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

Optimal. Leaf size=225 \[ \frac {1}{6} \sqrt [3]{x^3+x^2} (3 x+1)-\frac {17}{9} \log \left (\sqrt [3]{x^3+x^2}-x\right )+\sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{x^3+x^2}-2 x\right )+\frac {17}{18} \log \left (x^2+\sqrt [3]{x^3+x^2} x+\left (x^3+x^2\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3+x^2} x+\sqrt [3]{2} \left (x^3+x^2\right )^{2/3}\right )}{2^{2/3}}-\frac {17 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x^2}+x}\right )}{3 \sqrt {3}}+\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3+x^2}+x}\right ) \]

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Rubi [A]  time = 1.01, antiderivative size = 369, normalized size of antiderivative = 1.64, number of steps used = 36, number of rules used = 14, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {2056, 1586, 6733, 6725, 331, 292, 31, 634, 618, 204, 628, 321, 494, 617} \begin {gather*} \frac {1}{2} \sqrt [3]{x^3+x^2} x+\frac {1}{6} \sqrt [3]{x^3+x^2}-\frac {17 \sqrt [3]{x^3+x^2} \log \left (1-\frac {\sqrt [3]{x}}{\sqrt [3]{x+1}}\right )}{9 \sqrt [3]{x+1} x^{2/3}}+\frac {17 \sqrt [3]{x^3+x^2} \log \left (\frac {x^{2/3}}{(x+1)^{2/3}}+\frac {\sqrt [3]{x}}{\sqrt [3]{x+1}}+1\right )}{18 \sqrt [3]{x+1} x^{2/3}}+\frac {\sqrt [3]{2} \sqrt [3]{x^3+x^2} \log \left (1-\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x+1}}\right )}{\sqrt [3]{x+1} x^{2/3}}-\frac {\sqrt [3]{x^3+x^2} \log \left (\frac {2^{2/3} x^{2/3}}{(x+1)^{2/3}}+\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x+1}}+1\right )}{2^{2/3} \sqrt [3]{x+1} x^{2/3}}-\frac {17 \sqrt [3]{x^3+x^2} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{x+1} x^{2/3}}+\frac {\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3+x^2} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{x+1} x^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2 + x^3)^(1/3)/6 + (x*(x^2 + x^3)^(1/3))/2 - (17*(x^2 + x^3)^(1/3)*ArcTan[(1 + (2*x^(1/3))/(1 + x)^(1/3))/S
qrt[3]])/(3*Sqrt[3]*x^(2/3)*(1 + x)^(1/3)) + (2^(1/3)*Sqrt[3]*(x^2 + x^3)^(1/3)*ArcTan[(1 + (2*2^(1/3)*x^(1/3)
)/(1 + x)^(1/3))/Sqrt[3]])/(x^(2/3)*(1 + x)^(1/3)) - (17*(x^2 + x^3)^(1/3)*Log[1 - x^(1/3)/(1 + x)^(1/3)])/(9*
x^(2/3)*(1 + x)^(1/3)) + (17*(x^2 + x^3)^(1/3)*Log[1 + x^(2/3)/(1 + x)^(2/3) + x^(1/3)/(1 + x)^(1/3)])/(18*x^(
2/3)*(1 + x)^(1/3)) + (2^(1/3)*(x^2 + x^3)^(1/3)*Log[1 - (2^(1/3)*x^(1/3))/(1 + x)^(1/3)])/(x^(2/3)*(1 + x)^(1
/3)) - ((x^2 + x^3)^(1/3)*Log[1 + (2^(2/3)*x^(2/3))/(1 + x)^(2/3) + (2^(1/3)*x^(1/3))/(1 + x)^(1/3)])/(2^(2/3)
*x^(2/3)*(1 + x)^(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 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 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 494

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[(k*a^(p + (m + 1)/n))/n, Subst[Int[(x^((k*(m + 1))/n - 1)*(c - (b*c - a*d)*x^k)^q)/(1 - b*x^k)^(p
+ q + (m + 1)/n + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 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 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 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 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 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^2\right ) \sqrt [3]{x^2+x^3}}{-1+x^2} \, dx &=\frac {\sqrt [3]{x^2+x^3} \int \frac {x^{2/3} \sqrt [3]{1+x} \left (1+x^2\right )}{-1+x^2} \, dx}{x^{2/3} \sqrt [3]{1+x}}\\ &=\frac {\sqrt [3]{x^2+x^3} \int \frac {x^{2/3} \left (1+x^2\right )}{(-1+x) (1+x)^{2/3}} \, dx}{x^{2/3} \sqrt [3]{1+x}}\\ &=\frac {\left (3 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (1+x^6\right )}{\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{x^{2/3} \sqrt [3]{1+x}}\\ &=\frac {\left (3 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {2 x}{\left (1+x^3\right )^{2/3}}+\frac {x^4}{\left (1+x^3\right )^{2/3}}+\frac {x^7}{\left (1+x^3\right )^{2/3}}+\frac {2 x}{\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{x^{2/3} \sqrt [3]{1+x}}\\ &=\frac {\left (3 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{x^{2/3} \sqrt [3]{1+x}}+\frac {\left (3 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{x^{2/3} \sqrt [3]{1+x}}+\frac {\left (6 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{x^{2/3} \sqrt [3]{1+x}}+\frac {\left (6 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{x^{2/3} \sqrt [3]{1+x}}\\ &=\sqrt [3]{x^2+x^3}+\frac {1}{2} x \sqrt [3]{x^2+x^3}-\frac {\left (2 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{x^{2/3} \sqrt [3]{1+x}}-\frac {\left (5 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{2 x^{2/3} \sqrt [3]{1+x}}+\frac {\left (6 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}+\frac {\left (6 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{-1+2 x^3} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}\\ &=\frac {1}{6} \sqrt [3]{x^2+x^3}+\frac {1}{2} x \sqrt [3]{x^2+x^3}+\frac {\left (5 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{3 x^{2/3} \sqrt [3]{1+x}}+\frac {\left (2 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}-\frac {\left (2 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}-\frac {\left (2 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}+\frac {\left (2^{2/3} \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt [3]{2} x} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}-\frac {\left (2^{2/3} \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {-1+\sqrt [3]{2} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}\\ &=\frac {1}{6} \sqrt [3]{x^2+x^3}+\frac {1}{2} x \sqrt [3]{x^2+x^3}-\frac {2 \sqrt [3]{x^2+x^3} \log \left (1-\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}+\frac {\sqrt [3]{2} \sqrt [3]{x^2+x^3} \log \left (1-\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}-\frac {\left (2 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{3 x^{2/3} \sqrt [3]{1+x}}+\frac {\left (2 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{3 x^{2/3} \sqrt [3]{1+x}}+\frac {\sqrt [3]{x^2+x^3} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}+\frac {\left (5 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{3 x^{2/3} \sqrt [3]{1+x}}-\frac {\left (3 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}-\frac {\sqrt [3]{x^2+x^3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{2}+2\ 2^{2/3} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{2^{2/3} x^{2/3} \sqrt [3]{1+x}}+\frac {\left (3 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{\sqrt [3]{2} x^{2/3} \sqrt [3]{1+x}}\\ &=\frac {1}{6} \sqrt [3]{x^2+x^3}+\frac {1}{2} x \sqrt [3]{x^2+x^3}-\frac {4 \sqrt [3]{x^2+x^3} \log \left (1-\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{3 x^{2/3} \sqrt [3]{1+x}}+\frac {\sqrt [3]{x^2+x^3} \log \left (1+\frac {x^{2/3}}{(1+x)^{2/3}}+\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}+\frac {\sqrt [3]{2} \sqrt [3]{x^2+x^3} \log \left (1-\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}-\frac {\sqrt [3]{x^2+x^3} \log \left (1+\frac {2^{2/3} x^{2/3}}{(1+x)^{2/3}}+\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{2^{2/3} x^{2/3} \sqrt [3]{1+x}}-\frac {\sqrt [3]{x^2+x^3} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{3 x^{2/3} \sqrt [3]{1+x}}+\frac {\left (5 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{9 x^{2/3} \sqrt [3]{1+x}}-\frac {\left (5 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{9 x^{2/3} \sqrt [3]{1+x}}+\frac {\sqrt [3]{x^2+x^3} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}+\frac {\left (6 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}-\frac {\left (3 \sqrt [3]{2} \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}\\ &=\frac {1}{6} \sqrt [3]{x^2+x^3}+\frac {1}{2} x \sqrt [3]{x^2+x^3}-\frac {2 \sqrt {3} \sqrt [3]{x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{x^{2/3} \sqrt [3]{1+x}}+\frac {\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{x^{2/3} \sqrt [3]{1+x}}-\frac {17 \sqrt [3]{x^2+x^3} \log \left (1-\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{9 x^{2/3} \sqrt [3]{1+x}}+\frac {2 \sqrt [3]{x^2+x^3} \log \left (1+\frac {x^{2/3}}{(1+x)^{2/3}}+\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{3 x^{2/3} \sqrt [3]{1+x}}+\frac {\sqrt [3]{2} \sqrt [3]{x^2+x^3} \log \left (1-\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}-\frac {\sqrt [3]{x^2+x^3} \log \left (1+\frac {2^{2/3} x^{2/3}}{(1+x)^{2/3}}+\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{2^{2/3} x^{2/3} \sqrt [3]{1+x}}+\frac {\left (5 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{18 x^{2/3} \sqrt [3]{1+x}}-\frac {\left (5 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{6 x^{2/3} \sqrt [3]{1+x}}-\frac {\left (2 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}\\ &=\frac {1}{6} \sqrt [3]{x^2+x^3}+\frac {1}{2} x \sqrt [3]{x^2+x^3}+\frac {2 \sqrt [3]{x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{\sqrt {3} x^{2/3} \sqrt [3]{1+x}}-\frac {2 \sqrt {3} \sqrt [3]{x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{x^{2/3} \sqrt [3]{1+x}}+\frac {\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{x^{2/3} \sqrt [3]{1+x}}-\frac {17 \sqrt [3]{x^2+x^3} \log \left (1-\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{9 x^{2/3} \sqrt [3]{1+x}}+\frac {17 \sqrt [3]{x^2+x^3} \log \left (1+\frac {x^{2/3}}{(1+x)^{2/3}}+\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{18 x^{2/3} \sqrt [3]{1+x}}+\frac {\sqrt [3]{2} \sqrt [3]{x^2+x^3} \log \left (1-\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}-\frac {\sqrt [3]{x^2+x^3} \log \left (1+\frac {2^{2/3} x^{2/3}}{(1+x)^{2/3}}+\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{2^{2/3} x^{2/3} \sqrt [3]{1+x}}+\frac {\left (5 \sqrt [3]{x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{3 x^{2/3} \sqrt [3]{1+x}}\\ &=\frac {1}{6} \sqrt [3]{x^2+x^3}+\frac {1}{2} x \sqrt [3]{x^2+x^3}+\frac {\sqrt [3]{x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{3 \sqrt {3} x^{2/3} \sqrt [3]{1+x}}-\frac {2 \sqrt {3} \sqrt [3]{x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{x^{2/3} \sqrt [3]{1+x}}+\frac {\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^2+x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{x^{2/3} \sqrt [3]{1+x}}-\frac {17 \sqrt [3]{x^2+x^3} \log \left (1-\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{9 x^{2/3} \sqrt [3]{1+x}}+\frac {17 \sqrt [3]{x^2+x^3} \log \left (1+\frac {x^{2/3}}{(1+x)^{2/3}}+\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{18 x^{2/3} \sqrt [3]{1+x}}+\frac {\sqrt [3]{2} \sqrt [3]{x^2+x^3} \log \left (1-\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{x^{2/3} \sqrt [3]{1+x}}-\frac {\sqrt [3]{x^2+x^3} \log \left (1+\frac {2^{2/3} x^{2/3}}{(1+x)^{2/3}}+\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{2^{2/3} x^{2/3} \sqrt [3]{1+x}}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 84, normalized size = 0.37 \begin {gather*} \frac {3 \sqrt [3]{x^2 (x+1)} \left (x (x+1) \, _2F_1\left (-\frac {1}{3},\frac {5}{3};\frac {8}{3};-x\right )+5 (x+1) \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};-x\right )-5 \sqrt [3]{x+1} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 x}{x+1}\right )\right )}{5 (x+1)^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(x^2*(1 + x))^(1/3)*(x*(1 + x)*Hypergeometric2F1[-1/3, 5/3, 8/3, -x] + 5*(1 + x)*Hypergeometric2F1[2/3, 2/3
