Integrand size = 22, antiderivative size = 166 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{3 e^5}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}{5 e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{7/2}}{7 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{9/2}}{9 e^5}+\frac {2 c^2 (d+e x)^{11/2}}{11 e^5} \]
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Time = 0.05 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {712} \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{7/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{7 e^5}-\frac {4 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^5}+\frac {2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2}{3 e^5}-\frac {4 c (d+e x)^{9/2} (2 c d-b e)}{9 e^5}+\frac {2 c^2 (d+e x)^{11/2}}{11 e^5} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}{e^4}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}{e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{5/2}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{7/2}}{e^4}+\frac {c^2 (d+e x)^{9/2}}{e^4}\right ) \, dx \\ & = \frac {2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{3 e^5}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}{5 e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{7/2}}{7 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{9/2}}{9 e^5}+\frac {2 c^2 (d+e x)^{11/2}}{11 e^5} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.04 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{3/2} \left (c^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+33 e^2 \left (35 a^2 e^2+14 a b e (-2 d+3 e x)+b^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-22 c e \left (-3 a e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )\right )\right )}{3465 e^5} \]
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Time = 0.21 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{5}}\) | \(136\) |
default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{5}}\) | \(136\) |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (\left (\frac {3 c^{2} x^{4}}{11}+\frac {6 \left (\frac {7 b x}{9}+a \right ) x^{2} c}{7}+\frac {3 b^{2} x^{2}}{7}+\frac {6 a b x}{5}+a^{2}\right ) e^{4}-\frac {4 \left (\frac {10 c^{2} x^{3}}{33}+\frac {6 \left (\frac {5 b x}{6}+a \right ) x c}{7}+b \left (\frac {3 b x}{7}+a \right )\right ) d \,e^{3}}{5}+\frac {16 \left (\frac {5 c^{2} x^{2}}{11}+\left (b x +a \right ) c +\frac {b^{2}}{2}\right ) d^{2} e^{2}}{35}-\frac {32 \left (\frac {6 c x}{11}+b \right ) c \,d^{3} e}{105}+\frac {128 c^{2} d^{4}}{1155}\right )}{3 e^{5}}\) | \(138\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 c^{2} x^{4} e^{4}+770 b c \,e^{4} x^{3}-280 c^{2} d \,e^{3} x^{3}+990 a c \,e^{4} x^{2}+495 b^{2} e^{4} x^{2}-660 b c d \,e^{3} x^{2}+240 c^{2} d^{2} e^{2} x^{2}+1386 a b \,e^{4} x -792 a c d \,e^{3} x -396 b^{2} d \,e^{3} x +528 b c \,d^{2} e^{2} x -192 c^{2} d^{3} e x +1155 a^{2} e^{4}-924 a b d \,e^{3}+528 a c \,d^{2} e^{2}+264 b^{2} d^{2} e^{2}-352 b c \,d^{3} e +128 c^{2} d^{4}\right )}{3465 e^{5}}\) | \(194\) |
trager | \(\frac {2 \left (315 c^{2} e^{5} x^{5}+770 b c \,e^{5} x^{4}+35 c^{2} d \,e^{4} x^{4}+990 a c \,e^{5} x^{3}+495 b^{2} e^{5} x^{3}+110 b c d \,e^{4} x^{3}-40 c^{2} d^{2} e^{3} x^{3}+1386 a b \,e^{5} x^{2}+198 a c d \,e^{4} x^{2}+99 b^{2} d \,e^{4} x^{2}-132 b c \,d^{2} e^{3} x^{2}+48 c^{2} d^{3} e^{2} x^{2}+1155 a^{2} e^{5} x +462 a b d \,e^{4} x -264 a c \,d^{2} e^{3} x -132 b^{2} d^{2} e^{3} x +176 b c \,d^{3} e^{2} x -64 c^{2} d^{4} e x +1155 a^{2} d \,e^{4}-924 a b \,d^{2} e^{3}+528 a c \,d^{3} e^{2}+264 b^{2} d^{3} e^{2}-352 b c \,d^{4} e +128 c^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{5}}\) | \(271\) |
risch | \(\frac {2 \left (315 c^{2} e^{5} x^{5}+770 b c \,e^{5} x^{4}+35 c^{2} d \,e^{4} x^{4}+990 a c \,e^{5} x^{3}+495 b^{2} e^{5} x^{3}+110 b c d \,e^{4} x^{3}-40 c^{2} d^{2} e^{3} x^{3}+1386 a b \,e^{5} x^{2}+198 a c d \,e^{4} x^{2}+99 b^{2} d \,e^{4} x^{2}-132 b c \,d^{2} e^{3} x^{2}+48 c^{2} d^{3} e^{2} x^{2}+1155 a^{2} e^{5} x +462 a b d \,e^{4} x -264 a c \,d^{2} e^{3} x -132 b^{2} d^{2} e^{3} x +176 b c \,d^{3} e^{2} x -64 c^{2} d^{4} e x +1155 a^{2} d \,e^{4}-924 a b \,d^{2} e^{3}+528 a c \,d^{3} e^{2}+264 b^{2} d^{3} e^{2}-352 b c \,d^{4} e +128 c^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{5}}\) | \(271\) |
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Time = 0.27 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.43 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (315 \, c^{2} e^{5} x^{5} + 128 \, c^{2} d^{5} - 352 \, b c d^{4} e - 924 \, a b d^{2} e^{3} + 1155 \, a^{2} d e^{4} + 264 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} + 35 \, {\left (c^{2} d e^{4} + 22 \, b c e^{5}\right )} x^{4} - 5 \, {\left (8 \, c^{2} d^{2} e^{3} - 22 \, b c d e^{4} - 99 \, {\left (b^{2} + 2 \, a c\right )} e^{5}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{3} e^{2} - 44 \, b c d^{2} e^{3} + 462 \, a b e^{5} + 33 \, {\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} x^{2} - {\left (64 \, c^{2} d^{4} e - 176 \, b c d^{3} e^{2} - 462 \, a b d e^{4} - 1155 \, a^{2} e^{5} + 132 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{5}} \]
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Time = 0.90 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.67 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (2 b c e - 4 c^{2} d\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 a b e^{3} - 4 a c d e^{2} - 2 b^{2} d e^{2} + 6 b c d^{2} e - 4 c^{2} d^{3}\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{3 e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (a^{2} x + a b x^{2} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5} + \frac {x^{3} \cdot \left (2 a c + b^{2}\right )}{3}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.06 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{2} - 770 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 1386 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{3465 \, e^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (146) = 292\).
Time = 0.28 (sec) , antiderivative size = 547, normalized size of antiderivative = 3.30 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (3465 \, \sqrt {e x + d} a^{2} d + 1155 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{2} + \frac {2310 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a b d}{e} + \frac {231 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b^{2} d}{e^{2}} + \frac {462 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a c d}{e^{2}} + \frac {462 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a b}{e} + \frac {198 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b c d}{e^{3}} + \frac {99 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b^{2}}{e^{2}} + \frac {198 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a c}{e^{2}} + \frac {11 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} c^{2} d}{e^{4}} + \frac {22 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b c}{e^{3}} + \frac {5 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} c^{2}}{e^{4}}\right )}}{3465 \, e} \]
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Time = 0.05 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2\,c^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{7\,e^5}+\frac {2\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{3\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {4\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{5\,e^5} \]
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