Integrand size = 28, antiderivative size = 252 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^6}+\frac {4 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{9 e^6}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^6}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{13/2}}{13 e^6}-\frac {2 c^2 (2 c d-b e) (d+e x)^{15/2}}{3 e^6}+\frac {4 c^3 (d+e x)^{17/2}}{17 e^6} \]
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Time = 0.11 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {785} \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {8 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^6}-\frac {2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^6}+\frac {4 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{9 e^6}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^6}-\frac {2 c^2 (d+e x)^{15/2} (2 c d-b e)}{3 e^6}+\frac {4 c^3 (d+e x)^{17/2}}{17 e^6} \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{e^5}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^{7/2}}{e^5}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^{9/2}}{e^5}+\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{11/2}}{e^5}-\frac {5 c^2 (2 c d-b e) (d+e x)^{13/2}}{e^5}+\frac {2 c^3 (d+e x)^{15/2}}{e^5}\right ) \, dx \\ & = -\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^6}+\frac {4 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{9 e^6}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^6}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{13/2}}{13 e^6}-\frac {2 c^2 (2 c d-b e) (d+e x)^{15/2}}{3 e^6}+\frac {4 c^3 (d+e x)^{17/2}}{17 e^6} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.15 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{7/2} \left (-2 c^3 \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )+221 b e^3 \left (99 a^2 e^2+22 a b e (-2 d+7 e x)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-34 c e^2 \left (143 a^2 e^2 (2 d-7 e x)-39 a b e \left (8 d^2-28 d e x+63 e^2 x^2\right )+6 b^2 \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )+17 c^2 e \left (12 a e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )\right )}{153153 e^6} \]
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Time = 0.80 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (\left (\frac {14 c^{3} x^{5}}{17}+\left (\frac {7}{3} b \,x^{4}+\frac {28}{13} a \,x^{3}\right ) c^{2}+\left (\frac {42}{11} a b \,x^{2}+\frac {28}{13} b^{2} x^{3}+\frac {14}{9} a^{2} x \right ) c +b \left (\frac {14}{9} a b x +\frac {7}{11} b^{2} x^{2}+a^{2}\right )\right ) e^{5}-\frac {4 d \left (\frac {21 c^{3} x^{4}}{17}+\frac {378 \left (\frac {11 b x}{9}+a \right ) x^{2} c^{2}}{143}+\left (\frac {378}{143} b^{2} x^{2}+\frac {42}{11} a b x +a^{2}\right ) c +b^{2} \left (\frac {7 b x}{11}+a \right )\right ) e^{4}}{9}+\frac {16 d^{2} \left (\frac {154 c^{3} x^{3}}{221}+\frac {14 \left (\frac {3 b x}{2}+a \right ) x \,c^{2}}{13}+b \left (\frac {14 b x}{13}+a \right ) c +\frac {b^{3}}{6}\right ) e^{3}}{33}-\frac {64 d^{3} \left (\frac {21 c^{2} x^{2}}{17}+\left (\frac {7 b x}{3}+a \right ) c +b^{2}\right ) c \,e^{2}}{429}+\frac {128 d^{4} \left (\frac {14 c x}{17}+b \right ) c^{2} e}{1287}-\frac {512 c^{3} d^{5}}{21879}\right )}{7 e^{6}}\) | \(240\) |
