3.28.58 \(\int \frac {1-x^3 \sqrt [3]{5+4 x}-x^3 (5+4 x)^{2/3}}{1-x \sqrt {5+4 x}} \, dx\)

Optimal. Leaf size=258 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3-4\& ,\frac {2 \text {$\#$1}^5 \log \left (\sqrt [6]{4 x+5}-\text {$\#$1}\right )-\text {$\#$1}^4 \log \left (\sqrt [6]{4 x+5}-\text {$\#$1}\right )+2 \text {$\#$1}^3 \log \left (\sqrt [6]{4 x+5}-\text {$\#$1}\right )-8 \text {$\#$1}^2 \log \left (\sqrt [6]{4 x+5}-\text {$\#$1}\right )+3 \text {$\#$1} \log \left (\sqrt [6]{4 x+5}-\text {$\#$1}\right )-4 \log \left (\sqrt [6]{4 x+5}-\text {$\#$1}\right )}{2 \text {$\#$1}^5-\text {$\#$1}^2}\& \right ]+\frac {3}{748} (4 x+5)^{5/6} \left (22 x^2-30 x+45\right )+\frac {3 \sqrt [6]{4 x+5} \left (728 x^3+70 x^2-150 x+4583\right )}{6916}+\frac {3}{64} \sqrt [3]{4 x+5} (4 x-15)+\frac {3}{160} (4 x+5)^{2/3} (8 x-15)-3 \log \left (\sqrt [6]{4 x+5}+1\right ) \]

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Rubi [B]  time = 5.24, antiderivative size = 976, normalized size of antiderivative = 3.78, number of steps used = 40, number of rules used = 17, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.362, Rules used = {2055, 6741, 12, 6742, 2074, 1790, 1422, 200, 31, 634, 617, 204, 628, 1468, 632, 1510, 292} \begin {gather*} \frac {3}{608} (4 x+5)^{19/6}+\frac {3}{544} (4 x+5)^{17/6}-\frac {15}{208} (4 x+5)^{13/6}-\frac {15}{176} (4 x+5)^{11/6}+\frac {3}{80} (4 x+5)^{5/3}+\frac {3}{64} (4 x+5)^{4/3}+\frac {75}{224} (4 x+5)^{7/6}+\frac {15}{32} (4 x+5)^{5/6}-\frac {15}{32} (4 x+5)^{2/3}-\frac {15}{16} \sqrt [3]{4 x+5}+\frac {3}{2} \sqrt [6]{4 x+5}+\frac {1}{4} \sqrt {\frac {3}{17}} \sqrt [3]{243+59 \sqrt {17}} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\frac {2}{-1+\sqrt {17}}} \sqrt [6]{4 x+5}}{\sqrt {3}}\right )-\frac {\sqrt {\frac {3}{17}} \sqrt [3]{37+9 \sqrt {17}} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\frac {2}{-1+\sqrt {17}}} \sqrt [6]{4 x+5}}{\sqrt {3}}\right )}{2^{2/3}}+\frac {1}{4} \sqrt {\frac {3}{17}} \sqrt [3]{-243+59 \sqrt {17}} \tan ^{-1}\left (\frac {2 \sqrt [3]{\frac {2}{1+\sqrt {17}}} \sqrt [6]{4 x+5}+1}{\sqrt {3}}\right )-\frac {\sqrt {\frac {3}{17}} \sqrt [3]{-37+9 \sqrt {17}} \tan ^{-1}\left (\frac {2 \sqrt [3]{\frac {2}{1+\sqrt {17}}} \sqrt [6]{4 x+5}+1}{\sqrt {3}}\right )}{2^{2/3}}-3 \log \left (\sqrt [6]{4 x+5}+1\right )+\frac {\sqrt [3]{-243+59 \sqrt {17}} \log \left (\sqrt [3]{1+\sqrt {17}}-\sqrt [3]{2} \sqrt [6]{4 x+5}\right )}{4 \sqrt {17}}+\frac {\sqrt [3]{-37+9 \sqrt {17}} \log \left (\sqrt [3]{1+\sqrt {17}}-\sqrt [3]{2} \sqrt [6]{4 x+5}\right )}{2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{243+59 \sqrt {17}} \log \left (\sqrt [3]{2} \sqrt [6]{4 x+5}+\sqrt [3]{-1+\sqrt {17}}\right )}{4 \sqrt {17}}+\frac {\sqrt [3]{37+9 \sqrt {17}} \log \left (\sqrt [3]{2} \sqrt [6]{4 x+5}+\sqrt [3]{-1+\sqrt {17}}\right )}{2^{2/3} \sqrt {17}}-\frac {\sqrt [3]{243+59 \sqrt {17}} \log \left (2^{2/3} \sqrt [3]{4 x+5}-\sqrt [3]{2 \left (-1+\sqrt {17}\right )} \sqrt [6]{4 x+5}+\left (-1+\sqrt {17}\right )^{2/3}\right )}{8 \sqrt {17}}-\frac {\sqrt [3]{37+9 \sqrt {17}} \log \left (2^{2/3} \sqrt [3]{4 x+5}-\sqrt [3]{2 \left (-1+\sqrt {17}\right )} \sqrt [6]{4 x+5}+\left (-1+\sqrt {17}\right )^{2/3}\right )}{2\ 2^{2/3} \sqrt {17}}-\frac {\sqrt [3]{-243+59 \sqrt {17}} \log \left (2^{2/3} \sqrt [3]{4 x+5}+\sqrt [3]{2 \left (1+\sqrt {17}\right )} \sqrt [6]{4 x+5}+\left (1+\sqrt {17}\right )^{2/3}\right )}{8 \sqrt {17}}-\frac {\sqrt [3]{-37+9 \sqrt {17}} \log \left (2^{2/3} \sqrt [3]{4 x+5}+\sqrt [3]{2 \left (1+\sqrt {17}\right )} \sqrt [6]{4 x+5}+\left (1+\sqrt {17}\right )^{2/3}\right )}{2\ 2^{2/3} \sqrt {17}}+\frac {1}{34} \left (17+7 \sqrt {17}\right ) \log \left (-2 \sqrt {4 x+5}-\sqrt {17}+1\right )+\frac {1}{34} \left (17-7 \sqrt {17}\right ) \log \left (-2 \sqrt {4 x+5}+\sqrt {17}+1\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(1 - x^3*(5 + 4*x)^(1/3) - x^3*(5 + 4*x)^(2/3))/(1 - x*Sqrt[5 + 4*x]),x]

[Out]

