Integrand size = 28, antiderivative size = 187 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 (b d-a e)^6 (d+e x)^{9/2}}{9 e^7}-\frac {12 b (b d-a e)^5 (d+e x)^{11/2}}{11 e^7}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{13/2}}{13 e^7}-\frac {8 b^3 (b d-a e)^3 (d+e x)^{15/2}}{3 e^7}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{17/2}}{17 e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{19/2}}{19 e^7}+\frac {2 b^6 (d+e x)^{21/2}}{21 e^7} \]
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Time = 0.06 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 45} \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=-\frac {12 b^5 (d+e x)^{19/2} (b d-a e)}{19 e^7}+\frac {30 b^4 (d+e x)^{17/2} (b d-a e)^2}{17 e^7}-\frac {8 b^3 (d+e x)^{15/2} (b d-a e)^3}{3 e^7}+\frac {30 b^2 (d+e x)^{13/2} (b d-a e)^4}{13 e^7}-\frac {12 b (d+e x)^{11/2} (b d-a e)^5}{11 e^7}+\frac {2 (d+e x)^{9/2} (b d-a e)^6}{9 e^7}+\frac {2 b^6 (d+e x)^{21/2}}{21 e^7} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^6 (d+e x)^{7/2} \, dx \\ & = \int \left (\frac {(-b d+a e)^6 (d+e x)^{7/2}}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{9/2}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{11/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{13/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{15/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{17/2}}{e^6}+\frac {b^6 (d+e x)^{19/2}}{e^6}\right ) \, dx \\ & = \frac {2 (b d-a e)^6 (d+e x)^{9/2}}{9 e^7}-\frac {12 b (b d-a e)^5 (d+e x)^{11/2}}{11 e^7}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{13/2}}{13 e^7}-\frac {8 b^3 (b d-a e)^3 (d+e x)^{15/2}}{3 e^7}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{17/2}}{17 e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{19/2}}{19 e^7}+\frac {2 b^6 (d+e x)^{21/2}}{21 e^7} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.56 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 (d+e x)^{9/2} \left (323323 a^6 e^6+176358 a^5 b e^5 (-2 d+9 e x)+33915 a^4 b^2 e^4 \left (8 d^2-36 d e x+99 e^2 x^2\right )+9044 a^3 b^3 e^3 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+399 a^2 b^4 e^2 \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )+42 a b^5 e \left (-256 d^5+1152 d^4 e x-3168 d^3 e^2 x^2+6864 d^2 e^3 x^3-12870 d e^4 x^4+21879 e^5 x^5\right )+b^6 \left (1024 d^6-4608 d^5 e x+12672 d^4 e^2 x^2-27456 d^3 e^3 x^3+51480 d^2 e^4 x^4-87516 d e^5 x^5+138567 e^6 x^6\right )\right )}{2909907 e^7} \]
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Time = 2.39 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.48
