Integrand size = 28, antiderivative size = 146 \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {\sqrt {d+e x}}{3 b (a+b x)^3}-\frac {e \sqrt {d+e x}}{12 b (b d-a e) (a+b x)^2}+\frac {e^2 \sqrt {d+e x}}{8 b (b d-a e)^2 (a+b x)}-\frac {e^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 44, 65, 214} \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {e^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{5/2}}+\frac {e^2 \sqrt {d+e x}}{8 b (a+b x) (b d-a e)^2}-\frac {e \sqrt {d+e x}}{12 b (a+b x)^2 (b d-a e)}-\frac {\sqrt {d+e x}}{3 b (a+b x)^3} \]
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Rule 27
Rule 43
Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {d+e x}}{(a+b x)^4} \, dx \\ & = -\frac {\sqrt {d+e x}}{3 b (a+b x)^3}+\frac {e \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{6 b} \\ & = -\frac {\sqrt {d+e x}}{3 b (a+b x)^3}-\frac {e \sqrt {d+e x}}{12 b (b d-a e) (a+b x)^2}-\frac {e^2 \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{8 b (b d-a e)} \\ & = -\frac {\sqrt {d+e x}}{3 b (a+b x)^3}-\frac {e \sqrt {d+e x}}{12 b (b d-a e) (a+b x)^2}+\frac {e^2 \sqrt {d+e x}}{8 b (b d-a e)^2 (a+b x)}+\frac {e^3 \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b (b d-a e)^2} \\ & = -\frac {\sqrt {d+e x}}{3 b (a+b x)^3}-\frac {e \sqrt {d+e x}}{12 b (b d-a e) (a+b x)^2}+\frac {e^2 \sqrt {d+e x}}{8 b (b d-a e)^2 (a+b x)}+\frac {e^2 \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 b (b d-a e)^2} \\ & = -\frac {\sqrt {d+e x}}{3 b (a+b x)^3}-\frac {e \sqrt {d+e x}}{12 b (b d-a e) (a+b x)^2}+\frac {e^2 \sqrt {d+e x}}{8 b (b d-a e)^2 (a+b x)}-\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{5/2}} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\sqrt {d+e x} \left (-3 a^2 e^2+2 a b e (7 d+4 e x)+b^2 \left (-8 d^2-2 d e x+3 e^2 x^2\right )\right )}{24 b (b d-a e)^2 (a+b x)^3}+\frac {e^3 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{8 b^{3/2} (-b d+a e)^{5/2}} \]
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Time = 2.60 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(\frac {e^{3} \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )-\left (\left (-3 e x -4 d \right ) b +a e \right ) \sqrt {e x +d}\, \left (\frac {\left (e x -2 d \right ) b}{3}+a e \right ) \sqrt {\left (a e -b d \right ) b}}{8 \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )^{3} \left (a e -b d \right )^{2} b}\) | \(118\) |
derivativedivides | \(2 e^{3} \left (\frac {\frac {b \left (e x +d \right )^{\frac {5}{2}}}{16 a^{2} e^{2}-32 a b d e +16 b^{2} d^{2}}+\frac {\left (e x +d \right )^{\frac {3}{2}}}{6 a e -6 b d}-\frac {\sqrt {e x +d}}{16 b}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 b \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (a e -b d \right ) b}}\right )\) | \(152\) |
default | \(2 e^{3} \left (\frac {\frac {b \left (e x +d \right )^{\frac {5}{2}}}{16 a^{2} e^{2}-32 a b d e +16 b^{2} d^{2}}+\frac {\left (e x +d \right )^{\frac {3}{2}}}{6 a e -6 b d}-\frac {\sqrt {e x +d}}{16 b}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 b \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (a e -b d \right ) b}}\right )\) | \(152\) |
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Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (122) = 244\).
Time = 0.43 (sec) , antiderivative size = 785, normalized size of antiderivative = 5.38 \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\left [\frac {3 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (8 \, b^{4} d^{3} - 22 \, a b^{3} d^{2} e + 17 \, a^{2} b^{2} d e^{2} - 3 \, a^{3} b e^{3} - 3 \, {\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \, {\left (b^{4} d^{2} e - 5 \, a b^{3} d e^{2} + 4 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{48 \, {\left (a^{3} b^{5} d^{3} - 3 \, a^{4} b^{4} d^{2} e + 3 \, a^{5} b^{3} d e^{2} - a^{6} b^{2} e^{3} + {\left (b^{8} d^{3} - 3 \, a b^{7} d^{2} e + 3 \, a^{2} b^{6} d e^{2} - a^{3} b^{5} e^{3}\right )} x^{3} + 3 \, {\left (a b^{7} d^{3} - 3 \, a^{2} b^{6} d^{2} e + 3 \, a^{3} b^{5} d e^{2} - a^{4} b^{4} e^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{6} d^{3} - 3 \, a^{3} b^{5} d^{2} e + 3 \, a^{4} b^{4} d e^{2} - a^{5} b^{3} e^{3}\right )} x\right )}}, \frac {3 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (8 \, b^{4} d^{3} - 22 \, a b^{3} d^{2} e + 17 \, a^{2} b^{2} d e^{2} - 3 \, a^{3} b e^{3} - 3 \, {\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \, {\left (b^{4} d^{2} e - 5 \, a b^{3} d e^{2} + 4 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (a^{3} b^{5} d^{3} - 3 \, a^{4} b^{4} d^{2} e + 3 \, a^{5} b^{3} d e^{2} - a^{6} b^{2} e^{3} + {\left (b^{8} d^{3} - 3 \, a b^{7} d^{2} e + 3 \, a^{2} b^{6} d e^{2} - a^{3} b^{5} e^{3}\right )} x^{3} + 3 \, {\left (a b^{7} d^{3} - 3 \, a^{2} b^{6} d^{2} e + 3 \, a^{3} b^{5} d e^{2} - a^{4} b^{4} e^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{6} d^{3} - 3 \, a^{3} b^{5} d^{2} e + 3 \, a^{4} b^{4} d e^{2} - a^{5} b^{3} e^{3}\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.28 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {e^{3} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} \sqrt {-b^{2} d + a b e}} + \frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{2} e^{3} - 8 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} d e^{3} - 3 \, \sqrt {e x + d} b^{2} d^{2} e^{3} + 8 \, {\left (e x + d\right )}^{\frac {3}{2}} a b e^{4} + 6 \, \sqrt {e x + d} a b d e^{4} - 3 \, \sqrt {e x + d} a^{2} e^{5}}{24 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{3}} \]
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Time = 9.56 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\frac {e^3\,{\left (d+e\,x\right )}^{3/2}}{3\,\left (a\,e-b\,d\right )}-\frac {e^3\,\sqrt {d+e\,x}}{8\,b}+\frac {b\,e^3\,{\left (d+e\,x\right )}^{5/2}}{8\,{\left (a\,e-b\,d\right )}^2}}{\left (d+e\,x\right )\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )+b^3\,{\left (d+e\,x\right )}^3-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^2+a^3\,e^3-b^3\,d^3+3\,a\,b^2\,d^2\,e-3\,a^2\,b\,d\,e^2}+\frac {e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{8\,b^{3/2}\,{\left (a\,e-b\,d\right )}^{5/2}} \]
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