}\) | \(48\) |
risch | \(\frac {1}{2} a^{4} c^{3} x^{2}-\frac {2}{3} a^{3} b \,c^{3} x^{3}+\frac {2}{5} a \,b^{3} c^{3} x^{5}-\frac {1}{6} b^{4} c^{3} x^{6}\) | \(48\) |
parallelrisch | \(\frac {1}{2} a^{4} c^{3} x^{2}-\frac {2}{3} a^{3} b \,c^{3} x^{3}+\frac {2}{5} a \,b^{3} c^{3} x^{5}-\frac {1}{6} b^{4} c^{3} x^{6}\) | \(48\) |
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none
Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int x (a+b x) (a c-b c x)^3 \, dx=-\frac {1}{6} \, b^{4} c^{3} x^{6} + \frac {2}{5} \, a b^{3} c^{3} x^{5} - \frac {2}{3} \, a^{3} b c^{3} x^{3} + \frac {1}{2} \, a^{4} c^{3} x^{2} \]
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Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int x (a+b x) (a c-b c x)^3 \, dx=\frac {a^{4} c^{3} x^{2}}{2} - \frac {2 a^{3} b c^{3} x^{3}}{3} + \frac {2 a b^{3} c^{3} x^{5}}{5} - \frac {b^{4} c^{3} x^{6}}{6} \]
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none
Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int x (a+b x) (a c-b c x)^3 \, dx=-\frac {1}{6} \, b^{4} c^{3} x^{6} + \frac {2}{5} \, a b^{3} c^{3} x^{5} - \frac {2}{3} \, a^{3} b c^{3} x^{3} + \frac {1}{2} \, a^{4} c^{3} x^{2} \]
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none
Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int x (a+b x) (a c-b c x)^3 \, dx=-\frac {1}{6} \, b^{4} c^{3} x^{6} + \frac {2}{5} \, a b^{3} c^{3} x^{5} - \frac {2}{3} \, a^{3} b c^{3} x^{3} + \frac {1}{2} \, a^{4} c^{3} x^{2} \]
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Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int x (a+b x) (a c-b c x)^3 \, dx=\frac {a^4\,c^3\,x^2}{2}-\frac {2\,a^3\,b\,c^3\,x^3}{3}+\frac {2\,a\,b^3\,c^3\,x^5}{5}-\frac {b^4\,c^3\,x^6}{6} \]
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