method | result | size |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (\left (\frac {3}{7} b^{6} x^{6}+a^{6}+\frac {54}{19} a \,x^{5} b^{5}+\frac {135}{17} a^{2} x^{4} b^{4}+12 a^{3} x^{3} b^{3}+\frac {135}{13} a^{4} x^{2} b^{2}+\frac {54}{11} a^{5} x b \right ) e^{6}-\frac {12 b d \left (\frac {33}{133} b^{5} x^{5}+\frac {495}{323} a \,b^{4} x^{4}+\frac {66}{17} a^{2} b^{3} x^{3}+\frac {66}{13} a^{3} b^{2} x^{2}+\frac {45}{13} a^{4} b x +a^{5}\right ) e^{5}}{11}+\frac {120 b^{2} d^{2} \left (\frac {429}{2261} b^{4} x^{4}+\frac {1716}{1615} a \,b^{3} x^{3}+\frac {198}{85} a^{2} b^{2} x^{2}+\frac {12}{5} a^{3} b x +a^{4}\right ) e^{4}}{143}-\frac {64 b^{3} d^{3} \left (\frac {429}{2261} b^{3} x^{3}+\frac {297}{323} a \,b^{2} x^{2}+\frac {27}{17} a^{2} b x +a^{3}\right ) e^{3}}{143}+\frac {384 b^{4} \left (\frac {33}{133} b^{2} x^{2}+\frac {18}{19} a b x +a^{2}\right ) d^{4} e^{2}}{2431}-\frac {1536 b^{5} \left (\frac {3 b x}{7}+a \right ) d^{5} e}{46189}+\frac {1024 b^{6} d^{6}}{323323}\right )}{9 e^{7}}\) | \(276\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (138567 x^{6} b^{6} e^{6}+918918 x^{5} a \,b^{5} e^{6}-87516 x^{5} b^{6} d \,e^{5}+2567565 x^{4} a^{2} b^{4} e^{6}-540540 x^{4} a \,b^{5} d \,e^{5}+51480 x^{4} b^{6} d^{2} e^{4}+3879876 x^{3} a^{3} b^{3} e^{6}-1369368 x^{3} a^{2} b^{4} d \,e^{5}+288288 x^{3} a \,b^{5} d^{2} e^{4}-27456 x^{3} b^{6} d^{3} e^{3}+3357585 x^{2} a^{4} b^{2} e^{6}-1790712 x^{2} a^{3} b^{3} d \,e^{5}+632016 x^{2} a^{2} b^{4} d^{2} e^{4}-133056 x^{2} a \,b^{5} d^{3} e^{3}+12672 x^{2} b^{6} d^{4} e^{2}+1587222 x \,a^{5} b \,e^{6}-1220940 x \,a^{4} b^{2} d \,e^{5}+651168 x \,a^{3} b^{3} d^{2} e^{4}-229824 x \,a^{2} b^{4} d^{3} e^{3}+48384 x a \,b^{5} d^{4} e^{2}-4608 x \,b^{6} d^{5} e +323323 a^{6} e^{6}-352716 a^{5} b d \,e^{5}+271320 a^{4} b^{2} d^{2} e^{4}-144704 a^{3} b^{3} d^{3} e^{3}+51072 a^{2} b^{4} d^{4} e^{2}-10752 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right )}{2909907 e^{7}}\) | \(377\) |
derivativedivides | \(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {21}{2}}}{21}+\frac {6 \left (2 a e b -2 b^{2} d \right ) b^{4} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a e b -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) b^{2}+\left (2 a e b -2 b^{2} d \right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )+2 \left (2 a e b -2 b^{2} d \right )^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{7}}\) | \(457\) |
default | \(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {21}{2}}}{21}+\frac {6 \left (2 a e b -2 b^{2} d \right ) b^{4} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a e b -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) b^{2}+\left (2 a e b -2 b^{2} d \right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )+2 \left (2 a e b -2 b^{2} d \right )^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{7}}\) | \(457\) |
