Integrand size = 11, antiderivative size = 58 \[ \int \frac {(a+b x)^5}{x^2} \, dx=-\frac {a^5}{x}+10 a^3 b^2 x+5 a^2 b^3 x^2+\frac {5}{3} a b^4 x^3+\frac {b^5 x^4}{4}+5 a^4 b \log (x) \]
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Time = 0.01 (sec) , antiderivative size = 58, 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} \[ \int \frac {(a+b x)^5}{x^2} \, dx=-\frac {a^5}{x}+5 a^4 b \log (x)+10 a^3 b^2 x+5 a^2 b^3 x^2+\frac {5}{3} a b^4 x^3+\frac {b^5 x^4}{4} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (10 a^3 b^2+\frac {a^5}{x^2}+\frac {5 a^4 b}{x}+10 a^2 b^3 x+5 a b^4 x^2+b^5 x^3\right ) \, dx \\ & = -\frac {a^5}{x}+10 a^3 b^2 x+5 a^2 b^3 x^2+\frac {5}{3} a b^4 x^3+\frac {b^5 x^4}{4}+5 a^4 b \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{x^2} \, dx=-\frac {a^5}{x}+10 a^3 b^2 x+5 a^2 b^3 x^2+\frac {5}{3} a b^4 x^3+\frac {b^5 x^4}{4}+5 a^4 b \log (x) \]
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Time = 0.17 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {a^{5}}{x}+10 a^{3} b^{2} x +5 a^{2} b^{3} x^{2}+\frac {5 a \,b^{4} x^{3}}{3}+\frac {b^{5} x^{4}}{4}+5 a^{4} b \ln \left (x \right )\) | \(55\) |
risch | \(-\frac {a^{5}}{x}+10 a^{3} b^{2} x +5 a^{2} b^{3} x^{2}+\frac {5 a \,b^{4} x^{3}}{3}+\frac {b^{5} x^{4}}{4}+5 a^{4} b \ln \left (x \right )\) | \(55\) |
norman | \(\frac {-a^{5}+\frac {1}{4} b^{5} x^{5}+\frac {5}{3} a \,b^{4} x^{4}+5 a^{2} b^{3} x^{3}+10 a^{3} b^{2} x^{2}}{x}+5 a^{4} b \ln \left (x \right )\) | \(59\) |
parallelrisch | \(\frac {3 b^{5} x^{5}+20 a \,b^{4} x^{4}+60 a^{2} b^{3} x^{3}+60 a^{4} b \ln \left (x \right ) x +120 a^{3} b^{2} x^{2}-12 a^{5}}{12 x}\) | \(60\) |
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none
Time = 0.22 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^5}{x^2} \, dx=\frac {3 \, b^{5} x^{5} + 20 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} + 120 \, a^{3} b^{2} x^{2} + 60 \, a^{4} b x \log \left (x\right ) - 12 \, a^{5}}{12 \, x} \]
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Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^5}{x^2} \, dx=- \frac {a^{5}}{x} + 5 a^{4} b \log {\left (x \right )} + 10 a^{3} b^{2} x + 5 a^{2} b^{3} x^{2} + \frac {5 a b^{4} x^{3}}{3} + \frac {b^{5} x^{4}}{4} \]
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none
Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^5}{x^2} \, dx=\frac {1}{4} \, b^{5} x^{4} + \frac {5}{3} \, a b^{4} x^{3} + 5 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x + 5 \, a^{4} b \log \left (x\right ) - \frac {a^{5}}{x} \]
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Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^5}{x^2} \, dx=\frac {1}{4} \, b^{5} x^{4} + \frac {5}{3} \, a b^{4} x^{3} + 5 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x + 5 \, a^{4} b \log \left ({\left | x \right |}\right ) - \frac {a^{5}}{x} \]
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Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^5}{x^2} \, dx=\frac {b^5\,x^4}{4}-\frac {a^5}{x}+10\,a^3\,b^2\,x+\frac {5\,a\,b^4\,x^3}{3}+5\,a^4\,b\,\ln \left (x\right )+5\,a^2\,b^3\,x^2 \]
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