(3*(5 + 4*x)^(1/6))/2 - (15*(5 + 4*x)^(1/3))/16 - (15*(5 + 4*x)^(2/3))/32 + (15*(5 + 4*x)^(5/6))/32 + (75*(5 +
4*x)^(7/6))/224 + (3*(5 + 4*x)^(4/3))/64 + (3*(5 + 4*x)^(5/3))/80 - (15*(5 + 4*x)^(11/6))/176 - (15*(5 + 4*x)
^(13/6))/208 + (3*(5 + 4*x)^(17/6))/544 + (3*(5 + 4*x)^(19/6))/608 - (Sqrt[3/17]*(37 + 9*Sqrt[17])^(1/3)*ArcTa
n[(1 - 2*(2/(-1 + Sqrt[17]))^(1/3)*(5 + 4*x)^(1/6))/Sqrt[3]])/2^(2/3) + (Sqrt[3/17]*(243 + 59*Sqrt[17])^(1/3)*
ArcTan[(1 - 2*(2/(-1 + Sqrt[17]))^(1/3)*(5 + 4*x)^(1/6))/Sqrt[3]])/4 - (Sqrt[3/17]*(-37 + 9*Sqrt[17])^(1/3)*Ar
cTan[(1 + 2*(2/(1 + Sqrt[17]))^(1/3)*(5 + 4*x)^(1/6))/Sqrt[3]])/2^(2/3) + (Sqrt[3/17]*(-243 + 59*Sqrt[17])^(1/
3)*ArcTan[(1 + 2*(2/(1 + Sqrt[17]))^(1/3)*(5 + 4*x)^(1/6))/Sqrt[3]])/4 - 3*Log[1 + (5 + 4*x)^(1/6)] + ((-37 +
9*Sqrt[17])^(1/3)*Log[(1 + Sqrt[17])^(1/3) - 2^(1/3)*(5 + 4*x)^(1/6)])/(2^(2/3)*Sqrt[17]) + ((-243 + 59*Sqrt[1
7])^(1/3)*Log[(1 + Sqrt[17])^(1/3) - 2^(1/3)*(5 + 4*x)^(1/6)])/(4*Sqrt[17]) + ((37 + 9*Sqrt[17])^(1/3)*Log[(-1
+ Sqrt[17])^(1/3) + 2^(1/3)*(5 + 4*x)^(1/6)])/(2^(2/3)*Sqrt[17]) + ((243 + 59*Sqrt[17])^(1/3)*Log[(-1 + Sqrt[
17])^(1/3) + 2^(1/3)*(5 + 4*x)^(1/6)])/(4*Sqrt[17]) - ((37 + 9*Sqrt[17])^(1/3)*Log[(-1 + Sqrt[17])^(2/3) - (2*
(-1 + Sqrt[17]))^(1/3)*(5 + 4*x)^(1/6) + 2^(2/3)*(5 + 4*x)^(1/3)])/(2*2^(2/3)*Sqrt[17]) - ((243 + 59*Sqrt[17])
^(1/3)*Log[(-1 + Sqrt[17])^(2/3) - (2*(-1 + Sqrt[17]))^(1/3)*(5 + 4*x)^(1/6) + 2^(2/3)*(5 + 4*x)^(1/3)])/(8*Sq
rt[17]) - ((-37 + 9*Sqrt[17])^(1/3)*Log[(1 + Sqrt[17])^(2/3) + (2*(1 + Sqrt[17]))^(1/3)*(5 + 4*x)^(1/6) + 2^(2
/3)*(5 + 4*x)^(1/3)])/(2*2^(2/3)*Sqrt[17]) - ((-243 + 59*Sqrt[17])^(1/3)*Log[(1 + Sqrt[17])^(2/3) + (2*(1 + Sq
rt[17]))^(1/3)*(5 + 4*x)^(1/6) + 2^(2/3)*(5 + 4*x)^(1/3)])/(8*Sqrt[17]) + ((17 + 7*Sqrt[17])*Log[1 - Sqrt[17]
- 2*Sqrt[5 + 4*x]])/34 + ((17 - 7*Sqrt[17])*Log[1 + Sqrt[17] - 2*Sqrt[5 + 4*x]])/34

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 204

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

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1422

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] ||  !IGtQ[n/2, 0])

Rule 1468

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
Dist[1/n, Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x]
&& EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rule 1510

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Dist[e/
2 - (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2
, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]

Rule 1790

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[
Sum[x^j*Sum[Coeff[Pq, x, j + k*n]*x^(k*n), {k, 0, (q - j)/n + 1}]*(a + b*x^n + c*x^(2*n))^p, {j, 0, n - 1}], x
]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !PolyQ[P
q, x^n]

Rule 2055

Int[(u_.)*(P_)*(Q_)^(q_), x_Symbol] :> Module[{gcd = PolyGCD[P, Q, x]}, Int[u*gcd^(q + 1)*PolynomialQuotient[P
, gcd, x]*PolynomialQuotient[Q, gcd, x]^q, x] /; NeQ[gcd, 1]] /; ILtQ[q, 0] && PolyQ[P, x] && PolyQ[Q, x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 6741

