3.11.28 \(\int \frac {(-1-2 x+x^2+3 x^3)^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx\)

Optimal. Leaf size=78 \[ \frac {4 \left ((x-1)^3\right )^{3/4} \left (1949108765175 x^{12}[phone] x^{11}[phone] x^{10}-9283999210200 x^9-8805988591725 x^8+3131067556500 x^7+9260757242646 x^6+4070651298324 x^5-2008108342110 x^4-2834315032620 x^3-1158885626660 x^2+32327777464 x+1308401597431\right )}{1179090487575 (x-1)^2} \]

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Rubi [B]  time = 1.00, antiderivative size = 248, normalized size of antiderivative = 3.18, number of steps used = 53, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6742, 2067, 15, 30, 2081, 43} \begin {gather*} -\frac {324 (1-x)^{13}}{49 \sqrt [4]{(x-1)^3}}+\frac {96 (1-x)^{12}}{\sqrt [4]{(x-1)^3}}-\frac {25488 (1-x)^{11}}{41 \sqrt [4]{(x-1)^3}}+\frac {87312 (1-x)^{10}}{37 \sqrt [4]{(x-1)^3}}-\frac {191416 (1-x)^9}{33 \sqrt [4]{(x-1)^3}}+\frac {278928 (1-x)^8}{29 \sqrt [4]{(x-1)^3}}-\frac {271528 (1-x)^7}{25 \sqrt [4]{(x-1)^3}}+\frac {57648 (1-x)^6}{7 \sqrt [4]{(x-1)^3}}-\frac {68820 (1-x)^5}{17 \sqrt [4]{(x-1)^3}}+\frac {16032 (1-x)^4}{13 \sqrt [4]{(x-1)^3}}+\frac {144 (1-x)^2}{5 \sqrt [4]{(x-1)^3}}-\frac {4 (1-x)}{\sqrt [4]{(x-1)^3}}+\frac {2104}{9} \left ((x-1)^3\right )^{3/4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - 2*x + x^2 + 3*x^3)^4/(-1 + 3*x - 3*x^2 + x^3)^(1/4),x]

[Out]

(-4*(1 - x))/((-1 + x)^3)^(1/4) + (144*(1 - x)^2)/(5*((-1 + x)^3)^(1/4)) + (16032*(1 - x)^4)/(13*((-1 + x)^3)^
(1/4)) - (68820*(1 - x)^5)/(17*((-1 + x)^3)^(1/4)) + (57648*(1 - x)^6)/(7*((-1 + x)^3)^(1/4)) - (271528*(1 - x
)^7)/(25*((-1 + x)^3)^(1/4)) + (278928*(1 - x)^8)/(29*((-1 + x)^3)^(1/4)) - (191416*(1 - x)^9)/(33*((-1 + x)^3
)^(1/4)) + (87312*(1 - x)^10)/(37*((-1 + x)^3)^(1/4)) - (25488*(1 - x)^11)/(41*((-1 + x)^3)^(1/4)) + (96*(1 -
x)^12)/((-1 + x)^3)^(1/4) - (324*(1 - x)^13)/(49*((-1 + x)^3)^(1/4)) + (2104*((-1 + x)^3)^(3/4))/9

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2067

Int[(P3_)^(p_), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3
, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x,
x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[p, x] && PolyQ[P3, x, 3]

