3.17.52 \(\int x^6 \sqrt [3]{x+x^3} \, dx\)

Optimal. Leaf size=112 \[ \frac {5}{243} \log \left (\sqrt [3]{x^3+x}-x\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right )}{81 \sqrt {3}}-\frac {5}{486} \log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right )+\frac {1}{648} \sqrt [3]{x^3+x} \left (81 x^7+9 x^5-12 x^3+20 x\right ) \]

________________________________________________________________________________________

Rubi [B]  time = 0.24, antiderivative size = 226, normalized size of antiderivative = 2.02, number of steps used = 14, number of rules used = 12, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {2021, 2024, 2032, 329, 275, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} -\frac {1}{54} \sqrt [3]{x^3+x} x^3+\frac {5}{162} \sqrt [3]{x^3+x} x+\frac {1}{8} \sqrt [3]{x^3+x} x^7+\frac {1}{72} \sqrt [3]{x^3+x} x^5+\frac {5 \left (x^2+1\right )^{2/3} x^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2+1}}\right )}{243 \left (x^3+x\right )^{2/3}}-\frac {5 \left (x^2+1\right )^{2/3} x^{2/3} \log \left (\frac {x^{4/3}}{\left (x^2+1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2+1}}+1\right )}{486 \left (x^3+x\right )^{2/3}}+\frac {5 \left (x^2+1\right )^{2/3} x^{2/3} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{81 \sqrt {3} \left (x^3+x\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6*(x + x^3)^(1/3),x]

[Out]

(5*x*(x + x^3)^(1/3))/162 - (x^3*(x + x^3)^(1/3))/54 + (x^5*(x + x^3)^(1/3))/72 + (x^7*(x + x^3)^(1/3))/8 + (5
*x^(2/3)*(1 + x^2)^(2/3)*ArcTan[(1 + (2*x^(2/3))/(1 + x^2)^(1/3))/Sqrt[3]])/(81*Sqrt[3]*(x + x^3)^(2/3)) + (5*
x^(2/3)*(1 + x^2)^(2/3)*Log[1 - x^(2/3)/(1 + x^2)^(1/3)])/(243*(x + x^3)^(2/3)) - (5*x^(2/3)*(1 + x^2)^(2/3)*L
og[1 + x^(4/3)/(1 + x^2)^(2/3) + x^(2/3)/(1 + x^2)^(1/3)])/(486*(x + x^3)^(2/3))

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 275

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 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 329

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 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 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 2021

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 2024

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 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[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*} \int x^6 \sqrt [3]{x+x^3} \, dx &=\frac {1}{8} x^7 \sqrt [3]{x+x^3}+\frac {1}{12} \int \frac {x^7}{\left (x+x^3\right )^{2/3}} \, dx\\ &=\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {2}{27} \int \frac {x^5}{\left (x+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}+\frac {5}{81} \int \frac {x^3}{\left (x+x^3\right )^{2/3}} \, dx\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {10}{243} \int \frac {x}{\left (x+x^3\right )^{2/3}} \, dx\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {\left (10 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^2\right )^{2/3}} \, dx}{243 \left (x+x^3\right )^{2/3}}\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {\left (10 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{81 \left (x+x^3\right )^{2/3}}\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{81 \left (x+x^3\right )^{2/3}}\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{81 \left (x+x^3\right )^{2/3}}\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{243 \left (x+x^3\right )^{2/3}}+\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{243 \left (x+x^3\right )^{2/3}}\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}+\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{243 \left (x+x^3\right )^{2/3}}-\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{486 \left (x+x^3\right )^{2/3}}+\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{162 \left (x+x^3\right )^{2/3}}\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}+\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{243 \left (x+x^3\right )^{2/3}}-\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{486 \left (x+x^3\right )^{2/3}}-\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{81 \left (x+x^3\right )^{2/3}}\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}+\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{81 \sqrt {3} \left (x+x^3\right )^{2/3}}+\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{243 \left (x+x^3\right )^{2/3}}-\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{486 \left (x+x^3\right )^{2/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.03, size = 68, normalized size = 0.61 \begin {gather*} \frac {x \sqrt [3]{x^3+x} \left (\sqrt [3]{x^2+1} \left (27 x^6+3 x^4-4 x^2+20\right )-20 \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};-x^2\right )\right )}{216 \sqrt [3]{x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6*(x + x^3)^(1/3),x]

[Out]