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

Rule 6742

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

Rubi steps

\begin {align*} \int \frac {1-x^3 \sqrt [3]{5+4 x}-x^3 (5+4 x)^{2/3}}{1-x \sqrt {5+4 x}} \, dx &=6 \operatorname {Subst}\left (\int \frac {x^5 \left (1-\frac {1}{64} x^2 \left (-5+x^6\right )^3-\frac {1}{64} x^4 \left (-5+x^6\right )^3\right )}{4+5 x^3-x^9} \, dx,x,\sqrt [6]{5+4 x}\right )\\ &=6 \operatorname {Subst}\left (\int \frac {x^5 \left (64+64 x+125 x^2+61 x^3+61 x^4-61 x^6-61 x^7-75 x^8-14 x^9-14 x^{10}+14 x^{12}+14 x^{13}+15 x^{14}+x^{15}+x^{16}-x^{18}-x^{19}-x^{20}\right )}{256+256 x+64 x^3+64 x^4-64 x^6-64 x^7} \, dx,x,\sqrt [6]{5+4 x}\right )\\ &=6 \operatorname {Subst}\left (\int \frac {x^5 \left (64+64 x+125 x^2+61 x^3+61 x^4-61 x^6-61 x^7-75 x^8-14 x^9-14 x^{10}+14 x^{12}+14 x^{13}+15 x^{14}+x^{15}+x^{16}-x^{18}-x^{19}-x^{20}\right )}{64 \left (4+4 x+x^3+x^4-x^6-x^7\right )} \, dx,x,\sqrt [6]{5+4 x}\right )\\ &=\frac {3}{32} \operatorname {Subst}\left (\int \frac {x^5 \left (64+64 x+125 x^2+61 x^3+61 x^4-61 x^6-61 x^7-75 x^8-14 x^9-14 x^{10}+14 x^{12}+14 x^{13}+15 x^{14}+x^{15}+x^{16}-x^{18}-x^{19}-x^{20}\right )}{4+4 x+x^3+x^4-x^6-x^7} \, dx,x,\sqrt [6]{5+4 x}\right )\\ &=\frac {3}{32} \operatorname {Subst}\left (\int \left (16-20 x-20 x^3+25 x^4+25 x^6+4 x^7+4 x^9-10 x^{10}-10 x^{12}+x^{16}+x^{18}-\frac {16 \left (4-x-5 x^2-4 x^3+x^4+x^5\right )}{4+4 x+x^3+x^4-x^6-x^7}\right ) \, dx,x,\sqrt [6]{5+4 x}\right )\\ &=\frac {3}{2} \sqrt [6]{5+4 x}-\frac {15}{16} \sqrt [3]{5+4 x}-\frac {15}{32} (5+4 x)^{2/3}+\frac {15}{32} (5+4 x)^{5/6}+\frac {75}{224} (5+4 x)^{7/6}+\frac {3}{64} (5+4 x)^{4/3}+\frac {3}{80} (5+4 x)^{5/3}-\frac {15}{176} (5+4 x)^{11/6}-\frac {15}{208} (5+4 x)^{13/6}+\frac {3}{544} (5+4 x)^{17/6}+\frac {3}{608} (5+4 x)^{19/6}-\frac {3}{2} \operatorname {Subst}\left (\int \frac {4-x-5 x^2-4 x^3+x^4+x^5}{4+4 x+x^3+x^4-x^6-x^7} \, dx,x,\sqrt [6]{5+4 x}\right )\\ &=\frac {3}{2} \sqrt [6]{5+4 x}-\frac {15}{16} \sqrt [3]{5+4 x}-\frac {15}{32} (5+4 x)^{2/3}+\frac {15}{32} (5+4 x)^{5/6}+\frac {75}{224} (5+4 x)^{7/6}+\frac {3}{64} (5+4 x)^{4/3}+\frac {3}{80} (5+4 x)^{5/3}-\frac {15}{176} (5+4 x)^{11/6}-\frac {15}{208} (5+4 x)^{13/6}+\frac {3}{544} (5+4 x)^{17/6}+\frac {3}{608} (5+4 x)^{19/6}-\frac {3}{2} \operatorname {Subst}\left (\int \left (\frac {2}{1+x}+\frac {4-3 x+8 x^2-2 x^3+x^4-2 x^5}{-4-x^3+x^6}\right ) \, dx,x,\sqrt [6]{5+4 x}\right )\\ &=\frac {3}{2} \sqrt [6]{5+4 x}-\frac {15}{16} \sqrt [3]{5+4 x}-\frac {15}{32} (5+4 x)^{2/3}+\frac {15}{32} (5+4 x)^{5/6}+\frac {75}{224} (5+4 x)^{7/6}+\frac {3}{64} (5+4 x)^{4/3}+\frac {3}{80} (5+4 x)^{5/3}-\frac {15}{176} (5+4 x)^{11/6}-\frac {15}{208} (5+4 x)^{13/6}+\frac {3}{544} (5+4 x)^{17/6}+\frac {3}{608} (5+4 x)^{19/6}-3 \log \left (1+\sqrt [6]{5+4 x}\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {4-3 x+8 x^2-2 x^3+x^4-2 x^5}{-4-x^3+x^6} \, dx,x,\sqrt [6]{5+4 x}\right )\\ &=\frac {3}{2} \sqrt [6]{5+4 x}-\frac {15}{16} \sqrt [3]{5+4 x}-\frac {15}{32} (5+4 x)^{2/3}+\frac {15}{32} (5+4 x)^{5/6}+\frac {75}{224} (5+4 x)^{7/6}+\frac {3}{64} (5+4 x)^{4/3}+\frac {3}{80} (5+4 x)^{5/3}-\frac {15}{176} (5+4 x)^{11/6}-\frac {15}{208} (5+4 x)^{13/6}+\frac {3}{544} (5+4 x)^{17/6}+\frac {3}{608} (5+4 x)^{19/6}-3 \log \left (1+\sqrt [6]{5+4 x}\right )-\frac {3}{2} \operatorname {Subst}\left (\int \left (\frac {4-2 x^3}{-4-x^3+x^6}+\frac {x^2 \left (8-2 x^3\right )}{-4-x^3+x^6}+\frac {x \left (-3+x^3\right )}{-4-x^3+x^6}\right ) \, dx,x,\sqrt [6]{5+4 x}\right )\\ &=\frac {3}{2} \sqrt [6]{5+4 x}-\frac {15}{16} \sqrt [3]{5+4 x}-\frac {15}{32} (5+4 x)^{2/3}+\frac {15}{32} (5+4 x)^{5/6}+\frac {75}{224} (5+4 x)^{7/6}+\frac {3}{64} (5+4 x)^{4/3}+\frac {3}{80} (5+4 x)^{5/3}-\frac {15}{176} (5+4 x)^{11/6}-\frac {15}{208} (5+4 x)^{13/6}+\frac {3}{544} (5+4 x)^{17/6}+\frac {3}{608} (5+4 x)^{19/6}-3 \log \left (1+\sqrt [6]{5+4 x}\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {4-2 x^3}{-4-x^3+x^6} \, dx,x,\sqrt [6]{5+4 x}\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {x^2 \left (8-2 x^3\right )}{-4-x^3+x^6} \, dx,x,\sqrt [6]{5+4 x}\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {x \left (-3+x^3\right )}{-4-x^3+x^6} \, dx,x,\sqrt [6]{5+4 x}\right )\\ &=\frac {3}{2} \sqrt [6]{5+4 x}-\frac {15}{16} \sqrt [3]{5+4 x}-\frac {15}{32} (5+4 x)^{2/3}+\frac {15}{32} (5+4 x)^{5/6}+\frac {75}{224} (5+4 x)^{7/6}+\frac {3}{64} (5+4 x)^{4/3}+\frac {3}{80} (5+4 x)^{5/3}-\frac {15}{176} (5+4 x)^{11/6}-\frac {15}{208} (5+4 x)^{13/6}+\frac {3}{544} (5+4 x)^{17/6}+\frac {3}{608} (5+4 x)^{19/6}-3 \log \left (1+\sqrt [6]{5+4 x}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {8-2 x}{-4-x+x^2} \, dx,x,\sqrt {5+4 x}\right )-\frac {1}{68} \left (3 \left (17-5 \sqrt {17}\right )\right ) \operatorname {Subst}\left (\int \frac {x}{-\frac {1}{2}-\frac {\sqrt {17}}{2}+x^3} \, dx,x,\sqrt [6]{5+4 x}\right )+\frac {1}{34} \left (3 \left (17-3 \sqrt {17}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}-\frac {\sqrt {17}}{2}+x^3} \, dx,x,\sqrt [6]{5+4 x}\right )+\frac {1}{34} \left (3 \left (17+3 \sqrt {17}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {\sqrt {17}}{2}+x^3} \, dx,x,\sqrt [6]{5+4 x}\right )-\frac {1}{68} \left (3 \left (17+5 \sqrt {17}\right )\right ) \operatorname {Subst}\left (\int \frac {x}{-\frac {1}{2}+\frac {\sqrt {17}}{2}+x^3} \, dx,x,\sqrt [6]{5+4 x}\right )\\ &=\frac {3}{2} \sqrt [6]{5+4 x}-\frac {15}{16} \sqrt [3]{5+4 x}-\frac {15}{32} (5+4 x)^{2/3}+\frac {15}{32} (5+4 x)^{5/6}+\frac {75}{224} (5+4 x)^{7/6}+\frac {3}{64} (5+4 x)^{4/3}+\frac {3}{80} (5+4 x)^{5/3}-\frac {15}{176} (5+4 x)^{11/6}-\frac {15}{208} (5+4 x)^{13/6}+\frac {3}{544} (5+4 x)^{17/6}+\frac {3}{608} (5+4 x)^{19/6}-3 \log \left (1+\sqrt [6]{5+4 x}\right )-\frac {1}{34} \left (-17+7 \sqrt {17}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}-\frac {\sqrt {17}}{2}+x} \, dx,x,\sqrt {5+4 x}\right )+\frac {1}{34} \left (17+7 \sqrt {17}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {\sqrt {17}}{2}+x} \, dx,x,\sqrt {5+4 x}\right )+\frac {\sqrt [3]{-37+9 \sqrt {17}} \operatorname {Subst}\left (\int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {17}\right )}+x} \, dx,x,\sqrt [6]{5+4 x}\right )}{2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{-37+9 \sqrt {17}} \operatorname {Subst}\left (\int \frac {-2^{2/3} \sqrt [3]{1+\sqrt {17}}-x}{\left (\frac {1}{2} \left (1+\sqrt {17}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {17}\right )} x+x^2} \, dx,x,\sqrt [6]{5+4 x}\right )}{2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{37+9 \sqrt {17}} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {17}\right )}+x} \, dx,x,\sqrt [6]{5+4 x}\right )}{2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{37+9 \sqrt {17}} \operatorname {Subst}\left (\int \frac {2^{2/3} \sqrt [3]{-1+\sqrt {17}}-x}{\left (\frac {1}{2} \left (-1+\sqrt {17}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {17}\right )} x+x^2} \, dx,x,\sqrt [6]{5+4 x}\right )}{2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{-243+59 \sqrt {17}} \operatorname {Subst}\left (\int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {17}\right )}+x} \, dx,x,\sqrt [6]{5+4 x}\right )}{4 \sqrt {17}}-\frac {\sqrt [3]{-243+59 \sqrt {17}} \operatorname {Subst}\left (\int \frac {-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {17}\right )}+x}{\left (\frac {1}{2} \left (1+\sqrt {17}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {17}\right )} x+x^2} \, dx,x,\sqrt [6]{5+4 x}\right )}{4 \sqrt {17}}+\frac {\sqrt [3]{243+59 \sqrt {17}} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {17}\right )}+x} \, dx,x,\sqrt [6]{5+4 x}\right )}{4 \sqrt {17}}-\frac {\sqrt [3]{243+59 \sqrt {17}} \operatorname {Subst}\left (\int \frac {\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {17}\right )}+x}{\left (\frac {1}{2} \left (-1+\sqrt {17}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {17}\right )} x+x^2} \, dx,x,\sqrt [6]{5+4 x}\right )}{4 \sqrt {17}}\\ &=\frac {3}{2} \sqrt [6]{5+4 x}-\frac {15}{16} \sqrt [3]{5+4 x}-\frac {15}{32} (5+4 x)^{2/3}+\frac {15}{32} (5+4 x)^{5/6}+\frac {75}{224} (5+4 x)^{7/6}+\frac {3}{64} (5+4 x)^{4/3}+\frac {3}{80} (5+4 x)^{5/3}-\frac {15}{176} (5+4 x)^{11/6}-\frac {15}{208} (5+4 x)^{13/6}+\frac {3}{544} (5+4 x)^{17/6}+\frac {3}{608} (5+4 x)^{19/6}-3 \log \left (1+\sqrt [6]{5+4 x}\right )+\frac {\sqrt [3]{-37+9 \sqrt {17}} \log \left (\sqrt [3]{1+\sqrt {17}}-\sqrt [3]{2} \sqrt [6]{5+4 x}\right )}{2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{-243+59 \sqrt {17}} \log \left (\sqrt [3]{1+\sqrt {17}}-\sqrt [3]{2} \sqrt [6]{5+4 x}\right )}{4 \sqrt {17}}+\frac {\sqrt [3]{37+9 \sqrt {17}} \log \left (\sqrt [3]{-1+\sqrt {17}}+\sqrt [3]{2} \sqrt [6]{5+4 x}\right )}{2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{243+59 \sqrt {17}} \log \left (\sqrt [3]{-1+\sqrt {17}}+\sqrt [3]{2} \sqrt [6]{5+4 x}\right )}{4 \sqrt {17}}+\frac {1}{34} \left (17+7 \sqrt {17}\right ) \log \left (1-\sqrt {17}-2 \sqrt {5+4 x}\right )+\frac {1}{34} \left (17-7 \sqrt {17}\right ) \log \left (1+\sqrt {17}-2 \sqrt {5+4 x}\right )+\frac {\left (3 \sqrt [3]{\frac {1}{2} \left (95-23 \sqrt {17}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (1+\sqrt {17}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {17}\right )} x+x^2} \, dx,x,\sqrt [6]{5+4 x}\right )}{4 \sqrt {17}}-\frac {\left (3 \sqrt [3]{\frac {1}{2} \left (29-7 \sqrt {17}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (1+\sqrt {17}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {17}\right )} x+x^2} \, dx,x,\sqrt [6]{5+4 x}\right )}{2 \sqrt {17}}+\frac {\left (3 \sqrt [3]{\frac {1}{2} \left (29+7 \sqrt {17}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (-1+\sqrt {17}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {17}\right )} x+x^2} \, dx,x,\sqrt [6]{5+4 x}\right )}{2 \sqrt {17}}-\frac {\sqrt [3]{-37+9 \sqrt {17}} \operatorname {Subst}\left (\int \frac {\sqrt [3]{\frac {1}{2} \left (1+\sqrt {17}\right )}+2 x}{\left (\frac {1}{2} \left (1+\sqrt {17}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {17}\right )} x+x^2} \, dx,x,\sqrt [6]{5+4 x}\right )}{2\ 2^{2/3} \sqrt {17}}-\frac {\sqrt [3]{37+9 \sqrt {17}} \operatorname {Subst}\left (\int \frac {-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {17}\right )}+2 x}{\left (\frac {1}{2} \left (-1+\sqrt {17}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {17}\right )} x+x^2} \, dx,x,\sqrt [6]{5+4 x}\right )}{2\ 2^{2/3} \sqrt {17}}-\frac {\left (3 \sqrt [3]{\frac {1}{2} \left (95+23 \sqrt {17}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (-1+\sqrt {17}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {17}\right )} x+x^2} \, dx,x,\sqrt [6]{5+4 x}\right )}{4 \sqrt {17}}-\frac {\sqrt [3]{-243+59 \sqrt {17}} \operatorname {Subst}\left (\int \frac {\sqrt [3]{\frac {1}{2} \left (1+\sqrt {17}\right )}+2 x}{\left (\frac {1}{2} \left (1+\sqrt {17}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {17}\right )} x+x^2} \, dx,x,\sqrt [6]{5+4 x}\right )}{8 \sqrt {17}}-\frac {\sqrt [3]{243+59 \sqrt {17}} \operatorname {Subst}\left (\int \frac {-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {17}\right )}+2 x}{\left (\frac {1}{2} \left (-1+\sqrt {17}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {17}\right )} x+x^2} \, dx,x,\sqrt [6]{5+4 x}\right )}{8 \sqrt {17}}\\ &=\frac {3}{2} \sqrt [6]{5+4 x}-\frac {15}{16} \sqrt [3]{5+4 x}-\frac {15}{32} (5+4 x)^{2/3}+\frac {15}{32} (5+4 x)^{5/6}+\frac {75}{224} (5+4 x)^{7/6}+\frac {3}{64} (5+4 x)^{4/3}+\frac {3}{80} (5+4 x)^{5/3}-\frac {15}{176} (5+4 x)^{11/6}-\frac {15}{208} (5+4 x)^{13/6}+\frac {3}{544} (5+4 x)^{17/6}+\frac {3}{608} (5+4 x)^{19/6}-3 \log \left (1+\sqrt [6]{5+4 x}\right )+\frac {\sqrt [3]{-37+9 \sqrt {17}} \log \left (\sqrt [3]{1+\sqrt {17}}-\sqrt [3]{2} \sqrt [6]{5+4 x}\right )}{2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{-243+59 \sqrt {17}} \log \left (\sqrt [3]{1+\sqrt {17}}-\sqrt [3]{2} \sqrt [6]{5+4 x}\right )}{4 \sqrt {17}}+\frac {\sqrt [3]{37+9 \sqrt {17}} \log \left (\sqrt [3]{-1+\sqrt {17}}+\sqrt [3]{2} \sqrt [6]{5+4 x}\right )}{2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{243+59 \sqrt {17}} \log \left (\sqrt [3]{-1+\sqrt {17}}+\sqrt [3]{2} \sqrt [6]{5+4 x}\right )}{4 \sqrt {17}}-\frac {\sqrt [3]{37+9 \sqrt {17}} \log \left (\left (-1+\sqrt {17}\right )^{2/3}-\sqrt [3]{2 \left (-1+\sqrt {17}\right )} \sqrt [6]{5+4 x}+2^{2/3} \sqrt [3]{5+4 x}\right )}{2\ 2^{2/3} \sqrt {17}}-\frac {\sqrt [3]{243+59 \sqrt {17}} \log \left (\left (-1+\sqrt {17}\right )^{2/3}-\sqrt [3]{2 \left (-1+\sqrt {17}\right )} \sqrt [6]{5+4 x}+2^{2/3} \sqrt [3]{5+4 x}\right )}{8 \sqrt {17}}-\frac {\sqrt [3]{-37+9 \sqrt {17}} \log \left (\left (1+\sqrt {17}\right )^{2/3}+\sqrt [3]{2 \left (1+\sqrt {17}\right )} \sqrt [6]{5+4 x}+2^{2/3} \sqrt [3]{5+4 x}\right )}{2\ 2^{2/3} \sqrt {17}}-\frac {\sqrt [3]{-243+59 \sqrt {17}} \log \left (\left (1+\sqrt {17}\right )^{2/3}+\sqrt [3]{2 \left (1+\sqrt {17}\right )} \sqrt [6]{5+4 x}+2^{2/3} \sqrt [3]{5+4 x}\right )}{8 \sqrt {17}}+\frac {1}{34} \left (17+7 \sqrt {17}\right ) \log \left (1-\sqrt {17}-2 \sqrt {5+4 x}\right )+\frac {1}{34} \left (17-7 \sqrt {17}\right ) \log \left (1+\sqrt {17}-2 \sqrt {5+4 x}\right )+\frac {\left (3 \sqrt [3]{-37+9 \sqrt {17}}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\frac {2}{1+\sqrt {17}}} \sqrt [6]{5+4 x}\right )}{2^{2/3} \sqrt {17}}+\frac {\left (3 \sqrt [3]{37+9 \sqrt {17}}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{\frac {2}{-1+\sqrt {17}}} \sqrt [6]{5+4 x}\right )}{2^{2/3} \sqrt {17}}-\frac {\left (3 \sqrt [3]{-243+59 \sqrt {17}}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\frac {2}{1+\sqrt {17}}} \sqrt [6]{5+4 x}\right )}{4 \sqrt {17}}-\frac {\left (3 \sqrt [3]{243+59 \sqrt {17}}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{\frac {2}{-1+\sqrt {17}}} \sqrt [6]{5+4 x}\right )}{4 \sqrt {17}}\\ &=\frac {3}{2} \sqrt [6]{5+4 x}-\frac {15}{16} \sqrt [3]{5+4 x}-\frac {15}{32} (5+4 x)^{2/3}+\frac {15}{32} (5+4 x)^{5/6}+\frac {75}{224} (5+4 x)^{7/6}+\frac {3}{64} (5+4 x)^{4/3}+\frac {3}{80} (5+4 x)^{5/3}-\frac {15}{176} (5+4 x)^{11/6}-\frac {15}{208} (5+4 x)^{13/6}+\frac {3}{544} (5+4 x)^{17/6}+\frac {3}{608} (5+4 x)^{19/6}-\frac {\sqrt {\frac {3}{17}} \sqrt [3]{37+9 \sqrt {17}} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\frac {2}{-1+\sqrt {17}}} \sqrt [6]{5+4 x}}{\sqrt {3}}\right )}{2^{2/3}}+\frac {1}{4} \sqrt {\frac {3}{17}} \sqrt [3]{243+59 \sqrt {17}} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\frac {2}{-1+\sqrt {17}}} \sqrt [6]{5+4 x}}{\sqrt {3}}\right )-\frac {\sqrt {\frac {3}{17}} \sqrt [3]{-37+9 \sqrt {17}} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{\frac {2}{1+\sqrt {17}}} \sqrt [6]{5+4 x}}{\sqrt {3}}\right )}{2^{2/3}}+\frac {1}{4} \sqrt {\frac {3}{17}} \sqrt [3]{-243+59 \sqrt {17}} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{\frac {2}{1+\sqrt {17}}} \sqrt [6]{5+4 x}}{\sqrt {3}}\right )-3 \log \left (1+\sqrt [6]{5+4 x}\right )+\frac {\sqrt [3]{-37+9 \sqrt {17}} \log \left (\sqrt [3]{1+\sqrt {17}}-\sqrt [3]{2} \sqrt [6]{5+4 x}\right )}{2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{-243+59 \sqrt {17}} \log \left (\sqrt [3]{1+\sqrt {17}}-\sqrt [3]{2} \sqrt [6]{5+4 x}\right )}{4 \sqrt {17}}+\frac {\sqrt [3]{37+9 \sqrt {17}} \log \left (\sqrt [3]{-1+\sqrt {17}}+\sqrt [3]{2} \sqrt [6]{5+4 x}\right )}{2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{243+59 \sqrt {17}} \log \left (\sqrt [3]{-1+\sqrt {17}}+\sqrt [3]{2} \sqrt [6]{5+4 x}\right )}{4 \sqrt {17}}-\frac {\sqrt [3]{37+9 \sqrt {17}} \log \left (\left (-1+\sqrt {17}\right )^{2/3}-\sqrt [3]{2 \left (-1+\sqrt {17}\right )} \sqrt [6]{5+4 x}+2^{2/3} \sqrt [3]{5+4 x}\right )}{2\ 2^{2/3} \sqrt {17}}-\frac {\sqrt [3]{243+59 \sqrt {17}} \log \left (\left (-1+\sqrt {17}\right )^{2/3}-\sqrt [3]{2 \left (-1+\sqrt {17}\right )} \sqrt [6]{5+4 x}+2^{2/3} \sqrt [3]{5+4 x}\right )}{8 \sqrt {17}}-\frac {\sqrt [3]{-37+9 \sqrt {17}} \log \left (\left (1+\sqrt {17}\right )^{2/3}+\sqrt [3]{2 \left (1+\sqrt {17}\right )} \sqrt [6]{5+4 x}+2^{2/3} \sqrt [3]{5+4 x}\right )}{2\ 2^{2/3} \sqrt {17}}-\frac {\sqrt [3]{-243+59 \sqrt {17}} \log \left (\left (1+\sqrt {17}\right )^{2/3}+\sqrt [3]{2 \left (1+\sqrt {17}\right )} \sqrt [6]{5+4 x}+2^{2/3} \sqrt [3]{5+4 x}\right )}{8 \sqrt {17}}+\frac {1}{34} \left (17+7 \sqrt {17}\right ) \log \left (1-\sqrt {17}-2 \sqrt {5+4 x}\right )+\frac {1}{34} \left (17-7 \sqrt {17}\right ) \log \left (1+\sqrt {17}-2 \sqrt {5+4 x}\right )\\ \end {align*}

