Integrand size = 29, antiderivative size = 270 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=\frac {c^4 \left (c^3+4 a d^2\right )^4 x}{d^8}-\frac {8 c^5 \left (c^3+4 a d^2\right )^3 \left (\frac {c}{d}+x\right )^3}{3 d^6}+\frac {4 c^3 \left (c^3+4 a d^2\right )^2 \left (7 c^3+4 a d^2\right ) \left (\frac {c}{d}+x\right )^5}{5 d^4}-\frac {8 c^4 \left (c^3+4 a d^2\right ) \left (7 c^3+12 a d^2\right ) \left (\frac {c}{d}+x\right )^7}{7 d^2}+\frac {2}{9} c^2 \left (35 c^6+120 a c^3 d^2+48 a^2 d^4\right ) \left (\frac {c}{d}+x\right )^9-\frac {8}{11} c^3 d^2 \left (7 c^3+12 a d^2\right ) \left (\frac {c}{d}+x\right )^{11}+\frac {4}{13} c d^4 \left (7 c^3+4 a d^2\right ) \left (\frac {c}{d}+x\right )^{13}-\frac {8}{15} c^2 d^6 \left (\frac {c}{d}+x\right )^{15}+\frac {1}{17} d^8 \left (\frac {c}{d}+x\right )^{17} \]
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Time = 0.42 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1120, 1104} \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=\frac {2}{9} c^2 \left (48 a^2 d^4+120 a c^3 d^2+35 c^6\right ) \left (\frac {c}{d}+x\right )^9-\frac {8}{11} c^3 d^2 \left (12 a d^2+7 c^3\right ) \left (\frac {c}{d}+x\right )^{11}+\frac {4}{13} c d^4 \left (4 a d^2+7 c^3\right ) \left (\frac {c}{d}+x\right )^{13}+\frac {4 c^3 \left (4 a d^2+c^3\right )^2 \left (4 a d^2+7 c^3\right ) \left (\frac {c}{d}+x\right )^5}{5 d^4}-\frac {8 c^5 \left (4 a d^2+c^3\right )^3 \left (\frac {c}{d}+x\right )^3}{3 d^6}-\frac {8 c^4 \left (4 a d^2+c^3\right ) \left (12 a d^2+7 c^3\right ) \left (\frac {c}{d}+x\right )^7}{7 d^2}+\frac {c^4 x \left (4 a d^2+c^3\right )^4}{d^8}-\frac {8}{15} c^2 d^6 \left (\frac {c}{d}+x\right )^{15}+\frac {1}{17} d^8 \left (\frac {c}{d}+x\right )^{17} \]
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Rule 1104
Rule 1120
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4\right )^4 \, dx,x,\frac {c}{d}+x\right ) \\ & = \text {Subst}\left (\int \left (\frac {\left (c^4+4 a c d^2\right )^4}{d^8}-\frac {8 c^5 \left (c^3+4 a d^2\right )^3 x^2}{d^6}+\frac {24 c^6 \left (c^3+4 a d^2\right )^2 \left (\frac {7}{6}+\frac {2 a d^2}{3 c^3}\right ) x^4}{d^4}-\frac {32 c^7 \left (c^3+4 a d^2\right ) \left (\frac {7}{4}+\frac {3 a d^2}{c^3}\right ) x^6}{d^2}+16 c^8 \left (\frac {35}{8}+\frac {15 a d^2}{c^3}+\frac {6 a^2 d^4}{c^6}\right ) x^8-32 c^6 d^2 \left (\frac {7}{4}+\frac {3 a d^2}{c^3}\right ) x^{10}+24 c^4 d^4 \left (\frac {7}{6}+\frac {2 a d^2}{3 c^3}\right ) x^{12}-8 c^2 d^6 x^{14}+d^8 x^{16}\right ) \, dx,x,\frac {c}{d}+x\right ) \\ & = \frac {c^4 \left (c^3+4 a d^2\right )^4 x}{d^8}-\frac {8 c^5 \left (c^3+4 a d^2\right )^3 \left (\frac {c}{d}+x\right )^3}{3 d^6}+\frac {4 c^3 \left (c^3+4 a d^2\right )^2 \left (7 c^3+4 a d^2\right ) \left (\frac {c}{d}+x\right )^5}{5 