Integrand size = 17, antiderivative size = 38 \[ \int (a+b x) (a c-b c x)^3 \, dx=-\frac {a c^3 (a-b x)^4}{2 b}+\frac {c^3 (a-b x)^5}{5 b} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \[ \int (a+b x) (a c-b c x)^3 \, dx=\frac {c^3 (a-b x)^5}{5 b}-\frac {a c^3 (a-b x)^4}{2 b} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (2 a (a c-b c x)^3-\frac {(a c-b c x)^4}{c}\right ) \, dx \\ & = -\frac {a c^3 (a-b x)^4}{2 b}+\frac {c^3 (a-b x)^5}{5 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05 \[ \int (a+b x) (a c-b c x)^3 \, dx=c^3 \left (a^4 x-a^3 b x^2+\frac {1}{2} a b^3 x^4-\frac {b^4 x^5}{5}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97
method | result | size |
gosper | \(\frac {x \left (-2 b^{4} x^{4}+5 a \,b^{3} x^{3}-10 a^{3} b x +10 a^{4}\right ) c^{3}}{10}\) | \(37\) |
default | \(-\frac {1}{5} b^{4} c^{3} x^{5}+\frac {1}{2} a \,b^{3} c^{3} x^{4}-a^{3} c^{3} b \,x^{2}+a^{4} c^{3} x\) | \(45\) |
norman | \(-\frac {1}{5} b^{4} c^{3} x^{5}+\frac {1}{2} a \,b^{3} c^{3} x^{4}-a^{3} c^{3} b \,x^{2}+a^{4} c^{3} x\) | \(45\) |
risch | \(-\frac {1}{5} b^{4} c^{3} x^{5}+\frac {1}{2} a \,b^{3} c^{3} x^{4}-a^{3} c^{3} b \,x^{2}+a^{4} c^{3} x\) | \(45\) |
parallelrisch | \(-\frac {1}{5} b^{4} c^{3} x^{5}+\frac {1}{2} a \,b^{3} c^{3} x^{4}-a^{3} c^{3} b \,x^{2}+a^{4} c^{3} x\) | \(45\) |
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none
Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int (a+b x) (a c-b c x)^3 \, dx=-\frac {1}{5} \, b^{4} c^{3} x^{5} + \frac {1}{2} \, a b^{3} c^{3} x^{4} - a^{3} b c^{3} x^{2} + a^{4} c^{3} x \]
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Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int (a+b x) (a c-b c x)^3 \, dx=a^{4} c^{3} x - a^{3} b c^{3} x^{2} + \frac {a b^{3} c^{3} x^{4}}{2} - \frac {b^{4} c^{3} x^{5}}{5} \]
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none
Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int (a+b x) (a c-b c x)^3 \, dx=-\frac {1}{5} \, b^{4} c^{3} x^{5} + \frac {1}{2} \, a b^{3} c^{3} x^{4} - a^{3} b c^{3} x^{2} + a^{4} c^{3} x \]
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Time = 0.33 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int (a+b x) (a c-b c x)^3 \, dx=-\frac {1}{5} \, b^{4} c^{3} x^{5} + \frac {1}{2} \, a b^{3} c^{3} x^{4} - a^{3} b c^{3} x^{2} + a^{4} c^{3} x \]
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Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int (a+b x) (a c-b c x)^3 \, dx=a^4\,c^3\,x-a^3\,b\,c^3\,x^2+\frac {a\,b^3\,c^3\,x^4}{2}-\frac {b^4\,c^3\,x^5}{5} \]
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