Integrand size = 20, antiderivative size = 78 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^4} \, dx=-\frac {a^5 c^4}{3 x^3}+\frac {3 a^4 b c^4}{2 x^2}-\frac {2 a^3 b^2 c^4}{x}-3 a b^4 c^4 x+\frac {1}{2} b^5 c^4 x^2+2 a^2 b^3 c^4 \log (x) \]
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Time = 0.02 (sec) , antiderivative size = 78, 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} \[ \int \frac {(a+b x) (a c-b c x)^4}{x^4} \, dx=-\frac {a^5 c^4}{3 x^3}+\frac {3 a^4 b c^4}{2 x^2}-\frac {2 a^3 b^2 c^4}{x}+2 a^2 b^3 c^4 \log (x)-3 a b^4 c^4 x+\frac {1}{2} b^5 c^4 x^2 \]
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Rule 76
Rubi steps \begin{align*} \text {integral}& = \int \left (-3 a b^4 c^4+\frac {a^5 c^4}{x^4}-\frac {3 a^4 b c^4}{x^3}+\frac {2 a^3 b^2 c^4}{x^2}+\frac {2 a^2 b^3 c^4}{x}+b^5 c^4 x\right ) \, dx \\ & = -\frac {a^5 c^4}{3 x^3}+\frac {3 a^4 b c^4}{2 x^2}-\frac {2 a^3 b^2 c^4}{x}-3 a b^4 c^4 x+\frac {1}{2} b^5 c^4 x^2+2 a^2 b^3 c^4 \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^4} \, dx=-\frac {a^5 c^4}{3 x^3}+\frac {3 a^4 b c^4}{2 x^2}-\frac {2 a^3 b^2 c^4}{x}-3 a b^4 c^4 x+\frac {1}{2} b^5 c^4 x^2+2 a^2 b^3 c^4 \log (x) \]
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Time = 0.37 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.76
method | result | size |
default | \(c^{4} \left (\frac {b^{5} x^{2}}{2}-3 a \,b^{4} x +2 a^{2} b^{3} \ln \left (x \right )-\frac {a^{5}}{3 x^{3}}-\frac {2 a^{3} b^{2}}{x}+\frac {3 a^{4} b}{2 x^{2}}\right )\) | \(59\) |
risch | \(\frac {b^{5} c^{4} x^{2}}{2}-3 a \,b^{4} c^{4} x +\frac {-2 a^{3} b^{2} c^{4} x^{2}+\frac {3}{2} a^{4} b \,c^{4} x -\frac {1}{3} a^{5} c^{4}}{x^{3}}+2 a^{2} b^{3} c^{4} \ln \left (x \right )\) | \(73\) |
norman | \(\frac {-\frac {1}{3} a^{5} c^{4}+\frac {1}{2} b^{5} c^{4} x^{5}-3 a \,b^{4} c^{4} x^{4}-2 a^{3} b^{2} c^{4} x^{2}+\frac {3}{2} a^{4} b \,c^{4} x}{x^{3}}+2 a^{2} b^{3} c^{4} \ln \left (x \right )\) | \(75\) |
parallelrisch | \(\frac {3 b^{5} c^{4} x^{5}+12 a^{2} c^{4} b^{3} \ln \left (x \right ) x^{3}-18 a \,b^{4} c^{4} x^{4}-12 a^{3} b^{2} c^{4} x^{2}+9 a^{4} b \,c^{4} x -2 a^{5} c^{4}}{6 x^{3}}\) | \(78\) |
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Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^4} \, dx=\frac {3 \, b^{5} c^{4} x^{5} - 18 \, a b^{4} c^{4} x^{4} + 12 \, a^{2} b^{3} c^{4} x^{3} \log \left (x\right ) - 12 \, a^{3} b^{2} c^{4} x^{2} + 9 \, a^{4} b c^{4} x - 2 \, a^{5} c^{4}}{6 \, x^{3}} \]
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Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^4} \, dx=2 a^{2} b^{3} c^{4} \log {\left (x \right )} - 3 a b^{4} c^{4} x + \frac {b^{5} c^{4} x^{2}}{2} + \frac {- 2 a^{5} c^{4} + 9 a^{4} b c^{4} x - 12 a^{3} b^{2} c^{4} x^{2}}{6 x^{3}} \]
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Time = 0.22 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^4} \, dx=\frac {1}{2} \, b^{5} c^{4} x^{2} - 3 \, a b^{4} c^{4} x + 2 \, a^{2} b^{3} c^{4} \log \left (x\right ) - \frac {12 \, a^{3} b^{2} c^{4} x^{2} - 9 \, a^{4} b c^{4} x + 2 \, a^{5} c^{4}}{6 \, x^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^4} \, dx=\frac {1}{2} \, b^{5} c^{4} x^{2} - 3 \, a b^{4} c^{4} x + 2 \, a^{2} b^{3} c^{4} \log \left ({\left | x \right |}\right ) - \frac {12 \, a^{3} b^{2} c^{4} x^{2} - 9 \, a^{4} b c^{4} x + 2 \, a^{5} c^{4}}{6 \, x^{3}} \]
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Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^4} \, dx=\frac {b^5\,c^4\,x^2}{2}-\frac {\frac {a^5\,c^4}{3}-\frac {3\,a^4\,b\,c^4\,x}{2}+2\,a^3\,b^2\,c^4\,x^2}{x^3}+2\,a^2\,b^3\,c^4\,\ln \left (x\right )-3\,a\,b^4\,c^4\,x \]
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