trager | \(\frac {2 \left (138567 e^{10} b^{6} x^{10}+918918 a \,b^{5} e^{10} x^{9}+466752 b^{6} d \,e^{9} x^{9}+2567565 a^{2} b^{4} e^{10} x^{8}+3135132 a \,b^{5} d \,e^{9} x^{8}+532818 b^{6} d^{2} e^{8} x^{8}+3879876 a^{3} b^{3} e^{10} x^{7}+8900892 a^{2} b^{4} d \,e^{9} x^{7}+3639636 a \,b^{5} d^{2} e^{8} x^{7}+207636 b^{6} d^{3} e^{7} x^{7}+3357585 a^{4} b^{2} e^{10} x^{6}+13728792 a^{3} b^{3} d \,e^{9} x^{6}+10559934 a^{2} b^{4} d^{2} e^{8} x^{6}+1452528 a \,b^{5} d^{3} e^{7} x^{6}+231 b^{6} d^{4} e^{6} x^{6}+1587222 a^{5} b \,e^{10} x^{5}+12209400 a^{4} b^{2} d \,e^{9} x^{5}+16767576 a^{3} b^{3} d^{2} e^{8} x^{5}+4352292 a^{2} b^{4} d^{3} e^{7} x^{5}+2646 a \,b^{5} d^{4} e^{6} x^{5}-252 b^{6} d^{5} e^{5} x^{5}+323323 a^{6} e^{10} x^{4}+5996172 a^{5} b d \,e^{9} x^{4}+15533070 a^{4} b^{2} d^{2} e^{8} x^{4}+7235200 a^{3} b^{3} d^{3} e^{7} x^{4}+13965 a^{2} b^{4} d^{4} e^{6} x^{4}-2940 a \,b^{5} d^{5} e^{5} x^{4}+280 b^{6} d^{6} e^{4} x^{4}+1293292 a^{6} d \,e^{9} x^{3}+8112468 a^{5} b \,d^{2} e^{8} x^{3}+7189980 a^{4} b^{2} d^{3} e^{7} x^{3}+45220 a^{3} b^{3} d^{4} e^{6} x^{3}-15960 a^{2} b^{4} d^{5} e^{5} x^{3}+3360 a \,b^{5} d^{6} e^{4} x^{3}-320 b^{6} d^{7} e^{3} x^{3}+1939938 a^{6} d^{2} e^{8} x^{2}+4232592 a^{5} b \,d^{3} e^{7} x^{2}+101745 a^{4} b^{2} d^{4} e^{6} x^{2}-54264 a^{3} b^{3} d^{5} e^{5} x^{2}+19152 a^{2} b^{4} d^{6} e^{4} x^{2}-4032 a \,b^{5} d^{7} e^{3} x^{2}+384 b^{6} d^{8} e^{2} x^{2}+1293292 a^{6} d^{3} e^{7} x +176358 a^{5} b \,d^{4} e^{6} x -135660 a^{4} b^{2} d^{5} e^{5} x +72352 a^{3} b^{3} d^{6} e^{4} x -25536 a^{2} b^{4} d^{7} e^{3} x +5376 a \,b^{5} d^{8} e^{2} x -512 b^{6} d^{9} e x +323323 a^{6} d^{4} e^{6}-352716 a^{5} b \,d^{5} e^{5}+271320 a^{4} b^{2} d^{6} e^{4}-144704 a^{3} b^{3} d^{7} e^{3}+51072 a^{2} b^{4} d^{8} e^{2}-10752 a \,b^{5} d^{9} e +1024 b^{6} d^{10}\right ) \sqrt {e x +d}}{2909907 e^{7}}\) | \(809\) |
risch | \(\frac {2 \left (138567 e^{10} b^{6} x^{10}+918918 a \,b^{5} e^{10} x^{9}+466752 b^{6} d \,e^{9} x^{9}+2567565 a^{2} b^{4} e^{10} x^{8}+3135132 a \,b^{5} d \,e^{9} x^{8}+532818 b^{6} d^{2} e^{8} x^{8}+3879876 a^{3} b^{3} e^{10} x^{7}+8900892 a^{2} b^{4} d \,e^{9} x^{7}+3639636 a \,b^{5} d^{2} e^{8} x^{7}+207636 b^{6} d^{3} e^{7} x^{7}+3357585 a^{4} b^{2} e^{10} x^{6}+13728792 a^{3} b^{3} d \,e^{9} x^{6}+10559934 a^{2} b^{4} d^{2} e^{8} x^{6}+1452528 a \,b^{5} d^{3} e^{7} x^{6}+231 b^{6} d^{4} e^{6} x^{6}+1587222 a^{5} b \,e^{10} x^{5}+12209400 a^{4} b^{2} d \,e^{9} x^{5}+16767576 a^{3} b^{3} d^{2} e^{8} x^{5}+4352292 a^{2} b^{4} d^{3} e^{7} x^{5}+2646 a \,b^{5} d^{4} e^{6} x^{5}-252 b^{6} d^{5} e^{5} x^{5}+323323 a^{6} e^{10} x^{4}+5996172 a^{5} b d \,e^{9} x^{4}+15533070 a^{4} b^{2} d^{2} e^{8} x^{4}+7235200 a^{3} b^{3} d^{3} e^{7} x^{4}+13965 a^{2} b^{4} d^{4} e^{6} x^{4}-2940 a \,b^{5} d^{5} e^{5} x^{4}+280 b^{6} d^{6} e^{4} x^{4}+1293292 a^{6} d \,e^{9} x^{3}+8112468 a^{5} b \,d^{2} e^{8} x^{3}+7189980 a^{4} b^{2} d^{3} e^{7} x^{3}+45220 a^{3} b^{3} d^{4} e^{6} x^{3}-15960 a^{2} b^{4} d^{5} e^{5} x^{3}+3360 a \,b^{5} d^{6} e^{4} x^{3}-320 b^{6} d^{7} e^{3} x^{3}+1939938 a^{6} d^{2} e^{8} x^{2}+4232592 a^{5} b \,d^{3} e^{7} x^{2}+101745 a^{4} b^{2} d^{4} e^{6} x^{2}-54264 a^{3} b^{3} d^{5} e^{5} x^{2}+19152 