Integrand size = 33, antiderivative size = 41 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (b d-a e) (d+e x)^3} \]
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Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {781} \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (d+e x)^3 (b d-a e)} \]
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Rule 781
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (b d-a e) (d+e x)^3} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.73 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=-\frac {\sqrt {(a+b x)^2} \left (a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{3 e^3 (a+b x) (d+e x)^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.56 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.61
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (3 b^{2} e^{2} x^{2}+3 a b \,e^{2} x +3 b^{2} d e x +e^{2} a^{2}+a b d e +b^{2} d^{2}\right )}{3 e^{3} \left (e x +d \right )^{3}}\) | \(66\) |
gosper | \(-\frac {\left (3 b^{2} e^{2} x^{2}+3 a b \,e^{2} x +3 b^{2} d e x +e^{2} a^{2}+a b d e +b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{3 \left (e x +d \right )^{3} e^{3} \left (b x +a \right )}\) | \(76\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{2} x^{2}}{e}-\frac {b \left (a e +b d \right ) x}{e^{2}}-\frac {e^{2} a^{2}+a b d e +b^{2} d^{2}}{3 e^{3}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{3}}\) | \(76\) |
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (37) = 74\).
Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.05 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=-\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
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Timed out. \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (37) = 74\).
Time = 0.26 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.98 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\frac {b^{3} \mathrm {sgn}\left (b x + a\right )}{3 \, {\left (b d e^{3} - a e^{4}\right )}} - \frac {3 \, b^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, b^{2} d e x \mathrm {sgn}\left (b x + a\right ) + 3 \, a b e^{2} x \mathrm {sgn}\left (b x + a\right ) + b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) + a b d e \mathrm {sgn}\left (b x + a\right ) + a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )}{3 \, {\left (e x + d\right )}^{3} e^{3}} \]
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Time = 10.91 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=-\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (a^2\,e^2+a\,b\,d\,e+3\,a\,b\,e^2\,x+b^2\,d^2+3\,b^2\,d\,e\,x+3\,b^2\,e^2\,x^2\right )}{3\,e^3\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3} \]
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