Integrand size = 26, antiderivative size = 28 \[ \int \frac {(d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {(d+e x)^3}{3 (b d-a e) (a+b x)^3} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 37} \[ \int \frac {(d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {(d+e x)^3}{3 (a+b x)^3 (b d-a e)} \]
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Rule 27
Rule 37
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^2}{(a+b x)^4} \, dx \\ & = -\frac {(d+e x)^3}{3 (b d-a e) (a+b x)^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {(d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 b^3 (a+b x)^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(26)=52\).
Time = 2.30 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36
method | result | size |
norman | \(\frac {-\frac {e^{2} x^{2}}{b}+\frac {\left (-e^{2} a -b d e \right ) x}{b^{2}}+\frac {-a^{2} e^{2}-a b d e -b^{2} d^{2}}{3 b^{3}}}{\left (b x +a \right )^{3}}\) | \(66\) |
default | \(-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{3 b^{3} \left (b x +a \right )^{3}}+\frac {e \left (a e -b d \right )}{b^{3} \left (b x +a \right )^{2}}-\frac {e^{2}}{b^{3} \left (b x +a \right )}\) | \(70\) |
gosper | \(-\frac {3 x^{2} b^{2} e^{2}+3 x a b \,e^{2}+3 b^{2} d e x +a^{2} e^{2}+a b d e +b^{2} d^{2}}{3 \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right ) b^{3}}\) | \(78\) |
risch | \(\frac {-\frac {e^{2} x^{2}}{b}-\frac {e \left (a e +b d \right ) x}{b^{2}}-\frac {a^{2} e^{2}+a b d e +b^{2} d^{2}}{3 b^{3}}}{\left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )}\) | \(78\) |
parallelrisch | \(\frac {-3 x^{2} b^{2} e^{2}-3 x a b \,e^{2}-3 b^{2} d e x -a^{2} e^{2}-a b d e -b^{2} d^{2}}{3 b^{3} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )}\) | \(81\) |
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (26) = 52\).
Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00 \[ \int \frac {(d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (22) = 44\).
Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.14 \[ \int \frac {(d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {- a^{2} e^{2} - a b d e - b^{2} d^{2} - 3 b^{2} e^{2} x^{2} + x \left (- 3 a b e^{2} - 3 b^{2} d e\right )}{3 a^{3} b^{3} + 9 a^{2} b^{4} x + 9 a b^{5} x^{2} + 3 b^{6} x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00 \[ \int \frac {(d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {(d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {3 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d e x + 3 \, a b e^{2} x + b^{2} d^{2} + a b d e + a^{2} e^{2}}{3 \, {\left (b x + a\right )}^{3} b^{3}} \]
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Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86 \[ \int \frac {(d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {\frac {a^2\,e^2+a\,b\,d\,e+b^2\,d^2}{3\,b^3}+\frac {e^2\,x^2}{b}+\frac {e\,x\,\left (a\,e+b\,d\right )}{b^2}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]
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