\(\int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx\) [2875]

Optimal result

Integrand size = 38, antiderivative size = 308 \[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {5 \sqrt {3} \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}}{12 \sqrt [3]{10}-2 \sqrt [3]{10} x-2 \sqrt [3]{10} x^2+5 \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}}\right )}{5 \sqrt [3]{10}}+\frac {\log \left (-6 \sqrt [3]{10}+\sqrt [3]{10} x+\sqrt [3]{10} x^2+5 \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}\right )}{5 \sqrt [3]{10}}-\frac {\log \left (36\ 10^{2/3}-12\ 10^{2/3} x-11\ 10^{2/3} x^2+2\ 10^{2/3} x^3+10^{2/3} x^4+\left (30 \sqrt [3]{10}-5 \sqrt [3]{10} x-5 \sqrt [3]{10} x^2\right ) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}+25 \left (27+27 x+36 x^2+28 x^3+9 x^4+x^5\right )^{2/3}\right )}{10 \sqrt [3]{10}} \]

[Out]

1/50*3^(1/2)*arctan(5*3^(1/2)*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)/(12*10^(1/3)-2*10^(1/3)*x-2*10^(1/3)*x^2
+5*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)))*10^(2/3)+1/50*ln(-6*10^(1/3)+10^(1/3)*x+10^(1/3)*x^2+5*(x^5+9*x^4
+28*x^3+36*x^2+27*x+27)^(1/3))*10^(2/3)-1/100*ln(36*10^(2/3)-12*10^(2/3)*x-11*10^(2/3)*x^2+2*10^(2/3)*x^3+10^(
2/3)*x^4+(30*10^(1/3)-5*10^(1/3)*x-5*10^(1/3)*x^2)*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)+25*(x^5+9*x^4+28*x^
3+36*x^2+27*x+27)^(2/3))*10^(2/3)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.64 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.47, number of steps used = 20, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6820, 6851, 6874, 771, 440, 455, 57, 631, 210, 31} \[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=\frac {2 x (x+3) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{3},\frac {3}{2},\frac {x^2}{9},-x^2\right )}{15 \sqrt [3]{(x+3)^3 \left (x^2+1\right )}}+\frac {x (x+3) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{3},\frac {3}{2},4 x^2,-x^2\right )}{5 \sqrt [3]{(x+3)^3 \left (x^2+1\right )}}+\frac {\sqrt {3} (x+3) \sqrt [3]{x^2+1} \arctan \left (\frac {2^{2/3} \sqrt [3]{x^2+1}+\sqrt [3]{5}}{\sqrt {3} \sqrt [3]{5}}\right )}{5 \sqrt [3]{10} \sqrt [3]{(x+3)^3 \left (x^2+1\right )}}+\frac {\sqrt {3} (x+3) \sqrt [3]{x^2+1} \arctan \left (\frac {2\ 2^{2/3} \sqrt [3]{x^2+1}+\sqrt [3]{5}}{\sqrt {3} \sqrt [3]{5}}\right )}{10 \sqrt [3]{10} \sqrt [3]{(x+3)^3 \left (x^2+1\right )}}-\frac {(x+3) \sqrt [3]{x^2+1} \log \left (1-4 x^2\right )}{20 \sqrt [3]{10} \sqrt [3]{(x+3)^3 \left (x^2+1\right )}}-\frac {(x+3) \sqrt [3]{x^2+1} \log \left (9-x^2\right )}{10 \sqrt [3]{10} \sqrt [3]{(x+3)^3 \left (x^2+1\right )}}+\frac {3 (x+3) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{10}-2 \sqrt [3]{x^2+1}\right )}{20 \sqrt [3]{10} \sqrt [3]{(x+3)^3 \left (x^2+1\right )}}+\frac {3 (x+3) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{10}-\sqrt [3]{x^2+1}\right )}{10 \sqrt [3]{10} \sqrt [3]{(x+3)^3 \left (x^2+1\right )}} \]

