Integrand size = 18, antiderivative size = 59 \[ \int x (a+b x) (a c-b c x)^5 \, dx=-\frac {a^2 c^5 (a-b x)^6}{3 b^2}+\frac {3 a c^5 (a-b x)^7}{7 b^2}-\frac {c^5 (a-b x)^8}{8 b^2} \]
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Time = 0.02 (sec) , antiderivative size = 59, 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} \[ \int x (a+b x) (a c-b c x)^5 \, dx=-\frac {a^2 c^5 (a-b x)^6}{3 b^2}-\frac {c^5 (a-b x)^8}{8 b^2}+\frac {3 a c^5 (a-b x)^7}{7 b^2} \]
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Rule 76
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 a^2 (a c-b c x)^5}{b}-\frac {3 a (a c-b c x)^6}{b c}+\frac {(a c-b c x)^7}{b c^2}\right ) \, dx \\ & = -\frac {a^2 c^5 (a-b x)^6}{3 b^2}+\frac {3 a c^5 (a-b x)^7}{7 b^2}-\frac {c^5 (a-b x)^8}{8 b^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24 \[ \int x (a+b x) (a c-b c x)^5 \, dx=c^5 \left (\frac {a^6 x^2}{2}-\frac {4}{3} a^5 b x^3+\frac {5}{4} a^4 b^2 x^4-\frac {5}{6} a^2 b^4 x^6+\frac {4}{7} a b^5 x^7-\frac {b^6 x^8}{8}\right ) \]
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Time = 0.37 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03
method | result | size |
gosper | \(\frac {x^{2} \left (-21 b^{6} x^{6}+96 a \,x^{5} b^{5}-140 a^{2} x^{4} b^{4}+210 a^{4} x^{2} b^{2}-224 a^{5} x b +84 a^{6}\right ) c^{5}}{168}\) | \(61\) |
default | \(-\frac {1}{8} b^{6} c^{5} x^{8}+\frac {4}{7} a \,b^{5} c^{5} x^{7}-\frac {5}{6} a^{2} c^{5} b^{4} x^{6}+\frac {5}{4} a^{4} c^{5} b^{2} x^{4}-\frac {4}{3} a^{5} c^{5} b \,x^{3}+\frac {1}{2} a^{6} c^{5} x^{2}\) | \(76\) |
norman | \(-\frac {1}{8} b^{6} c^{5} x^{8}+\frac {4}{7} a \,b^{5} c^{5} x^{7}-\frac {5}{6} a^{2} c^{5} b^{4} x^{6}+\frac {5}{4} a^{4} c^{5} b^{2} x^{4}-\frac {4}{3} a^{5} c^{5} b \,x^{3}+\frac {1}{2} a^{6} c^{5} x^{2}\) | \(76\) |
risch | \(-\frac {1}{8} b^{6} c^{5} x^{8}+\frac {4}{7} a \,b^{5} c^{5} x^{7}-\frac {5}{6} a^{2} c^{5} b^{4} x^{6}+\frac {5}{4} a^{4} c^{5} b^{2} x^{4}-\frac {4}{3} a^{5} c^{5} b \,x^{3}+\frac {1}{2} a^{6} c^{5} x^{2}\) | \(76\) |
parallelrisch | \(-\frac {1}{8} b^{6} c^{5} x^{8}+\frac {4}{7} a \,b^{5} c^{5} x^{7}-\frac {5}{6} a^{2} c^{5} b^{4} x^{6}+\frac {5}{4} a^{4} c^{5} b^{2} x^{4}-\frac {4}{3} a^{5} c^{5} b \,x^{3}+\frac {1}{2} a^{6} c^{5} x^{2}\) | \(76\) |
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Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int x (a+b x) (a c-b c x)^5 \, dx=-\frac {1}{8} \, b^{6} c^{5} x^{8} + \frac {4}{7} \, a b^{5} c^{5} x^{7} - \frac {5}{6} \, a^{2} b^{4} c^{5} x^{6} + \frac {5}{4} \, a^{4} b^{2} c^{5} x^{4} - \frac {4}{3} \, a^{5} b c^{5} x^{3} + \frac {1}{2} \, a^{6} c^{5} x^{2} \]
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Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.47 \[ \int x (a+b x) (a c-b c x)^5 \, dx=\frac {a^{6} c^{5} x^{2}}{2} - \frac {4 a^{5} b c^{5} x^{3}}{3} + \frac {5 a^{4} b^{2} c^{5} x^{4}}{4} - \frac {5 a^{2} b^{4} c^{5} x^{6}}{6} + \frac {4 a b^{5} c^{5} x^{7}}{7} - \frac {b^{6} c^{5} x^{8}}{8} \]
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Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int x (a+b x) (a c-b c x)^5 \, dx=-\frac {1}{8} \, b^{6} c^{5} x^{8} + \frac {4}{7} \, a b^{5} c^{5} x^{7} - \frac {5}{6} \, a^{2} b^{4} c^{5} x^{6} + \frac {5}{4} \, a^{4} b^{2} c^{5} x^{4} - \frac {4}{3} \, a^{5} b c^{5} x^{3} + \frac {1}{2} \, a^{6} c^{5} x^{2} \]
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Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int x (a+b x) (a c-b c x)^5 \, dx=-\frac {1}{8} \, b^{6} c^{5} x^{8} + \frac {4}{7} \, a b^{5} c^{5} x^{7} - \frac {5}{6} \, a^{2} b^{4} c^{5} x^{6} + \frac {5}{4} \, a^{4} b^{2} c^{5} x^{4} - \frac {4}{3} \, a^{5} b c^{5} x^{3} + \frac {1}{2} \, a^{6} c^{5} x^{2} \]
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Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int x (a+b x) (a c-b c x)^5 \, dx=\frac {a^6\,c^5\,x^2}{2}-\frac {4\,a^5\,b\,c^5\,x^3}{3}+\frac {5\,a^4\,b^2\,c^5\,x^4}{4}-\frac {5\,a^2\,b^4\,c^5\,x^6}{6}+\frac {4\,a\,b^5\,c^5\,x^7}{7}-\frac {b^6\,c^5\,x^8}{8} \]
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