Integrand size = 33, antiderivative size = 78 \[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {4 \left ((-1+x)^3\right )^{3/4} \left (1308401597431+32327777464 x-1158885626660 x^2-2834315032620 x^3-2008108342110 x^4+4070651298324 x^5+9260757242646 x^6+3131067556500 x^7-8805988591725 x^8-9283999210200 x^9+609206533650 x^{10}+4908866519700 x^{11}+1949108765175 x^{12}\right )}{1179090487575 (-1+x)^2} \] Output:
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 \[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {4 (-1+x) \left (1308401597431+32327777464 x-1158885626660 x^2-2834315032620 x^3-2008108342110 x^4+4070651298324 x^5+9260757242646 x^6+3131067556500 x^7-8805988591725 x^8-9283999210200 x^9+609206533650 x^{10}+4908866519700 x^{11}+1949108765175 x^{12}\right )}{1179090487575 \sqrt [4]{(-1+x)^3}} \] Input:
Integrate[(-1 - 2*x + x^2 + 3*x^3)^4/(-1 + 3*x - 3*x^2 + x^3)^(1/4),x]
Output:
(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 = 0.31 (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 |
| \(\displaystyle \frac {\left (\frac {324}{49} (x-1)^{49/4}+96 (x-1)^{45/4}+\frac {25488}{41} (x-1)^{41/4}+\frac {87312}{37} (x-1)^{37/4}+\frac {191416}{33} (x-1)^{33/4}+\frac {278928}{29} (x-1)^{29/4}+\frac {271528}{25} (x-1)^{25/4}+\frac {57648}{7} (x-1)^{21/4}+\frac {68820}{17} (x-1)^{17/4}+\frac {16032}{13} (x-1)^{13/4}+\frac {2104}{9} (x-1)^{9/4}+\frac {144}{5} (x-1)^{5/4}+4 \sqrt [4]{x-1}\right ) (x-1)^{3/4}}{\sqrt [4]{(x-1)^3}}\) |
Input:
Int[(-1 - 2*x + x^2 + 3*x^3)^4/(-1 + 3*x - 3*x^2 + x^3)^(1/4),x]
Output:
((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)
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 = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(\frac {4 \left (-1+x \right ) \left (1949108765175 x^{12}+4908866519700 x^{11}+609206533650 x^{10}-9283999210200 x^{9}-8805988591725 x^{8}+3131067556500 x^{7}+9260757242646 x^{6}+4070651298324 x^{5}-2008108342110 x^{4}-2834315032620 x^{3}-1158885626660 x^{2}+32327777464 x +1308401597431\right )}{1179090487575 \left (\left (-1+x \right )^{3}\right )^{\frac {1}{4}}}\) | \(73\) |
| gosper | \(\frac {4 \left (-1+x \right ) \left (1949108765175 x^{12}+4908866519700 x^{11}+609206533650 x^{10}-9283999210200 x^{9}-8805988591725 x^{8}+3131067556500 x^{7}+9260757242646 x^{6}+4070651298324 x^{5}-2008108342110 x^{4}-2834315032620 x^{3}-1158885626660 x^{2}+32327777464 x +1308401597431\right )}{1179090487575 \left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}\) | \(81\) |
| orering | \(\frac {4 \left (-1+x \right ) \left (1949108765175 x^{12}+4908866519700 x^{11}+609206533650 x^{10}-9283999210200 x^{9}-8805988591725 x^{8}+3131067556500 x^{7}+9260757242646 x^{6}+4070651298324 x^{5}-2008108342110 x^{4}-2834315032620 x^{3}-1158885626660 x^{2}+32327777464 x +1308401597431\right )}{1179090487575 \left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}\) | \(81\) |
| trager | \(\frac {4 \left (1949108765175 x^{12}+4908866519700 x^{11}+609206533650 x^{10}-9283999210200 x^{9}-8805988591725 x^{8}+3131067556500 x^{7}+9260757242646 x^{6}+4070651298324 x^{5}-2008108342110 x^{4}-2834315032620 x^{3}-1158885626660 x^{2}+32327777464 x +1308401597431\right ) \left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {3}{4}}}{1179090487575 \left (-1+x \right )^{2}}\) | \(83\) |
Input:
int((3*x^3+x^2-2*x-1)^4/(x^3-3*x^2+3*x-1)^(1/4),x,method=_RETURNVERBOSE)
Output:
4/1179090487575*(-1+x)*(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)/((-1+x)^3)^(1/4)
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.12 \[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {4 \, {\left (1949108765175 \, x^{12} + 4908866519700 \, x^{11} + 609206533650 \, x^{10} - 9283999210200 \, x^{9} - 8805988591725 \, x^{8} + 3131067556500 \, x^{7} + 9260757242646 \, x^{6} + 4070651298324 \, x^{5} - 2008108342110 \, x^{4} - 2834315032620 \, x^{3} - 1158885626660 \, x^{2} + 32327777464 \, x + 1308401597431\right )} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {3}{4}}}{1179090487575 \, {\left (x^{2} - 2 \, x + 1\right )}} \] Input:
integrate((3*x^3+x^2-2*x-1)^4/(x^3-3*x^2+3*x-1)^(1/4),x, algorithm="fricas ")
Output:
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)
\[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\int \frac {\left (3 x^{3} + x^{2} - 2 x - 1\right )^{4}}{\sqrt [4]{\left (x - 1\right )^{3}}}\, dx \] Input:
integrate((3*x**3+x**2-2*x-1)**4/(x**3-3*x**2+3*x-1)**(1/4),x)
Output:
Integral((3*x**3 + x**2 - 2*x - 1)**4/((x - 1)**3)**(1/4), x)
Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (74) = 148\).
