Integrand size = 33, antiderivative size = 78 \[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {4 \left ((-1+x)^3\right )^{3/4} \left (1308401597431+32327777464 x-1158885626660 x^2-2834315032620 x^3-2008108342110 x^4+4070651298324 x^5+9260757242646 x^6+3131067556500 x^7-8805988591725 x^8-9283999210200 x^9+609206533650 x^{10}+4908866519700 x^{11}+1949108765175 x^{12}\right )}{1179090487575 (-1+x)^2} \]
4/1179090487575*((-1+x)^3)^(3/4)*(1949108765175*x^12+4908866519700*x^11+60 9206533650*x^10-9283999210200*x^9-8805988591725*x^8+3131067556500*x^7+9260 757242646*x^6+4070651298324*x^5-2008108342110*x^4-2834315032620*x^3-115888 5626660*x^2+32327777464*x+1308401597431)/(-1+x)^2
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {4 (-1+x) \left (1308401597431+32327777464 x-1158885626660 x^2-2834315032620 x^3-2008108342110 x^4+4070651298324 x^5+9260757242646 x^6+3131067556500 x^7-8805988591725 x^8-9283999210200 x^9+609206533650 x^{10}+4908866519700 x^{11}+1949108765175 x^{12}\right )}{1179090487575 \sqrt [4]{(-1+x)^3}} \]
(4*(-1 + x)*(1308401597431 + 32327777464*x - 1158885626660*x^2 - 283431503 2620*x^3 - 2008108342110*x^4 + 4070651298324*x^5 + 9260757242646*x^6 + 313 1067556500*x^7 - 8805988591725*x^8 - 9283999210200*x^9 + 609206533650*x^10 + 4908866519700*x^11 + 1949108765175*x^12))/(1179090487575*((-1 + x)^3)^( 1/4))
Leaf count is larger than twice the leaf count of optimal. \(157\) vs. \(2(78)=156\).
Time = 0.32 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2008, 2389, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (3 x^3+x^2-2 x-1\right )^4}{\sqrt [4]{x^3-3 x^2+3 x-1}} \, dx\) |
| \(\Big \downarrow \) 2008 |
| \(\displaystyle \frac {(x-1)^{3/4} \int \frac {\left (-3 x^3-x^2+2 x+1\right )^4}{(x-1)^{3/4}}dx}{\sqrt [4]{(x-1)^3}}\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \frac {(x-1)^{3/4} \int \left (81 (x-1)^{45/4}+1080 (x-1)^{41/4}+6372 (x-1)^{37/4}+21828 (x-1)^{33/4}+47854 (x-1)^{29/4}+69732 (x-1)^{25/4}+67882 (x-1)^{21/4}+43236 (x-1)^{17/4}+17205 (x-1)^{13/4}+4008 (x-1)^{9/4}+526 (x-1)^{5/4}+36 \sqrt [4]{x-1}+\frac {1}{(x-1)^{3/4}}\right )dx}{\sqrt [4]{(x-1)^3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (\frac {324}{49} (x-1)^{49/4}+96 (x-1)^{45/4}+\frac {25488}{41} (x-1)^{41/4}+\frac {87312}{37} (x-1)^{37/4}+\frac {191416}{33} (x-1)^{33/4}+\frac {278928}{29} (x-1)^{29/4}+\frac {271528}{25} (x-1)^{25/4}+\frac {57648}{7} (x-1)^{21/4}+\frac {68820}{17} (x-1)^{17/4}+\frac {16032}{13} (x-1)^{13/4}+\frac {2104}{9} (x-1)^{9/4}+\frac {144}{5} (x-1)^{5/4}+4 \sqrt [4]{x-1}\right ) (x-1)^{3/4}}{\sqrt [4]{(x-1)^3}}\) |
((4*(-1 + x)^(1/4) + (144*(-1 + x)^(5/4))/5 + (2104*(-1 + x)^(9/4))/9 + (1 6032*(-1 + x)^(13/4))/13 + (68820*(-1 + x)^(17/4))/17 + (57648*(-1 + x)^(2 1/4))/7 + (271528*(-1 + x)^(25/4))/25 + (278928*(-1 + x)^(29/4))/29 + (191 416*(-1 + x)^(33/4))/33 + (87312*(-1 + x)^(37/4))/37 + (25488*(-1 + x)^(41 /4))/41 + 96*(-1 + x)^(45/4) + (324*(-1 + x)^(49/4))/49)*(-1 + x)^(3/4))/( (-1 + x)^3)^(1/4)
