3.29.75 \(\int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx\)
Optimal. Leaf size=308 \[ \frac {\log \left (\sqrt [3]{10} x^2+5 \sqrt [3]{x^5+9 x^4+28 x^3+36 x^2+27 x+27}+\sqrt [3]{10} x-6 \sqrt [3]{10}\right )}{5 \sqrt [3]{10}}-\frac {\log \left (10^{2/3} x^4+2\ 10^{2/3} x^3-11\ 10^{2/3} x^2+25 \left (x^5+9 x^4+28 x^3+36 x^2+27 x+27\right )^{2/3}+\left (-5 \sqrt [3]{10} x^2-5 \sqrt [3]{10} x+30 \sqrt [3]{10}\right ) \sqrt [3]{x^5+9 x^4+28 x^3+36 x^2+27 x+27}-12\ 10^{2/3} x+36\ 10^{2/3}\right )}{10 \sqrt [3]{10}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {5 \sqrt {3} \sqrt [3]{x^5+9 x^4+28 x^3+36 x^2+27 x+27}}{-2 \sqrt [3]{10} x^2+5 \sqrt [3]{x^5+9 x^4+28 x^3+36 x^2+27 x+27}-2 \sqrt [3]{10} x+12 \sqrt [3]{10}}\right )}{5 \sqrt [3]{10}} \]
________________________________________________________________________________________
Rubi [C] time = 1.02, antiderivative size = 454, normalized size of antiderivative = 1.47,
number of steps used = 20, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used
= {6688, 6719, 6742, 757, 429, 444, 55, 617, 204, 31} \begin {gather*} \frac {2 x (x+3) \sqrt [3]{x^2+1} F_1\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} F_1\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 {(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 )}}+\frac {\sqrt {3} (x+3) \sqrt [3]{x^2+1} \tan ^{-1}\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} \tan ^{-1}\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 )}} \end {gather*}
Warning: Unable to verify antiderivative.
[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 55
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 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 429
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 444
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 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 757
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 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rule 6719
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[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 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rubi steps
\begin {align*} \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 {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} F_1\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} F_1\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 ) \operatorname {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 ) \operatorname {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} F_1\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} F_1\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} F_1\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} F_1\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 ) \operatorname {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 ) \operatorname {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} F_1\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} F_1\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} \tan ^{-1}\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} \tan ^{-1}\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 [F] time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
[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]
Integrate[(1 + x)/((1 + 2*x)*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3)), x]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.40, size = 308, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\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}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(1 + x)/((1 + 2*x)*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3)),x]
[Out]
(Sqrt[3]*ArcTan[(5*Sqrt[3]*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3))/(12*10^(1/3) - 2*10^(1/3)*x - 2*
10^(1/3)*x^2 + 5*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3))])/(5*10^(1/3)) + Log[-6*10^(1/3) + 10^(1/3
)*x + 10^(1/3)*x^2 + 5*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3)]/(5*10^(1/3)) - Log[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)*(
27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3) + 25*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(2/3)]/(10*1
0^(1/3))
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
maple [C] time = 20.26, size = 3730, normalized size = 12.11
| | |
| method |
result |
size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(3730\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[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]
1/9*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*ln(-(-42103828584900*(x^5+9*x^4+28*x^3+36*
x^2+27*x+27)^(2/3)+634969321830*RootOf(_Z^3-100)*x^5-89275137943635*RootOf(_Z^3-100)-4839705196875*RootOf(_Z^3
-100)*x^4-38562771008700*RootOf(_Z^3-100)*x^3-42047358750450*RootOf(_Z^3-100)*x^2+9059928128550*RootOf(_Z^3-10
0)*x-640746115263450*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)-34735564406250*RootOf(81*
RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^4-276772977189000*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*
RootOf(_Z^3-100)+2500*_Z^2)*x^3-301782583561500*RootOf(81*RootOf(_Z^3-100)^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+4557306050100*RootOf(81
