Integrand size = 16, antiderivative size = 85 \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=-\frac {a^5 B}{5 x^5}-\frac {5 a^4 b B}{4 x^4}-\frac {10 a^3 b^2 B}{3 x^3}-\frac {5 a^2 b^3 B}{x^2}-\frac {5 a b^4 B}{x}-\frac {A (a+b x)^6}{6 a x^6}+b^5 B \log (x) \]
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Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 45} \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=-\frac {a^5 B}{5 x^5}-\frac {5 a^4 b B}{4 x^4}-\frac {10 a^3 b^2 B}{3 x^3}-\frac {5 a^2 b^3 B}{x^2}-\frac {A (a+b x)^6}{6 a x^6}-\frac {5 a b^4 B}{x}+b^5 B \log (x) \]
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Rule 45
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {A (a+b x)^6}{6 a x^6}+B \int \frac {(a+b x)^5}{x^6} \, dx \\ & = -\frac {A (a+b x)^6}{6 a x^6}+B \int \left (\frac {a^5}{x^6}+\frac {5 a^4 b}{x^5}+\frac {10 a^3 b^2}{x^4}+\frac {10 a^2 b^3}{x^3}+\frac {5 a b^4}{x^2}+\frac {b^5}{x}\right ) \, dx \\ & = -\frac {a^5 B}{5 x^5}-\frac {5 a^4 b B}{4 x^4}-\frac {10 a^3 b^2 B}{3 x^3}-\frac {5 a^2 b^3 B}{x^2}-\frac {5 a b^4 B}{x}-\frac {A (a+b x)^6}{6 a x^6}+b^5 B \log (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.28 \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=-\frac {60 A b^5 x^5+150 a b^4 x^4 (A+2 B x)+100 a^2 b^3 x^3 (2 A+3 B x)+50 a^3 b^2 x^2 (3 A+4 B x)+15 a^4 b x (4 A+5 B x)+2 a^5 (5 A+6 B x)-60 b^5 B x^6 \log (x)}{60 x^6} \]
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Time = 0.40 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.20
method | result | size |
default | \(b^{5} B \ln \left (x \right )-\frac {a^{5} A}{6 x^{6}}-\frac {10 a^{2} b^{2} \left (A b +B a \right )}{3 x^{3}}-\frac {b^{4} \left (A b +5 B a \right )}{x}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{2 x^{2}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{4 x^{4}}-\frac {a^{4} \left (5 A b +B a \right )}{5 x^{5}}\) | \(102\) |
norman | \(\frac {\left (-\frac {5}{2} a \,b^{4} A -5 a^{2} b^{3} B \right ) x^{4}+\left (-\frac {10}{3} a^{2} b^{3} A -\frac {10}{3} a^{3} b^{2} B \right ) x^{3}+\left (-\frac {5}{2} a^{3} b^{2} A -\frac {5}{4} a^{4} b B \right ) x^{2}+\left (-a^{4} b A -\frac {1}{5} a^{5} B \right ) x +\left (-b^{5} A -5 a \,b^{4} B \right ) x^{5}-\frac {a^{5} A}{6}}{x^{6}}+b^{5} B \ln \left (x \right )\) | \(119\) |
risch | \(\frac {\left (-\frac {5}{2} a \,b^{4} A -5 a^{2} b^{3} B \right ) x^{4}+\left (-\frac {10}{3} a^{2} b^{3} A -\frac {10}{3} a^{3} b^{2} B \right ) x^{3}+\left (-\frac {5}{2} a^{3} b^{2} A -\frac {5}{4} a^{4} b B \right ) x^{2}+\left (-a^{4} b A -\frac {1}{5} a^{5} B \right ) x +\left (-b^{5} A -5 a \,b^{4} B \right ) x^{5}-\frac {a^{5} A}{6}}{x^{6}}+b^{5} B \ln \left (x \right )\) | \(119\) |
parallelrisch | \(-\frac {-60 b^{5} B \ln \left (x \right ) x^{6}+60 A \,b^{5} x^{5}+300 B a \,b^{4} x^{5}+150 a A \,b^{4} x^{4}+300 B \,a^{2} b^{3} x^{4}+200 a^{2} A \,b^{3} x^{3}+200 B \,a^{3} b^{2} x^{3}+150 a^{3} A \,b^{2} x^{2}+75 B \,a^{4} b \,x^{2}+60 a^{4} A b x +12 a^{5} B x +10 a^{5} A}{60 x^{6}}\) | \(126\) |
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Time = 0.22 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.42 \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=\frac {60 \, B b^{5} x^{6} \log \left (x\right ) - 10 \, A a^{5} - 60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} - 150 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} - 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \]
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Time = 1.54 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.54 \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=B b^{5} \log {\left (x \right )} + \frac {- 10 A a^{5} + x^{5} \left (- 60 A b^{5} - 300 B a b^{4}\right ) + x^{4} \left (- 150 A a b^{4} - 300 B a^{2} b^{3}\right ) + x^{3} \left (- 200 A a^{2} b^{3} - 200 B a^{3} b^{2}\right ) + x^{2} \left (- 150 A a^{3} b^{2} - 75 B a^{4} b\right ) + x \left (- 60 A a^{4} b - 12 B a^{5}\right )}{60 x^{6}} \]
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Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=B b^{5} \log \left (x\right ) - \frac {10 \, A a^{5} + 60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 150 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \]
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Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=B b^{5} \log \left ({\left | x \right |}\right ) - \frac {10 \, A a^{5} + 60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 150 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \]
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Time = 0.36 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.38 \[ \int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx=B\,b^5\,\ln \left (x\right )-\frac {x\,\left (\frac {B\,a^5}{5}+A\,b\,a^4\right )+\frac {A\,a^5}{6}+x^4\,\left (5\,B\,a^2\,b^3+\frac {5\,A\,a\,b^4}{2}\right )+x^2\,\left (\frac {5\,B\,a^4\,b}{4}+\frac {5\,A\,a^3\,b^2}{2}\right )+x^5\,\left (A\,b^5+5\,B\,a\,b^4\right )+x^3\,\left (\frac {10\,B\,a^3\,b^2}{3}+\frac {10\,A\,a^2\,b^3}{3}\right )}{x^6} \]
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