derivativedivides | \(\frac {\frac {4 c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {15}{2}}}{3}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} c +2 c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+2 c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (b e -2 c d \right ) \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{6}}\) | \(265\) |
default | \(\frac {\frac {4 c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {15}{2}}}{3}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} c +2 c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+2 c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (b e -2 c d \right ) \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{6}}\) | \(265\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (18018 x^{5} c^{3} e^{5}+51051 x^{4} b \,c^{2} e^{5}-12012 x^{4} c^{3} d \,e^{4}+47124 x^{3} a \,c^{2} e^{5}+47124 x^{3} b^{2} c \,e^{5}-31416 x^{3} b \,c^{2} d \,e^{4}+7392 x^{3} c^{3} d^{2} e^{3}+83538 x^{2} a b c \,e^{5}-25704 x^{2} a \,c^{2} d \,e^{4}+13923 x^{2} b^{3} e^{5}-25704 x^{2} b^{2} c d \,e^{4}+17136 x^{2} b \,c^{2} d^{2} e^{3}-4032 x^{2} c^{3} d^{3} e^{2}+34034 x \,a^{2} c \,e^{5}+34034 x a \,b^{2} e^{5}-37128 x a b c d \,e^{4}+11424 x a \,c^{2} d^{2} e^{3}-6188 x \,b^{3} d \,e^{4}+11424 x \,b^{2} c \,d^{2} e^{3}-7616 x b \,c^{2} d^{3} e^{2}+1792 x \,c^{3} d^{4} e +21879 a^{2} b \,e^{5}-9724 a^{2} c d \,e^{4}-9724 a \,b^{2} d \,e^{4}+10608 a b c \,d^{2} e^{3}-3264 a \,c^{2} d^{3} e^{2}+1768 b^{3} d^{2} e^{3}-3264 b^{2} c \,d^{3} e^{2}+2176 b \,c^{2} d^{4} e -512 c^{3} d^{5}\right )}{153153 e^{6}}\) | \(359\) |
trager | \(\frac {2 \left (18018 c^{3} e^{8} x^{8}+51051 b \,c^{2} e^{8} x^{7}+42042 c^{3} d \,e^{7} x^{7}+47124 a \,c^{2} e^{8} x^{6}+47124 b^{2} c \,e^{8} x^{6}+121737 b \,c^{2} d \,e^{7} x^{6}+25410 c^{3} d^{2} e^{6} x^{6}+83538 a b c \,e^{8} x^{5}+115668 a \,c^{2} d \,e^{7} x^{5}+13923 b^{3} e^{8} x^{5}+115668 b^{2} c d \,e^{7} x^{5}+76041 b \,c^{2} d^{2} e^{6} x^{5}+126 c^{3} d^{3} e^{5} x^{5}+34034 a^{2} c \,e^{8} x^{4}+34034 a \,b^{2} e^{8} x^{4}+213486 a b c d \,e^{7} x^{4}+75684 a \,c^{2} d^{2} e^{6} x^{4}+35581 b^{3} d \,e^{7} x^{4}+75684 b^{2} c \,d^{2} e^{6} x^{4}+595 b \,c^{2} d^{3} e^{5} x^{4}-140 c^{3} d^{4} e^{4} x^{4}+21879 a^{2} b \,e^{8} x^{3}+92378 a^{2} c d \,e^{7} x^{3}+92378 a \,b^{2} d \,e^{7} x^{3}+149838 a b c \,d^{2} e^{6} x^{3}+1020 a \,c^{2} d^{3} e^{5} x^{3}+24973 b^{3} d^{2} e^{6} x^{3}+1020 b^{2} c \,d^{3} e^{5} x^{3}-680 b \,c^{2} d^{4} e^{4} x^{3}+160 c^{3} d^{5} e^{3} x^{3}+65637 a^{2} b d \,e^{7} x^{2}+72930 a^{2} c \,d^{2} e^{6} x^{2}+72930 a \,b^{2} d^{2} e^{6} x^{2}+3978 a b c \,d^{3} e^{5} x^{2}-1224 a \,c^{2} d^{4} e^{4} x^{2}+663 b^{3} d^{3} e^{5} x^{2}-1224 b^{2} c \,d^{4} e^{4} x^{2}+816 b \,c^{2} d^{5} e^{3} x^{2}-192 c^{3} d^{6} e^{2} x^{2}+65637 a^{2} b \,d^{2} e^{6} x +4862 a^{2} c \,d^{3} e^{5} x +4862 a \,b^{2} d^{3} e^{5} x -5304 a b c \,d^{4} e^{4} x +1632 a \,c^{2} d^{5} e^{3} x -884 b^{3} d^{4} e^{4} x +1632 b^{2} c \,d^{5} e^{3} x -1088 b \,c^{2} d^{6} e^{2} x +256 c^{3} d^{7} e x +21879 a^{2} b \,d^{3} e^{5}-9724 a^{2} c \,d^{4} e^{4}-9724 a \,b^{2} d^{4} e^{4}+10608 a b c \,d^{5} e^{3}-3264 a \,c^{2} d^{6} e^{2}+1768 b^{3} d^{5} e^{3}-3264 b^{2} c \,d^{6} e^{2}+2176 b \,c^{2} d^{7} e -512 c^{3} d^{8}\right ) \sqrt {e x +d}}{153153 e^{6}}\) | \(751\) |
risch | \(\frac {2 \left (18018 c^{3} e^{8} x^{8}+51051 b \,c^{2} e^{8} x^{7}+42042 c^{3} d \,e^{7} x^{7}+47124 a \,c^{2} e^{8} x^{6}+47124 b^{2} c \,e^{8} x^{6}+121737 b \,c^{2} d \,e^{7} x^{6}+25410 c^{3} d^{2} e^{6} x^{6}+83538 a b c \,e^{8} x^{5}+115668 a \,c^{2} d \,e^{7} x^{5}+13923 b^{3} e^{8} x^{5}+115668 b^{2} c d \,e^{7} x^{5}+76041 b \,c^{2} d^{2} e^{6} x^{5}+126 c^{3} d^{3} e^{5} x^{5}+34034 a^{2} c \,e^{8} x^{4}+34034 a \,b^{2} e^{8} x^{4}+213486 a b c d \,e^{7} x^{4}+75684 a \,c^{2} d^{2} e^{6} x^{4}+35581 b^{3} d \,e^{7} x^{4}+75684 b^{2} c \,d^{2} e^{6} x^{4}+595 b \,c^{2} d^{3} e^{5} x^{4}-140 c^{3} d^{4} e^{4} x^{4}+21879 a^{2} b \,e^{8} x^{3}+92378 a^{2} c d \,e^{7} x^{3}+92378 a \,b^{2} d \,e^{7} x^{3}+149838 a b c \,d^{2} e^{6} x^{3}+1020 a \,c^{2} d^{3} e^{5} x^{3}+24973 b^{3} d^{2} e^{6} x^{3}+1020 b^{2} c \,d^{3} e^{5} x^{3}-680 b \,c^{2} d^{4} e^{4} x^{3}+160 c^{3} d^{5} e^{3} x^{3}+65637 a^{2} b d \,e^{7} x^{2}+72930 a^{2} c \,d^{2} e^{6} x^{2}+72930 a \,b^{2} d^{2} e^{6} x^{2}+3978 a b c \,d^{3} e^{5} x^{2}-1224 a \,c^{2} d^{4} e^{4} x^{2}+663 b^{3} d^{3} e^{5} x^{2}-1224 b^{2} c \,d^{4} e^{4} x^{2}+816 b \,c^{2} d^{5} e^{3} x^{2}-192 c^{3} d^{6} e^{2} x^{2}+65637 a^{2} b \,d^{2} e^{6} x +4862 a^{2} c \,d^{3} e^{5} x +4862 a \,b^{2} d^{3} e^{5} x -5304 a b c \,d^{4} e^{4} x +1632 a \,c^{2} d^{5} e^{3} x -884 b^{3} d^{4} e^{4} x +1632 b^{2} c \,d^{5} e^{3} x -1088 b \,c^{2} d^{6} e^{2} x +256 c^{3} d^{7} e x +21879 a^{2} b \,d^{3} e^{5}-9724 a^{2} c \,d^{4} e^{4}-9724 a \,b^{2} d^{4} e^{4}+10608 a b c \,d^{5} e^{3}-3264 a \,c^{2} d^{6} e^{2}+1768 b^{3} d^{5} e^{3}-3264 b^{2} c \,d^{6} e^{2}+2176 b \,c^{2} d^{7} e -512 c^{3} d^{8}\right ) \sqrt {e x +d}}{153153 e^{6}}\) | \(751\) |
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Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (228) = 456\).