*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^5+1235042290000*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*R
ootOf(_Z^3-100)+2500*_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*RootOf(_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^3-100)^3*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+21051914292450*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(2/3)
*x-5052459430188*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(_Z^3-100)^2+154380286250*RootOf(81*RootOf(_Z^3
-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x^5+21509800875*RootOf(81*RootOf(_Z^3-100)^2+4
50*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^5-3555059889600*(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*RootOf(_Z^3-10
0)+2500*_Z^2)*RootOf(_Z^3-100)*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+16673070915000*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-1
00)^2*x+2323058494500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x-421
038285849*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(_Z^3-100)^2*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+4210382858
49*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(_Z^3-100)^2*x^2+3368306286792*(x^5+9*x^4+28*x^3+36*x^2+27*x+
27)^(1/3)*RootOf(_Z^3-100)^2*x-42660718675200*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(81*RootOf(_Z^3-10
0)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100))/(1+2*x)/(3+x)^4)-1/50*ln((-21887249427900*(x^5+9*x^4
+28*x^3+36*x^2+27*x+27)^(2/3)+72330055488*RootOf(_Z^3-100)*x^5-26059162803786*RootOf(_Z^3-100)-2316822089850*R
ootOf(_Z^3-100)*x^4-14985883371420*RootOf(_Z^3-100)*x^3-22444920343620*RootOf(_Z^3-100)*x^2-9561129209820*Root
Of(_Z^3-100)*x-640746115263450*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)-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^2-235090299901500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x+177846089760
0*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^5-1235042290000*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*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*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+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+10943624713950*(x^5+9*x^4+28*x^3+36*x^2+27*x+
27)^(2/3)*x-2626469931348*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(_Z^3-100)^2-154380286250*RootOf(81*Ro
otOf(_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^5-3555059889600*(x^5+9*x^4+28*x^3+36*x^2+27*x+2
7)^(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*RootOf
(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)*x-5094549446250*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+25
00*_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*RootO
f(_Z^3-100)^2*x-678094270200*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^
3*x-218872494279*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(_Z^3-100)^2*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+218
872494279*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(_Z^3-100)^2*x^2+1750979954232*(x^5+9*x^4+28*x^3+36*x^
2+27*x+27)^(1/3)*RootOf(_Z^3-100)^2*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))/(1+2*x)/(3+x)^4)*RootOf(_Z^3-100)-1/9*ln((-21
887249427900*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(2/3)+72330055488*RootOf(_Z^3-100)*x^5-26059162803786*RootOf(_Z
^3-100)-2316822089850*RootOf(_Z^3-100)*x^4-14985883371420*RootOf(_Z^3-100)*x^3-22444920343620*RootOf(_Z^3-100)
*x^2-9561129209820*RootOf(_Z^3-100)*x-640746115263450*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+250
0*_Z^2)-56966325626250*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^4-368474867221500*Roo
tOf(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^2-235090299901500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+25
00*_Z^2)*x+1778460897600*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^5-1235042290000*Roo
tOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x^4-50229205200*RootOf(81*Ro
otOf(_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+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+10943624713950*(x^5+9*
x^4+28*x^3+36*x^2+27*x+27)^(2/3)*x-2626469931348*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(_Z^3-100)^2-15
4380286250*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^5-3555059889600*(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^3-10
0)*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^3-1
00)+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*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)*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*Root
Of(_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-678094270200*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*
_Z^2)*RootOf(_Z^3-100)^3*x-218872494279*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(_Z^3-100)^2*x^3-3555059
889600*(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*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(_Z^3-100)^2*x^2+1750979954232*
(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)*RootOf(_Z^3-100)^2*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))/(1+2*x)/(3+x)^4)*RootO
f(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\sqrt [3]{\left (x + 3\right )^{3} \left (x^{2} + 1\right )} \left (2 x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________