Rule 2081

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C
oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2
)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x
] && PolyQ[P3, x, 3]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \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 \left (\frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3}}+\frac {8 x}{\sqrt [4]{-1+3 x-3 x^2+x^3}}+\frac {20 x^2}{\sqrt [4]{-1+3 x-3 x^2+x^3}}-\frac {4 x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}}-\frac {98 x^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}}-\frac {116 x^5}{\sqrt [4]{-1+3 x-3 x^2+x^3}}+\frac {122 x^6}{\sqrt [4]{-1+3 x-3 x^2+x^3}}+\frac {316 x^7}{\sqrt [4]{-1+3 x-3 x^2+x^3}}+\frac {37 x^8}{\sqrt [4]{-1+3 x-3 x^2+x^3}}-\frac {312 x^9}{\sqrt [4]{-1+3 x-3 x^2+x^3}}-\frac {162 x^{10}}{\sqrt [4]{-1+3 x-3 x^2+x^3}}+\frac {108 x^{11}}{\sqrt [4]{-1+3 x-3 x^2+x^3}}+\frac {81 x^{12}}{\sqrt [4]{-1+3 x-3 x^2+x^3}}\right ) \, dx\\ &=-\left (4 \int \frac {x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx\right )+8 \int \frac {x}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx+20 \int \frac {x^2}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx+37 \int \frac {x^8}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx+81 \int \frac {x^{12}}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx-98 \int \frac {x^4}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx+108 \int \frac {x^{11}}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx-116 \int \frac {x^5}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx+122 \int \frac {x^6}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx-162 \int \frac {x^{10}}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx-312 \int \frac {x^9}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx+316 \int \frac {x^7}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx+\int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {(1+x)^3}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )\right )+8 \operatorname {Subst}\left (\int \frac {1+x}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )+20 \operatorname {Subst}\left (\int \frac {(1+x)^2}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )+37 \operatorname {Subst}\left (\int \frac {(1+x)^8}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )+81 \operatorname {Subst}\left (\int \frac {(1+x)^{12}}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )-98 \operatorname {Subst}\left (\int \frac {(1+x)^4}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )+108 \operatorname {Subst}\left (\int \frac {(1+x)^{11}}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )-116 \operatorname {Subst}\left (\int \frac {(1+x)^5}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )+122 \operatorname {Subst}\left (\int \frac {(1+x)^6}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )-162 \operatorname {Subst}\left (\int \frac {(1+x)^{10}}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )-312 \operatorname {Subst}\left (\int \frac {(1+x)^9}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )+316 \operatorname {Subst}\left (\int \frac {(1+x)^7}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )+\operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )\\ &=\frac {(-1+x)^{3/4} \operatorname {Subst}\left (\int \frac {1}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}-\frac {\left (4 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(1+x)^3}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (8 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1+x}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (20 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(1+x)^2}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (37 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(1+x)^8}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (81 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(1+x)^{12}}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}-\frac {\left (98 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(1+x)^4}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (108 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(1+x)^{11}}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}-\frac {\left (116 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(1+x)^5}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (122 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(1+x)^6}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}-\frac {\left (162 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(1+x)^{10}}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}-\frac {\left (312 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(1+x)^9}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (316 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(1+x)^7}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}\\ &=-\frac {4 (1-x)}{\sqrt [4]{(-1+x)^3}}-\frac {\left (4 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+3 \sqrt [4]{x}+3 x^{5/4}+x^{9/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (8 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+\sqrt [4]{x}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (20 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+2 \sqrt [4]{x}+x^{5/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (37 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+8 \sqrt [4]{x}+28 x^{5/4}+56 x^{9/4}+70 x^{13/4}+56 x^{17/4}+28 x^{21/4}+8 x^{25/4}+x^{29/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (81 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+12 \sqrt [4]{x}+66 x^{5/4}+220 x^{9/4}+495 x^{13/4}+792 x^{17/4}+924 x^{21/4}+792 x^{25/4}+495 x^{29/4}+220 x^{33/4}+66 x^{37/4}+12 x^{41/4}+x^{45/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}-\frac {\left (98 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+4 \sqrt [4]{x}+6 x^{5/4}+4 x^{9/4}+x^{13/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (108 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+11 \sqrt [4]{x}+55 x^{5/4}+165 x^{9/4}+330 x^{13/4}+462 x^{17/4}+462 x^{21/4}+330 x^{25/4}+165 x^{29/4}+55 x^{33/4}+11 x^{37/4}+x^{41/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}-\frac {\left (116 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+5 \sqrt [4]{x}+10 x^{5/4}+10 x^{9/4}+5 x^{13/4}+x^{17/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (122 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+6 \sqrt [4]{x}+15 x^{5/4}+20 x^{9/4}+15 x^{13/4}+6 x^{17/4}+x^{21/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}-\frac {\left (162 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+10 \sqrt [4]{x}+45 x^{5/4}+120 x^{9/4}+210 x^{13/4}+252 x^{17/4}+210 x^{21/4}+120 x^{25/4}+45 x^{29/4}+10 x^{33/4}+x^{37/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}-\frac {\left (312 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+9 \sqrt [4]{x}+36 x^{5/4}+84 x^{9/4}+126 x^{13/4}+126 x^{17/4}+84 x^{21/4}+36 x^{25/4}+9 x^{29/4}+x^{33/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (316 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+7 \sqrt [4]{x}+21 x^{5/4}+35 x^{9/4}+35 x^{13/4}+21 x^{17/4}+7 x^{21/4}+x^{25/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}\\ &=-\frac {4 (1-x)}{\sqrt [4]{(-1+x)^3}}+\frac {144 (1-x)^2}{5 \sqrt [4]{(-1+x)^3}}+\frac {16032 (1-x)^4}{13 \sqrt [4]{(-1+x)^3}}-\frac {68820 (1-x)^5}{17 \sqrt [4]{(-1+x)^3}}+\frac {57648 (1-x)^6}{7 \sqrt [4]{(-1+x)^3}}-\frac {271528 (1-x)^7}{25 \sqrt [4]{(-1+x)^3}}+\frac {278928 (1-x)^8}{29 \sqrt [4]{(-1+x)^3}}-\frac {191416 (1-x)^9}{33 \sqrt [4]{(-1+x)^3}}+\frac {87312 (1-x)^{10}}{37 \sqrt [4]{(-1+x)^3}}-\frac {25488 (1-x)^{11}}{41 \sqrt [4]{(-1+x)^3}}+\frac {96 (1-x)^{12}}{\sqrt [4]{(-1+x)^3}}-\frac {324 (1-x)^{13}}{49 \sqrt [4]{(-1+x)^3}}+\frac {2104}{9} \left ((-1+x)^3\right )^{3/4}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 156, normalized size = 2.00 \begin {gather*} \frac {4 \left (\frac {81}{49} (x-1)^{49/4}+24 (x-1)^{45/4}+\frac {6372}{41} (x-1)^{41/4}+\frac {21828}{37} (x-1)^{37/4}+\frac {47854}{33} (x-1)^{33/4}+\frac {69732}{29} (x-1)^{29/4}+\frac {67882}{25} (x-1)^{25/4}+\frac {14412}{7} (x-1)^{21/4}+\frac {17205}{17} (x-1)^{17/4}+\frac {4008}{13} (x-1)^{13/4}+\frac {526}{9} (x-1)^{9/4}+\frac {36}{5} (x-1)^{5/4}+\sqrt [4]{x-1}\right ) (x-1)^{3/4}}{\sqrt [4]{(x-1)^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 2*x + x^2 + 3*x^3)^4/(-1 + 3*x - 3*x^2 + x^3)^(1/4),x]

