Integrand size = 22, antiderivative size = 123 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=-\frac {8}{3} (3+a)^3 (-1+x)^3+\frac {4}{5} (3-a) (3+a)^2 (-1+x)^5+\frac {8}{7} (3+a) (5+3 a) (-1+x)^7-\frac {2}{9} \left (37+6 a-3 a^2\right ) (-1+x)^9-\frac {8}{11} (5+3 a) (-1+x)^{11}+\frac {4}{13} (3-a) (-1+x)^{13}+\frac {8}{15} (-1+x)^{15}+\frac {1}{17} (-1+x)^{17}+(3+a)^4 x \]
[Out]
Time = 0.18 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1120, 1104} \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=-\frac {2}{9} \left (-3 a^2+6 a+37\right ) (x-1)^9+\frac {4}{13} (3-a) (x-1)^{13}-\frac {8}{11} (3 a+5) (x-1)^{11}+\frac {8}{7} (a+3) (3 a+5) (x-1)^7+\frac {4}{5} (3-a) (a+3)^2 (x-1)^5-\frac {8}{3} (a+3)^3 (x-1)^3+(a+3)^4 x+\frac {1}{17} (x-1)^{17}+\frac {8}{15} (x-1)^{15} \]
[In]
[Out]
Rule 1104
Rule 1120
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \left (3+a-2 x^2-x^4\right )^4 \, dx,x,-1+x\right ) \\ & = \text {Subst}\left (\int \left (81 \left (1+\frac {1}{81} a \left (108+54 a+12 a^2+a^3\right )\right )-216 \left (1+a \left (1+\frac {1}{27} a (9+a)\right )\right ) x^2+108 \left (1-\frac {1}{27} a \left (-9+3 a+a^2\right )\right ) x^4+120 \left (1+\frac {1}{15} a (14+3 a)\right ) x^6-74 \left (1-\frac {3}{37} (-2+a) a\right ) x^8-40 \left (1+\frac {3 a}{5}\right ) x^{10}+12 \left (1-\frac {a}{3}\right ) x^{12}+8 x^{14}+x^{16}\right ) \, dx,x,-1+x\right ) \\ & = -\frac {8}{3} (3+a)^3 (-1+x)^3+\frac {4}{5} (3-a) (3+a)^2 (-1+x)^5+\frac {8}{7} (3+a) (5+3 a) (-1+x)^7-\frac {2}{9} \left (37+6 a-3 a^2\right ) (-1+x)^9-\frac {8}{11} (5+3 a) (-1+x)^{11}+\frac {4}{13} (3-a) (-1+x)^{13}+\frac {8}{15} (-1+x)^{15}+\frac {1}{17} (-1+x)^{17}+(3+a)^4 x \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.59 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=a^4 x+16 a^3 x^2-\frac {32}{3} (-12+a) a^2 x^3+4 a \left (128-48 a+a^2\right ) x^4-\frac {4}{5} \left (-1024+1536 a-192 a^2+a^3\right ) x^5-\frac {16}{3} \left (512-288 a+15 a^2\right ) x^6+\frac {64}{7} \left (512-140 a+3 a^2\right ) x^7-6 \left (896-128 a+a^2\right ) x^8+\frac {2}{9} \left (20480-1536 a+3 a^2\right ) x^9+\frac {16}{5} (-928+35 a) x^{10}-\frac {32}{11} (-524+9 a) x^{11}+\frac {4}{3} (-464+3 a) x^{12}-\frac {4}{13} (-640+a) x^{13}-48 x^{14}+\frac {128 x^{15}}{15}-x^{16}+\frac {x^{17}}{17} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.46
method | result | size |