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Mathematica [A]  time = 10.10, size = 314, normalized size = 1.22 \begin {gather*} \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3-4\&,\frac {2 \text {$\#$1}^5 \log \left (\sqrt [6]{4 x+5}-\text {$\#$1}\right )-\text {$\#$1}^4 \log \left (\sqrt [6]{4 x+5}-\text {$\#$1}\right )+2 \text {$\#$1}^3 \log \left (\sqrt [6]{4 x+5}-\text {$\#$1}\right )-8 \text {$\#$1}^2 \log \left (\sqrt [6]{4 x+5}-\text {$\#$1}\right )+3 \text {$\#$1} \log \left (\sqrt [6]{4 x+5}-\text {$\#$1}\right )-4 \log \left (\sqrt [6]{4 x+5}-\text {$\#$1}\right )}{2 \text {$\#$1}^5-\text {$\#$1}^2}\&\right ]+\frac {3}{608} (4 x+5)^{19/6}+\frac {3}{544} (4 x+5)^{17/6}-\frac {15}{208} (4 x+5)^{13/6}-\frac {15}{176} (4 x+5)^{11/6}+\frac {3}{80} (4 x+5)^{5/3}+\frac {3}{64} (4 x+5)^{4/3}+\frac {75}{224} (4 x+5)^{7/6}+\frac {15}{32} (4 x+5)^{5/6}-\frac {15}{32} (4 x+5)^{2/3}-\frac {15}{16} \sqrt [3]{4 x+5}+\frac {3}{2} \sqrt [6]{4 x+5}-3 \log \left (\sqrt [6]{4 x+5}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^3*(5 + 4*x)^(1/3) - x^3*(5 + 4*x)^(2/3))/(1 - x*Sqrt[5 + 4*x]),x]