[Out]

(4*((-1 + x)^(1/4) + (36*(-1 + x)^(5/4))/5 + (526*(-1 + x)^(9/4))/9 + (4008*(-1 + x)^(13/4))/13 + (17205*(-1 +
x)^(17/4))/17 + (14412*(-1 + x)^(21/4))/7 + (67882*(-1 + x)^(25/4))/25 + (69732*(-1 + x)^(29/4))/29 + (47854*
(-1 + x)^(33/4))/33 + (21828*(-1 + x)^(37/4))/37 + (6372*(-1 + x)^(41/4))/41 + 24*(-1 + x)^(45/4) + (81*(-1 +
x)^(49/4))/49)*(-1 + x)^(3/4))/((-1 + x)^3)^(1/4)

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IntegrateAlgebraic [A]  time = 5.08, size = 138, normalized size = 1.77 \begin {gather*} \frac {4 \left (1179090487575 \sqrt [4]{-1+x}[phone] (-1+x)^{5/4}[phone] (-1+x)^{9/4}[phone] (-1+x)^{13/4}+1193308931689875 (-1+x)^{17/4}+2427578872418700 (-1+x)^{21/4}+3201560819102646 (-1+x)^{25/4}+2835184064813100 (-1+x)^{29/4}+1709824127042850 (-1+x)^{33/4}[phone] (-1+x)^{37/4}[phone] (-1+x)^{41/4}[phone] (-1+x)^{45/4}[phone] (-1+x)^{49/4}\right ) \left ((-1+x)^3\right )^{3/4}}{1179090487575 (-1+x)^{9/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 - 2*x + x^2 + 3*x^3)^4/(-1 + 3*x - 3*x^2 + x^3)^(1/4),x]

[Out]

(4*(1179090487575*(-1 + x)^(1/4) + 8489451510540*(-1 + x)^(5/4) + 68911288496050*(-1 + x)^(9/4) + 363522667246
200*(-1 + x)^(13/4) + 1193308931689875*(-1 + x)^(17/4) + 2427578872418700*(-1 + x)^(21/4) + 3201560819102646*(
-1 + x)^(25/4) + 2835184064813100*(-1 + x)^(29/4) + 1709824127042850*(-1 + x)^(33/4) + 695599653048300*(-1 + x
)^(37/4) + 183247916751900*(-1 + x)^(41/4) + 28298171701800*(-1 + x)^(45/4) + 1949108765175*(-1 + x)^(49/4))*(
(-1 + x)^3)^(3/4))/(1179090487575*(-1 + x)^(9/4))

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fricas [A]  time = 0.49, size = 87, normalized size = 1.12 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3+x^2-2*x-1)^4/(x^3-3*x^2+3*x-1)^(1/4),x, algorithm="fricas")

