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

Optimal. Leaf size=134 \[ -\frac {9869 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )}{4096}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^3}}\right )+\frac {9869 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )}{4096}-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^3}}\right )+\frac {\sqrt [4]{x^4-x^3} \left (6144 x^4-8064 x^3+10400 x^2-1060 x-32575\right )}{30720} \]

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Rubi [C]  time = 0.78, antiderivative size = 223, normalized size of antiderivative = 1.66, number of steps used = 35, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2056, 6733, 6725, 279, 331, 298, 203, 206, 321, 511, 510} \begin {gather*} \frac {4 \sqrt [4]{x^4-x^3} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}+\frac {1}{5} \sqrt [4]{x^4-x^3} x^4-\frac {21}{80} \sqrt [4]{x^4-x^3} x^3-\frac {53 \sqrt [4]{x^4-x^3} x}{1536}-\frac {6515 \sqrt [4]{x^4-x^3}}{6144}-\frac {1677 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{4096 \sqrt [4]{x-1} x^{3/4}}+\frac {1677 \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{4096 \sqrt [4]{x-1} x^{3/4}}+\frac {65}{192} \sqrt [4]{x^4-x^3} x^2 \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-6515*(-x^3 + x^4)^(1/4))/6144 - (53*x*(-x^3 + x^4)^(1/4))/1536 + (65*x^2*(-x^3 + x^4)^(1/4))/192 - (21*x^3*(
-x^3 + x^4)^(1/4))/80 + (x^4*(-x^3 + x^4)^(1/4))/5 + (4*(-x^3 + x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7/4, x, -x])
/(3*(1 - x)^(1/4)) - (1677*(-x^3 + x^4)^(1/4)*ArcTan[x^(1/4)/(-1 + x)^(1/4)])/(4096*(-1 + x)^(1/4)*x^(3/4)) +
(1677*(-x^3 + x^4)^(1/4)*ArcTanh[x^(1/4)/(-1 + x)^(1/4)])/(4096*(-1 + x)^(1/4)*x^(3/4))

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6733

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rubi steps

\begin {align*} \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} \left (-1+x+x^4\right )}{1+x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [4]{-1+x^4} \left (-1+x^4+x^{16}\right )}{1+x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (-x^2 \sqrt [4]{-1+x^4}+x^{10} \sqrt [4]{-1+x^4}-x^{14} \sqrt [4]{-1+x^4}+x^{18} \sqrt [4]{-1+x^4}+\frac {x^2 \sqrt [4]{-1+x^4}}{1+x^4}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int x^2 \sqrt [4]{-1+x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int x^{10} \sqrt [4]{-1+x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int x^{14} \sqrt [4]{-1+x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int x^{18} \sqrt [4]{-1+x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{1+x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\sqrt [4]{-x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1-x^4}}{1+x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{1-x} x^{3/4}}-\frac {\sqrt [4]{-x^3+x^4} \operatorname {Subst}\left (\int \frac {x^{18}}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{5 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \operatorname {Subst}\left (\int \frac {x^{14}}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{-x^3+x^4} \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\sqrt [4]{-x^3+x^4}-\frac {1}{24} x \sqrt [4]{-x^3+x^4}+\frac {17}{48} x^2 \sqrt [4]{-x^3+x^4}-\frac {21}{80} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {4 \sqrt [4]{-x^3+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}-\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{16 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (11 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{48 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (7 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{24 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {103}{96} \sqrt [4]{-x^3+x^4}-\frac {5}{384} x \sqrt [4]{-x^3+x^4}+\frac {65}{192} x^2 \sqrt [4]{-x^3+x^4}-\frac {21}{80} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {4 \sqrt [4]{-x^3+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}-\frac {\left (11 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (77 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{384 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (7 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{-x^3+x^4} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1571 \sqrt [4]{-x^3+x^4}}{1536}-\frac {53 x \sqrt [4]{-x^3+x^4}}{1536}+\frac {65}{192} x^2 \sqrt [4]{-x^3+x^4}-\frac {21}{80} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {4 \sqrt [4]{-x^3+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}-\frac {\sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (77 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{512 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (77 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{512 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (7 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {6515 \sqrt [4]{-x^3+x^4}}{6144}-\frac {53 x \sqrt [4]{-x^3+x^4}}{1536}+\frac {65}{192} x^2 \sqrt [4]{-x^3+x^4}-\frac {21}{80} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {4 \sqrt [4]{-x^3+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}-\frac {\sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (7 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (7 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (231 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{2048 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (77 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{512 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {6515 \sqrt [4]{-x^3+x^4}}{6144}-\frac {53 x \sqrt [4]{-x^3+x^4}}{1536}+\frac {65}{192} x^2 \sqrt [4]{-x^3+x^4}-\frac {21}{80} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {4 \sqrt [4]{-x^3+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}-\frac {25 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}+\frac {25 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (77 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{1024 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (77 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{1024 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (231 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2048 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {6515 \sqrt [4]{-x^3+x^4}}{6144}-\frac {53 x \sqrt [4]{-x^3+x^4}}{1536}+\frac {65}{192} x^2 \sqrt [4]{-x^3+x^4}-\frac {21}{80} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {4 \sqrt [4]{-x^3+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}-\frac {477 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{1024 \sqrt [4]{-1+x} x^{3/4}}+\frac {477 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{1024 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (231 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{4096 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (231 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{4096 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {6515 \sqrt [4]{-x^3+x^4}}{6144}-\frac {53 x \sqrt [4]{-x^3+x^4}}{1536}+\frac {65}{192} x^2 \sqrt [4]{-x^3+x^4}-\frac {21}{80} x^3 \sqrt [4]{-x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{-x^3+x^4}+\frac {4 \sqrt [4]{-x^3+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x,-x\right )}{3 \sqrt [4]{1-x}}-\frac {1677 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{4096 \sqrt [4]{-1+x} x^{3/4}}+\frac {1677 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{4096 \sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}

