Integrand size = 11, antiderivative size = 60 \[ \int \frac {(a+b x)^5}{x^4} \, dx=-\frac {a^5}{3 x^3}-\frac {5 a^4 b}{2 x^2}-\frac {10 a^3 b^2}{x}+5 a b^4 x+\frac {b^5 x^2}{2}+10 a^2 b^3 \log (x) \]
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Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {(a+b x)^5}{x^4} \, dx=-\frac {a^5}{3 x^3}-\frac {5 a^4 b}{2 x^2}-\frac {10 a^3 b^2}{x}+10 a^2 b^3 \log (x)+5 a b^4 x+\frac {b^5 x^2}{2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (5 a b^4+\frac {a^5}{x^4}+\frac {5 a^4 b}{x^3}+\frac {10 a^3 b^2}{x^2}+\frac {10 a^2 b^3}{x}+b^5 x\right ) \, dx \\ & = -\frac {a^5}{3 x^3}-\frac {5 a^4 b}{2 x^2}-\frac {10 a^3 b^2}{x}+5 a b^4 x+\frac {b^5 x^2}{2}+10 a^2 b^3 \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{x^4} \, dx=-\frac {a^5}{3 x^3}-\frac {5 a^4 b}{2 x^2}-\frac {10 a^3 b^2}{x}+5 a b^4 x+\frac {b^5 x^2}{2}+10 a^2 b^3 \log (x) \]
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Time = 0.17 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {a^{5}}{3 x^{3}}-\frac {5 a^{4} b}{2 x^{2}}-\frac {10 a^{3} b^{2}}{x}+5 a \,b^{4} x +\frac {b^{5} x^{2}}{2}+10 a^{2} b^{3} \ln \left (x \right )\) | \(55\) |
risch | \(\frac {b^{5} x^{2}}{2}+5 a \,b^{4} x +\frac {-10 a^{3} b^{2} x^{2}-\frac {5}{2} a^{4} b x -\frac {1}{3} a^{5}}{x^{3}}+10 a^{2} b^{3} \ln \left (x \right )\) | \(55\) |
norman | \(\frac {-\frac {1}{3} a^{5}+\frac {1}{2} b^{5} x^{5}+5 a \,b^{4} x^{4}-10 a^{3} b^{2} x^{2}-\frac {5}{2} a^{4} b x}{x^{3}}+10 a^{2} b^{3} \ln \left (x \right )\) | \(57\) |
parallelrisch | \(\frac {3 b^{5} x^{5}+60 a^{2} b^{3} \ln \left (x \right ) x^{3}+30 a \,b^{4} x^{4}-60 a^{3} b^{2} x^{2}-15 a^{4} b x -2 a^{5}}{6 x^{3}}\) | \(60\) |
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none
Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x)^5}{x^4} \, dx=\frac {3 \, b^{5} x^{5} + 30 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} \log \left (x\right ) - 60 \, a^{3} b^{2} x^{2} - 15 \, a^{4} b x - 2 \, a^{5}}{6 \, x^{3}} \]
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Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{x^4} \, dx=10 a^{2} b^{3} \log {\left (x \right )} + 5 a b^{4} x + \frac {b^{5} x^{2}}{2} + \frac {- 2 a^{5} - 15 a^{4} b x - 60 a^{3} b^{2} x^{2}}{6 x^{3}} \]
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none
Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^5}{x^4} \, dx=\frac {1}{2} \, b^{5} x^{2} + 5 \, a b^{4} x + 10 \, a^{2} b^{3} \log \left (x\right ) - \frac {60 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x + 2 \, a^{5}}{6 \, x^{3}} \]
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none
Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^5}{x^4} \, dx=\frac {1}{2} \, b^{5} x^{2} + 5 \, a b^{4} x + 10 \, a^{2} b^{3} \log \left ({\left | x \right |}\right ) - \frac {60 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x + 2 \, a^{5}}{6 \, x^{3}} \]
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Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^5}{x^4} \, dx=\frac {b^5\,x^2}{2}-\frac {\frac {a^5}{3}+\frac {5\,a^4\,b\,x}{2}+10\,a^3\,b^2\,x^2}{x^3}+10\,a^2\,b^3\,\ln \left (x\right )+5\,a\,b^4\,x \]
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