Integrand size = 20, antiderivative size = 79 \[ \int \frac {(a+b x) (a c-b c x)^5}{x} \, dx=-4 a^5 b c^5 x+\frac {5}{2} a^4 b^2 c^5 x^2-\frac {5}{4} a^2 b^4 c^5 x^4+\frac {4}{5} a b^5 c^5 x^5-\frac {1}{6} b^6 c^5 x^6+a^6 c^5 \log (x) \]
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Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {76} \[ \int \frac {(a+b x) (a c-b c x)^5}{x} \, dx=a^6 c^5 \log (x)-4 a^5 b c^5 x+\frac {5}{2} a^4 b^2 c^5 x^2-\frac {5}{4} a^2 b^4 c^5 x^4+\frac {4}{5} a b^5 c^5 x^5-\frac {1}{6} b^6 c^5 x^6 \]
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Rule 76
Rubi steps \begin{align*} \text {integral}& = \int \left (-4 a^5 b c^5+\frac {a^6 c^5}{x}+5 a^4 b^2 c^5 x-5 a^2 b^4 c^5 x^3+4 a b^5 c^5 x^4-b^6 c^5 x^5\right ) \, dx \\ & = -4 a^5 b c^5 x+\frac {5}{2} a^4 b^2 c^5 x^2-\frac {5}{4} a^2 b^4 c^5 x^4+\frac {4}{5} a b^5 c^5 x^5-\frac {1}{6} b^6 c^5 x^6+a^6 c^5 \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x) (a c-b c x)^5}{x} \, dx=c^5 \left (\frac {127 a^6}{60}-4 a^5 b x+\frac {5}{2} a^4 b^2 x^2-\frac {5}{4} a^2 b^4 x^4+\frac {4}{5} a b^5 x^5-\frac {b^6 x^6}{6}+a^6 \log (-b x)\right ) \]
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Time = 0.37 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.73
method | result | size |
default | \(c^{5} \left (-\frac {b^{6} x^{6}}{6}+\frac {4 a \,x^{5} b^{5}}{5}-\frac {5 a^{2} x^{4} b^{4}}{4}+\frac {5 a^{4} x^{2} b^{2}}{2}-4 a^{5} x b +a^{6} \ln \left (x \right )\right )\) | \(58\) |
norman | \(-4 a^{5} b \,c^{5} x +\frac {5 a^{4} b^{2} c^{5} x^{2}}{2}-\frac {5 a^{2} b^{4} c^{5} x^{4}}{4}+\frac {4 a \,b^{5} c^{5} x^{5}}{5}-\frac {b^{6} c^{5} x^{6}}{6}+a^{6} c^{5} \ln \left (x \right )\) | \(72\) |
risch | \(-4 a^{5} b \,c^{5} x +\frac {5 a^{4} b^{2} c^{5} x^{2}}{2}-\frac {5 a^{2} b^{4} c^{5} x^{4}}{4}+\frac {4 a \,b^{5} c^{5} x^{5}}{5}-\frac {b^{6} c^{5} x^{6}}{6}+a^{6} c^{5} \ln \left (x \right )\) | \(72\) |
parallelrisch | \(-4 a^{5} b \,c^{5} x +\frac {5 a^{4} b^{2} c^{5} x^{2}}{2}-\frac {5 a^{2} b^{4} c^{5} x^{4}}{4}+\frac {4 a \,b^{5} c^{5} x^{5}}{5}-\frac {b^{6} c^{5} x^{6}}{6}+a^{6} c^{5} \ln \left (x \right )\) | \(72\) |
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none
Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x) (a c-b c x)^5}{x} \, dx=-\frac {1}{6} \, b^{6} c^{5} x^{6} + \frac {4}{5} \, a b^{5} c^{5} x^{5} - \frac {5}{4} \, a^{2} b^{4} c^{5} x^{4} + \frac {5}{2} \, a^{4} b^{2} c^{5} x^{2} - 4 \, a^{5} b c^{5} x + a^{6} c^{5} \log \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x) (a c-b c x)^5}{x} \, dx=a^{6} c^{5} \log {\left (x \right )} - 4 a^{5} b c^{5} x + \frac {5 a^{4} b^{2} c^{5} x^{2}}{2} - \frac {5 a^{2} b^{4} c^{5} x^{4}}{4} + \frac {4 a b^{5} c^{5} x^{5}}{5} - \frac {b^{6} c^{5} x^{6}}{6} \]
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none
Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x) (a c-b c x)^5}{x} \, dx=-\frac {1}{6} \, b^{6} c^{5} x^{6} + \frac {4}{5} \, a b^{5} c^{5} x^{5} - \frac {5}{4} \, a^{2} b^{4} c^{5} x^{4} + \frac {5}{2} \, a^{4} b^{2} c^{5} x^{2} - 4 \, a^{5} b c^{5} x + a^{6} c^{5} \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x) (a c-b c x)^5}{x} \, dx=-\frac {1}{6} \, b^{6} c^{5} x^{6} + \frac {4}{5} \, a b^{5} c^{5} x^{5} - \frac {5}{4} \, a^{2} b^{4} c^{5} x^{4} + \frac {5}{2} \, a^{4} b^{2} c^{5} x^{2} - 4 \, a^{5} b c^{5} x + a^{6} c^{5} \log \left ({\left | x \right |}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x) (a c-b c x)^5}{x} \, dx=a^6\,c^5\,\ln \left (x\right )-\frac {b^6\,c^5\,x^6}{6}+\frac {4\,a\,b^5\,c^5\,x^5}{5}+\frac {5\,a^4\,b^2\,c^5\,x^2}{2}-\frac {5\,a^2\,b^4\,c^5\,x^4}{4}-4\,a^5\,b\,c^5\,x \]
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