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Mathematica [C]  time = 0.20, size = 118, normalized size = 0.88 \begin {gather*} \frac {x^3 \left (148035 x \left (1-x^2\right )^{3/4} F_1\left (\frac {7}{4};\frac {3}{4},1;\frac {11}{4};x,-x\right )-228025 (1-x)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {2 x}{x+1}\right )+7 (x+1)^{3/4} \left (6144 x^5-14208 x^4+18464 x^3-11460 x^2-31515 x+32575\right )\right )}{215040 \left ((x-1) x^3\right )^{3/4} (x+1)^{3/4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^3*(7*(1 + x)^(3/4)*(32575 - 31515*x - 11460*x^2 + 18464*x^3 - 14208*x^4 + 6144*x^5) + 148035*x*(1 - x^2)^(3
/4)*AppellF1[7/4, 3/4, 1, 11/4, x, -x] - 228025*(1 - x)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, (2*x)/(1 + x)])
)/(215040*((-1 + x)*x^3)^(3/4)*(1 + x)^(3/4))

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IntegrateAlgebraic [A]  time = 0.76, size = 134, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-x^3+x^4} \left (-32575-1060 x+10400 x^2-8064 x^3+6144 x^4\right )}{30720}-\frac {9869 \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{4096}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {9869 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{4096}-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-x^3 + x^4)^(1/4)*(-32575 - 1060*x + 10400*x^2 - 8064*x^3 + 6144*x^4))/30720 - (9869*ArcTan[x/(-x^3 + x^4)^(
1/4)])/4096 + 2*2^(1/4)*ArcTan[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)] + (9869*ArcTanh[x/(-x^3 + x^4)^(1/4)])/4096 - 2
*2^(1/4)*ArcTanh[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)]

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fricas [A]  time = 0.58, size = 209, normalized size = 1.56 \begin {gather*} \frac {1}{30720} \, {\left (6144 \, x^{4} - 8064 \, x^{3} + 10400 \, x^{2} - 1060 \, x - 32575\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} + 4 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} x \sqrt {\frac {\sqrt {2} x^{2} + \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2^{\frac {3}{4}} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{2 \, x}\right ) - 2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {9869}{4096} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {9869}{8192} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {9869}{8192} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30720*(6144*x^4 - 8064*x^3 + 10400*x^2 - 1060*x - 32575)*(x^4 - x^3)^(1/4) + 4*2^(1/4)*arctan(1/2*(2^(3/4)*x
*sqrt((sqrt(2)*x^2 + sqrt(x^4 - x^3))/x^2) - 2^(3/4)*(x^4 - x^3)^(1/4))/x) - 2^(1/4)*log((2^(1/4)*x + (x^4 - x
^3)^(1/4))/x) + 2^(1/4)*log(-(2^(1/4)*x - (x^4 - x^3)^(1/4))/x) + 9869/4096*arctan((x^4 - x^3)^(1/4)/x) + 9869
/8192*log((x + (x^4 - x^3)^(1/4))/x) - 9869/8192*log(-(x - (x^4 - x^3)^(1/4))/x)

