\(\int x^6 \sqrt [3]{x+x^3} \, dx\) [1652]

Optimal result

Integrand size = 13, antiderivative size = 112 \[ \int x^6 \sqrt [3]{x+x^3} \, dx=\frac {1}{648} \sqrt [3]{x+x^3} \left (20 x-12 x^3+9 x^5+81 x^7\right )+\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{81 \sqrt {3}}+\frac {5}{243} \log \left (-x+\sqrt [3]{x+x^3}\right )-\frac {5}{486} \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.47, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2046, 2049, 2057, 335, 281, 337} \[ \int x^6 \sqrt [3]{x+x^3} \, dx=\frac {5 \left (x^2+1\right )^{2/3} x^{2/3} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{81 \sqrt {3} \left (x^3+x\right )^{2/3}}-\frac {1}{54} \sqrt [3]{x^3+x} x^3+\frac {5}{162} \sqrt [3]{x^3+x} x+\frac {1}{8} \sqrt [3]{x^3+x} x^7+\frac {1}{72} \sqrt [3]{x^3+x} x^5+\frac {5 \left (x^2+1\right )^{2/3} x^{2/3} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{162 \left (x^3+x\right )^{2/3}} \]

[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[x^(2/3) - (1 + x^2)^(1/3)])/(162*(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}{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 ) \text {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 ) \text {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 {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 )}{81 \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 )}{162 \left (x+x^3\right )^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.23 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.63 \[ \int x^6 \sqrt [3]{x+x^3} \, dx=\frac {\sqrt [3]{x+x^3} \left (60 x^{4/3} \sqrt [3]{1+x^2}-36 x^{10/3} \sqrt [3]{1+x^2}+27 x^{16/3} \sqrt [3]{1+x^2}+243 x^{22/3} \sqrt [3]{1+x^2}+40 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2}}\right )+40 \log \left (-x^{2/3}+\sqrt [3]{1+x^2}\right )-20 \log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )}{1944 \sqrt [3]{x} \sqrt [3]{1+x^2}} \]

[In]

Integrate[x^6*(x + x^3)^(1/3),x]

[Out]

((x + x^3)^(1/3)*(60*x^(4/3)*(1 + x^2)^(1/3) - 36*x^(10/3)*(1 + x^2)^(1/3) + 27*x^(16/3)*(1 + x^2)^(1/3) + 243
*x^(22/3)*(1 + x^2)^(1/3) + 40*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2*(1 + x^2)^(1/3))] + 40*Log[-x^(2/
3) + (1 + x^2)^(1/3)] - 20*Log[x^(4/3) + x^(2/3)*(1 + x^2)^(1/3) + (1 + x^2)^(2/3)]))/(1944*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 = 1.95 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.15

method	result	size
meijerg	\(\frac {3 x^{\frac {22}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {11}{3}\right ], \left [\frac {14}{3}\right ], -x^{2}\right )}{22}\)	\(17\)
pseudoelliptic	\(-\frac {x^{4} \left (-243 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x^{7}-27 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x^{5}+36 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x^{3}+40 \sqrt {3}\, \arctan \left (\frac {\left (2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )-60 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +20 \ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )-40 \ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}-x}{x}\right )\right )}{1944 {\left ({\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}\right )}^{4} {\left ({\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}-x \right )}^{4}}\)	\(180\)
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 (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}+15 \left (x^{3}+x \right )^{\frac {2}{3}}+15 x \left (x^{3}+x \right )^{\frac {1}{3}}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+20 x^{2}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+8\right )}{243}-\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}+15 \left (x^{3}+x \right )^{\frac {2}{3}}+15 x \left (x^{3}+x \right )^{\frac {1}{3}}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+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 )}{81}+\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}-9 \left (x^{3}+x \right )^{\frac {2}{3}}-9 x \left (x^{3}+x \right )^{\frac {1}{3}}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-4 x^{2}-48 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-3\right )}{81}\)	\(447\)
risch	\(\text {Expression too large to display}\)	\(746\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.96 \[ \int x^6 \sqrt [3]{x+x^3} \, dx=\frac {5}{243} \, \sqrt {3} \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (539 \, x^{2} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2205 \, x^{2} + 2197}\right ) + \frac {1}{648} \, {\left (81 \, x^{7} + 9 \, x^{5} - 12 \, x^{3} + 20 \, x\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}} + \frac {5}{486} \, \log \left (3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) \]

[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)

Sympy [F]

\[ \int x^6 \sqrt [3]{x+x^3} \, dx=\int x^{6} \sqrt [3]{x \left (x^{2} + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x^6 \sqrt [3]{x+x^3} \, dx=\int { {\left (x^{3} + x\right )}^{\frac {1}{3}} x^{6} \,d x } \]

[In]

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

[Out]

integrate((x^3 + x)^(1/3)*x^6, x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.87 \[ \int x^6 \sqrt [3]{x+x^3} \, dx=\frac {1}{648} \, {\left (20 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {10}{3}} - 72 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {7}{3}} + 93 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} + 40 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )} x^{8} - \frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {5}{486} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{243} \, \log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]

[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))

Mupad [F(-1)]

Timed out. \[ \int x^6 \sqrt [3]{x+x^3} \, dx=\int x^6\,{\left (x^3+x\right )}^{1/3} \,d x \]

[In]

int(x^6*(x + x^3)^(1/3),x)

[Out]

int(x^6*(x + x^3)^(1/3), x)