Integrand size = 28, antiderivative size = 147 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac {5 e^2 \sqrt {d+e x}}{8 (b d-a e)^3 (a+b x)}+\frac {5 e^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {b} (b d-a e)^{7/2}} \]
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Time = 0.06 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 44, 65, 214} \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {5 e^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {b} (b d-a e)^{7/2}}-\frac {5 e^2 \sqrt {d+e x}}{8 (a+b x) (b d-a e)^3}+\frac {5 e \sqrt {d+e x}}{12 (a+b x)^2 (b d-a e)^2}-\frac {\sqrt {d+e x}}{3 (a+b x)^3 (b d-a e)} \]
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Rule 27
Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x)^4 \sqrt {d+e x}} \, dx \\ & = -\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}-\frac {(5 e) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{6 (b d-a e)} \\ & = -\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}+\frac {\left (5 e^2\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{8 (b d-a e)^2} \\ & = -\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac {5 e^2 \sqrt {d+e x}}{8 (b d-a e)^3 (a+b x)}-\frac {\left (5 e^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 (b d-a e)^3} \\ & = -\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac {5 e^2 \sqrt {d+e x}}{8 (b d-a e)^3 (a+b x)}-\frac {\left (5 e^2\right ) \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 d-a e)^3} \\ & = -\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac {5 e^2 \sqrt {d+e x}}{8 (b d-a e)^3 (a+b x)}+\frac {5 e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {b} (b d-a e)^{7/2}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\sqrt {d+e x} \left (33 a^2 e^2+2 a b e (-13 d+20 e x)+b^2 \left (8 d^2-10 d e x+15 e^2 x^2\right )\right )}{24 (-b d+a e)^3 (a+b x)^3}+\frac {5 e^3 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{8 \sqrt {b} (-b d+a e)^{7/2}} \]
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Time = 3.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {33 \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, \left (\left (\frac {5}{11} x^{2} e^{2}-\frac {10}{33} d e x +\frac {8}{33} d^{2}\right ) b^{2}-\frac {26 \left (-\frac {20 e x}{13}+d \right ) e a b}{33}+a^{2} e^{2}\right )+15 e^{3} \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{24 \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{3} \left (b x +a \right )^{3}}\) | \(130\) |
derivativedivides | \(2 e^{3} \left (\frac {\sqrt {e x +d}}{6 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {\frac {5 \sqrt {e x +d}}{24 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {e x +d}}{8 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}}\right )}{6 \left (a e -b d \right )}}{a e -b d}\right )\) | \(187\) |
default | \(2 e^{3} \left (\frac {\sqrt {e x +d}}{6 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {\frac {5 \sqrt {e x +d}}{24 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {e x +d}}{8 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}}\right )}{6 \left (a e -b d \right )}}{a e -b d}\right )\) | \(187\) |
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Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (123) = 246\).
Time = 0.41 (sec) , antiderivative size = 884, normalized size of antiderivative = 6.01 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\left [-\frac {15 \, {\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} - 34 \, a b^{3} d^{2} e + 59 \, a^{2} b^{2} d e^{2} - 33 \, a^{3} b e^{3} + 15 \, {\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} - 10 \, {\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^{4} - 4 \, a^{4} b^{4} d^{3} e + 6 \, a^{5} b^{3} d^{2} e^{2} - 4 \, a^{6} b^{2} d e^{3} + a^{7} b e^{4} + {\left (b^{8} d^{4} - 4 \, a b^{7} d^{3} e + 6 \, a^{2} b^{6} d^{2} e^{2} - 4 \, a^{3} b^{5} d e^{3} + a^{4} b^{4} e^{4}\right )} x^{3} + 3 \, {\left (a b^{7} d^{4} - 4 \, a^{2} b^{6} d^{3} e + 6 \, a^{3} b^{5} d^{2} e^{2} - 4 \, a^{4} b^{4} d e^{3} + a^{5} b^{3} e^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{6} d^{4} - 4 \, a^{3} b^{5} d^{3} e + 6 \, a^{4} b^{4} d^{2} e^{2} - 4 \, a^{5} b^{3} d e^{3} + a^{6} b^{2} e^{4}\right )} x\right )}}, -\frac {15 \, {\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} - 34 \, a b^{3} d^{2} e + 59 \, a^{2} b^{2} d e^{2} - 33 \, a^{3} b e^{3} + 15 \, {\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} - 10 \, {\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^{4} - 4 \, a^{4} b^{4} d^{3} e + 6 \, a^{5} b^{3} d^{2} e^{2} - 4 \, a^{6} b^{2} d e^{3} + a^{7} b e^{4} + {\left (b^{8} d^{4} - 4 \, a b^{7} d^{3} e + 6 \, a^{2} b^{6} d^{2} e^{2} - 4 \, a^{3} b^{5} d e^{3} + a^{4} b^{4} e^{4}\right )} x^{3} + 3 \, {\left (a b^{7} d^{4} - 4 \, a^{2} b^{6} d^{3} e + 6 \, a^{3} b^{5} d^{2} e^{2} - 4 \, a^{4} b^{4} d e^{3} + a^{5} b^{3} e^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{6} d^{4} - 4 \, a^{3} b^{5} d^{3} e + 6 \, a^{4} b^{4} d^{2} e^{2} - 4 \, a^{5} b^{3} d e^{3} + a^{6} b^{2} e^{4}\right )} x\right )}}\right ] \]
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\[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\int \frac {1}{\left (a + b x\right )^{4} \sqrt {d + e x}}\, dx \]
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Exception generated. \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.27 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {5 \, e^{3} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e}} - \frac {15 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{2} e^{3} - 40 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} d e^{3} + 33 \, \sqrt {e x + d} b^{2} d^{2} e^{3} + 40 \, {\left (e x + d\right )}^{\frac {3}{2}} a b e^{4} - 66 \, \sqrt {e x + d} a b d e^{4} + 33 \, \sqrt {e x + d} a^{2} e^{5}}{24 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{3}} \]
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Time = 9.64 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.48 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\frac {11\,e^3\,\sqrt {d+e\,x}}{8\,\left (a\,e-b\,d\right )}+\frac {5\,b^2\,e^3\,{\left (d+e\,x\right )}^{5/2}}{8\,{\left (a\,e-b\,d\right )}^3}+\frac {5\,b\,e^3\,{\left (d+e\,x\right )}^{3/2}}{3\,{\left (a\,e-b\,d\right )}^2}}{\left (d+e\,x\right )\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )+b^3\,{\left (d+e\,x\right )}^3-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^2+a^3\,e^3-b^3\,d^3+3\,a\,b^2\,d^2\,e-3\,a^2\,b\,d\,e^2}+\frac {5\,e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{8\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{7/2}} \]
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