(x*(x + x^3)^(1/3)*((1 + x^2)^(1/3)*(20 - 4*x^2 + 3*x^4 + 27*x^6) - 20*Hypergeometric2F1[-1/3, 2/3, 5/3, -x^2]
))/(216*(1 + x^2)^(1/3))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.40, size = 112, normalized size = 1.00 \begin {gather*} \frac {1}{648} \sqrt [3]{x+x^3} \left (20 x-12 x^3+9 x^5+81 x^7\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{81 \sqrt {3}}+\frac {5}{243} \log \left (-x+\sqrt [3]{x+x^3}\right )-\frac {5}{486} \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^6*(x + x^3)^(1/3),x]

[Out]

((x + x^3)^(1/3)*(20*x - 12*x^3 + 9*x^5 + 81*x^7))/648 + (5*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/(81*S
qrt[3]) + (5*Log[-x + (x + x^3)^(1/3)])/243 - (5*Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)])/486

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 108, normalized size = 0.96 \begin {gather*} \frac {5}{243} \, \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}{648} \, {\left (81 \, x^{7} + 9 \, x^{5} - 12 \, x^{3} + 20 \, x\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}} + \frac {5}{486} \, \log \left (3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^3+x)^(1/3),x, algorithm="fricas")

[Out]

5/243*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)) + 1/648*(81*x^7 + 9*x^5 - 12*x^3 + 20*x)*(x^3 + x)^(1/3) + 5/486*log(3*(x^3 + x)^(1/3)*x -
3*(x^3 + x)^(2/3) + 1)

________________________________________________________________________________________

giac [A]  time = 1.00, size = 97, normalized size = 0.87 \begin {gather*} \frac {1}{648} \, {\left (20 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {10}{3}} - 72 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {7}{3}} + 93 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} + 40 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )} x^{8} - \frac {5}{243} \, \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}{486} \, \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}{243} \, \log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^3+x)^(1/3),x, algorithm="giac")

[Out]

1/648*(20*(1/x^2 + 1)^(10/3) - 72*(1/x^2 + 1)^(7/3) + 93*(1/x^2 + 1)^(4/3) + 40*(1/x^2 + 1)^(1/3))*x^8 - 5/243
*sqrt(3)*arctan(1/3*sqrt(3)*(2*(1/x^2 + 1)^(1/3) + 1)) - 5/486*log((1/x^2 + 1)^(2/3) + (1/x^2 + 1)^(1/3) + 1)
+ 5/243*log(abs((1/x^2 + 1)^(1/3) - 1))

________________________________________________________________________________________

maple [C]  time = 3.87, size = 17, normalized size = 0.15

method	result	size
meijerg	\(\frac {3 x^{\frac {22}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {11}{3}\right ], \left [\frac {14}{3}\right ], -x^{2}\right )}{22}\)	\(17\)
trager	\(\frac {x \left (81 x^{6}+9 x^{4}-12 x^{2}+20\right ) \left (x^{3}+x \right )^{\frac {1}{3}}}{648}+\frac {5 \ln \left (-36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-72 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}+27 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {1}{3}} x +33 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-9 \left (x^{3}+x \right )^{\frac {2}{3}}-15 x \left (x^{3}+x \right )^{\frac {1}{3}}+20 x^{2}+51 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+15\right )}{243}+\frac {5 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+72 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {1}{3}} x -30 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+15 \left (x^{3}+x \right )^{\frac {2}{3}}+9 x \left (x^{3}+x \right )^{\frac {1}{3}}-25 x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-10\right )}{81}\)	\(304\)
risch	\(\frac {x \left (81 x^{6}+9 x^{4}-12 x^{2}+20\right ) \left (x \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{648}+\frac {\left (\frac {5 \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2} x^{4}-38 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{4}+18 \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 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}-70 \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 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}+18 \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 \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 )}{243}+\frac {5 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2} x^{4}+20 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{4}-48 \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 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+14 \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}-\RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}-48 \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 \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 )}{486}\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 )}\)	\(514\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(x^3+x)^(1/3),x,method=_RETURNVERBOSE)

[Out]

3/22*x^(22/3)*hypergeom([-1/3,11/3],[14/3],-x^2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (x^{3} + x\right )}^{\frac {1}{3}} x^{6}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^3+x)^(1/3),x, algorithm="maxima")

[Out]

integrate((x^3 + x)^(1/3)*x^6, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^6\,{\left (x^3+x\right )}^{1/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(x + x^3)^(1/3),x)

[Out]

int(x^6*(x + x^3)^(1/3), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{6} \sqrt [3]{x \left (x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(x**3+x)**(1/3),x)

[Out]

Integral(x**6*(x*(x**2 + 1))**(1/3), x)

________________________________________________________________________________________