[Out]

(3*(5 + 4*x)^(1/6))/2 - (15*(5 + 4*x)^(1/3))/16 - (15*(5 + 4*x)^(2/3))/32 + (15*(5 + 4*x)^(5/6))/32 + (75*(5 +
4*x)^(7/6))/224 + (3*(5 + 4*x)^(4/3))/64 + (3*(5 + 4*x)^(5/3))/80 - (15*(5 + 4*x)^(11/6))/176 - (15*(5 + 4*x)
^(13/6))/208 + (3*(5 + 4*x)^(17/6))/544 + (3*(5 + 4*x)^(19/6))/608 - 3*Log[1 + (5 + 4*x)^(1/6)] + RootSum[-4 -
#1^3 + #1^6 & , (-4*Log[(5 + 4*x)^(1/6) - #1] + 3*Log[(5 + 4*x)^(1/6) - #1]*#1 - 8*Log[(5 + 4*x)^(1/6) - #1]*
#1^2 + 2*Log[(5 + 4*x)^(1/6) - #1]*#1^3 - Log[(5 + 4*x)^(1/6) - #1]*#1^4 + 2*Log[(5 + 4*x)^(1/6) - #1]*#1^5)/(
-#1^2 + 2*#1^5) & ]/2

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IntegrateAlgebraic [A]  time = 0.14, size = 297, normalized size = 1.15 \begin {gather*} \frac {3 \left (51731680 \sqrt [6]{5+4 x}-32332300 \sqrt [3]{5+4 x}-16166150 (5+4 x)^{2/3}[phone] (5+4 x)^{5/6}[phone] (5+4 x)^{7/6}+1616615 (5+4 x)^{4/3}+1293292 (5+4 x)^{5/3}-2939300 (5+4 x)^{11/6}-2487100 (5+4 x)^{13/6}+190190 (5+4 x)^{17/6}+170170 (5+4 x)^{19/6}\right )}{103463360}-3 \log \left (1+\sqrt [6]{5+4 x}\right )+\frac {1}{2} \text {RootSum}\left [-4-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-4 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right )+3 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}-8 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}^2+2 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}^3-\log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}^5}{-\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - x^3*(5 + 4*x)^(1/3) - x^3*(5 + 4*x)^(2/3))/(1 - x*Sqrt[5 + 4*x]),x]

[Out]

(3*(51731680*(5 + 4*x)^(1/6) - 32332300*(5 + 4*x)^(1/3) - 16166150*(5 + 4*x)^(2/3) + 16166150*(5 + 4*x)^(5/6)
+ 11547250*(5 + 4*x)^(7/6) + 1616615*(5 + 4*x)^(4/3) + 1293292*(5 + 4*x)^(5/3) - 2939300*(5 + 4*x)^(11/6) - 24
87100*(5 + 4*x)^(13/6) + 190190*(5 + 4*x)^(17/6) + 170170*(5 + 4*x)^(19/6)))/103463360 - 3*Log[1 + (5 + 4*x)^(
1/6)] + RootSum[-4 - #1^3 + #1^6 & , (-4*Log[(5 + 4*x)^(1/6) - #1] + 3*Log[(5 + 4*x)^(1/6) - #1]*#1 - 8*Log[(5
+ 4*x)^(1/6) - #1]*#1^2 + 2*Log[(5 + 4*x)^(1/6) - #1]*#1^3 - Log[(5 + 4*x)^(1/6) - #1]*#1^4 + 2*Log[(5 + 4*x)
^(1/6) - #1]*#1^5)/(-#1^2 + 2*#1^5) & ]/2

