Integrand size = 24, antiderivative size = 28 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=\frac {(a+b x)^3}{3 (b d-a e) (d+e x)^3} \]
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Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 37} \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=\frac {(a+b x)^3}{3 (d+e x)^3 (b d-a e)} \]
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Rule 27
Rule 37
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^2}{(d+e x)^4} \, dx \\ & = \frac {(a+b x)^3}{3 (b d-a e) (d+e x)^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=-\frac {a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(26)=52\).
Time = 2.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14
method | result | size |
gosper | \(-\frac {3 x^{2} b^{2} e^{2}+3 x a b \,e^{2}+3 b^{2} d e x +a^{2} e^{2}+a b d e +b^{2} d^{2}}{3 e^{3} \left (e x +d \right )^{3}}\) | \(60\) |
risch | \(\frac {-\frac {b^{2} x^{2}}{e}-\frac {b \left (a e +b d \right ) x}{e^{2}}-\frac {a^{2} e^{2}+a b d e +b^{2} d^{2}}{3 e^{3}}}{\left (e x +d \right )^{3}}\) | \(60\) |
norman | \(\frac {-\frac {b^{2} x^{2}}{e}-\frac {\left (a e b +b^{2} d \right ) x}{e^{2}}-\frac {a^{2} e^{2}+a b d e +b^{2} d^{2}}{3 e^{3}}}{\left (e x +d \right )^{3}}\) | \(62\) |
parallelrisch | \(\frac {-3 x^{2} b^{2} e^{2}-3 x a b \,e^{2}-3 b^{2} d e x -a^{2} e^{2}-a b d e -b^{2} d^{2}}{3 e^{3} \left (e x +d \right )^{3}}\) | \(63\) |
default | \(-\frac {b^{2}}{e^{3} \left (e x +d \right )}-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{3 e^{3} \left (e x +d \right )^{3}}-\frac {b \left (a e -b d \right )}{e^{3} \left (e x +d \right )^{2}}\) | \(71\) |
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (26) = 52\).
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=-\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (20) = 40\).
Time = 0.32 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.14 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=\frac {- a^{2} e^{2} - a b d e - b^{2} d^{2} - 3 b^{2} e^{2} x^{2} + x \left (- 3 a b e^{2} - 3 b^{2} d e\right )}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (26) = 52\).
Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=-\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=-\frac {3 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d e x + 3 \, a b e^{2} x + b^{2} d^{2} + a b d e + a^{2} e^{2}}{3 \, {\left (e x + d\right )}^{3} e^{3}} \]
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Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=-\frac {\frac {a^2\,e^2+a\,b\,d\,e+b^2\,d^2}{3\,e^3}+\frac {b^2\,x^2}{e}+\frac {b\,x\,\left (a\,e+b\,d\right )}{e^2}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]
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