[In]

Int[(1 + x)/((1 + 2*x)*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3)),x]

[Out]

(2*x*(3 + x)*(1 + x^2)^(1/3)*AppellF1[1/2, 1, 1/3, 3/2, x^2/9, -x^2])/(15*((3 + x)^3*(1 + x^2))^(1/3)) + (x*(3
+ x)*(1 + x^2)^(1/3)*AppellF1[1/2, 1, 1/3, 3/2, 4*x^2, -x^2])/(5*((3 + x)^3*(1 + x^2))^(1/3)) + (Sqrt[3]*(3 +
x)*(1 + x^2)^(1/3)*ArcTan[(5^(1/3) + 2^(2/3)*(1 + x^2)^(1/3))/(Sqrt[3]*5^(1/3))])/(5*10^(1/3)*((3 + x)^3*(1 +
x^2))^(1/3)) + (Sqrt[3]*(3 + x)*(1 + x^2)^(1/3)*ArcTan[(5^(1/3) + 2*2^(2/3)*(1 + x^2)^(1/3))/(Sqrt[3]*5^(1/3)
)])/(10*10^(1/3)*((3 + x)^3*(1 + x^2))^(1/3)) - ((3 + x)*(1 + x^2)^(1/3)*Log[1 - 4*x^2])/(20*10^(1/3)*((3 + x)
^3*(1 + x^2))^(1/3)) - ((3 + x)*(1 + x^2)^(1/3)*Log[9 - x^2])/(10*10^(1/3)*((3 + x)^3*(1 + x^2))^(1/3)) + (3*(
3 + x)*(1 + x^2)^(1/3)*Log[10^(1/3) - 2*(1 + x^2)^(1/3)])/(20*10^(1/3)*((3 + x)^3*(1 + x^2))^(1/3)) + (3*(3 +
x)*(1 + x^2)^(1/3)*Log[10^(1/3) - (1 + x^2)^(1/3)])/(10*10^(1/3)*((3 + x)^3*(1 + x^2))^(1/3))

Rule 31

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

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 210

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

Rule 440

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

Rule 455

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

Rule 631

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 771

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^2)^p, (d/(d
^2 - e^2*x^2) - e*(x/(d^2 - e^2*x^2)))^(-m), x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&
!IntegerQ[p] && ILtQ[m, 0]