a^{2} b^{4} d^{6} e^{4} x^{2}-4032 a \,b^{5} d^{7} e^{3} x^{2}+384 b^{6} d^{8} e^{2} x^{2}+1293292 a^{6} d^{3} e^{7} x +176358 a^{5} b \,d^{4} e^{6} x -135660 a^{4} b^{2} d^{5} e^{5} x +72352 a^{3} b^{3} d^{6} e^{4} x -25536 a^{2} b^{4} d^{7} e^{3} x +5376 a \,b^{5} d^{8} e^{2} x -512 b^{6} d^{9} e x +323323 a^{6} d^{4} e^{6}-352716 a^{5} b \,d^{5} e^{5}+271320 a^{4} b^{2} d^{6} e^{4}-144704 a^{3} b^{3} d^{7} e^{3}+51072 a^{2} b^{4} d^{8} e^{2}-10752 a \,b^{5} d^{9} e +1024 b^{6} d^{10}\right ) \sqrt {e x +d}}{2909907 e^{7}}\) | \(809\) |
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Leaf count of result is larger than twice the leaf count of optimal. 729 vs. \(2 (159) = 318\).
Time = 0.38 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.90 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (138567 \, b^{6} e^{10} x^{10} + 1024 \, b^{6} d^{10} - 10752 \, a b^{5} d^{9} e + 51072 \, a^{2} b^{4} d^{8} e^{2} - 144704 \, a^{3} b^{3} d^{7} e^{3} + 271320 \, a^{4} b^{2} d^{6} e^{4} - 352716 \, a^{5} b d^{5} e^{5} + 323323 \, a^{6} d^{4} e^{6} + 14586 \, {\left (32 \, b^{6} d e^{9} + 63 \, a b^{5} e^{10}\right )} x^{9} + 3861 \, {\left (138 \, b^{6} d^{2} e^{8} + 812 \, a b^{5} d e^{9} + 665 \, a^{2} b^{4} e^{10}\right )} x^{8} + 1716 \, {\left (121 \, b^{6} d^{3} e^{7} + 2121 \, a b^{5} d^{2} e^{8} + 5187 \, a^{2} b^{4} d e^{9} + 2261 \, a^{3} b^{3} e^{10}\right )} x^{7} + 231 \, {\left (b^{6} d^{4} e^{6} + 6288 \, a b^{5} d^{3} e^{7} + 45714 \, a^{2} b^{4} d^{2} e^{8} + 59432 \, a^{3} b^{3} d e^{9} + 14535 \, a^{4} b^{2} e^{10}\right )} x^{6} - 126 \, {\left (2 \, b^{6} d^{5} e^{5} - 21 \, a b^{5} d^{4} e^{6} - 34542 \, a^{2} b^{4} d^{3} e^{7} - 133076 \, a^{3} b^{3} d^{2} e^{8} - 96900 \, a^{4} b^{2} d e^{9} - 12597 \, a^{5} b e^{10}\right )} x^{5} + 7 \, {\left (40 \, b^{6} d^{6} e^{4} - 420 \, a b^{5} d^{5} e^{5} + 1995 \, a^{2} b^{4} d^{4} e^{6} + 1033600 \, a^{3} b^{3} d^{3} e^{7} + 2219010 \, a^{4} b^{2} d^{2} e^{8} + 856596 \, a^{5} b d e^{9} + 46189 \, a^{6} e^{10}\right )} x^{4} - 4 \, {\left (80 \, b^{6} d^{7} e^{3} - 840 \, a b^{5} d^{6} e^{4} + 3990 \, a^{2} b^{4} d^{5} e^{5} - 11305 \, a^{3} b^{3} d^{4} e^{6} - 1797495 \, a^{4} b^{2} d^{3} e^{7} - 2028117 \, a^{5} b d^{2} e^{8} - 323323 \, a^{6} d e^{9}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{8} e^{2} - 1344 \, a b^{5} d^{7} e^{3} + 6384 \, a^{2} b^{4} d^{6} e^{4} - 18088 \, a^{3} b^{3} d^{5} e^{5} + 33915 \, a^{4} b^{2} d^{4} e^{6} + 1410864 \, a^{5} b d^{3} e^{7} + 646646 \, a^{6} d^{2} e^{8}\right )} x^{2} - 2 \, {\left (256 \, b^{6} d^{9} e - 2688 \, a b^{5} d^{8} e^{2} + 12768 \, a^{2} b^{4} d^{7} e^{3} - 36176 \, a^{3} b^{3} d^{6} e^{4} + 67830 \, a^{4} b^{2} d^{5} e^{5} - 88179 \, a^{5} b d^{4} e^{6} - 646646 \, a^{6} d^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{2909907 \, e^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (173) = 346\).
