\(\int x^4 \sqrt [3]{x+x^3} \, dx\) [1550]

Optimal result

Integrand size = 13, antiderivative size = 107 \[ \int x^4 \sqrt [3]{x+x^3} \, dx=\frac {1}{108} \sqrt [3]{x+x^3} \left (-5 x+3 x^3+18 x^5\right )-\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{54 \sqrt {3}}-\frac {5}{162} \log \left (-x+\sqrt [3]{x+x^3}\right )+\frac {5}{324} \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right ) \]

[Out]

1/108*(x^3+x)^(1/3)*(18*x^5+3*x^3-5*x)-5/162*arctan(3^(1/2)*x/(x+2*(x^3+x)^(1/3)))*3^(1/2)-5/162*ln(-x+(x^3+x)
^(1/3))+5/324*ln(x^2+x*(x^3+x)^(1/3)+(x^3+x)^(2/3))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.39, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2046, 2049, 2057, 335, 281, 337} \[ \int x^4 \sqrt [3]{x+x^3} \, dx=-\frac {5 \left (x^2+1\right )^{2/3} x^{2/3} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{54 \sqrt {3} \left (x^3+x\right )^{2/3}}+\frac {1}{36} \sqrt [3]{x^3+x} x^3-\frac {5}{108} \sqrt [3]{x^3+x} x+\frac {1}{6} \sqrt [3]{x^3+x} x^5-\frac {5 \left (x^2+1\right )^{2/3} x^{2/3} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{108 \left (x^3+x\right )^{2/3}} \]

[In]

Int[x^4*(x + x^3)^(1/3),x]

[Out]

(-5*x*(x + x^3)^(1/3))/108 + (x^3*(x + x^3)^(1/3))/36 + (x^5*(x + x^3)^(1/3))/6 - (5*x^(2/3)*(1 + x^2)^(2/3)*A
rcTan[(1 + (2*x^(2/3))/(1 + x^2)^(1/3))/Sqrt[3]])/(54*Sqrt[3]*(x + x^3)^(2/3)) - (5*x^(2/3)*(1 + x^2)^(2/3)*Lo
g[x^(2/3) - (1 + x^2)^(1/3)])/(108*(x + x^3)^(2/3))

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 337

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(
1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]

Rule 2046

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a*x^j + b
*x^n)^p/(c*(m + n*p + 1))), x] + Dist[a*(n - j)*(p/(c^j*(m + n*p + 1))), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2049

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

Rule 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^5 \sqrt [3]{x+x^3}+\frac {1}{9} \int \frac {x^5}{\left (x+x^3\right )^{2/3}} \, dx \\ & = \frac {1}{36} x^3 \sqrt [3]{x+x^3}+\frac {1}{6} x^5 \sqrt [3]{x+x^3}-\frac {5}{54} \int \frac {x^3}{\left (x+x^3\right )^{2/3}} \, dx \\ & = -\frac {5}{108} x \sqrt [3]{x+x^3}+\frac {1}{36} x^3 \sqrt [3]{x+x^3}+\frac {1}{6} x^5 \sqrt [3]{x+x^3}+\frac {5}{81} \int \frac {x}{\left (x+x^3\right )^{2/3}} \, dx \\ & = -\frac {5}{108} x \sqrt [3]{x+x^3}+\frac {1}{36} x^3 \sqrt [3]{x+x^3}+\frac {1}{6} x^5 \sqrt [3]{x+x^3}+\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^2\right )^{2/3}} \, dx}{81 \left (x+x^3\right )^{2/3}} \\ & = -\frac {5}{108} x \sqrt [3]{x+x^3}+\frac {1}{36} x^3 \sqrt [3]{x+x^3}+\frac {1}{6} x^5 \sqrt [3]{x+x^3}+\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{27 \left (x+x^3\right )^{2/3}} \\ & = -\frac {5}{108} x \sqrt [3]{x+x^3}+\frac {1}{36} x^3 \sqrt [3]{x+x^3}+\frac {1}{6} x^5 \sqrt [3]{x+x^3}+\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{54 \left (x+x^3\right )^{2/3}} \\ & = -\frac {5}{108} x \sqrt [3]{x+x^3}+\frac {1}{36} x^3 \sqrt [3]{x+x^3}+\frac {1}{6} x^5 \sqrt [3]{x+x^3}-\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{54 \sqrt {3} \left (x+x^3\right )^{2/3}}-\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{108 \left (x+x^3\right )^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.56 \[ \int x^4 \sqrt [3]{x+x^3} \, dx=\frac {\sqrt [3]{x+x^3} \left (-15 x^{4/3} \sqrt [3]{1+x^2}+9 x^{10/3} \sqrt [3]{1+x^2}+54 x^{16/3} \sqrt [3]{1+x^2}-10 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2}}\right )-10 \log \left (-x^{2/3}+\sqrt [3]{1+x^2}\right )+5 \log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )}{324 \sqrt [3]{x} \sqrt [3]{1+x^2}} \]

[In]

Integrate[x^4*(x + x^3)^(1/3),x]

[Out]

((x + x^3)^(1/3)*(-15*x^(4/3)*(1 + x^2)^(1/3) + 9*x^(10/3)*(1 + x^2)^(1/3) + 54*x^(16/3)*(1 + x^2)^(1/3) - 10*
Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2*(1 + x^2)^(1/3))] - 10*Log[-x^(2/3) + (1 + x^2)^(1/3)] + 5*Log[x
^(4/3) + x^(2/3)*(1 + x^2)^(1/3) + (1 + x^2)^(2/3)]))/(324*x^(1/3)*(1 + x^2)^(1/3))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3.