Rule 6820

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

Rule 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[p])*w^(n*FracPart[p]))), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1+x}{(1+2 x) \sqrt [3]{(3+x)^3 \left (1+x^2\right )}} \, dx \\ & = \frac {\left ((3+x) \sqrt [3]{1+x^2}\right ) \int \frac {1+x}{(3+x) (1+2 x) \sqrt [3]{1+x^2}} \, dx}{\sqrt [3]{(3+x)^3 \left (1+x^2\right )}} \\ & = \frac {\left ((3+x) \sqrt [3]{1+x^2}\right ) \int \left (\frac {2}{5 (3+x) \sqrt [3]{1+x^2}}+\frac {1}{5 (1+2 x) \sqrt [3]{1+x^2}}\right ) \, dx}{\sqrt [3]{(3+x)^3 \left (1+x^2\right )}} \\ & = \frac {\left ((3+x) \sqrt [3]{1+x^2}\right ) \int \frac {1}{(1+2 x) \sqrt [3]{1+x^2}} \, dx}{5 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {\left (2 (3+x) \sqrt [3]{1+x^2}\right ) \int \frac {1}{(3+x) \sqrt [3]{1+x^2}} \, dx}{5 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}} \\ & = \frac {\left ((3+x) \sqrt [3]{1+x^2}\right ) \int \left (\frac {1}{\left (1-4 x^2\right ) \sqrt [3]{1+x^2}}+\frac {2 x}{\sqrt [3]{1+x^2} \left (-1+4 x^2\right )}\right ) \, dx}{5 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {\left (2 (3+x) \sqrt [3]{1+x^2}\right ) \int \left (-\frac {3}{\left (-9+x^2\right ) \sqrt [3]{1+x^2}}+\frac {x}{\left (-9+x^2\right ) \sqrt [3]{1+x^2}}\right ) \, dx}{5 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}} \\ & = \frac {\left ((3+x) \sqrt [3]{1+x^2}\right ) \int \frac {1}{\left (1-4 x^2\right ) \sqrt [3]{1+x^2}} \, dx}{5 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {\left (2 (3+x) \sqrt [3]{1+x^2}\right ) \int \frac {x}{\left (-9+x^2\right ) \sqrt [3]{1+x^2}} \, dx}{5 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {\left (2 (3+x) \sqrt [3]{1+x^2}\right ) \int \frac {x}{\sqrt [3]{1+x^2} \left (-1+4 x^2\right )} \, dx}{5 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}-\frac {\left (6 (3+x) \sqrt [3]{1+x^2}\right ) \int \frac {1}{\left (-9+x^2\right ) \sqrt [3]{1+x^2}} \, dx}{5 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}} \\ & = \frac {2 x (3+x) \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{3},\frac {3}{2},\frac {x^2}{9},-x^2\right )}{15 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {x (3+x) \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{3},\frac {3}{2},4 x^2,-x^2\right )}{5 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {\left ((3+x) \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-9+x) \sqrt [3]{1+x}} \, dx,x,x^2\right )}{5 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {\left ((3+x) \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x} (-1+4 x)} \, dx,x,x^2\right )}{5 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}} \\ & = \frac {2 x (3+x) \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{3},\frac {3}{2},\frac {x^2}{9},-x^2\right )}{15 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {x (3+x) \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{3},\frac {3}{2},4 x^2,-x^2\right )}{5 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}-\frac {(3+x) \sqrt [3]{1+x^2} \log \left (1-4 x^2\right )}{20 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}-\frac {(3+x) \sqrt [3]{1+x^2} \log \left (9-x^2\right )}{10 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {\left (3 (3+x) \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {5^{2/3}}{2 \sqrt [3]{2}}+\frac {\sqrt [3]{5} x}{2^{2/3}}+x^2} \, dx,x,\sqrt [3]{1+x^2}\right )}{40 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {\left (3 (3+x) \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{10^{2/3}+\sqrt [3]{10} x+x^2} \, dx,x,\sqrt [3]{1+x^2}\right )}{10 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}-\frac {\left (3 (3+x) \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{5}}{2^{2/3}}-x} \, dx,x,\sqrt [3]{1+x^2}\right )}{20 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}-\frac {\left (3 (3+x) \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{10}-x} \, dx,x,\sqrt [3]{1+x^2}\right )}{10 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}} \\ & = \frac {2 x (3+x) \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{3},\frac {3}{2},\frac {x^2}{9},-x^2\right )}{15 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {x (3+x) \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{3},\frac {3}{2},4 x^2,-x^2\right )}{5 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}-\frac {(3+x) \sqrt [3]{1+x^2} \log \left (1-4 x^2\right )}{20 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}-\frac {(3+x) \sqrt [3]{1+x^2} \log \left (9-x^2\right )}{10 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {3 (3+x) \sqrt [3]{1+x^2} \log \left (\sqrt [3]{10}-2 \sqrt [3]{1+x^2}\right )}{20 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {3 (3+x) \sqrt [3]{1+x^2} \log \left (\sqrt [3]{10}-\sqrt [3]{1+x^2}\right )}{10 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}-\frac {\left (3 (3+x) \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2\ 2^{2/3} \sqrt [3]{1+x^2}}{\sqrt [3]{5}}\right )}{10 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}-\frac {\left (3 (3+x) \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{1+x^2}}{\sqrt [3]{5}}\right )}{5 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}} \\ & = \frac {2 x (3+x) \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{3},\frac {3}{2},\frac {x^2}{9},-x^2\right )}{15 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {x (3+x) \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{3},\frac {3}{2},4 x^2,-x^2\right )}{5 \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {\sqrt {3} (3+x) \sqrt [3]{1+x^2} \arctan \left (\frac {5+10^{2/3} \sqrt [3]{1+x^2}}{5 \sqrt {3}}\right )}{5 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {\sqrt {3} (3+x) \sqrt [3]{1+x^2} \arctan \left (\frac {5+2\ 10^{2/3} \sqrt [3]{1+x^2}}{5 \sqrt {3}}\right )}{10 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}-\frac {(3+x) \sqrt [3]{1+x^2} \log \left (1-4 x^2\right )}{20 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}-\frac {(3+x) \sqrt [3]{1+x^2} \log \left (9-x^2\right )}{10 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {3 (3+x) \sqrt [3]{1+x^2} \log \left (\sqrt [3]{10}-2 \sqrt [3]{1+x^2}\right )}{20 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}}+\frac {3 (3+x) \sqrt [3]{1+x^2} \log \left (\sqrt [3]{10}-\sqrt [3]{1+x^2}\right )}{10 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.57 \[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=-\frac {(3+x) \sqrt [3]{1+x^2} \left (2 \sqrt {3} \arctan \left (\frac {4 \sqrt [3]{10}-2 \sqrt [3]{10} x+5 \sqrt [3]{1+x^2}}{5 \sqrt {3} \sqrt [3]{1+x^2}}\right )-2 \log \left (-2 \sqrt [3]{10}+\sqrt [3]{10} x+5 \sqrt [3]{1+x^2}\right )+\log \left (4\ 10^{2/3}-4\ 10^{2/3} x+10^{2/3} x^2-5 \sqrt [3]{10} (-2+x) \sqrt [3]{1+x^2}+25 \left (1+x^2\right )^{2/3}\right )\right )}{10 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}} \]

