Integrand size = 20, antiderivative size = 75 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^6} \, dx=-\frac {a^6 c^5}{5 x^5}+\frac {a^5 b c^5}{x^4}-\frac {5 a^4 b^2 c^5}{3 x^3}+\frac {5 a^2 b^4 c^5}{x}-b^6 c^5 x+4 a b^5 c^5 \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 75, 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} \[ \int \frac {(a+b x) (a c-b c x)^5}{x^6} \, dx=-\frac {a^6 c^5}{5 x^5}+\frac {a^5 b c^5}{x^4}-\frac {5 a^4 b^2 c^5}{3 x^3}+\frac {5 a^2 b^4 c^5}{x}+4 a b^5 c^5 \log (x)-b^6 c^5 x \]
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Rule 76
Rubi steps \begin{align*} \text {integral}& = \int \left (-b^6 c^5+\frac {a^6 c^5}{x^6}-\frac {4 a^5 b c^5}{x^5}+\frac {5 a^4 b^2 c^5}{x^4}-\frac {5 a^2 b^4 c^5}{x^2}+\frac {4 a b^5 c^5}{x}\right ) \, dx \\ & = -\frac {a^6 c^5}{5 x^5}+\frac {a^5 b c^5}{x^4}-\frac {5 a^4 b^2 c^5}{3 x^3}+\frac {5 a^2 b^4 c^5}{x}-b^6 c^5 x+4 a b^5 c^5 \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^6} \, dx=c^5 \left (-\frac {a^6}{5 x^5}+\frac {a^5 b}{x^4}-\frac {5 a^4 b^2}{3 x^3}+\frac {5 a^2 b^4}{x}-b^6 x+4 a b^5 \log (x)\right ) \]
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Time = 0.38 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.77
method | result | size |
default | \(c^{5} \left (-b^{6} x +4 a \,b^{5} \ln \left (x \right )-\frac {5 a^{4} b^{2}}{3 x^{3}}+\frac {5 a^{2} b^{4}}{x}+\frac {a^{5} b}{x^{4}}-\frac {a^{6}}{5 x^{5}}\right )\) | \(58\) |
risch | \(-b^{6} c^{5} x +\frac {5 a^{2} b^{4} c^{5} x^{4}-\frac {5}{3} a^{4} b^{2} c^{5} x^{2}+a^{5} b \,c^{5} x -\frac {1}{5} a^{6} c^{5}}{x^{5}}+4 a \,b^{5} c^{5} \ln \left (x \right )\) | \(72\) |
norman | \(\frac {a^{5} b \,c^{5} x -\frac {1}{5} a^{6} c^{5}-b^{6} c^{5} x^{6}+5 a^{2} b^{4} c^{5} x^{4}-\frac {5}{3} a^{4} b^{2} c^{5} x^{2}}{x^{5}}+4 a \,b^{5} c^{5} \ln \left (x \right )\) | \(74\) |
parallelrisch | \(\frac {60 a \,b^{5} c^{5} \ln \left (x \right ) x^{5}-15 b^{6} c^{5} x^{6}+75 a^{2} b^{4} c^{5} x^{4}-25 a^{4} b^{2} c^{5} x^{2}+15 a^{5} b \,c^{5} x -3 a^{6} c^{5}}{15 x^{5}}\) | \(78\) |
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Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^6} \, dx=-\frac {15 \, b^{6} c^{5} x^{6} - 60 \, a b^{5} c^{5} x^{5} \log \left (x\right ) - 75 \, a^{2} b^{4} c^{5} x^{4} + 25 \, a^{4} b^{2} c^{5} x^{2} - 15 \, a^{5} b c^{5} x + 3 \, a^{6} c^{5}}{15 \, x^{5}} \]
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Time = 0.17 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^6} \, dx=4 a b^{5} c^{5} \log {\left (x \right )} - b^{6} c^{5} x - \frac {3 a^{6} c^{5} - 15 a^{5} b c^{5} x + 25 a^{4} b^{2} c^{5} x^{2} - 75 a^{2} b^{4} c^{5} x^{4}}{15 x^{5}} \]
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Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^6} \, dx=-b^{6} c^{5} x + 4 \, a b^{5} c^{5} \log \left (x\right ) + \frac {75 \, a^{2} b^{4} c^{5} x^{4} - 25 \, a^{4} b^{2} c^{5} x^{2} + 15 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{15 \, x^{5}} \]
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Time = 0.31 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^6} \, dx=-b^{6} c^{5} x + 4 \, a b^{5} c^{5} \log \left ({\left | x \right |}\right ) + \frac {75 \, a^{2} b^{4} c^{5} x^{4} - 25 \, a^{4} b^{2} c^{5} x^{2} + 15 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{15 \, x^{5}} \]
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Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^6} \, dx=-\frac {c^5\,\left (\frac {a^6}{5}+b^6\,x^6+\frac {5\,a^4\,b^2\,x^2}{3}-5\,a^2\,b^4\,x^4-a^5\,b\,x-4\,a\,b^5\,x^5\,\ln \left (x\right )\right )}{x^5} \]
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