d^4}-\frac {8 c^4 \left (c^3+4 a d^2\right ) \left (7 c^3+12 a d^2\right ) \left (\frac {c}{d}+x\right )^7}{7 d^2}+\frac {2}{9} c^2 \left (35 c^6+120 a c^3 d^2+48 a^2 d^4\right ) \left (\frac {c}{d}+x\right )^9-\frac {8}{11} c^3 d^2 \left (7 c^3+12 a d^2\right ) \left (\frac {c}{d}+x\right )^{11}+\frac {4}{13} c d^4 \left (7 c^3+4 a d^2\right ) \left (\frac {c}{d}+x\right )^{13}-\frac {8}{15} c^2 d^6 \left (\frac {c}{d}+x\right )^{15}+\frac {1}{17} d^8 \left (\frac {c}{d}+x\right )^{17} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.06 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=256 a^4 c^4 x+\frac {1024}{3} a^3 c^5 x^3+256 a^3 c^4 d x^4+\frac {256}{5} a^2 c^3 \left (6 c^3+a d^2\right ) x^5+512 a^2 c^5 d x^6+\frac {256}{7} a c^4 \left (4 c^3+9 a d^2\right ) x^7+96 a c^3 d \left (4 c^3+a d^2\right ) x^8+\frac {32}{9} c^2 \left (8 c^6+120 a c^3 d^2+3 a^2 d^4\right ) x^9+\frac {256}{5} c^4 d \left (2 c^3+5 a d^2\right ) x^{10}+\frac {64}{11} c^3 d^2 \left (28 c^3+15 a d^2\right ) x^{11}+\frac {16}{3} c^2 d^3 \left (28 c^3+3 a d^2\right ) x^{12}+\frac {16}{13} c d^4 \left (70 c^3+a d^2\right ) x^{13}+32 c^3 d^5 x^{14}+\frac {112}{15} c^2 d^6 x^{15}+c d^7 x^{16}+\frac {d^8 x^{17}}{17} \]
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Time = 0.05 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.99
method | result | size |
norman | \(256 a^{4} c^{4} x +\frac {1024 a^{3} c^{5} x^{3}}{3}+256 a^{3} c^{4} d \,x^{4}+\left (\frac {256}{5} a^{3} c^{3} d^{2}+\frac {1536}{5} a^{2} c^{6}\right ) x^{5}+512 a^{2} c^{5} d \,x^{6}+\left (\frac {2304}{7} a^{2} c^{4} d^{2}+\frac {1024}{7} a \,c^{7}\right ) x^{7}+\left (96 a^{2} c^{3} d^{3}+384 a \,c^{6} d \right ) x^{8}+\left (\frac {32}{3} a^{2} c^{2} d^{4}+\frac {1280}{3} a \,c^{5} d^{2}+\frac {256}{9} c^{8}\right ) x^{9}+\left (256 a \,c^{4} d^{3}+\frac {512}{5} c^{7} d \right ) x^{10}+\left (\frac {960}{11} a \,c^{3} d^{4}+\frac {1792}{11} c^{6} d^{2}\right ) x^{11}+\left (16 a \,c^{2} d^{5}+\frac {448}{3} c^{5} d^{3}\right ) x^{12}+\left (\frac {16}{13} a c \,d^{6}+\frac {1120}{13} c^{4} d^{4}\right ) x^{13}+32 c^{3} d^{5} x^{14}+\frac {112 c^{2} d^{6} x^{15}}{15}+d^{7} c \,x^{16}+\frac {d^{8} x^{17}}{17}\) | \(267\) |
gosper | \(384 a \,c^{6} d \,x^{8}+96 a^{2} c^{3} d^{3} x^{8}+\frac {960}{11} x^{11} a \,c^{3} d^{4}+16 x^{12} a \,c^{2} d^{5}+\frac {16}{13} x^{13} a c \,d^{6}+\frac {32}{3} x^{9} a^{2} c^{2} d^{4}+\frac {1280}{3} x^{9} a \,c^{5} d^{2}+256 x^{10} a \,c^{4} d^{3}+\frac {2304}{7} x^{7} a^{2} c^{4} d^{2}+512 a^{2} c^{5} d \,x^{6}+\frac {256}{5} x^{5} a^{3} c^{3} d^{2}+256 a^{3} c^{4} d \,x^{4}+256 a^{4} c^{4} x +\frac {1024}{3} a^{3} c^{5} x^{3}+32 c^{3} d^{5} x^{14}+\frac {112}{15} c^{2} d^{6} x^{15}+d^{7} c \,x^{16}+\frac {1536}{5} x^{5} a^{2} c^{6}+\frac {1024}{7} x^{7} a \,c^{7}+\frac {512}{5} x^{10} c^{7} d +\frac {1792}{11} x^{11} c^{6} d^{2}+\frac {448}{3} x^{12} c^{5} d^{3}+\frac {1120}{13} x^{13} c^{4} d^{4}+\frac {1}{17} d^{8} x^{17}+\frac {256}{9} x^{9} c^{8}\) | \(278\) |