Time = 0.05 (sec) , antiderivative size = 540, normalized size of antiderivative = 6.92 \[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx =\text {Too large to display} \] Input:
integrate((3*x^3+x^2-2*x-1)^4/(x^3-3*x^2+3*x-1)^(1/4),x, algorithm="maxima ")
Output:
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...
\[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\int { \frac {{\left (3 \, x^{3} + x^{2} - 2 \, x - 1\right )}^{4}}{{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((3*x^3+x^2-2*x-1)^4/(x^3-3*x^2+3*x-1)^(1/4),x, algorithm="giac")
Output:
integrate((3*x^3 + x^2 - 2*x - 1)^4/(x^3 - 3*x^2 + 3*x - 1)^(1/4), x)
Time = 7.84 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {{\left (x^3-3\,x^2+3\,x-1\right )}^{3/4}\,\left (\frac {324\,x^{12}}{49}+\frac {816\,x^{11}}{49}+\frac {4152\,x^{10}}{2009}-\frac {2341152\,x^9}{74333}-\frac {73280188\,x^8}{2452989}+\frac {755612080\,x^7}{71136681}+\frac {7981691224\,x^6}{254059575}+\frac {24558982192\,x^5}{1778417025}-\frac {41191965992\,x^4}{6046617885}-\frac {755817342032\,x^3}{78606032505}-\frac {927108501328\,x^2}{235818097515}+\frac {129311109856\,x}{1179090487575}+\frac {5233606389724}{1179090487575}\right )}{x^2-2\,x+1} \] Input:
int((2*x - x^2 - 3*x^3 + 1)^4/(3*x - 3*x^2 + x^3 - 1)^(1/4),x)
Output:
((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)
\[ \int \frac {\left (-1-2 x+x^2+3 x^3\right )^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=81 \left (\int \frac {x^{12}}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}d x \right )+108 \left (\int \frac {x^{11}}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}d x \right )-162 \left (\int \frac {x^{10}}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}d x \right )-312 \left (\int \frac {x^{9}}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}d x \right )+37 \left (\int \frac {x^{8}}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}d x \right )+316 \left (\int \frac {x^{7}}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}d x \right )+122 \left (\int \frac {x^{6}}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}d x \right )-116 \left (\int \frac {x^{5}}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}d x \right )-98 \left (\int \frac {x^{4}}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}d x \right )-4 \left (\int \frac {x^{3}}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}d x \right )+20 \left (\int \frac {x^{2}}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}d x \right )+8 \left (\int \frac {x}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}d x \right )+\int \frac {1}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}d x \] Input:
int((3*x^3+x^2-2*x-1)^4/(x^3-3*x^2+3*x-1)^(1/4),x)
Output:
81*int(x**12/(x**3 - 3*x**2 + 3*x - 1)**(1/4),x) + 108*int(x**11/(x**3 - 3 *x**2 + 3*x - 1)**(1/4),x) - 162*int(x**10/(x**3 - 3*x**2 + 3*x - 1)**(1/4 ),x) - 312*int(x**9/(x**3 - 3*x**2 + 3*x - 1)**(1/4),x) + 37*int(x**8/(x** 3 - 3*x**2 + 3*x - 1)**(1/4),x) + 316*int(x**7/(x**3 - 3*x**2 + 3*x - 1)** (1/4),x) + 122*int(x**6/(x**3 - 3*x**2 + 3*x - 1)**(1/4),x) - 116*int(x**5 /(x**3 - 3*x**2 + 3*x - 1)**(1/4),x) - 98*int(x**4/(x**3 - 3*x**2 + 3*x - 1)**(1/4),x) - 4*int(x**3/(x**3 - 3*x**2 + 3*x - 1)**(1/4),x) + 20*int(x** 2/(x**3 - 3*x**2 + 3*x - 1)**(1/4),x) + 8*int(x/(x**3 - 3*x**2 + 3*x - 1)* *(1/4),x) + int(1/(x**3 - 3*x**2 + 3*x - 1)**(1/4),x)