[In]

Integrate[(1 + x)/((1 + 2*x)*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3)),x]

[Out]

-1/10*((3 + x)*(1 + x^2)^(1/3)*(2*Sqrt[3]*ArcTan[(4*10^(1/3) - 2*10^(1/3)*x + 5*(1 + x^2)^(1/3))/(5*Sqrt[3]*(1
+ x^2)^(1/3))] - 2*Log[-2*10^(1/3) + 10^(1/3)*x + 5*(1 + x^2)^(1/3)] + Log[4*10^(2/3) - 4*10^(2/3)*x + 10^(2/
3)*x^2 - 5*10^(1/3)*(-2 + x)*(1 + x^2)^(1/3) + 25*(1 + x^2)^(2/3)]))/(10^(1/3)*((3 + x)^3*(1 + x^2))^(1/3))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 10.52 (sec) , antiderivative size = 3732, normalized size of antiderivative = 12.12

method	result	size
trager	\(\text {Expression too large to display}\)	\(3732\)

[In]

int((1+x)/(1+2*x)/(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3),x,method=_RETURNVERBOSE)

[Out]

-1/50*ln(-(640746115263450*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)[phone]*(x^5
+9*x^4+28*x^3+36*x^2+27*x+27)^(2/3)-1778460897600*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z
^2)*x^5-72330055488*RootOf(_Z^3-100)*x^5+9561129209820*RootOf(_Z^3-100)*x+56966325626250*RootOf(81*RootOf(_Z^3
-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^4+368474867221500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3
-100)+2500*_Z^2)*x^3+551878647286500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^2+23509
0299901500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x+2316822089850*RootOf(_Z^3-100)*x^
4+14985883371420*RootOf(_Z^3-100)*x^3+22444920343620*RootOf(_Z^3-100)*x^2-1777529944800*(x^5+9*x^4+28*x^3+36*x
^2+27*x+27)^(2/3)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^2*x+1235042
290000*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x^4+50229205200*Ro
otOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^4+13894225762500*RootOf(81*
RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x^2+565078558500*RootOf(81*RootOf(_
Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^2+26059162803786*RootOf(_Z^3-100)[phone]
250*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x^5+6278650650*RootOf
(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^5+678094270200*RootOf(81*RootOf
(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x+5094549446250*RootOf(81*RootOf(_Z^3-100)^
2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x^3+207195471450*RootOf(81*RootOf(_Z^3-100)^2+450*_Z
*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^3+16673070915000*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(
_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x+3555059889600*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(81*Ro
otOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)*x^3-3555059889600*(x^5+9*x^4+28*x^3+36*x^
2+27*x+27)^(1/3)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)*x^2-28440479
116800*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)
*RootOf(_Z^3-100)*x-1750979954232*RootOf(_Z^3-100)^2*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*x+42660718675200*
(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf
(_Z^3-100)[phone]*RootOf(_Z^3-100)^2*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*x^3+3555059889600*(x^5+9*x^
4+28*x^3+36*x^2+27*x+27)^(2/3)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100
)^2-218872494279*RootOf(_Z^3-100)^2*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*x^2+2626469931348*RootOf(_Z^3-100)
^2*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)-10943624713950*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(2/3)*x)/(1+2*x)/(
3+x)^4)*RootOf(_Z^3-100)-1/9*ln(-(640746115263450*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z
^2)[phone]*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(2/3)-1778460897600*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*R
ootOf(_Z^3-100)+2500*_Z^2)*x^5-72330055488*RootOf(_Z^3-100)*x^5+9561129209820*RootOf(_Z^3-100)*x+5696632562625
0*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^4+368474867221500*RootOf(81*RootOf(_Z^3-10
0)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^3+551878647286500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-10
0)+2500*_Z^2)*x^2+235090299901500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x+2316822089
850*RootOf(_Z^3-100)*x^4+14985883371420*RootOf(_Z^3-100)*x^3+22444920343620*RootOf(_Z^3-100)*x^2-1777529944800
*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(2/3)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootO
f(_Z^3-100)^2*x+1235042290000*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-10
0)^2*x^4+50229205200*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^4+13
894225762500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x^2+56507855
8500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^2+26059162803786*Roo