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giac [A]  time = 0.22, size = 184, normalized size = 1.37 \begin {gather*} \frac {1}{30720} \, {\left (32575 \, {\left (\frac {1}{x} - 1\right )}^{4} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 131360 \, {\left (\frac {1}{x} - 1\right )}^{3} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 188230 \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 120744 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 25155 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{5} + 2 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) - \frac {9869}{4096} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {9869}{8192} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {9869}{8192} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30720*(32575*(1/x - 1)^4*(-1/x + 1)^(1/4) + 131360*(1/x - 1)^3*(-1/x + 1)^(1/4) + 188230*(1/x - 1)^2*(-1/x +
1)^(1/4) - 120744*(-1/x + 1)^(5/4) + 25155*(-1/x + 1)^(1/4))*x^5 + 2*2^(1/4)*arctan(1/2*2^(3/4)*(-1/x + 1)^(1
/4)) + 2^(1/4)*log(2^(1/4) + (-1/x + 1)^(1/4)) - 2^(1/4)*log(abs(-2^(1/4) + (-1/x + 1)^(1/4))) - 9869/4096*arc
tan((-1/x + 1)^(1/4)) - 9869/8192*log((-1/x + 1)^(1/4) + 1) + 9869/8192*log(abs((-1/x + 1)^(1/4) - 1))

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maple [C]  time = 3.58, size = 480, normalized size = 3.58

method	result	size
trager	\(\left (\frac {1}{5} x^{4}-\frac {21}{80} x^{3}+\frac {65}{192} x^{2}-\frac {53}{1536} x -\frac {6515}{6144}\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-\frac {9869 \ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-2 x^{3}+x^{2}}{x^{2}}\right )}{8192}-\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{3}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}} x^{2}+4 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-2\right ) x +4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}} x^{2}-4 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )+\frac {9869 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {2 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x -2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-4 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2}}\right )}{16384}\)	\(480\)
risch	\(\frac {\left (6144 x^{4}-8064 x^{3}+10400 x^{2}-1060 x -32575\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{30720}+\frac {\left (-\frac {9869 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right )^{2}}\right )}{8192}+\frac {9869 \ln \left (\frac {2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}+2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, x +2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+2 x^{3}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -5 x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+4 x -1}{\left (-1+x \right )^{2}}\right )}{8192}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )+2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x -3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{3}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}+7 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-5 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{\left (-1+x \right )^{2} \left (1+x \right )}\right )-\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x -2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3}+2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x +3 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{3}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}-7 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}+4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}+5 \RootOf \left (\textit {\_Z}^{4}-2\right ) x -\RootOf \left (\textit {\_Z}^{4}-2\right )}{\left (-1+x \right )^{2} \left (1+x \right )}\right )\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x \left (-1+x \right )^{3}\right )^{\frac {1}{4}}}{x \left (-1+x \right )}\)	\(934\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/5*x^4-21/80*x^3+65/192*x^2-53/1536*x-6515/6144)*(x^4-x^3)^(1/4)-9869/8192*ln((2*(x^4-x^3)^(3/4)-2*(x^4-x^3)
^(1/2)*x+2*x^2*(x^4-x^3)^(1/4)-2*x^3+x^2)/x^2)-RootOf(_Z^4-2)*ln((3*RootOf(_Z^4-2)^3*x^3-RootOf(_Z^4-2)^3*x^2+
4*RootOf(_Z^4-2)^2*(x^4-x^3)^(1/4)*x^2+4*(x^4-x^3)^(1/2)*RootOf(_Z^4-2)*x+4*(x^4-x^3)^(3/4))/x^2/(1+x))+RootOf
(_Z^2+RootOf(_Z^4-2)^2)*ln((3*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^2*x^3-RootOf(_Z^2+RootOf(_Z^4-2)^2)
*RootOf(_Z^4-2)^2*x^2-4*RootOf(_Z^4-2)^2*(x^4-x^3)^(1/4)*x^2-4*(x^4-x^3)^(1/2)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*x
+4*(x^4-x^3)^(3/4))/x^2/(1+x))+9869/16384*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)*ln((2*(x^4-x^3)^(1/2)
*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^3*x-2*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^3*x^3+RootOf(
_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^3*x^2+4*(x^4-x^3)^(3/4)-4*x^2*(x^4-x^3)^(1/4))/x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^4-x^3\right )}^{1/4}\,\left (x^4+x-1\right )}{x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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