Integrand size = 11, antiderivative size = 87 \[ \int \frac {(a+b x)^7}{x} \, dx=7 a^6 b x+\frac {21}{2} a^5 b^2 x^2+\frac {35}{3} a^4 b^3 x^3+\frac {35}{4} a^3 b^4 x^4+\frac {21}{5} a^2 b^5 x^5+\frac {7}{6} a b^6 x^6+\frac {b^7 x^7}{7}+a^7 \log (x) \]
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Time = 0.02 (sec) , antiderivative size = 87, 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} \[ \int \frac {(a+b x)^7}{x} \, dx=a^7 \log (x)+7 a^6 b x+\frac {21}{2} a^5 b^2 x^2+\frac {35}{3} a^4 b^3 x^3+\frac {35}{4} a^3 b^4 x^4+\frac {21}{5} a^2 b^5 x^5+\frac {7}{6} a b^6 x^6+\frac {b^7 x^7}{7} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (7 a^6 b+\frac {a^7}{x}+21 a^5 b^2 x+35 a^4 b^3 x^2+35 a^3 b^4 x^3+21 a^2 b^5 x^4+7 a b^6 x^5+b^7 x^6\right ) \, dx \\ & = 7 a^6 b x+\frac {21}{2} a^5 b^2 x^2+\frac {35}{3} a^4 b^3 x^3+\frac {35}{4} a^3 b^4 x^4+\frac {21}{5} a^2 b^5 x^5+\frac {7}{6} a b^6 x^6+\frac {b^7 x^7}{7}+a^7 \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^7}{x} \, dx=7 a^6 b x+\frac {21}{2} a^5 b^2 x^2+\frac {35}{3} a^4 b^3 x^3+\frac {35}{4} a^3 b^4 x^4+\frac {21}{5} a^2 b^5 x^5+\frac {7}{6} a b^6 x^6+\frac {b^7 x^7}{7}+a^7 \log (x) \]
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Time = 0.17 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87
method | result | size |
default | \(7 a^{6} b x +\frac {21 a^{5} b^{2} x^{2}}{2}+\frac {35 a^{4} b^{3} x^{3}}{3}+\frac {35 a^{3} b^{4} x^{4}}{4}+\frac {21 a^{2} b^{5} x^{5}}{5}+\frac {7 a \,b^{6} x^{6}}{6}+\frac {b^{7} x^{7}}{7}+a^{7} \ln \left (x \right )\) | \(76\) |
norman | \(7 a^{6} b x +\frac {21 a^{5} b^{2} x^{2}}{2}+\frac {35 a^{4} b^{3} x^{3}}{3}+\frac {35 a^{3} b^{4} x^{4}}{4}+\frac {21 a^{2} b^{5} x^{5}}{5}+\frac {7 a \,b^{6} x^{6}}{6}+\frac {b^{7} x^{7}}{7}+a^{7} \ln \left (x \right )\) | \(76\) |
risch | \(7 a^{6} b x +\frac {21 a^{5} b^{2} x^{2}}{2}+\frac {35 a^{4} b^{3} x^{3}}{3}+\frac {35 a^{3} b^{4} x^{4}}{4}+\frac {21 a^{2} b^{5} x^{5}}{5}+\frac {7 a \,b^{6} x^{6}}{6}+\frac {b^{7} x^{7}}{7}+a^{7} \ln \left (x \right )\) | \(76\) |
parallelrisch | \(7 a^{6} b x +\frac {21 a^{5} b^{2} x^{2}}{2}+\frac {35 a^{4} b^{3} x^{3}}{3}+\frac {35 a^{3} b^{4} x^{4}}{4}+\frac {21 a^{2} b^{5} x^{5}}{5}+\frac {7 a \,b^{6} x^{6}}{6}+\frac {b^{7} x^{7}}{7}+a^{7} \ln \left (x \right )\) | \(76\) |
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none
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^7}{x} \, dx=\frac {1}{7} \, b^{7} x^{7} + \frac {7}{6} \, a b^{6} x^{6} + \frac {21}{5} \, a^{2} b^{5} x^{5} + \frac {35}{4} \, a^{3} b^{4} x^{4} + \frac {35}{3} \, a^{4} b^{3} x^{3} + \frac {21}{2} \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x + a^{7} \log \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b x)^7}{x} \, dx=a^{7} \log {\left (x \right )} + 7 a^{6} b x + \frac {21 a^{5} b^{2} x^{2}}{2} + \frac {35 a^{4} b^{3} x^{3}}{3} + \frac {35 a^{3} b^{4} x^{4}}{4} + \frac {21 a^{2} b^{5} x^{5}}{5} + \frac {7 a b^{6} x^{6}}{6} + \frac {b^{7} x^{7}}{7} \]
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Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^7}{x} \, dx=\frac {1}{7} \, b^{7} x^{7} + \frac {7}{6} \, a b^{6} x^{6} + \frac {21}{5} \, a^{2} b^{5} x^{5} + \frac {35}{4} \, a^{3} b^{4} x^{4} + \frac {35}{3} \, a^{4} b^{3} x^{3} + \frac {21}{2} \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x + a^{7} \log \left (x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^7}{x} \, dx=\frac {1}{7} \, b^{7} x^{7} + \frac {7}{6} \, a b^{6} x^{6} + \frac {21}{5} \, a^{2} b^{5} x^{5} + \frac {35}{4} \, a^{3} b^{4} x^{4} + \frac {35}{3} \, a^{4} b^{3} x^{3} + \frac {21}{2} \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x + a^{7} \log \left ({\left | x \right |}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^7}{x} \, dx=a^7\,\ln \left (x\right )+\frac {b^7\,x^7}{7}+\frac {7\,a\,b^6\,x^6}{6}+\frac {21\,a^5\,b^2\,x^2}{2}+\frac {35\,a^4\,b^3\,x^3}{3}+\frac {35\,a^3\,b^4\,x^4}{4}+\frac {21\,a^2\,b^5\,x^5}{5}+7\,a^6\,b\,x \]
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