norman | \(a^{4} x +16 a^{3} x^{2}+\left (-\frac {32}{3} a^{3}+128 a^{2}\right ) x^{3}+\left (4 a^{3}-192 a^{2}+512 a \right ) x^{4}+\left (-\frac {4}{5} a^{3}+\frac {768}{5} a^{2}-\frac {6144}{5} a +\frac {4096}{5}\right ) x^{5}+\left (-80 a^{2}+1536 a -\frac {8192}{3}\right ) x^{6}+\left (\frac {192}{7} a^{2}-1280 a +\frac {32768}{7}\right ) x^{7}+\left (-6 a^{2}+768 a -5376\right ) x^{8}+\left (\frac {2}{3} a^{2}-\frac {1024}{3} a +\frac {40960}{9}\right ) x^{9}+\left (112 a -\frac {14848}{5}\right ) x^{10}+\left (-\frac {288 a}{11}+\frac {16768}{11}\right ) x^{11}+\left (4 a -\frac {1856}{3}\right ) x^{12}+\left (-\frac {4 a}{13}+\frac {2560}{13}\right ) x^{13}-48 x^{14}+\frac {128 x^{15}}{15}-x^{16}+\frac {x^{17}}{17}\) | \(179\) |
gosper | \(-\frac {14848}{5} x^{10}+\frac {16768}{11} x^{11}-\frac {1856}{3} x^{12}+\frac {2560}{13} x^{13}+\frac {4096}{5} x^{5}+\frac {40960}{9} x^{9}+\frac {32768}{7} x^{7}+512 a \,x^{4}-48 x^{14}-5376 x^{8}-\frac {8192}{3} x^{6}-\frac {32}{3} a^{3} x^{3}+16 a^{3} x^{2}-192 a^{2} x^{4}+128 a^{2} x^{3}-1280 a \,x^{7}+a^{4} x +1536 a \,x^{6}-\frac {6144}{5} a \,x^{5}-\frac {4}{5} x^{5} a^{3}+\frac {1}{17} x^{17}-x^{16}+\frac {128}{15} x^{15}+4 a^{3} x^{4}-\frac {4}{13} x^{13} a -6 a^{2} x^{8}+\frac {2}{3} x^{9} a^{2}-\frac {1024}{3} x^{9} a +112 x^{10} a -\frac {288}{11} x^{11} a +4 x^{12} a +768 a \,x^{8}+\frac {192}{7} a^{2} x^{7}-80 a^{2} x^{6}+\frac {768}{5} a^{2} x^{5}\) | \(220\) |
risch | \(-\frac {14848}{5} x^{10}+\frac {16768}{11} x^{11}-\frac {1856}{3} x^{12}+\frac {2560}{13} x^{13}+\frac {4096}{5} x^{5}+\frac {40960}{9} x^{9}+\frac {32768}{7} x^{7}+512 a \,x^{4}-48 x^{14}-5376 x^{8}-\frac {8192}{3} x^{6}-\frac {32}{3} a^{3} x^{3}+16 a^{3} x^{2}-192 a^{2} x^{4}+128 a^{2} x^{3}-1280 a \,x^{7}+a^{4} x +1536 a \,x^{6}-\frac {6144}{5} a \,x^{5}-\frac {4}{5} x^{5} a^{3}+\frac {1}{17} x^{17}-x^{16}+\frac {128}{15} x^{15}+4 a^{3} x^{4}-\frac {4}{13} x^{13} a -6 a^{2} x^{8}+\frac {2}{3} x^{9} a^{2}-\frac {1024}{3} x^{9} a +112 x^{10} a -\frac {288}{11} x^{11} a +4 x^{12} a +768 a \,x^{8}+\frac {192}{7} a^{2} x^{7}-80 a^{2} x^{6}+\frac {768}{5} a^{2} x^{5}\) | \(220\) |
parallelrisch | \(-\frac {14848}{5} x^{10}+\frac {16768}{11} x^{11}-\frac {1856}{3} x^{12}+\frac {2560}{13} x^{13}+\frac {4096}{5} x^{5}+\frac {40960}{9} x^{9}+\frac {32768}{7} x^{7}+512 a \,x^{4}-48 x^{14}-5376 x^{8}-\frac {8192}{3} x^{6}-\frac {32}{3} a^{3} x^{3}+16 a^{3} x^{2}-192 a^{2} x^{4}+128 a^{2} x^{3}-1280 a \,x^{7}+a^{4} x +1536 a \,x^{6}-\frac {6144}{5} a \,x^{5}-\frac {4}{5} x^{5} a^{3}+\frac {1}{17} x^{17}-x^{16}+\frac {128}{15} x^{15}+4 a^{3} x^{4}-\frac {4}{13} x^{13} a -6 a^{2} x^{8}+\frac {2}{3} x^{9} a^{2}-\frac {1024}{3} x^{9} a +112 x^{10} a -\frac {288}{11} x^{11} a +4 x^{12} a +768 a \,x^{8}+\frac {192}{7} a^{2} x^{7}-80 a^{2} x^{6}+\frac {768}{5} a^{2} x^{5}\) | \(220\) |