3.11.28.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Simp[((a + b*x)^Exp on[Px, x])^p/(a + b*x)^(Expon[Px, x]*p) Int[u*(a + b*x)^(Expon[Px, x]*p), x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; !IntegerQ[p] && PolyQ[Px, x ] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(\frac {4 \left (x -1\right ) \left (1949108765175 x^{12}+4908866519700 x^{11}+609206533650 x^{10}-9283999210200 x^{9}-8805988591725 x^{8}+3131067556500 x^{7}+9260757242646 x^{6}+4070651298324 x^{5}-2008108342110 x^{4}-2834315032620 x^{3}-1158885626660 x^{2}+32327777464 x +1308401597431\right )}{1179090487575 \left (\left (x -1\right )^{3}\right )^{\frac {1}{4}}}\) | \(73\) |
| gosper | \(\frac {4 \left (x -1\right ) \left (1949108765175 x^{12}+4908866519700 x^{11}+609206533650 x^{10}-9283999210200 x^{9}-8805988591725 x^{8}+3131067556500 x^{7}+9260757242646 x^{6}+4070651298324 x^{5}-2008108342110 x^{4}-2834315032620 x^{3}-1158885626660 x^{2}+32327777464 x +1308401597431\right )}{1179090487575 \left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}\) | \(81\) |
| trager | \(\frac {4 \left (1949108765175 x^{12}+4908866519700 x^{11}+609206533650 x^{10}-9283999210200 x^{9}-8805988591725 x^{8}+3131067556500 x^{7}+9260757242646 x^{6}+4070651298324 x^{5}-2008108342110 x^{4}-2834315032620 x^{3}-1158885626660 x^{2}+32327777464 x +1308401597431\right ) \left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {3}{4}}}{1179090487575 \left (x -1\right )^{2}}\) | \(83\) |
4/1179090487575*(x-1)*(1949108765175*x^12+4908866519700*x^11+609206533650* x^10-9283999210200*x^9-8805988591725*x^8+3131067556500*x^7+9260757242646*x ^6+4070651298324*x^5-2008108342110*x^4-2834315032620*x^3-1158885626660*x^2 +32327777464*x+1308401597431)/((x-1)^3)^(1/4)
Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.12 \[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {4 \, {\left (1949108765175 \, x^{12} + 4908866519700 \, x^{11} + 609206533650 \, x^{10} - 9283999210200 \, x^{9} - 8805988591725 \, x^{8} + 3131067556500 \, x^{7} + 9260757242646 \, x^{6} + 4070651298324 \, x^{5} - 2008108342110 \, x^{4} - 2834315032620 \, x^{3} - 1158885626660 \, x^{2} + 32327777464 \, x + 1308401597431\right )} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {3}{4}}}{1179090487575 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
4/1179090487575*(1949108765175*x^12 + 4908866519700*x^11 + 609206533650*x^ 10 - 9283999210200*x^9 - 8805988591725*x^8 + 3131067556500*x^7 + 926075724 2646*x^6 + 4070651298324*x^5 - 2008108342110*x^4 - 2834315032620*x^3 - 115 8885626660*x^2 + 32327777464*x + 1308401597431)*(x^3 - 3*x^2 + 3*x - 1)^(3 /4)/(x^2 - 2*x + 1)
\[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\int \frac {\left (3 x^{3} + x^{2} - 2 x - 1\right )^{4}}{\sqrt [4]{\left (x - 1\right )^{3}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (74) = 148\).