risch | \(384 a \,c^{6} d \,x^{8}+96 a^{2} c^{3} d^{3} x^{8}+\frac {960}{11} x^{11} a \,c^{3} d^{4}+16 x^{12} a \,c^{2} d^{5}+\frac {16}{13} x^{13} a c \,d^{6}+\frac {32}{3} x^{9} a^{2} c^{2} d^{4}+\frac {1280}{3} x^{9} a \,c^{5} d^{2}+256 x^{10} a \,c^{4} d^{3}+\frac {2304}{7} x^{7} a^{2} c^{4} d^{2}+512 a^{2} c^{5} d \,x^{6}+\frac {256}{5} x^{5} a^{3} c^{3} d^{2}+256 a^{3} c^{4} d \,x^{4}+256 a^{4} c^{4} x +\frac {1024}{3} a^{3} c^{5} x^{3}+32 c^{3} d^{5} x^{14}+\frac {112}{15} c^{2} d^{6} x^{15}+d^{7} c \,x^{16}+\frac {1536}{5} x^{5} a^{2} c^{6}+\frac {1024}{7} x^{7} a \,c^{7}+\frac {512}{5} x^{10} c^{7} d +\frac {1792}{11} x^{11} c^{6} d^{2}+\frac {448}{3} x^{12} c^{5} d^{3}+\frac {1120}{13} x^{13} c^{4} d^{4}+\frac {1}{17} d^{8} x^{17}+\frac {256}{9} x^{9} c^{8}\) | \(278\) |
parallelrisch | \(384 a \,c^{6} d \,x^{8}+96 a^{2} c^{3} d^{3} x^{8}+\frac {960}{11} x^{11} a \,c^{3} d^{4}+16 x^{12} a \,c^{2} d^{5}+\frac {16}{13} x^{13} a c \,d^{6}+\frac {32}{3} x^{9} a^{2} c^{2} d^{4}+\frac {1280}{3} x^{9} a \,c^{5} d^{2}+256 x^{10} a \,c^{4} d^{3}+\frac {2304}{7} x^{7} a^{2} c^{4} d^{2}+512 a^{2} c^{5} d \,x^{6}+\frac {256}{5} x^{5} a^{3} c^{3} d^{2}+256 a^{3} c^{4} d \,x^{4}+256 a^{4} c^{4} x +\frac {1024}{3} a^{3} c^{5} x^{3}+32 c^{3} d^{5} x^{14}+\frac {112}{15} c^{2} d^{6} x^{15}+d^{7} c \,x^{16}+\frac {1536}{5} x^{5} a^{2} c^{6}+\frac {1024}{7} x^{7} a \,c^{7}+\frac {512}{5} x^{10} c^{7} d +\frac {1792}{11} x^{11} c^{6} d^{2}+\frac {448}{3} x^{12} c^{5} d^{3}+\frac {1120}{13} x^{13} c^{4} d^{4}+\frac {1}{17} d^{8} x^{17}+\frac {256}{9} x^{9} c^{8}\) | \(278\) |
default | \(\frac {d^{8} x^{17}}{17}+d^{7} c \,x^{16}+\frac {112 c^{2} d^{6} x^{15}}{15}+32 c^{3} d^{5} x^{14}+\frac {\left (2 \left (8 d^{2} a c +16 c^{4}\right ) d^{4}+1088 c^{4} d^{4}\right ) x^{13}}{13}+\frac {\left (64 a \,c^{2} d^{5}+16 \left (8 d^{2} a c +16 c^{4}\right ) d^{3} c +1536 c^{5} d^{3}\right ) x^{12}}{12}+\frac {\left (576 a \,c^{3} d^{4}+48 \left (8 d^{2} a c +16 c^{4}\right ) c^{2} d^{2}+1024 c^{6} d^{2}\right ) x^{11}}{11}+\frac {\left (2048 a \,c^{4} d^{3}+64 \left (8 d^{2} a c +16 c^{4}\right ) c^{3} d \right ) x^{10}}{10}+\frac {\left (32 a^{2} c^{2} d^{4}+3584 a \,c^{5} d^{2}+\left (8 d^{2} a c +16 c^{4}\right )^{2}\right ) x^{9}}{9}+\frac {\left (256 a^{2} c^{3} d^{3}+2048 a \,c^{6} d +64 a \,c^{2} d \left (8 d^{2} a c +16 c^{4}\right )\right ) x^{8}}{8}+\frac {\left (1792 a^{2} c^{4} d^{2}+64 a \,c^{3} \left (8 d^{2} a c +16 c^{4}\right )\right ) x^{7}}{7}+512 a^{2} c^{5} d \,x^{6}+\frac {\left (32 a^{2} c^{2} \left (8 d^{2} a c +16 c^{4}\right )+1024 a^{2} c^{6}\right ) x^{5}}{5}+256 a^{3} c^{4} d \,x^{4}+\frac {1024 a^{3} c^{5} x^{3}}{3}+256 a^{4} c^{4} x\) | \(392\) |
