Integrand size = 11, antiderivative size = 86 \[ \int \frac {(a+b x)^7}{x^2} \, dx=-\frac {a^7}{x}+21 a^5 b^2 x+\frac {35}{2} a^4 b^3 x^2+\frac {35}{3} a^3 b^4 x^3+\frac {21}{4} a^2 b^5 x^4+\frac {7}{5} a b^6 x^5+\frac {b^7 x^6}{6}+7 a^6 b \log (x) \]
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Time = 0.02 (sec) , antiderivative size = 86, 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} \[ \int \frac {(a+b x)^7}{x^2} \, dx=-\frac {a^7}{x}+7 a^6 b \log (x)+21 a^5 b^2 x+\frac {35}{2} a^4 b^3 x^2+\frac {35}{3} a^3 b^4 x^3+\frac {21}{4} a^2 b^5 x^4+\frac {7}{5} a b^6 x^5+\frac {b^7 x^6}{6} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (21 a^5 b^2+\frac {a^7}{x^2}+\frac {7 a^6 b}{x}+35 a^4 b^3 x+35 a^3 b^4 x^2+21 a^2 b^5 x^3+7 a b^6 x^4+b^7 x^5\right ) \, dx \\ & = -\frac {a^7}{x}+21 a^5 b^2 x+\frac {35}{2} a^4 b^3 x^2+\frac {35}{3} a^3 b^4 x^3+\frac {21}{4} a^2 b^5 x^4+\frac {7}{5} a b^6 x^5+\frac {b^7 x^6}{6}+7 a^6 b \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^7}{x^2} \, dx=-\frac {a^7}{x}+21 a^5 b^2 x+\frac {35}{2} a^4 b^3 x^2+\frac {35}{3} a^3 b^4 x^3+\frac {21}{4} a^2 b^5 x^4+\frac {7}{5} a b^6 x^5+\frac {b^7 x^6}{6}+7 a^6 b \log (x) \]
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Time = 0.16 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {a^{7}}{x}+21 a^{5} b^{2} x +\frac {35 a^{4} b^{3} x^{2}}{2}+\frac {35 a^{3} b^{4} x^{3}}{3}+\frac {21 a^{2} b^{5} x^{4}}{4}+\frac {7 a \,b^{6} x^{5}}{5}+\frac {b^{7} x^{6}}{6}+7 a^{6} b \ln \left (x \right )\) | \(77\) |
risch | \(-\frac {a^{7}}{x}+21 a^{5} b^{2} x +\frac {35 a^{4} b^{3} x^{2}}{2}+\frac {35 a^{3} b^{4} x^{3}}{3}+\frac {21 a^{2} b^{5} x^{4}}{4}+\frac {7 a \,b^{6} x^{5}}{5}+\frac {b^{7} x^{6}}{6}+7 a^{6} b \ln \left (x \right )\) | \(77\) |
norman | \(\frac {-a^{7}+\frac {1}{6} b^{7} x^{7}+\frac {7}{5} a \,b^{6} x^{6}+\frac {21}{4} a^{2} b^{5} x^{5}+\frac {35}{3} a^{3} b^{4} x^{4}+\frac {35}{2} a^{4} b^{3} x^{3}+21 a^{5} b^{2} x^{2}}{x}+7 a^{6} b \ln \left (x \right )\) | \(81\) |
parallelrisch | \(\frac {10 b^{7} x^{7}+84 a \,b^{6} x^{6}+315 a^{2} b^{5} x^{5}+700 a^{3} b^{4} x^{4}+1050 a^{4} b^{3} x^{3}+420 a^{6} b \ln \left (x \right ) x +1260 a^{5} b^{2} x^{2}-60 a^{7}}{60 x}\) | \(82\) |
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none
Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^7}{x^2} \, dx=\frac {10 \, b^{7} x^{7} + 84 \, a b^{6} x^{6} + 315 \, a^{2} b^{5} x^{5} + 700 \, a^{3} b^{4} x^{4} + 1050 \, a^{4} b^{3} x^{3} + 1260 \, a^{5} b^{2} x^{2} + 420 \, a^{6} b x \log \left (x\right ) - 60 \, a^{7}}{60 \, x} \]
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Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^7}{x^2} \, dx=- \frac {a^{7}}{x} + 7 a^{6} b \log {\left (x \right )} + 21 a^{5} b^{2} x + \frac {35 a^{4} b^{3} x^{2}}{2} + \frac {35 a^{3} b^{4} x^{3}}{3} + \frac {21 a^{2} b^{5} x^{4}}{4} + \frac {7 a b^{6} x^{5}}{5} + \frac {b^{7} x^{6}}{6} \]
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none
Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^7}{x^2} \, dx=\frac {1}{6} \, b^{7} x^{6} + \frac {7}{5} \, a b^{6} x^{5} + \frac {21}{4} \, a^{2} b^{5} x^{4} + \frac {35}{3} \, a^{3} b^{4} x^{3} + \frac {35}{2} \, a^{4} b^{3} x^{2} + 21 \, a^{5} b^{2} x + 7 \, a^{6} b \log \left (x\right ) - \frac {a^{7}}{x} \]
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none
Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^7}{x^2} \, dx=\frac {1}{6} \, b^{7} x^{6} + \frac {7}{5} \, a b^{6} x^{5} + \frac {21}{4} \, a^{2} b^{5} x^{4} + \frac {35}{3} \, a^{3} b^{4} x^{3} + \frac {35}{2} \, a^{4} b^{3} x^{2} + 21 \, a^{5} b^{2} x + 7 \, a^{6} b \log \left ({\left | x \right |}\right ) - \frac {a^{7}}{x} \]
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Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^7}{x^2} \, dx=\frac {b^7\,x^6}{6}-\frac {a^7}{x}+21\,a^5\,b^2\,x+\frac {7\,a\,b^6\,x^5}{5}+7\,a^6\,b\,\ln \left (x\right )+\frac {35\,a^4\,b^3\,x^2}{2}+\frac {35\,a^3\,b^4\,x^3}{3}+\frac {21\,a^2\,b^5\,x^4}{4} \]
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