Integrand size = 37, antiderivative size = 171 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^4}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 \left (c d^2-a e^2\right )^3 (d+e x)^3} \]
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Time = 0.05 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 664} \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^3 \left (c d^2-a e^2\right )^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^5 \left (c d^2-a e^2\right )} \]
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Rule 664
Rule 672
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {(4 c d) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx}{7 \left (c d^2-a e^2\right )} \\ & = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^4}+\frac {\left (8 c^2 d^2\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^3} \, dx}{35 \left (c d^2-a e^2\right )^2} \\ & = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^4}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 \left (c d^2-a e^2\right )^3 (d+e x)^3} \\ \end{align*}
Time = 1.38 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (15 a^3 e^5+3 a^2 c d e^3 (-14 d+e x)+a c^2 d^2 e \left (35 d^2-14 d e x-4 e^2 x^2\right )+c^3 d^3 x \left (35 d^2+28 d e x+8 e^2 x^2\right )\right )}{105 \left (c d^2-a e^2\right )^3 (d+e x)^4} \]
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Time = 3.24 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.85
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (8 x^{2} c^{2} d^{2} e^{2}-12 x a c d \,e^{3}+28 x \,c^{2} d^{3} e +15 a^{2} e^{4}-42 a c \,d^{2} e^{2}+35 c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{105 \left (e x +d \right )^{4} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}\) | \(146\) |
trager | \(-\frac {2 \left (8 e^{2} c^{3} d^{3} x^{3}-4 e^{3} a \,c^{2} d^{2} x^{2}+28 d^{4} e \,c^{3} x^{2}+3 d \,e^{4} a^{2} c x -14 d^{3} e^{2} c^{2} a x +35 d^{5} c^{3} x +15 a^{3} e^{5}-42 d^{2} e^{3} a^{2} c +35 d^{4} a \,c^{2} e \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{105 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right ) \left (e x +d \right )^{4}}\) | \(183\) |
default | \(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{5}}-\frac {4 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{4}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{7 \left (e^{2} a -c \,d^{2}\right )}}{e^{5}}\) | \(212\) |
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Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (159) = 318\).
Time = 2.51 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.18 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=\frac {2 \, {\left (8 \, c^{3} d^{3} e^{2} x^{3} + 35 \, a c^{2} d^{4} e - 42 \, a^{2} c d^{2} e^{3} + 15 \, a^{3} e^{5} + 4 \, {\left (7 \, c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} + {\left (35 \, c^{3} d^{5} - 14 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{105 \, {\left (c^{3} d^{10} - 3 \, a c^{2} d^{8} e^{2} + 3 \, a^{2} c d^{6} e^{4} - a^{3} d^{4} e^{6} + {\left (c^{3} d^{6} e^{4} - 3 \, a c^{2} d^{4} e^{6} + 3 \, a^{2} c d^{2} e^{8} - a^{3} e^{10}\right )} x^{4} + 4 \, {\left (c^{3} d^{7} e^{3} - 3 \, a c^{2} d^{5} e^{5} + 3 \, a^{2} c d^{3} e^{7} - a^{3} d e^{9}\right )} x^{3} + 6 \, {\left (c^{3} d^{8} e^{2} - 3 \, a c^{2} d^{6} e^{4} + 3 \, a^{2} c d^{4} e^{6} - a^{3} d^{2} e^{8}\right )} x^{2} + 4 \, {\left (c^{3} d^{9} e - 3 \, a c^{2} d^{7} e^{3} + 3 \, a^{2} c d^{5} e^{5} - a^{3} d^{3} e^{7}\right )} x\right )}} \]
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Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 711 vs. \(2 (159) = 318\).
Time = 0.33 (sec) , antiderivative size = 711, normalized size of antiderivative = 4.16 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=-\frac {2}{105} \, {\left (\frac {8 \, \sqrt {c d e} c^{3} d^{3} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c^{3} d^{6} e^{2} {\left | e \right |} - 3 \, a c^{2} d^{4} e^{4} {\left | e \right |} + 3 \, a^{2} c d^{2} e^{6} {\left | e \right |} - a^{3} e^{8} {\left | e \right |}} + \frac {\frac {3 \, {\left (35 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c^{3} d^{3} e^{3} - 35 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} c^{2} d^{2} e^{2} + 21 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {5}{2}} c d e - 5 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {7}{2}}\right )} c d^{2} e \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c^{3} d^{6} e^{6} - 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} - a^{3} e^{12}} - \frac {3 \, {\left (35 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c^{3} d^{3} e^{3} - 35 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} c^{2} d^{2} e^{2} + 21 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {5}{2}} c d e - 5 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {7}{2}}\right )} a e^{3} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c^{3} d^{6} e^{6} - 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} - a^{3} e^{12}} - \frac {7 \, {\left (15 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c^{2} d^{2} e^{2} - 10 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} c d e + 3 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {5}{2}}\right )} c d \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8}}}{c d^{2} {\left | e \right |} - a e^{2} {\left | e \right |}}\right )} {\left | e \right |} \]
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Time = 11.31 (sec) , antiderivative size = 877, normalized size of antiderivative = 5.13 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=\frac {\left (\frac {4\,c^2\,d^3}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}-\frac {4\,a\,c\,d\,e^2}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {2\,a\,e^2}{7\,a\,e^3-7\,c\,d^2\,e}-\frac {2\,c\,d^2}{7\,a\,e^3-7\,c\,d^2\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^4}+\frac {\left (\frac {4\,c^3\,d^4+4\,a\,c^2\,d^2\,e^2}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {8\,c^3\,d^4}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {8\,c^4\,d^5+8\,a\,c^3\,d^3\,e^2}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}-\frac {16\,c^4\,d^5}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}+\frac {\left (\frac {2\,c^2\,d^3+2\,a\,c\,d\,e^2}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}-\frac {4\,c^2\,d^3}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}+\frac {\left (\frac {8\,c^3\,d^4}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {8\,a\,c^2\,d^2\,e^2}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {16\,c^4\,d^5}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}-\frac {16\,a\,c^3\,d^3\,e}{105\,{\left (a\,e^2-c\,d^2\right )}^4}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}+\frac {12\,c^2\,d^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{35\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )\,{\left (d+e\,x\right )}^2}-\frac {8\,c^3\,d^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^3\,\left (d+e\,x\right )} \]
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