Time = 0.22 (sec) , antiderivative size = 540, normalized size of antiderivative = 6.92 \[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {108 \, {\left (729183975 \, x^{13} + 48612265 \, x^{12} + 56911920 \, x^{11} + 67679040 \, x^{10} + 82035200 \, x^{9} + 101836800 \, x^{8} + 130351104 \, x^{7} + 173801472 \, x^{6} + 245366784 \, x^{5} + 377487360 \, x^{4} + 671088640 \, x^{3} + 1610612736 \, x^{2} + 12884901888 \, x - 17179869184\right )}}{11910004925 \, {\left (x - 1\right )}^{\frac {3}{4}}} + \frac {48 \, {\left (48612265 \, x^{12} + 3556995 \, x^{11} + 4229940 \, x^{10} + 5127200 \, x^{9} + 6364800 \, x^{8} + 8146944 \, x^{7} + 10862592 \, x^{6} + 15335424 \, x^{5} + 23592960 \, x^{4} + 41943040 \, x^{3} + 100663296 \, x^{2} + 805306368 \, x - 1073741824\right )}}{243061325 \, {\left (x - 1\right )}^{\frac {3}{4}}} - \frac {648 \, {\left (13042315 \, x^{11} + 1057485 \, x^{10} + 1281800 \, x^{9} + 1591200 \, x^{8} + 2036736 \, x^{7} + 2715648 \, x^{6} + 3833856 \, x^{5} + 5898240 \, x^{4} + 10485760 \, x^{3} + 25165824 \, x^{2} + 201326592 \, x - 268435456\right )}}{534734915 \, {\left (x - 1\right )}^{\frac {3}{4}}} - \frac {96 \, {\left (1762475 \, x^{10} + 160225 \, x^{9} + 198900 \, x^{8} + 254592 \, x^{7} + 339456 \, x^{6} + 479232 \, x^{5} + 737280 \, x^{4} + 1310720 \, x^{3} + 3145728 \, x^{2} + 25165824 \, x - 33554432\right )}}{5016275 \, {\left (x - 1\right )}^{\frac {3}{4}}} + \frac {148 \, {\left (480675 \, x^{9} + 49725 \, x^{8} + 63648 \, x^{7} + 84864 \, x^{6} + 119808 \, x^{5} + 184320 \, x^{4} + 327680 \, x^{3} + 786432 \, x^{2} + 6291456 \, x - 8388608\right )}}{15862275 \, {\left (x - 1\right )}^{\frac {3}{4}}} + \frac {1264 \, {\left (16575 \, x^{8} + 1989 \, x^{7} + 2652 \, x^{6} + 3744 \, x^{5} + 5760 \, x^{4} + 10240 \, x^{3} + 24576 \, x^{2} + 196608 \, x - 262144\right )}}{480675 \, {\left (x - 1\right )}^{\frac {3}{4}}} + \frac {488 \, {\left (4641 \, x^{7} + 663 \, x^{6} + 936 \, x^{5} + 1440 \, x^{4} + 2560 \, x^{3} + 6144 \, x^{2} + 49152 \, x - 65536\right )}}{116025 \, {\left (x - 1\right )}^{\frac {3}{4}}} - \frac {464 \, {\left (663 \, x^{6} + 117 \, x^{5} + 180 \, x^{4} + 320 \, x^{3} + 768 \, x^{2} + 6144 \, x - 8192\right )}}{13923 \, {\left (x - 1\right )}^{\frac {3}{4}}} - \frac {392 \, {\left (195 \, x^{5} + 45 \, x^{4} + 80 \, x^{3} + 192 \, x^{2} + 1536 \, x - 2048\right )}}{3315 \, {\left (x - 1\right )}^{\frac {3}{4}}} - \frac {16 \, {\left (15 \, x^{4} + 5 \, x^{3} + 12 \, x^{2} + 96 \, x - 128\right )}}{195 \, {\left (x - 1\right )}^{\frac {3}{4}}} + \frac {16 \, {\left (5 \, x^{3} + 3 \, x^{2} + 24 \, x - 32\right )}}{9 \, {\left (x - 1\right )}^{\frac {3}{4}}} + \frac {32 \, {\left (x^{2} + 3 \, x - 4\right )}}{5 \, {\left (x - 1\right )}^{\frac {3}{4}}} + 4 \, {\left (x - 1\right )}^{\frac {1}{4}} \]