, 5/3, -x] - 5*(1 + x)^(1/3)*Hypergeometric2F1[2/3, 1, 5/3, (2*x)/(1 + x)]))/(5*(1 + x)^(4/3))

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IntegrateAlgebraic [A]  time = 0.62, size = 225, normalized size = 1.00 \begin {gather*} \frac {1}{6} (1+3 x) \sqrt [3]{x^2+x^3}-\frac {17 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )}{3 \sqrt {3}}+\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^3}}\right )-\frac {17}{9} \log \left (-x+\sqrt [3]{x^2+x^3}\right )+\sqrt [3]{2} \log \left (-2 x+2^{2/3} \sqrt [3]{x^2+x^3}\right )+\frac {17}{18} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x^2+x^3}+\sqrt [3]{2} \left (x^2+x^3\right )^{2/3}\right )}{2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + 3*x)*(x^2 + x^3)^(1/3))/6 - (17*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))])/(3*Sqrt[3]) + 2^(1/3)*Sqr
t[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x^2 + x^3)^(1/3))] - (17*Log[-x + (x^2 + x^3)^(1/3)])/9 + 2^(1/3)*Log[-2
*x + 2^(2/3)*(x^2 + x^3)^(1/3)] + (17*Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)])/18 - Log[2*x^2 + 2^(
2/3)*x*(x^2 + x^3)^(1/3) + 2^(1/3)*(x^2 + x^3)^(2/3)]/2^(2/3)

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fricas [A]  time = 0.49, size = 204, normalized size = 0.91 \begin {gather*} -\sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} + \sqrt {3} x}{3 \, x}\right ) + \frac {17}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{6} \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (3 \, x + 1\right )} + 2^{\frac {1}{3}} \log \left (-\frac {2^{\frac {1}{3}} x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left (\frac {2^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {17}{9} \, \log \left (-\frac {x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {17}{18} \, \log \left (\frac {x^{2} + {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*2^(1/3)*arctan(1/3*(sqrt(3)*2^(2/3)*(x^3 + x^2)^(1/3) + sqrt(3)*x)/x) + 17/9*sqrt(3)*arctan(1/3*(sqrt
(3)*x + 2*sqrt(3)*(x^3 + x^2)^(1/3))/x) + 1/6*(x^3 + x^2)^(1/3)*(3*x + 1) + 2^(1/3)*log(-(2^(1/3)*x - (x^3 + x
^2)^(1/3))/x) - 1/2*2^(1/3)*log((2^(2/3)*x^2 + 2^(1/3)*(x^3 + x^2)^(1/3)*x + (x^3 + x^2)^(2/3))/x^2) - 17/9*lo
g(-(x - (x^3 + x^2)^(1/3))/x) + 17/18*log((x^2 + (x^3 + x^2)^(1/3)*x + (x^3 + x^2)^(2/3))/x^2)

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giac [A]  time = 0.32, size = 154, normalized size = 0.68 \begin {gather*} \frac {1}{6} \, {\left ({\left (\frac {1}{x} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )} x^{2} - \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {17}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + {\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}}\right ) + 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} \right |}\right ) + \frac {17}{18} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {17}{9} \, \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((1/x + 1)^(4/3) + 2*(1/x + 1)^(1/3))*x^2 - sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(1/x +
1)^(1/3))) + 17/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(1/x + 1)^(1/3) + 1)) - 1/2*2^(1/3)*log(2^(2/3) + 2^(1/3)*(1/
x + 1)^(1/3) + (1/x + 1)^(2/3)) + 2^(1/3)*log(abs(-2^(1/3) + (1/x + 1)^(1/3))) + 17/18*log((1/x + 1)^(2/3) + (
1/x + 1)^(1/3) + 1) - 17/9*log(abs((1/x + 1)^(1/3) - 1))

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maple [C]  time = 12.54, size = 1603, normalized size = 7.12

method	result	size