default | \(\frac {x^{17}}{17}-x^{16}+\frac {128 x^{15}}{15}-48 x^{14}+\frac {\left (-4 a +2560\right ) x^{13}}{13}+\frac {\left (48 a -7424\right ) x^{12}}{12}+\frac {\left (-288 a +16768\right ) x^{11}}{11}+\frac {\left (1120 a -29696\right ) x^{10}}{10}+\frac {\left (2 a^{2}-2560 a +24576+\left (-2 a +128\right )^{2}\right ) x^{9}}{9}+\frac {\left (-16 a^{2}+3584 a -10240+2 \left (8 a -128\right ) \left (-2 a +128\right )\right ) x^{8}}{8}+\frac {\left (64 a^{2}-2560 a +2 \left (-16 a +64\right ) \left (-2 a +128\right )+\left (8 a -128\right )^{2}\right ) x^{7}}{7}+\frac {\left (-160 a^{2}+32 a \left (-2 a +128\right )+2 \left (-16 a +64\right ) \left (8 a -128\right )\right ) x^{6}}{6}+\frac {\left (2 a^{2} \left (-2 a +128\right )+32 a \left (8 a -128\right )+\left (-16 a +64\right )^{2}\right ) x^{5}}{5}+\frac {\left (2 a^{2} \left (8 a -128\right )+32 a \left (-16 a +64\right )\right ) x^{4}}{4}+\frac {\left (2 a^{2} \left (-16 a +64\right )+256 a^{2}\right ) x^{3}}{3}+16 a^{3} x^{2}+a^{4} x\) | \(264\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.46 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=\frac {1}{17} \, x^{17} - x^{16} + \frac {128}{15} \, x^{15} - \frac {4}{13} \, {\left (a - 640\right )} x^{13} - 48 \, x^{14} + \frac {4}{3} \, {\left (3 \, a - 464\right )} x^{12} - \frac {32}{11} \, {\left (9 \, a - 524\right )} x^{11} + \frac {16}{5} \, {\left (35 \, a - 928\right )} x^{10} + \frac {2}{9} \, {\left (3 \, a^{2} - 1536 \, a + 20480\right )} x^{9} - 6 \, {\left (a^{2} - 128 \, a + 896\right )} x^{8} + \frac {64}{7} \, {\left (3 \, a^{2} - 140 \, a + 512\right )} x^{7} - \frac {16}{3} \, {\left (15 \, a^{2} - 288 \, a + 512\right )} x^{6} - \frac {4}{5} \, {\left (a^{3} - 192 \, a^{2} + 1536 \, a - 1024\right )} x^{5} + a^{4} x + 16 \, a^{3} x^{2} + 4 \, {\left (a^{3} - 48 \, a^{2} + 128 \, a\right )} x^{4} - \frac {32}{3} \, {\left (a^{3} - 12 \, a^{2}\right )} x^{3} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.62 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=a^{4} x + 16 a^{3} x^{2} + \frac {x^{17}}{17} - x^{16} + \frac {128 x^{15}}{15} - 48 x^{14} + x^{13} \cdot \left (\frac {2560}{13} - \frac {4 a}{13}\right ) + x^{12} \cdot \left (4 a - \frac {1856}{3}\right ) + x^{11} \cdot \left (\frac {16768}{11} - \frac {288 a}{11}\right ) + x^{10} \cdot \left (112 a - \frac {14848}{5}\right ) + x^{9} \cdot \left (\frac {2 a^{2}}{3} - \frac {1024 a}{3} + \frac {40960}{9}\right ) + x^{8} \left (- 6 a^{2} + 768 a - 5376\right ) + x^{7} \cdot \left (\frac {192 a^{2}}{7} - 1280 a + \frac {32768}{7}\right ) + x^{6} \left (- 80 a^{2} + 1536 a - \frac {8192}{3}\right ) + x^{5} \left (- \frac {4 a^{3}}{5} + \frac {768 a^{2}}{5} - \frac {6144 a}{5} + \frac {4096}{5}\right ) + x^{4} \cdot \left (4 a^{3} - 192 a^{2} + 512 a\right ) + x^{3} \left (- \frac {32 a^{3}}{3} + 128 a^{2}\right ) \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.56 