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x^3*(5+4*x)^(1/3)-x^3*(5+4*x)^(2/3))/(1-x*(5+4*x)^(1/2)),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x + 5\right )}^{\frac {2}{3}} x^{3} + {\left (4 \, x + 5\right )}^{\frac {1}{3}} x^{3} - 1}{\sqrt {4 \, x + 5} x - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x^3*(5+4*x)^(1/3)-x^3*(5+4*x)^(2/3))/(1-x*(5+4*x)^(1/2)),x, algorithm="giac")

[Out]

integrate(((4*x + 5)^(2/3)*x^3 + (4*x + 5)^(1/3)*x^3 - 1)/(sqrt(4*x + 5)*x - 1), x)

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maple [A]  time = 0.18, size = 180, normalized size = 0.70

method	result	size
derivativedivides	\(\frac {3 \left (5+4 x \right )^{\frac {19}{6}}}{608}+\frac {3 \left (5+4 x \right )^{\frac {17}{6}}}{544}-\frac {15 \left (5+4 x \right )^{\frac {13}{6}}}{208}-\frac {15 \left (5+4 x \right )^{\frac {11}{6}}}{176}+\frac {3 \left (5+4 x \right )^{\frac {5}{3}}}{80}+\frac {3 \left (5+4 x \right )^{\frac {4}{3}}}{64}+\frac {75 \left (5+4 x \right )^{\frac {7}{6}}}{224}+\frac {15 \left (5+4 x \right )^{\frac {5}{6}}}{32}-\frac {15 \left (5+4 x \right )^{\frac {2}{3}}}{32}-\frac {15 \left (5+4 x \right )^{\frac {1}{3}}}{16}+\frac {3 \left (5+4 x \right )^{\frac {1}{6}}}{2}-3 \ln \left (1+\left (5+4 x \right )^{\frac {1}{6}}\right )+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}-4\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-\textit {\_R}^{4}+2 \textit {\_R}^{3}-8 \textit {\_R}^{2}+3 \textit {\_R} -4\right ) \ln \left (\left (5+4 x \right )^{\frac {1}{6}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{2}\)	\(180\)
default	\(\frac {3 \left (5+4 x \right )^{\frac {19}{6}}}{608}+\frac {3 \left (5+4 x \right )^{\frac {17}{6}}}{544}-\frac {15 \left (5+4 x \right )^{\frac {13}{6}}}{208}-\frac {15 \left (5+4 x \right )^{\frac {11}{6}}}{176}+\frac {3 \left (5+4 x \right )^{\frac {5}{3}}}{80}+\frac {3 \left (5+4 x \right )^{\frac {4}{3}}}{64}+\frac {75 \left (5+4 x \right )^{\frac {7}{6}}}{224}+\frac {15 \left (5+4 x \right )^{\frac {5}{6}}}{32}-\frac {15 \left (5+4 x \right )^{\frac {2}{3}}}{32}-\frac {15 \left (5+4 x \right )^{\frac {1}{3}}}{16}+\frac {3 \left (5+4 x \right )^{\frac {1}{6}}}{2}-3 \ln \left (1+\left (5+4 x \right )^{\frac {1}{6}}\right )+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}-4\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-\textit {\_R}^{4}+2 \textit {\_R}^{3}-8 \textit {\_R}^{2}+3 \textit {\_R} -4\right ) \ln \left (\left (5+4 x \right )^{\frac {1}{6}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{2}\)	\(180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x^3*(5+4*x)^(1/3)-x^3*(5+4*x)^(2/3))/(1-x*(5+4*x)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

3/608*(5+4*x)^(19/6)+3/544*(5+4*x)^(17/6)-15/208*(5+4*x)^(13/6)-15/176*(5+4*x)^(11/6)+3/80*(5+4*x)^(5/3)+3/64*
(5+4*x)^(4/3)+75/224*(5+4*x)^(7/6)+15/32*(5+4*x)^(5/6)-15/32*(5+4*x)^(2/3)-15/16*(5+4*x)^(1/3)+3/2*(5+4*x)^(1/
6)-3*ln(1+(5+4*x)^(1/6))+1/2*sum((2*_R^5-_R^4+2*_R^3-8*_R^2+3*_R-4)/(2*_R^5-_R^2)*ln((5+4*x)^(1/6)-_R),_R=Root
Of(_Z^6-_Z^3-4))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {3}{315392} \, {\left (11264 \, x^{4} + 2560 \, x^{3} - 3600 \, x^{2} + 5400 \, x - 10125\right )} {\left (4 \, x + 5\right )}^{\frac {2}{3}} - \frac {7}{34} \, \sqrt {17} \log \left (-\frac {\sqrt {17} - 2 \, \sqrt {4 \, x + 5} + 1}{\sqrt {17} + 2 \, \sqrt {4 \, x + 5} - 1}\right ) - \frac {3}{186368} \, {\left (7168 \, x^{4} + 896 \, x^{3} - 1440 \, x^{2} + 2700 \, x - 10125\right )} {\left (4 \, x + 5\right )}^{\frac {1}{3}} - \frac {3}{15865304} \, {\left (857584 \, x^{6} + 1314040 \, x^{5} + 12103 \, x^{4} - 436618 \, x^{3} - 7070 \, x^{2} + 15150 \, x - 113625\right )} {\left (4 \, x + 5\right )}^{\frac {1}{6}} - \int -\frac {2 \, {\left (4 \, x^{6} + 5 \, x^{5} - x^{3}\right )} {\left (4 \, x + 5\right )}^{\frac {1}{3}} + {\left (16 \, x^{8} + 40 \, x^{7} + 25 \, x^{6} - x^{2}\right )} {\left (4 \, x + 5\right )}^{\frac {1}{6}}}{4 \, {\left (4 \, x^{3} + 5 \, x^{2} - 2 \, \sqrt {4 \, x + 5} x + 1\right )}}\,{d x} + \frac {1}{2} \, \log \left (4 \, x - \sqrt {4 \, x + 5} + 1\right ) - \log \left (\sqrt {4 \, x + 5} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x^3*(5+4*x)^(1/3)-x^3*(5+4*x)^(2/3))/(1-x*(5+4*x)^(1/2)),x, algorithm="maxima")

[Out]

-3/315392*(11264*x^4 + 2560*x^3 - 3600*x^2 + 5400*x - 10125)*(4*x + 5)^(2/3) - 7/34*sqrt(17)*log(-(sqrt(17) -
2*sqrt(4*x + 5) + 1)/(sqrt(17) + 2*sqrt(4*x + 5) - 1)) - 3/186368*(7168*x^4 + 896*x^3 - 1440*x^2 + 2700*x - 10
125)*(4*x + 5)^(1/3) - 3/15865304*(857584*x^6 + 1314040*x^5 + 12103*x^4 - 436618*x^3 - 7070*x^2 + 15150*x - 11
3625)*(4*x + 5)^(1/6) - integrate(-1/4*(2*(4*x^6 + 5*x^5 - x^3)*(4*x + 5)^(1/3) + (16*x^8 + 40*x^7 + 25*x^6 -
x^2)*(4*x + 5)^(1/6))/(4*x^3 + 5*x^2 - 2*sqrt(4*x + 5)*x + 1), x) + 1/2*log(4*x - sqrt(4*x + 5) + 1) - log(sqr
t(4*x + 5) + 1)

