Integrand size = 22, antiderivative size = 166 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^5}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}{9 e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{11/2}}{11 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{13/2}}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \]
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Time = 0.06 (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} \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{11/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{11 e^5}-\frac {4 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{9 e^5}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2}{7 e^5}-\frac {4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \]
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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 (d+e x)^{5/2}}{e^4}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}{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)^{9/2}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{11/2}}{e^4}+\frac {c^2 (d+e x)^{13/2}}{e^4}\right ) \, dx \\ & = \frac {2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^5}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}{9 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)^{11/2}}{11 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{13/2}}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.04 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{7/2} \left (c^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+65 e^2 \left (99 a^2 e^2+22 a b e (-2 d+7 e x)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-10 c e \left (-13 a e \left (8 d^2-28 d e x+63 e^2 x^2\right )+3 b \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )\right )}{45045 e^5} \]
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Time = 0.25 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {4 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\) | \(136\) |
default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {4 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\) | \(136\) |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (\left (\frac {7 c^{2} x^{4}}{15}+\frac {14 \left (\frac {11 b x}{13}+a \right ) x^{2} c}{11}+\frac {7 b^{2} x^{2}}{11}+\frac {14 a b x}{9}+a^{2}\right ) e^{4}-\frac {4 \left (\frac {42 c^{2} x^{3}}{65}+\frac {14 x \left (\frac {27 b x}{26}+a \right ) c}{11}+b \left (\frac {7 b x}{11}+a \right )\right ) d \,e^{3}}{9}+\frac {16 \left (\frac {63 c^{2} x^{2}}{65}+\left (\frac {21 b x}{13}+a \right ) c +\frac {b^{2}}{2}\right ) d^{2} e^{2}}{99}-\frac {32 c \left (\frac {14 c x}{15}+b \right ) d^{3} e}{429}+\frac {128 c^{2} d^{4}}{6435}\right )}{7 e^{5}}\) | \(139\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 c^{2} x^{4} e^{4}+6930 b c \,e^{4} x^{3}-1848 c^{2} d \,e^{3} x^{3}+8190 a c \,e^{4} x^{2}+4095 b^{2} e^{4} x^{2}-3780 b c d \,e^{3} x^{2}+1008 c^{2} d^{2} e^{2} x^{2}+10010 a b \,e^{4} x -3640 a c d \,e^{3} x -1820 b^{2} d \,e^{3} x +1680 b c \,d^{2} e^{2} x -448 c^{2} d^{3} e x +6435 a^{2} e^{4}-2860 a b d \,e^{3}+1040 a c \,d^{2} e^{2}+520 b^{2} d^{2} e^{2}-480 b c \,d^{3} e +128 c^{2} d^{4}\right )}{45045 e^{5}}\) | \(194\) |
trager | \(\frac {2 \left (3003 c^{2} e^{7} x^{7}+6930 b c \,e^{7} x^{6}+7161 c^{2} d \,e^{6} x^{6}+8190 a c \,e^{7} x^{5}+4095 b^{2} e^{7} x^{5}+17010 b c d \,e^{6} x^{5}+4473 c^{2} d^{2} e^{5} x^{5}+10010 a b \,e^{7} x^{4}+20930 a c d \,e^{6} x^{4}+10465 b^{2} d \,e^{6} x^{4}+11130 b c \,d^{2} e^{5} x^{4}+35 c^{2} d^{3} e^{4} x^{4}+6435 a^{2} e^{7} x^{3}+27170 a b d \,e^{6} x^{3}+14690 a c \,d^{2} e^{5} x^{3}+7345 b^{2} d^{2} e^{5} x^{3}+150 b c \,d^{3} e^{4} x^{3}-40 c^{2} d^{4} e^{3} x^{3}+19305 a^{2} d \,e^{6} x^{2}+21450 a b \,d^{2} e^{5} x^{2}+390 a c \,d^{3} e^{4} x^{2}+195 b^{2} d^{3} e^{4} x^{2}-180 b c \,d^{4} e^{3} x^{2}+48 c^{2} d^{5} e^{2} x^{2}+19305 a^{2} d^{2} e^{5} x +1430 a b \,d^{3} e^{4} x -520 a c \,d^{4} e^{3} x -260 b^{2} d^{4} e^{3} x +240 b c \,d^{5} e^{2} x -64 c^{2} d^{6} e x +6435 a^{2} d^{3} e^{4}-2860 a b \,d^{4} e^{3}+1040 a c \,d^{5} e^{2}+520 b^{2} d^{5} e^{2}-480 b c \,d^{6} e +128 c^{2} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\) | \(433\) |