[Out]

4/1179090487575*(1949108765175*x^12 + 4908866519700*x^11 + 609206533650*x^10 - 9283999210200*x^9 - 88059885917
25*x^8 + 3131067556500*x^7 + 9260757242646*x^6 + 4070651298324*x^5 - 2008108342110*x^4 - 2834315032620*x^3 - 1
158885626660*x^2 + 32327777464*x + 1308401597431)*(x^3 - 3*x^2 + 3*x - 1)^(3/4)/(x^2 - 2*x + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3+x^2-2*x-1)^4/(x^3-3*x^2+3*x-1)^(1/4),x, algorithm="giac")

[Out]

integrate((3*x^3 + x^2 - 2*x - 1)^4/(x^3 - 3*x^2 + 3*x - 1)^(1/4), x)

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maple [A]  time = 0.08, size = 73, normalized size = 0.94

method	result	size
risch	\(\frac {4 \left (-1+x \right ) \left (1949108765175 x^{12}[phone] x^{11}[phone] x^{10}-9283999210200 x^{9}-8805988591725 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}-2008108342110 x^{4}-2834315032620 x^{3}-1158885626660 x^{2}[phone] x [phone]\right )}{1179090487575 \left (\left (-1+x \right )^{3}\right )^{\frac {1}{4}}}\)	\(73\)
gosper	\(\frac {4 \left (-1+x \right ) \left (1949108765175 x^{12}[phone] x^{11}[phone] x^{10}-9283999210200 x^{9}-8805988591725 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}-2008108342110 x^{4}-2834315032620 x^{3}-1158885626660 x^{2}[phone] x [phone]\right )}{1179090487575 \left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}\)	\(81\)
trager	\(\frac {4 \left (1949108765175 x^{12}[phone] x^{11}[phone] x^{10}-9283999210200 x^{9}-8805988591725 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}-2008108342110 x^{4}-2834315032620 x^{3}-1158885626660 x^{2}[phone] x [phone]\right ) \left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {3}{4}}}{1179090487575 \left (-1+x \right )^{2}}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^3+x^2-2*x-1)^4/(x^3-3*x^2+3*x-1)^(1/4),x,method=_RETURNVERBOSE)

[Out]

4/1179090487575*(-1+x)*(1949108765175*x^12+4908866519700*x^11+609206533650*x^10-9283999210200*x^9-880598859172
5*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)

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maxima [B]  time = 0.36, size = 540, normalized size = 6.92 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3+x^2-2*x-1)^4/(x^3-3*x^2+3*x-1)^(1/4),x, algorithm="maxima")

[Out]

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 + 245366784*x^5 + 377487360*x^4 + 671088640*x^3 + 1610612736*x^2 + 1288490188
8*x - 17179869184)/(x - 1)^(3/4) + 48/243061325*(48612265*x^12 + 3556995*x^11 + 4229940*x^10 + 5127200*x^9 + 6
364800*x^8 + 8146944*x^7 + 10862592*x^6 + 15335424*x^5 + 23592960*x^4 + 41943040*x^3 + 100663296*x^2 + 8053063
68*x - 1073741824)/(x - 1)^(3/4) - 648/534734915*(13042315*x^11 + 1057485*x^10 + 1281800*x^9 + 1591200*x^8 + 2
036736*x^7 + 2715648*x^6 + 3833856*x^5 + 5898240*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 + 73
7280*x^4 + 1310720*x^3 + 3145728*x^2 + 25165824*x - 33554432)/(x - 1)^(3/4) + 148/15862275*(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)/(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 + 2560*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*x^3 + 3*x^2 + 24*x - 32)/(x - 1)^(3/4) + 32/5*(x^2 + 3*x - 4)/
(x - 1)^(3/4) + 4*(x - 1)^(1/4)

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mupad [B]  time = 0.90, size = 86, normalized size = 1.10 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - x^2 - 3*x^3 + 1)^4/(3*x - 3*x^2 + x^3 - 1)^(1/4),x)

[Out]

((3*x - 3*x^2 + x^3 - 1)^(3/4)*((129311109856*x)/1179090487575 - (927108501328*x^2)/235818097515 - (7558173420
32*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 + 5233606389724/1179090487575))/(x^2 - 2*x + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x^{3} + x^{2} - 2 x - 1\right )^{4}}{\sqrt [4]{\left (x - 1\right )^{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**3+x**2-2*x-1)**4/(x**3-3*x**2+3*x-1)**(1/4),x)

[Out]

Integral((3*x**3 + x**2 - 2*x - 1)**4/((x - 1)**3)**(1/4), x)

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