Time = 1.69 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.65 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {b^{6} \left (d + e x\right )^{\frac {21}{2}}}{21 e^{6}} + \frac {\left (d + e x\right )^{\frac {19}{2}} \cdot \left (6 a b^{5} e - 6 b^{6} d\right )}{19 e^{6}} + \frac {\left (d + e x\right )^{\frac {17}{2}} \cdot \left (15 a^{2} b^{4} e^{2} - 30 a b^{5} d e + 15 b^{6} d^{2}\right )}{17 e^{6}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (20 a^{3} b^{3} e^{3} - 60 a^{2} b^{4} d e^{2} + 60 a b^{5} d^{2} e - 20 b^{6} d^{3}\right )}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (15 a^{4} b^{2} e^{4} - 60 a^{3} b^{3} d e^{3} + 90 a^{2} b^{4} d^{2} e^{2} - 60 a b^{5} d^{3} e + 15 b^{6} d^{4}\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (6 a^{5} b e^{5} - 30 a^{4} b^{2} d e^{4} + 60 a^{3} b^{3} d^{2} e^{3} - 60 a^{2} b^{4} d^{3} e^{2} + 30 a b^{5} d^{4} e - 6 b^{6} d^{5}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (a^{6} e^{6} - 6 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} + 15 a^{2} b^{4} d^{4} e^{2} - 6 a b^{5} d^{5} e + b^{6} d^{6}\right )}{9 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (a^{6} x + 3 a^{5} b x^{2} + 5 a^{4} b^{2} x^{3} + 5 a^{3} b^{3} x^{4} + 3 a^{2} b^{4} x^{5} + a b^{5} x^{6} + \frac {b^{6} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (159) = 318\).
Time = 0.20 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.87 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (138567 \, {\left (e x + d\right )}^{\frac {21}{2}} b^{6} - 918918 \, {\left (b^{6} d - a b^{5} e\right )} {\left (e x + d\right )}^{\frac {19}{2}} + 2567565 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (e x + d\right )}^{\frac {17}{2}} - 3879876 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 3357585 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 1587222 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 323323 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{2909907 \, e^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2771 vs. \(2 (159) = 318\).
Time = 0.33 (sec) , antiderivative size = 2771, normalized size of antiderivative = 14.82 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
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Time = 9.59 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.87 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2\,b^6\,{\left (d+e\,x\right )}^{21/2}}{21\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{19/2}}{19\,e^7}+\frac {2\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}+\frac {30\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {8\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{15/2}}{3\,e^7}+\frac {30\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {12\,b\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7} \]
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