Time = 2.36 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.16

method	result	size
meijerg	\(\frac {3 x^{\frac {16}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], -x^{2}\right )}{16}\)	\(17\)
pseudoelliptic	\(\frac {x^{3} \left (54 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x^{5}+9 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x^{3}+10 \sqrt {3}\, \arctan \left (\frac {\left (2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )-15 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +5 \ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )-10 \ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}-x}{x}\right )\right )}{324 {\left ({\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}\right )}^{3} {\left ({\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}-x \right )}^{3}}\)	\(166\)
trager	\(\frac {x \left (18 x^{4}+3 x^{2}-5\right ) \left (x^{3}+x \right )^{\frac {1}{3}}}{108}+\frac {5 \ln \left (45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-72 \left (x^{3}+x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -87 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+15 \left (x^{3}+x \right )^{\frac {2}{3}}+15 x \left (x^{3}+x \right )^{\frac {1}{3}}+20 x^{2}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+8\right )}{162}-\frac {5 \ln \left (45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-72 \left (x^{3}+x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -87 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+15 \left (x^{3}+x \right )^{\frac {2}{3}}+15 x \left (x^{3}+x \right )^{\frac {1}{3}}+20 x^{2}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+8\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{54}+\frac {5 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}+72 \left (x^{3}+x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-9 \left (x^{3}+x \right )^{\frac {2}{3}}-9 x \left (x^{3}+x \right )^{\frac {1}{3}}-4 x^{2}+48 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-3\right )}{54}\)	\(442\)
risch	\(\frac {x \left (18 x^{4}+3 x^{2}-5\right ) {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{108}+\frac {\left (-\frac {5 \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}+38 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-18 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-16 x^{4}-30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+70 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}-18 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+96 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}-28 x^{2}+32 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}-12}{x^{2}+1}\right )}{162}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}-20 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}+48 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+100 x^{4}-30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}-14 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}+48 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}-36 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+140 x^{2}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+40}{x^{2}+1}\right )}{324}\right ) {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )^{2}\right )^{\frac {1}{3}}}{x \left (x^{2}+1\right )}\)	\(509\)

[In]

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

[Out]

3/16*x^(16/3)*hypergeom([-1/3,8/3],[11/3],-x^2)

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.96 \[ \int x^4 \sqrt [3]{x+x^3} \, dx=-\frac {5}{162} \, \sqrt {3} \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (539 \, x^{2} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2205 \, x^{2} + 2197}\right ) + \frac {1}{108} \, {\left (18 \, x^{5} + 3 \, x^{3} - 5 \, x\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}} - \frac {5}{324} \, \log \left (3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) \]

[In]

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

[Out]

-5/162*sqrt(3)*arctan(-(196*sqrt(3)*(x^3 + x)^(1/3)*x - sqrt(3)*(539*x^2 + 507) - 1274*sqrt(3)*(x^3 + x)^(2/3)
)/(2205*x^2 + 2197)) + 1/108*(18*x^5 + 3*x^3 - 5*x)*(x^3 + x)^(1/3) - 5/324*log(3*(x^3 + x)^(1/3)*x - 3*(x^3 +
x)^(2/3) + 1)

Sympy [F]

\[ \int x^4 \sqrt [3]{x+x^3} \, dx=\int x^{4} \sqrt [3]{x \left (x^{2} + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x^4 \sqrt [3]{x+x^3} \, dx=\int { {\left (x^{3} + x\right )}^{\frac {1}{3}} x^{4} \,d x } \]

[In]

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

[Out]

integrate((x^3 + x)^(1/3)*x^4, x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.82 \[ \int x^4 \sqrt [3]{x+x^3} \, dx=-\frac {1}{108} \, {\left (5 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {7}{3}} - 13 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} - 10 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )} x^{6} + \frac {5}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {5}{324} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {5}{162} \, \log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]

[In]

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

[Out]

-1/108*(5*(1/x^2 + 1)^(7/3) - 13*(1/x^2 + 1)^(4/3) - 10*(1/x^2 + 1)^(1/3))*x^6 + 5/162*sqrt(3)*arctan(1/3*sqrt
(3)*(2*(1/x^2 + 1)^(1/3) + 1)) + 5/324*log((1/x^2 + 1)^(2/3) + (1/x^2 + 1)^(1/3) + 1) - 5/162*log(abs((1/x^2 +
1)^(1/3) - 1))

Mupad [F(-1)]

Timed out. \[ \int x^4 \sqrt [3]{x+x^3} \, dx=\int x^4\,{\left (x^3+x\right )}^{1/3} \,d x \]

[In]

int(x^4*(x + x^3)^(1/3),x)

[Out]

int(x^4*(x + x^3)^(1/3), x)