Integrand size = 24, antiderivative size = 38 \[ \int \frac {d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {b d-a e}{3 b^2 (a+b x)^3}-\frac {e}{2 b^2 (a+b x)^2} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 45} \[ \int \frac {d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {b d-a e}{3 b^2 (a+b x)^3}-\frac {e}{2 b^2 (a+b x)^2} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {d+e x}{(a+b x)^4} \, dx \\ & = \int \left (\frac {b d-a e}{b (a+b x)^4}+\frac {e}{b (a+b x)^3}\right ) \, dx \\ & = -\frac {b d-a e}{3 b^2 (a+b x)^3}-\frac {e}{2 b^2 (a+b x)^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.71 \[ \int \frac {d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {2 b d+a e+3 b e x}{6 b^2 (a+b x)^3} \]
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Time = 2.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89
method | result | size |
norman | \(\frac {-\frac {e x}{2 b}+\frac {-a e b -2 b^{2} d}{6 b^{3}}}{\left (b x +a \right )^{3}}\) | \(34\) |
default | \(-\frac {-a e +b d}{3 b^{2} \left (b x +a \right )^{3}}-\frac {e}{2 b^{2} \left (b x +a \right )^{2}}\) | \(35\) |
gosper | \(-\frac {3 b e x +a e +2 b d}{6 b^{2} \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )}\) | \(44\) |
risch | \(\frac {-\frac {e x}{2 b}-\frac {a e +2 b d}{6 b^{2}}}{\left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )}\) | \(48\) |
parallelrisch | \(\frac {-3 b^{2} e x -a e b -2 b^{2} d}{6 b^{3} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )}\) | \(50\) |
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Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32 \[ \int \frac {d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {3 \, b e x + 2 \, b d + a e}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]
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Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.39 \[ \int \frac {d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {- a e - 2 b d - 3 b e x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32 \[ \int \frac {d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {3 \, b e x + 2 \, b d + a e}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66 \[ \int \frac {d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {3 \, b e x + 2 \, b d + a e}{6 \, {\left (b x + a\right )}^{3} b^{2}} \]
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Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.37 \[ \int \frac {d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {\frac {a\,e+2\,b\,d}{6\,b^2}+\frac {e\,x}{2\,b}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]
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