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mupad [B]  time = 1.90, size = 423, normalized size = 1.64 \begin {gather*} \left (\sum _{k=1}^6\ln \left (\mathrm {root}\left (z^6-3\,z^5+\frac {45\,z^4}{68}+\frac {957\,z^3}{544}-\frac {1863\,z^2}{9248}-\frac {225\,z}{578}-\frac {2439}{39304},z,k\right )\,\left (-\mathrm {root}\left (z^6-3\,z^5+\frac {45\,z^4}{68}+\frac {957\,z^3}{544}-\frac {1863\,z^2}{9248}-\frac {225\,z}{578}-\frac {2439}{39304},z,k\right )\,\left (\mathrm {root}\left (z^6-3\,z^5+\frac {45\,z^4}{68}+\frac {957\,z^3}{544}-\frac {1863\,z^2}{9248}-\frac {225\,z}{578}-\frac {2439}{39304},z,k\right )\,\left (-\mathrm {root}\left (z^6-3\,z^5+\frac {45\,z^4}{68}+\frac {957\,z^3}{544}-\frac {1863\,z^2}{9248}-\frac {225\,z}{578}-\frac {2439}{39304},z,k\right )\,\left (\mathrm {root}\left (z^6-3\,z^5+\frac {45\,z^4}{68}+\frac {957\,z^3}{544}-\frac {1863\,z^2}{9248}-\frac {225\,z}{578}-\frac {2439}{39304},z,k\right )\,\left (-\mathrm {root}\left (z^6-3\,z^5+\frac {45\,z^4}{68}+\frac {957\,z^3}{544}-\frac {1863\,z^2}{9248}-\frac {225\,z}{578}-\frac {2439}{39304},z,k\right )\,\left (89890560\,{\left (4\,x+5\right )}^{1/6}[phone]\right )[phone]\,{\left (4\,x+5\right )}^{1/6}[phone]\right )[phone]\,{\left (4\,x+5\right )}^{1/6}[phone]\right )[phone]\,{\left (4\,x+5\right )}^{1/6}[phone]\right )+8945559\,{\left (4\,x+5\right )}^{1/6}[phone]\right )+5031558\,{\left (4\,x+5\right )}^{1/6}+2990358\right )+1119744\,{\left (4\,x+5\right )}^{1/6}+874800\right )\,\mathrm {root}\left (z^6-3\,z^5+\frac {45\,z^4}{68}+\frac {957\,z^3}{544}-\frac {1863\,z^2}{9248}-\frac {225\,z}{578}-\frac {2439}{39304},z,k\right )\right )-3\,\ln \left (-83860333479\,{\left (4\,x+5\right )}^{1/6}-83860333479\right )-\frac {15\,{\left (4\,x+5\right )}^{1/3}}{16}-\frac {15\,{\left (4\,x+5\right )}^{2/3}}{32}+\frac {3\,{\left (4\,x+5\right )}^{1/6}}{2}+\frac {3\,{\left (4\,x+5\right )}^{4/3}}{64}+\frac {3\,{\left (4\,x+5\right )}^{5/3}}{80}+\frac {15\,{\left (4\,x+5\right )}^{5/6}}{32}+\frac {75\,{\left (4\,x+5\right )}^{7/6}}{224}-\frac {15\,{\left (4\,x+5\right )}^{11/6}}{176}-\frac {15\,{\left (4\,x+5\right )}^{13/6}}{208}+\frac {3\,{\left (4\,x+5\right )}^{17/6}}{544}+\frac {3\,{\left (4\,x+5\right )}^{19/6}}{608} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(4*x + 5)^(1/3) + x^3*(4*x + 5)^(2/3) - 1)/(x*(4*x + 5)^(1/2) - 1),x)

[Out]

symsum(log(root(z^6 - 3*z^5 + (45*z^4)/68 + (957*z^3)/544 - (1863*z^2)/9248 - (225*z)/578 - 2439/39304, z, k)*
(5031558*(4*x + 5)^(1/6) - root(z^6 - 3*z^5 + (45*z^4)/68 + (957*z^3)/544 - (1863*z^2)/9248 - (225*z)/578 - 24
39/39304, z, k)*(root(z^6 - 3*z^5 + (45*z^4)/68 + (957*z^3)/544 - (1863*z^2)/9248 - (225*z)/578 - 2439/39304,
z, k)*(42123807*(4*x + 5)^(1/6) - root(z^6 - 3*z^5 + (45*z^4)/68 + (957*z^3)/544 - (1863*z^2)/9248 - (225*z)/5
78 - 2439/39304, z, k)*(root(z^6 - 3*z^5 + (45*z^4)/68 + (957*z^3)/544 - (1863*z^2)/9248 - (225*z)/578 - 2439/
39304, z, k)*(92137824*(4*x + 5)^(1/6) - root(z^6 - 3*z^5 + (45*z^4)/68 + (957*z^3)/544 - (1863*z^2)/9248 - (2
25*z)/578 - 2439/39304, z, k)*(89890560*(4*x + 5)^(1/6) + 95508720) + 79777872) + 37240965*(4*x + 5)^(1/6) + 5
2777656) + 37377288) + 8945559*(4*x + 5)^(1/6) + 13837149) + 2990358) + 1119744*(4*x + 5)^(1/6) + 874800)*root
(z^6 - 3*z^5 + (45*z^4)/68 + (957*z^3)/544 - (1863*z^2)/9248 - (225*z)/578 - 2439/39304, z, k), k, 1, 6) - 3*l
og(- 83860333479*(4*x + 5)^(1/6) - 83860333479) - (15*(4*x + 5)^(1/3))/16 - (15*(4*x + 5)^(2/3))/32 + (3*(4*x
+ 5)^(1/6))/2 + (3*(4*x + 5)^(4/3))/64 + (3*(4*x + 5)^(5/3))/80 + (15*(4*x + 5)^(5/6))/32 + (75*(4*x + 5)^(7/6
))/224 - (15*(4*x + 5)^(11/6))/176 - (15*(4*x + 5)^(13/6))/208 + (3*(4*x + 5)^(17/6))/544 + (3*(4*x + 5)^(19/6
))/608

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \left (4 x + 5\right )^{\frac {2}{3}} + x^{3} \sqrt [3]{4 x + 5} - 1}{x \sqrt {4 x + 5} - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x**3*(5+4*x)**(1/3)-x**3*(5+4*x)**(2/3))/(1-x*(5+4*x)**(1/2)),x)

[Out]

Integral((x**3*(4*x + 5)**(2/3) + x**3*(4*x + 5)**(1/3) - 1)/(x*sqrt(4*x + 5) - 1), x)

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