tOf(_Z^3-100)[phone]*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^
2*x^5+6278650650*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^5+678094
270200*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x+5094549446250*Root
Of(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x^3+207195471450*RootOf(81*Ro
otOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^3+16673070915000*RootOf(81*RootOf(_Z^
3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x+3555059889600*(x^5+9*x^4+28*x^3+36*x^2+27*x
+27)^(1/3)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)*x^3-3555059889600*
(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf
(_Z^3-100)*x^2-28440479116800*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*Root
Of(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)*x-1750979954232*RootOf(_Z^3-100)^2*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^
(1/3)*x+42660718675200*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3
-100)+2500*_Z^2)*RootOf(_Z^3-100)[phone]*RootOf(_Z^3-100)^2*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*x^3+
3555059889600*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(2/3)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+250
0*_Z^2)*RootOf(_Z^3-100)^2-218872494279*RootOf(_Z^3-100)^2*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*x^2+2626469
931348*RootOf(_Z^3-100)^2*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)-10943624713950*(x^5+9*x^4+28*x^3+36*x^2+27*x
+27)^(2/3)*x)/(1+2*x)/(3+x)^4)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)+1/9*RootOf(81*R
ootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*ln(-(-640746115263450*RootOf(81*RootOf(_Z^3-100)^2+450*_Z
*RootOf(_Z^3-100)+2500*_Z^2)-42103828584900*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(2/3)[phone]*RootOf(81*Ro
otOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^5+634969321830*RootOf(_Z^3-100)*x^5+9059928128550*RootOf
(_Z^3-100)*x-34735564406250*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^4-27677297718900
0*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^3-301782583561500*RootOf(81*RootOf(_Z^3-10
0)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^2+65024976568500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100
)+2500*_Z^2)*x-4839705196875*RootOf(_Z^3-100)*x^4-38562771008700*RootOf(_Z^3-100)*x^3-42047358750450*RootOf(_Z
^3-100)*x^2+1777529944800*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(2/3)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_
Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^2*x+1235042290000*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+25
00*_Z^2)^2*RootOf(_Z^3-100)^2*x^4+172078407000*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)
*RootOf(_Z^3-100)^3*x^4+13894225762500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootO
f(_Z^3-100)^2*x^2+1935882078750*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-10
0)^3*x^2-89275137943635*RootOf(_Z^3-100)[phone]*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+250
0*_Z^2)^2*RootOf(_Z^3-100)^2*x^5+21509800875*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*R
ootOf(_Z^3-100)^3*x^5+2323058494500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^
3-100)^3*x+5094549446250*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*
x^3+709823428875*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^3+166730
70915000*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x-3555059889600*
(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf
(_Z^3-100)*x^3+3555059889600*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootO
f(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)*x^2+28440479116800*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(81*R
ootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)*x+3368306286792*RootOf(_Z^3-100)^2*(x^5+
9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*x-42660718675200*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(81*RootOf(_
Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)-421038285849*RootOf(_Z^3-100)^2*(x^5+9*x^4+28*x
^3+36*x^2+27*x+27)^(1/3)*x^3-3555059889600*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(2/3)*RootOf(81*RootOf(_Z^3-100)^
2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^2+421038285849*RootOf(_Z^3-100)^2*(x^5+9*x^4+28*x^3+36*x
^2+27*x+27)^(1/3)*x^2-5052459430188*RootOf(_Z^3-100)^2*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)[phone]*
(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(2/3)*x)/(1+2*x)/(3+x)^4)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((1+x)/(1+2*x)/(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