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Time = 0.27 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=\frac {1}{17} \, d^{8} x^{17} + c d^{7} x^{16} + \frac {112}{15} \, c^{2} d^{6} x^{15} + 32 \, c^{3} d^{5} x^{14} + 512 \, a^{2} c^{5} d x^{6} + \frac {16}{13} \, {\left (70 \, c^{4} d^{4} + a c d^{6}\right )} x^{13} + \frac {16}{3} \, {\left (28 \, c^{5} d^{3} + 3 \, a c^{2} d^{5}\right )} x^{12} + 256 \, a^{3} c^{4} d x^{4} + \frac {64}{11} \, {\left (28 \, c^{6} d^{2} + 15 \, a c^{3} d^{4}\right )} x^{11} + \frac {1024}{3} \, a^{3} c^{5} x^{3} + \frac {256}{5} \, {\left (2 \, c^{7} d + 5 \, a c^{4} d^{3}\right )} x^{10} + \frac {32}{9} \, {\left (8 \, c^{8} + 120 \, a c^{5} d^{2} + 3 \, a^{2} c^{2} d^{4}\right )} x^{9} + 256 \, a^{4} c^{4} x + 96 \, {\left (4 \, a c^{6} d + a^{2} c^{3} d^{3}\right )} x^{8} + \frac {256}{7} \, {\left (4 \, a c^{7} + 9 \, a^{2} c^{4} d^{2}\right )} x^{7} + \frac {256}{5} \, {\left (6 \, a^{2} c^{6} + a^{3} c^{3} d^{2}\right )} x^{5} \]
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Time = 0.05 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.11 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=256 a^{4} c^{4} x + \frac {1024 a^{3} c^{5} x^{3}}{3} + 256 a^{3} c^{4} d x^{4} + 512 a^{2} c^{5} d x^{6} + 32 c^{3} d^{5} x^{14} + \frac {112 c^{2} d^{6} x^{15}}{15} + c d^{7} x^{16} + \frac {d^{8} x^{17}}{17} + x^{13} \cdot \left (\frac {16 a c d^{6}}{13} + \frac {1120 c^{4} d^{4}}{13}\right ) + x^{12} \cdot \left (16 a c^{2} d^{5} + \frac {448 c^{5} d^{3}}{3}\right ) + x^{11} \cdot \left (\frac {960 a c^{3} d^{4}}{11} + \frac {1792 c^{6} d^{2}}{11}\right ) + x^{10} \cdot \left (256 a c^{4} d^{3} + \frac {512 c^{7} d}{5}\right ) + x^{9} \cdot \left (\frac {32 a^{2} c^{2} d^{4}}{3} + \frac {1280 a c^{5} d^{2}}{3} + \frac {256 c^{8}}{9}\right ) + x^{8} \cdot \left (96 a^{2} c^{3} d^{3} + 384 a c^{6} d\right ) + x^{7} \cdot \left (\frac {2304 a^{2} c^{4} d^{2}}{7} + \frac {1024 a c^{7}}{7}\right ) + x^{5} \cdot \left (\frac {256 a^{3} c^{3} d^{2}}{5} + \frac {1536 a^{2} c^{6}}{5}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.38 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=\frac {1}{17} \, d^{8} x^{17} + c d^{7} x^{16} + \frac {32}{5} \, c^{2} d^{6} x^{15} + \frac {128}{7} \, c^{3} d^{5} x^{14} + \frac {256}{13} \, c^{4} d^{4} x^{13} + \frac {256}{9} \, c^{8} x^{9} + 256 \, a^{4} c^{4} x + \frac {256}{15} \, {\left (3 \, d^{2} x^{5} + 15 \, c d x^{4} + 20 \, c^{2} x^{3}\right )} a^{3} c^{3} + \frac {256}{55} \, {\left (5 \, d^{2} x^{11} + 22 \, c d x^{10}\right )} c^{6} + \frac {32}{105} \, {\left (35 \, d^{4} x^{9} + 315 \, c d^{3} x^{8} + 720 \, c^{2} d^{2} x^{7} + 1008 \, c^{4} x^{5} + 120 \, {\left (3 \, d^{2} x^{7} + 14 \, c d x^{6}\right )} c^{2}\right )} a^{2} c^{2} + \frac {32}{143} \, {\left (33 \, d^{4} x^{13} + 286 \, c d^{3} x^{12} + 624 \, c^{2} d^{2} x^{11}\right )} c^{4} + \frac {16}{15015} \, {\left (1155 \, d^{6} x^{13} + 15015 \, c d^{5} x^{12} + 65520 \, c^{2} d^{4} x^{11} + 96096 \, c^{3} d^{3} x^{10} + 137280 \, c^{6} x^{7} + 40040 \, {\left (2 \, d^{2} x^{9} + 9 \, c d x^{8}\right )} c^{4} + 364 \, {\left (45 \, d^{4} x^{11} + 396 \, c d^{3} x^{10} + 880 \, c^{2} d^{2} x^{9}\right )} c^{2}\right )} a c + \frac {16}{1365} \, {\left (91 \, d^{6} x^{15} + 1170 \, c d^{5} x^{14} + 5040 \, c^{2} d^{4} x^{13} + 7280 \, c^{3} d^{3} x^{12}\right )} c^{2} \]
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Time = 0.28 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.03 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=\frac {1}{17} \, d^{8} x^{17} + c d^{7} x^{16} + \frac {112}{15} \, c^{2} d^{6} x^{15} + 32 \, c^{3} d^{5} x^{14} + \frac {1120}{13} \, c^{4} d^{4} x^{13} + \frac {16}{13} \, a c d^{6} x^{13} + \frac {448}{3} \, c^{5} d^{3} x^{12} + 16 \, a c^{2} d^{5} x^{12} + \frac {1792}{11} \, c^{6} d^{2} x^{11} + \frac {960}{11} \, a c^{3} d^{4} x^{11} + \frac {512}{5} \, c^{7} d x^{10} + 256 \, a c^{4} d^{3} x^{10} + \frac {256}{9} \, c^{8} x^{9} + \frac {1280}{3} \, a c^{5} d^{2} x^{9} + \frac {32}{3} \, a^{2} c^{2} d^{4} x^{9} + 384 \, a c^{6} d x^{8} + 96 \, a^{2} c^{3} d^{3} x^{8} + \frac {1024}{7} \, a c^{7} x^{7} + \frac {2304}{7} \, a^{2} c^{4} d^{2} x^{7} + 512 \, a^{2} c^{5} d x^{6} + \frac {1536}{5} \, a^{2} c^{6} x^{5} + \frac {256}{5} \, a^{3} c^{3} d^{2} x^{5} + 256 \, a^{3} c^{4} d x^{4} + \frac {1024}{3} \, a^{3} c^{5} x^{3} + 256 \, a^{4} c^{4} x \]
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Time = 10.57 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.97 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^4 \, dx=x^{10}\,\left (\frac {512\,c^7\,d}{5}+256\,a\,c^4\,d^3\right )+x^{13}\,\left (\frac {1120\,c^4\,d^4}{13}+\frac {16\,a\,c\,d^6}{13}\right )+x^9\,\left (\frac {32\,a^2\,c^2\,d^4}{3}+\frac {1280\,a\,c^5\,d^2}{3}+\frac {256\,c^8}{9}\right )+x^{12}\,\left (\frac {448\,c^5\,d^3}{3}+16\,a\,c^2\,d^5\right )+x^{11}\,\left (\frac {1792\,c^6\,d^2}{11}+\frac {960\,a\,c^3\,d^4}{11}\right )+\frac {d^8\,x^{17}}{17}+256\,a^4\,c^4\,x+c\,d^7\,x^{16}+\frac {1024\,a^3\,c^5\,x^3}{3}+32\,c^3\,d^5\,x^{14}+\frac {112\,c^2\,d^6\,x^{15}}{15}+256\,a^3\,c^4\,d\,x^4+512\,a^2\,c^5\,d\,x^6+\frac {256\,a\,c^4\,x^7\,\left (4\,c^3+9\,a\,d^2\right )}{7}+\frac {256\,a^2\,c^3\,x^5\,\left (6\,c^3+a\,d^2\right )}{5}+96\,a\,c^3\,d\,x^8\,\left (4\,c^3+a\,d^2\right ) \]
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