Integrand size = 16, antiderivative size = 44 \[ \int \frac {(a+b x)^2 (A+B x)}{x^5} \, dx=-\frac {A (a+b x)^3}{4 a x^4}+\frac {(A b-4 a B) (a+b x)^3}{12 a^2 x^3} \]
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Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 37} \[ \int \frac {(a+b x)^2 (A+B x)}{x^5} \, dx=\frac {(a+b x)^3 (A b-4 a B)}{12 a^2 x^3}-\frac {A (a+b x)^3}{4 a x^4} \]
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Rule 37
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {A (a+b x)^3}{4 a x^4}+\frac {(-A b+4 a B) \int \frac {(a+b x)^2}{x^4} \, dx}{4 a} \\ & = -\frac {A (a+b x)^3}{4 a x^4}+\frac {(A b-4 a B) (a+b x)^3}{12 a^2 x^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b x)^2 (A+B x)}{x^5} \, dx=-\frac {6 b^2 x^2 (A+2 B x)+4 a b x (2 A+3 B x)+a^2 (3 A+4 B x)}{12 x^4} \]
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Time = 0.38 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.09
method | result | size |
default | \(-\frac {a \left (2 A b +B a \right )}{3 x^{3}}-\frac {b^{2} B}{x}-\frac {b \left (A b +2 B a \right )}{2 x^{2}}-\frac {a^{2} A}{4 x^{4}}\) | \(48\) |
norman | \(\frac {-b^{2} B \,x^{3}+\left (-\frac {1}{2} b^{2} A -a b B \right ) x^{2}+\left (-\frac {2}{3} a b A -\frac {1}{3} a^{2} B \right ) x -\frac {a^{2} A}{4}}{x^{4}}\) | \(51\) |
risch | \(\frac {-b^{2} B \,x^{3}+\left (-\frac {1}{2} b^{2} A -a b B \right ) x^{2}+\left (-\frac {2}{3} a b A -\frac {1}{3} a^{2} B \right ) x -\frac {a^{2} A}{4}}{x^{4}}\) | \(51\) |
gosper | \(-\frac {12 b^{2} B \,x^{3}+6 A \,b^{2} x^{2}+12 B a b \,x^{2}+8 a A b x +4 a^{2} B x +3 a^{2} A}{12 x^{4}}\) | \(52\) |
parallelrisch | \(-\frac {12 b^{2} B \,x^{3}+6 A \,b^{2} x^{2}+12 B a b \,x^{2}+8 a A b x +4 a^{2} B x +3 a^{2} A}{12 x^{4}}\) | \(52\) |
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Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x)^2 (A+B x)}{x^5} \, dx=-\frac {12 \, B b^{2} x^{3} + 3 \, A a^{2} + 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \]
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Time = 0.36 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b x)^2 (A+B x)}{x^5} \, dx=\frac {- 3 A a^{2} - 12 B b^{2} x^{3} + x^{2} \left (- 6 A b^{2} - 12 B a b\right ) + x \left (- 8 A a b - 4 B a^{2}\right )}{12 x^{4}} \]
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Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x)^2 (A+B x)}{x^5} \, dx=-\frac {12 \, B b^{2} x^{3} + 3 \, A a^{2} + 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x)^2 (A+B x)}{x^5} \, dx=-\frac {12 \, B b^{2} x^{3} + 12 \, B a b x^{2} + 6 \, A b^{2} x^{2} + 4 \, B a^{2} x + 8 \, A a b x + 3 \, A a^{2}}{12 \, x^{4}} \]
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Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b x)^2 (A+B x)}{x^5} \, dx=-\frac {x^2\,\left (\frac {A\,b^2}{2}+B\,a\,b\right )+\frac {A\,a^2}{4}+x\,\left (\frac {B\,a^2}{3}+\frac {2\,A\,b\,a}{3}\right )+B\,b^2\,x^3}{x^4} \]
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