risch | \(\frac {2 \left (3003 c^{2} e^{7} x^{7}+6930 b c \,e^{7} x^{6}+7161 c^{2} d \,e^{6} x^{6}+8190 a c \,e^{7} x^{5}+4095 b^{2} e^{7} x^{5}+17010 b c d \,e^{6} x^{5}+4473 c^{2} d^{2} e^{5} x^{5}+10010 a b \,e^{7} x^{4}+20930 a c d \,e^{6} x^{4}+10465 b^{2} d \,e^{6} x^{4}+11130 b c \,d^{2} e^{5} x^{4}+35 c^{2} d^{3} e^{4} x^{4}+6435 a^{2} e^{7} x^{3}+27170 a b d \,e^{6} x^{3}+14690 a c \,d^{2} e^{5} x^{3}+7345 b^{2} d^{2} e^{5} x^{3}+150 b c \,d^{3} e^{4} x^{3}-40 c^{2} d^{4} e^{3} x^{3}+19305 a^{2} d \,e^{6} x^{2}+21450 a b \,d^{2} e^{5} x^{2}+390 a c \,d^{3} e^{4} x^{2}+195 b^{2} d^{3} e^{4} x^{2}-180 b c \,d^{4} e^{3} x^{2}+48 c^{2} d^{5} e^{2} x^{2}+19305 a^{2} d^{2} e^{5} x +1430 a b \,d^{3} e^{4} x -520 a c \,d^{4} e^{3} x -260 b^{2} d^{4} e^{3} x +240 b c \,d^{5} e^{2} x -64 c^{2} d^{6} e x +6435 a^{2} d^{3} e^{4}-2860 a b \,d^{4} e^{3}+1040 a c \,d^{5} e^{2}+520 b^{2} d^{5} e^{2}-480 b c \,d^{6} e +128 c^{2} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\) | \(433\) |
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Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (146) = 292\).
Time = 0.33 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.20 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (3003 \, c^{2} e^{7} x^{7} + 128 \, c^{2} d^{7} - 480 \, b c d^{6} e - 2860 \, a b d^{4} e^{3} + 6435 \, a^{2} d^{3} e^{4} + 520 \, {\left (b^{2} + 2 \, a c\right )} d^{5} e^{2} + 231 \, {\left (31 \, c^{2} d e^{6} + 30 \, b c e^{7}\right )} x^{6} + 63 \, {\left (71 \, c^{2} d^{2} e^{5} + 270 \, b c d e^{6} + 65 \, {\left (b^{2} + 2 \, a c\right )} e^{7}\right )} x^{5} + 35 \, {\left (c^{2} d^{3} e^{4} + 318 \, b c d^{2} e^{5} + 286 \, a b e^{7} + 299 \, {\left (b^{2} + 2 \, a c\right )} d e^{6}\right )} x^{4} - 5 \, {\left (8 \, c^{2} d^{4} e^{3} - 30 \, b c d^{3} e^{4} - 5434 \, a b d e^{6} - 1287 \, a^{2} e^{7} - 1469 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{5}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{5} e^{2} - 60 \, b c d^{4} e^{3} + 7150 \, a b d^{2} e^{5} + 6435 \, a^{2} d e^{6} + 65 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{4}\right )} x^{2} - {\left (64 \, c^{2} d^{6} e - 240 \, b c d^{5} e^{2} - 1430 \, a b d^{3} e^{4} - 19305 \, a^{2} d^{2} e^{5} + 260 \, {\left (b^{2} + 2 \, a c\right )} d^{4} e^{3}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \]
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Time = 1.08 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.67 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{4}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (2 b c e - 4 c^{2} d\right )}{13 e^{4}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (2 a b e^{3} - 4 a c d e^{2} - 2 b^{2} d e^{2} + 6 b c d^{2} e - 4 c^{2} d^{3}\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{7 e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a^{2} x + a b x^{2} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5} + \frac {x^{3} \cdot \left (2 a c + b^{2}\right )}{3}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.06 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{2} - 6930 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 4095 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 10010 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{45045 \, e^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1411 vs. \(2 (146) = 292\).
Time = 0.29 (sec) , antiderivative size = 1411, normalized size of antiderivative = 8.50 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
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Time = 9.78 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2\,c^2\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{11\,e^5}+\frac {2\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{7\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}+\frac {4\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{9\,e^5} \]
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