Sympy [F]

\[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=\int \frac {x + 1}{\sqrt [3]{\left (x + 3\right )^{3} \left (x^{2} + 1\right )} \left (2 x + 1\right )}\, dx \]

[In]

integrate((1+x)/(1+2*x)/(x**5+9*x**4+28*x**3+36*x**2+27*x+27)**(1/3),x)

[Out]

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

Maxima [F]

\[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=\int { \frac {x + 1}{{\left (x^{5} + 9 \, x^{4} + 28 \, x^{3} + 36 \, x^{2} + 27 \, x + 27\right )}^{\frac {1}{3}} {\left (2 \, x + 1\right )}} \,d x } \]

[In]

integrate((1+x)/(1+2*x)/(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3),x, algorithm="maxima")

[Out]

integrate((x + 1)/((x^5 + 9*x^4 + 28*x^3 + 36*x^2 + 27*x + 27)^(1/3)*(2*x + 1)), x)

Giac [F]

\[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=\int { \frac {x + 1}{{\left (x^{5} + 9 \, x^{4} + 28 \, x^{3} + 36 \, x^{2} + 27 \, x + 27\right )}^{\frac {1}{3}} {\left (2 \, x + 1\right )}} \,d x } \]

[In]

integrate((1+x)/(1+2*x)/(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3),x, algorithm="giac")

[Out]

integrate((x + 1)/((x^5 + 9*x^4 + 28*x^3 + 36*x^2 + 27*x + 27)^(1/3)*(2*x + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=\int \frac {x+1}{\left (2\,x+1\right )\,{\left (x^5+9\,x^4+28\,x^3+36\,x^2+27\,x+27\right )}^{1/3}} \,d x \]

[In]

int((x + 1)/((2*x + 1)*(27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5 + 27)^(1/3)),x)

[Out]

int((x + 1)/((2*x + 1)*(27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5 + 27)^(1/3)), x)