trager	\(\text {Expression too large to display}\)	\(1603\)
risch	\(\text {Expression too large to display}\)	\(2565\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/6+1/2*x)*(x^3+x^2)^(1/3)+RootOf(_Z^3-2)*ln(-(-RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3
-2)^3*x^2+10*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^4*x^2+2*RootOf(RootOf(_Z^3-2)^2+_Z
*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3*x-20*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^4
*x+120*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*(x^3+x^2)^(2/3)-17*RootOf(RootOf(_Z^3-
2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^2+170*RootOf(_Z^3-2)^2*x^2-198*(x^3+x^2)^(1/3)*RootOf(RootOf(_Z^
3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x-240*(x^3+x^2)^(1/3)*RootOf(_Z^3-2)*x-3*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3
-2)+_Z^2)*RootOf(_Z^3-2)*x+30*RootOf(_Z^3-2)^2*x+42*(x^3+x^2)^(2/3))/x/(-1+x))-ln((10*RootOf(RootOf(_Z^3-2)^2+
_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3*x^2-RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^
4*x^2-20*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3*x+2*RootOf(RootOf(_Z^3-2)^2+_Z*Roo
tOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^4*x+120*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*(x^3
+x^2)^(2/3)-150*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^2+15*RootOf(_Z^3-2)^2*x^2-42*
(x^3+x^2)^(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x-240*(x^3+x^2)^(1/3)*RootOf(_Z^3-2)*x-70*Root
Of(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x+7*RootOf(_Z^3-2)^2*x+198*(x^3+x^2)^(2/3))/x/(-1+x
))*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)-ln((10*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*Ro
otOf(_Z^3-2)^3*x^2-RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^4*x^2-20*RootOf(RootOf(_Z^3-
2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3*x+2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^
3-2)^4*x+120*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*(x^3+x^2)^(2/3)-150*RootOf(RootO
f(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^2+15*RootOf(_Z^3-2)^2*x^2-42*(x^3+x^2)^(1/3)*RootOf(RootO
f(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x-240*(x^3+x^2)^(1/3)*RootOf(_Z^3-2)*x-70*RootOf(RootOf(_Z^3-2)^2+_Z*RootO
f(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x+7*RootOf(_Z^3-2)^2*x+198*(x^3+x^2)^(2/3))/x/(-1+x))*RootOf(_Z^3-2)-17/9*ln((-
RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^2+2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^
3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x+96*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*(x^3+x^2)^
(2/3)-126*(x^3+x^2)^(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x+28*RootOf(RootOf(
_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^2-18*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*Root
Of(_Z^3-2)^2*x+252*(x^3+x^2)^(2/3)-60*x*(x^3+x^2)^(1/3)-196*x^2-140*x)/x)-17/18*RootOf(RootOf(_Z^3-2)^2+_Z*Roo
tOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*ln(-(-4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*
x^2+8*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x+48*RootOf(RootOf(_Z^3-2)^2+_Z*RootO
f(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*(x^3+x^2)^(2/3)+15*(x^3+x^2)^(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+
_Z^2)*RootOf(_Z^3-2)^2*x-71*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^2-10*RootOf(Roo
tOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x-30*(x^3+x^2)^(2/3)+126*x*(x^3+x^2)^(1/3)-112*x^2-42*x
)/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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