Time = 0.28 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.34 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (18018 \, c^{3} e^{8} x^{8} - 512 \, c^{3} d^{8} + 2176 \, b c^{2} d^{7} e + 21879 \, a^{2} b d^{3} e^{5} - 3264 \, {\left (b^{2} c + a c^{2}\right )} d^{6} e^{2} + 1768 \, {\left (b^{3} + 6 \, a b c\right )} d^{5} e^{3} - 9724 \, {\left (a b^{2} + a^{2} c\right )} d^{4} e^{4} + 3003 \, {\left (14 \, c^{3} d e^{7} + 17 \, b c^{2} e^{8}\right )} x^{7} + 231 \, {\left (110 \, c^{3} d^{2} e^{6} + 527 \, b c^{2} d e^{7} + 204 \, {\left (b^{2} c + a c^{2}\right )} e^{8}\right )} x^{6} + 63 \, {\left (2 \, c^{3} d^{3} e^{5} + 1207 \, b c^{2} d^{2} e^{6} + 1836 \, {\left (b^{2} c + a c^{2}\right )} d e^{7} + 221 \, {\left (b^{3} + 6 \, a b c\right )} e^{8}\right )} x^{5} - 7 \, {\left (20 \, c^{3} d^{4} e^{4} - 85 \, b c^{2} d^{3} e^{5} - 10812 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{6} - 5083 \, {\left (b^{3} + 6 \, a b c\right )} d e^{7} - 4862 \, {\left (a b^{2} + a^{2} c\right )} e^{8}\right )} x^{4} + {\left (160 \, c^{3} d^{5} e^{3} - 680 \, b c^{2} d^{4} e^{4} + 21879 \, a^{2} b e^{8} + 1020 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{5} + 24973 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{6} + 92378 \, {\left (a b^{2} + a^{2} c\right )} d e^{7}\right )} x^{3} - 3 \, {\left (64 \, c^{3} d^{6} e^{2} - 272 \, b c^{2} d^{5} e^{3} - 21879 \, a^{2} b d e^{7} + 408 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{4} - 221 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{5} - 24310 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{6}\right )} x^{2} + {\left (256 \, c^{3} d^{7} e - 1088 \, b c^{2} d^{6} e^{2} + 65637 \, a^{2} b d^{2} e^{6} + 1632 \, {\left (b^{2} c + a c^{2}\right )} d^{5} e^{3} - 884 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} e^{4} + 4862 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e^{5}\right )} x\right )} \sqrt {e x + d}}{153153 \, e^{6}} \]
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Time = 1.60 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.68 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \cdot \left (\frac {2 c^{3} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{5}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (5 b c^{2} e - 10 c^{3} d\right )}{15 e^{5}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (4 a c^{2} e^{2} + 4 b^{2} c e^{2} - 20 b c^{2} d e + 20 c^{3} d^{2}\right )}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (2 a^{2} c e^{4} + 2 a b^{2} e^{4} - 12 a b c d e^{3} + 12 a c^{2} d^{2} e^{2} - 2 b^{3} d e^{3} + 12 b^{2} c d^{2} e^{2} - 20 b c^{2} d^{3} e + 10 c^{3} d^{4}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{2} b e^{5} - 2 a^{2} c d e^{4} - 2 a b^{2} d e^{4} + 6 a b c d^{2} e^{3} - 4 a c^{2} d^{3} e^{2} + b^{3} d^{2} e^{3} - 4 b^{2} c d^{3} e^{2} + 5 b c^{2} d^{4} e - 2 c^{3} d^{5}\right )}{7 e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {d^{\frac {5}{2}} \left (a + b x + c x^{2}\right )^{3}}{3} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.22 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (18018 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{3} - 51051 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 47124 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 13923 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 34034 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 21879 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{153153 \, e^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2306 vs. \(2 (228) = 456\).
Time = 0.29 (sec) , antiderivative size = 2306, normalized size of antiderivative = 9.15 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
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Time = 0.10 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.06 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {{\left (d+e\,x\right )}^{9/2}\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4-24\,a\,b\,c\,d\,e^3+24\,a\,c^2\,d^2\,e^2-4\,b^3\,d\,e^3+24\,b^2\,c\,d^2\,e^2-40\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )}{9\,e^6}+\frac {4\,c^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^6}-\frac {\left (20\,c^3\,d-10\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^6}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (8\,b^2\,c\,e^2-40\,b\,c^2\,d\,e+40\,c^3\,d^2+8\,a\,c^2\,e^2\right )}{13\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{11\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{7\,e^6} \]
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