108/11910004925*(729183975*x^13 + 48612265*x^12 + 56911920*x^11 + 67679040 *x^10 + 82035200*x^9 + 101836800*x^8 + 130351104*x^7 + 173801472*x^6 + 245 366784*x^5 + 377487360*x^4 + 671088640*x^3 + 1610612736*x^2 + 12884901888* x - 17179869184)/(x - 1)^(3/4) + 48/243061325*(48612265*x^12 + 3556995*x^1 1 + 4229940*x^10 + 5127200*x^9 + 6364800*x^8 + 8146944*x^7 + 10862592*x^6 + 15335424*x^5 + 23592960*x^4 + 41943040*x^3 + 100663296*x^2 + 805306368*x - 1073741824)/(x - 1)^(3/4) - 648/534734915*(13042315*x^11 + 1057485*x^10 + 1281800*x^9 + 1591200*x^8 + 2036736*x^7 + 2715648*x^6 + 3833856*x^5 + 5 898240*x^4 + 10485760*x^3 + 25165824*x^2 + 201326592*x - 268435456)/(x - 1 )^(3/4) - 96/5016275*(1762475*x^10 + 160225*x^9 + 198900*x^8 + 254592*x^7 + 339456*x^6 + 479232*x^5 + 737280*x^4 + 1310720*x^3 + 3145728*x^2 + 25165 824*x - 33554432)/(x - 1)^(3/4) + 148/15862275*(480675*x^9 + 49725*x^8 + 6 3648*x^7 + 84864*x^6 + 119808*x^5 + 184320*x^4 + 327680*x^3 + 786432*x^2 + 6291456*x - 8388608)/(x - 1)^(3/4) + 1264/480675*(16575*x^8 + 1989*x^7 + 2652*x^6 + 3744*x^5 + 5760*x^4 + 10240*x^3 + 24576*x^2 + 196608*x - 262144 )/(x - 1)^(3/4) + 488/116025*(4641*x^7 + 663*x^6 + 936*x^5 + 1440*x^4 + 25 60*x^3 + 6144*x^2 + 49152*x - 65536)/(x - 1)^(3/4) - 464/13923*(663*x^6 + 117*x^5 + 180*x^4 + 320*x^3 + 768*x^2 + 6144*x - 8192)/(x - 1)^(3/4) - 392 /3315*(195*x^5 + 45*x^4 + 80*x^3 + 192*x^2 + 1536*x - 2048)/(x - 1)^(3/4) - 16/195*(15*x^4 + 5*x^3 + 12*x^2 + 96*x - 128)/(x - 1)^(3/4) + 16/9*(5...
\[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\int { \frac {{\left (3 \, x^{3} + x^{2} - 2 \, x - 1\right )}^{4}}{{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {1}{4}}} \,d x } \]
Time = 5.95 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {{\left (x^3-3\,x^2+3\,x-1\right )}^{3/4}\,\left (\frac {324\,x^{12}}{49}+\frac {816\,x^{11}}{49}+\frac {4152\,x^{10}}{2009}-\frac {2341152\,x^9}{74333}-\frac {73280188\,x^8}{2452989}+\frac {755612080\,x^7}{71136681}+\frac {7981691224\,x^6}{254059575}+\frac {24558982192\,x^5}{1778417025}-\frac {41191965992\,x^4}{6046617885}-\frac {755817342032\,x^3}{78606032505}-\frac {927108501328\,x^2}{235818097515}+\frac {129311109856\,x}{1179090487575}+\frac {5233606389724}{1179090487575}\right )}{x^2-2\,x+1} \]
((3*x - 3*x^2 + x^3 - 1)^(3/4)*((129311109856*x)/1179090487575 - (92710850 1328*x^2)/235818097515 - (755817342032*x^3)/78606032505 - (41191965992*x^4 )/6046617885 + (24558982192*x^5)/1778417025 + (7981691224*x^6)/254059575 + (755612080*x^7)/71136681 - (73280188*x^8)/2452989 - (2341152*x^9)/74333 + (4152*x^10)/2009 + (816*x^11)/49 + (324*x^12)/49 + 5233606389724/11790904 87575))/(x^2 - 2*x + 1)