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=\frac {1}{17} \, x^{17} - x^{16} + \frac {128}{15} \, x^{15} - 48 \, x^{14} + \frac {2560}{13} \, x^{13} - \frac {1856}{3} \, x^{12} + \frac {16768}{11} \, x^{11} - \frac {14848}{5} \, x^{10} + \frac {40960}{9} \, x^{9} - 5376 \, x^{8} + \frac {32768}{7} \, x^{7} - \frac {8192}{3} \, x^{6} + a^{4} x + \frac {4096}{5} \, x^{5} - \frac {4}{15} \, {\left (3 \, x^{5} - 15 \, x^{4} + 40 \, x^{3} - 60 \, x^{2}\right )} a^{3} + \frac {2}{105} \, {\left (35 \, x^{9} - 315 \, x^{8} + 1440 \, x^{7} - 4200 \, x^{6} + 8064 \, x^{5} - 10080 \, x^{4} + 6720 \, x^{3}\right )} a^{2} - \frac {4}{2145} \, {\left (165 \, x^{13} - 2145 \, x^{12} + 14040 \, x^{11} - 60060 \, x^{10} + 183040 \, x^{9} - 411840 \, x^{8} + 686400 \, x^{7} - 823680 \, x^{6} + 658944 \, x^{5} - 274560 \, x^{4}\right )} a \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (103) = 206\).
Time = 0.28 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.78 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=\frac {1}{17} \, x^{17} - x^{16} + \frac {128}{15} \, x^{15} - \frac {4}{13} \, a x^{13} - 48 \, x^{14} + 4 \, a x^{12} + \frac {2560}{13} \, x^{13} - \frac {288}{11} \, a x^{11} - \frac {1856}{3} \, x^{12} + \frac {2}{3} \, a^{2} x^{9} + 112 \, a x^{10} + \frac {16768}{11} \, x^{11} - 6 \, a^{2} x^{8} - \frac {1024}{3} \, a x^{9} - \frac {14848}{5} \, x^{10} + \frac {192}{7} \, a^{2} x^{7} + 768 \, a x^{8} + \frac {40960}{9} \, x^{9} - \frac {4}{5} \, a^{3} x^{5} - 80 \, a^{2} x^{6} - 1280 \, a x^{7} - 5376 \, x^{8} + 4 \, a^{3} x^{4} + \frac {768}{5} \, a^{2} x^{5} + 1536 \, a x^{6} + \frac {32768}{7} \, x^{7} - \frac {32}{3} \, a^{3} x^{3} - 192 \, a^{2} x^{4} - \frac {6144}{5} \, a x^{5} - \frac {8192}{3} \, x^{6} + a^{4} x + 16 \, a^{3} x^{2} + 128 \, a^{2} x^{3} + 512 \, a x^{4} + \frac {4096}{5} \, x^{5} \]
[In]
[Out]
Time = 0.20 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.42 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=x^{12}\,\left (4\,a-\frac {1856}{3}\right )-x^{13}\,\left (\frac {4\,a}{13}-\frac {2560}{13}\right )+x^{10}\,\left (112\,a-\frac {14848}{5}\right )-x^{11}\,\left (\frac {288\,a}{11}-\frac {16768}{11}\right )-x^8\,\left (6\,a^2-768\,a+5376\right )-x^6\,\left (80\,a^2-1536\,a+\frac {8192}{3}\right )+x^7\,\left (\frac {192\,a^2}{7}-1280\,a+\frac {32768}{7}\right )+x^9\,\left (\frac {2\,a^2}{3}-\frac {1024\,a}{3}+\frac {40960}{9}\right )-x^5\,\left (\frac {4\,a^3}{5}-\frac {768\,a^2}{5}+\frac {6144\,a}{5}-\frac {4096}{5}\right )+a^4\,x-48\,x^{14}+\frac {128\,x^{15}}{15}-x^{16}+\frac {x^{17}}{17}+16\,a^3\,x^2+4\,a\,x^4\,\left (a^2-48\,a+128\right )-\frac {32\,a^2\,x^3\,\left (a-12\right )}{3} \]
[In]
[Out]