Integrand size = 11, antiderivative size = 86 \[ \int \frac {(a+b x)^7}{x^4} \, dx=-\frac {a^7}{3 x^3}-\frac {7 a^6 b}{2 x^2}-\frac {21 a^5 b^2}{x}+35 a^3 b^4 x+\frac {21}{2} a^2 b^5 x^2+\frac {7}{3} a b^6 x^3+\frac {b^7 x^4}{4}+35 a^4 b^3 \log (x) \]
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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} \[ \int \frac {(a+b x)^7}{x^4} \, dx=-\frac {a^7}{3 x^3}-\frac {7 a^6 b}{2 x^2}-\frac {21 a^5 b^2}{x}+35 a^4 b^3 \log (x)+35 a^3 b^4 x+\frac {21}{2} a^2 b^5 x^2+\frac {7}{3} a b^6 x^3+\frac {b^7 x^4}{4} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (35 a^3 b^4+\frac {a^7}{x^4}+\frac {7 a^6 b}{x^3}+\frac {21 a^5 b^2}{x^2}+\frac {35 a^4 b^3}{x}+21 a^2 b^5 x+7 a b^6 x^2+b^7 x^3\right ) \, dx \\ & = -\frac {a^7}{3 x^3}-\frac {7 a^6 b}{2 x^2}-\frac {21 a^5 b^2}{x}+35 a^3 b^4 x+\frac {21}{2} a^2 b^5 x^2+\frac {7}{3} a b^6 x^3+\frac {b^7 x^4}{4}+35 a^4 b^3 \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^7}{x^4} \, dx=-\frac {a^7}{3 x^3}-\frac {7 a^6 b}{2 x^2}-\frac {21 a^5 b^2}{x}+35 a^3 b^4 x+\frac {21}{2} a^2 b^5 x^2+\frac {7}{3} a b^6 x^3+\frac {b^7 x^4}{4}+35 a^4 b^3 \log (x) \]
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Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {a^{7}}{3 x^{3}}-\frac {7 a^{6} b}{2 x^{2}}-\frac {21 a^{5} b^{2}}{x}+35 a^{3} b^{4} x +\frac {21 a^{2} b^{5} x^{2}}{2}+\frac {7 a \,b^{6} x^{3}}{3}+\frac {b^{7} x^{4}}{4}+35 a^{4} b^{3} \ln \left (x \right )\) | \(77\) |
risch | \(\frac {b^{7} x^{4}}{4}+\frac {7 a \,b^{6} x^{3}}{3}+\frac {21 a^{2} b^{5} x^{2}}{2}+35 a^{3} b^{4} x +\frac {-21 a^{5} b^{2} x^{2}-\frac {7}{2} a^{6} b x -\frac {1}{3} a^{7}}{x^{3}}+35 a^{4} b^{3} \ln \left (x \right )\) | \(77\) |
norman | \(\frac {-\frac {1}{3} a^{7}+\frac {1}{4} b^{7} x^{7}+\frac {7}{3} a \,b^{6} x^{6}+\frac {21}{2} a^{2} b^{5} x^{5}+35 a^{3} b^{4} x^{4}-21 a^{5} b^{2} x^{2}-\frac {7}{2} a^{6} b x}{x^{3}}+35 a^{4} b^{3} \ln \left (x \right )\) | \(79\) |
parallelrisch | \(\frac {3 b^{7} x^{7}+28 a \,b^{6} x^{6}+126 a^{2} b^{5} x^{5}+420 a^{4} b^{3} \ln \left (x \right ) x^{3}+420 a^{3} b^{4} x^{4}-252 a^{5} b^{2} x^{2}-42 a^{6} b x -4 a^{7}}{12 x^{3}}\) | \(82\) |
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Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^7}{x^4} \, dx=\frac {3 \, b^{7} x^{7} + 28 \, a b^{6} x^{6} + 126 \, a^{2} b^{5} x^{5} + 420 \, a^{3} b^{4} x^{4} + 420 \, a^{4} b^{3} x^{3} \log \left (x\right ) - 252 \, a^{5} b^{2} x^{2} - 42 \, a^{6} b x - 4 \, a^{7}}{12 \, x^{3}} \]
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Time = 0.12 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b x)^7}{x^4} \, dx=35 a^{4} b^{3} \log {\left (x \right )} + 35 a^{3} b^{4} x + \frac {21 a^{2} b^{5} x^{2}}{2} + \frac {7 a b^{6} x^{3}}{3} + \frac {b^{7} x^{4}}{4} + \frac {- 2 a^{7} - 21 a^{6} b x - 126 a^{5} b^{2} x^{2}}{6 x^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^7}{x^4} \, dx=\frac {1}{4} \, b^{7} x^{4} + \frac {7}{3} \, a b^{6} x^{3} + \frac {21}{2} \, a^{2} b^{5} x^{2} + 35 \, a^{3} b^{4} x + 35 \, a^{4} b^{3} \log \left (x\right ) - \frac {126 \, a^{5} b^{2} x^{2} + 21 \, a^{6} b x + 2 \, a^{7}}{6 \, x^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^7}{x^4} \, dx=\frac {1}{4} \, b^{7} x^{4} + \frac {7}{3} \, a b^{6} x^{3} + \frac {21}{2} \, a^{2} b^{5} x^{2} + 35 \, a^{3} b^{4} x + 35 \, a^{4} b^{3} \log \left ({\left | x \right |}\right ) - \frac {126 \, a^{5} b^{2} x^{2} + 21 \, a^{6} b x + 2 \, a^{7}}{6 \, x^{3}} \]
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Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^7}{x^4} \, dx=\frac {b^7\,x^4}{4}-\frac {\frac {a^7}{3}+\frac {7\,a^6\,b\,x}{2}+21\,a^5\,b^2\,x^2}{x^3}+35\,a^3\,b^4\,x+\frac {7\,a\,b^6\,x^3}{3}+\frac {21\,a^2\,b^5\,x^2}{2}+35\,a^4\,b^3\,\ln \left (x\right ) \]
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