Integrand size = 16, antiderivative size = 55 \[ \int \frac {(a+b x)^2 (A+B x)}{x^6} \, dx=-\frac {a^2 A}{5 x^5}-\frac {a (2 A b+a B)}{4 x^4}-\frac {b (A b+2 a B)}{3 x^3}-\frac {b^2 B}{2 x^2} \]
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Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \int \frac {(a+b x)^2 (A+B x)}{x^6} \, dx=-\frac {a^2 A}{5 x^5}-\frac {a (a B+2 A b)}{4 x^4}-\frac {b (2 a B+A b)}{3 x^3}-\frac {b^2 B}{2 x^2} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 A}{x^6}+\frac {a (2 A b+a B)}{x^5}+\frac {b (A b+2 a B)}{x^4}+\frac {b^2 B}{x^3}\right ) \, dx \\ & = -\frac {a^2 A}{5 x^5}-\frac {a (2 A b+a B)}{4 x^4}-\frac {b (A b+2 a B)}{3 x^3}-\frac {b^2 B}{2 x^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^2 (A+B x)}{x^6} \, dx=-\frac {10 b^2 x^2 (2 A+3 B x)+10 a b x (3 A+4 B x)+3 a^2 (4 A+5 B x)}{60 x^5} \]
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Time = 0.38 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87
method | result | size |
default | \(-\frac {a^{2} A}{5 x^{5}}-\frac {a \left (2 A b +B a \right )}{4 x^{4}}-\frac {b \left (A b +2 B a \right )}{3 x^{3}}-\frac {b^{2} B}{2 x^{2}}\) | \(48\) |
norman | \(\frac {-\frac {b^{2} B \,x^{3}}{2}+\left (-\frac {1}{3} b^{2} A -\frac {2}{3} a b B \right ) x^{2}+\left (-\frac {1}{2} a b A -\frac {1}{4} a^{2} B \right ) x -\frac {a^{2} A}{5}}{x^{5}}\) | \(51\) |
risch | \(\frac {-\frac {b^{2} B \,x^{3}}{2}+\left (-\frac {1}{3} b^{2} A -\frac {2}{3} a b B \right ) x^{2}+\left (-\frac {1}{2} a b A -\frac {1}{4} a^{2} B \right ) x -\frac {a^{2} A}{5}}{x^{5}}\) | \(51\) |
gosper | \(-\frac {30 b^{2} B \,x^{3}+20 A \,b^{2} x^{2}+40 B a b \,x^{2}+30 a A b x +15 a^{2} B x +12 a^{2} A}{60 x^{5}}\) | \(52\) |
parallelrisch | \(-\frac {30 b^{2} B \,x^{3}+20 A \,b^{2} x^{2}+40 B a b \,x^{2}+30 a A b x +15 a^{2} B x +12 a^{2} A}{60 x^{5}}\) | \(52\) |
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Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^2 (A+B x)}{x^6} \, dx=-\frac {30 \, B b^{2} x^{3} + 12 \, A a^{2} + 20 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{5}} \]
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Time = 0.45 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^2 (A+B x)}{x^6} \, dx=\frac {- 12 A a^{2} - 30 B b^{2} x^{3} + x^{2} \left (- 20 A b^{2} - 40 B a b\right ) + x \left (- 30 A a b - 15 B a^{2}\right )}{60 x^{5}} \]
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Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^2 (A+B x)}{x^6} \, dx=-\frac {30 \, B b^{2} x^{3} + 12 \, A a^{2} + 20 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^2 (A+B x)}{x^6} \, dx=-\frac {30 \, B b^{2} x^{3} + 40 \, B a b x^{2} + 20 \, A b^{2} x^{2} + 15 \, B a^{2} x + 30 \, A a b x + 12 \, A a^{2}}{60 \, x^{5}} \]
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Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^2 (A+B x)}{x^6} \, dx=-\frac {x^2\,\left (\frac {A\,b^2}{3}+\frac {2\,B\,a\,b}{3}\right )+\frac {A\,a^2}{5}+x\,\left (\frac {B\,a^2}{4}+\frac {A\,b\,a}{2}\right )+\frac {B\,b^2\,x^3}{2}}{x^5} \]
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