\(\int \frac {b+a x^6}{\sqrt [3]{-x+x^3}} \, dx\) [2038]

Optimal result

Integrand size = 19, antiderivative size = 145 \[ \int \frac {b+a x^6}{\sqrt [3]{-x+x^3}} \, dx=\frac {1}{108} \left (-x+x^3\right )^{2/3} \left (28 a+21 a x^2+18 a x^4\right )+\frac {1}{162} \left (14 \sqrt {3} a+81 \sqrt {3} b\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )+\frac {1}{162} (-14 a-81 b) \log \left (-x+\sqrt [3]{-x+x^3}\right )+\frac {1}{324} (14 a+81 b) \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \]

[Out]

1/108*(x^3-x)^(2/3)*(18*a*x^4+21*a*x^2+28*a)+1/162*(14*3^(1/2)*a+81*3^(1/2)*b)*arctan(3^(1/2)*x/(x+2*(x^3-x)^(
1/3)))+1/162*(-14*a-81*b)*ln(-x+(x^3-x)^(1/3))+1/324*(14*a+81*b)*ln(x^2+x*(x^3-x)^(1/3)+(x^3-x)^(2/3))

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.87, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2078, 2036, 335, 281, 245, 2049} \[ \int \frac {b+a x^6}{\sqrt [3]{-x+x^3}} \, dx=\frac {7 a \sqrt [3]{x^2-1} \sqrt [3]{x} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{27 \sqrt {3} \sqrt [3]{x^3-x}}+\frac {7}{27} a \left (x^3-x\right )^{2/3}+\frac {1}{6} a \left (x^3-x\right )^{2/3} x^4+\frac {7}{36} a \left (x^3-x\right )^{2/3} x^2-\frac {7 a \sqrt [3]{x^2-1} \sqrt [3]{x} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{54 \sqrt [3]{x^3-x}}+\frac {\sqrt {3} b \sqrt [3]{x^2-1} \sqrt [3]{x} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{x^3-x}}-\frac {3 b \sqrt [3]{x^2-1} \sqrt [3]{x} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{x^3-x}} \]

[In]

Int[(b + a*x^6)/(-x + x^3)^(1/3),x]

[Out]

(7*a*(-x + x^3)^(2/3))/27 + (7*a*x^2*(-x + x^3)^(2/3))/36 + (a*x^4*(-x + x^3)^(2/3))/6 + (7*a*x^(1/3)*(-1 + x^
2)^(1/3)*ArcTan[(1 + (2*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(27*Sqrt[3]*(-x + x^3)^(1/3)) + (Sqrt[3]*b*x^(1/3
)*(-1 + x^2)^(1/3)*ArcTan[(1 + (2*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(2*(-x + x^3)^(1/3)) - (7*a*x^(1/3)*(-1
+ x^2)^(1/3)*Log[x^(2/3) - (-1 + x^2)^(1/3)])/(54*(-x + x^3)^(1/3)) - (3*b*x^(1/3)*(-1 + x^2)^(1/3)*Log[x^(2/
3) - (-1 + x^2)^(1/3)])/(4*(-x + x^3)^(1/3))

Rule 245

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

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 2036

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

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 2078

Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Pq*(a*x^j + b*x^n)^p, x]
, x] /; FreeQ[{a, b, j, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IntegerQ[p] && NeQ[n, j]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b}{\sqrt [3]{-x+x^3}}+\frac {a x^6}{\sqrt [3]{-x+x^3}}\right ) \, dx \\ & = a \int \frac {x^6}{\sqrt [3]{-x+x^3}} \, dx+b \int \frac {1}{\sqrt [3]{-x+x^3}} \, dx \\ & = \frac {1}{6} a x^4 \left (-x+x^3\right )^{2/3}+\frac {1}{9} (7 a) \int \frac {x^4}{\sqrt [3]{-x+x^3}} \, dx+\frac {\left (b \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x+x^3}} \\ & = \frac {7}{36} a x^2 \left (-x+x^3\right )^{2/3}+\frac {1}{6} a x^4 \left (-x+x^3\right )^{2/3}+\frac {1}{27} (14 a) \int \frac {x^2}{\sqrt [3]{-x+x^3}} \, dx+\frac {\left (3 b \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}} \\ & = \frac {7}{27} a \left (-x+x^3\right )^{2/3}+\frac {7}{36} a x^2 \left (-x+x^3\right )^{2/3}+\frac {1}{6} a x^4 \left (-x+x^3\right )^{2/3}+\frac {1}{81} (14 a) \int \frac {1}{\sqrt [3]{-x+x^3}} \, dx+\frac {\left (3 b \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}} \\ & = \frac {7}{27} a \left (-x+x^3\right )^{2/3}+\frac {7}{36} a x^2 \left (-x+x^3\right )^{2/3}+\frac {1}{6} a x^4 \left (-x+x^3\right )^{2/3}+\frac {\sqrt {3} b \sqrt [3]{x} \sqrt [3]{-1+x^2} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-x+x^3}}-\frac {3 b \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 \sqrt [3]{-x+x^3}}+\frac {\left (14 a \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \, dx}{81 \sqrt [3]{-x+x^3}} \\ & = \frac {7}{27} a \left (-x+x^3\right )^{2/3}+\frac {7}{36} a x^2 \left (-x+x^3\right )^{2/3}+\frac {1}{6} a x^4 \left (-x+x^3\right )^{2/3}+\frac {\sqrt {3} b \sqrt [3]{x} \sqrt [3]{-1+x^2} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-x+x^3}}-\frac {3 b \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 \sqrt [3]{-x+x^3}}+\frac {\left (14 a \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{27 \sqrt [3]{-x+x^3}} \\ & = \frac {7}{27} a \left (-x+x^3\right )^{2/3}+\frac {7}{36} a x^2 \left (-x+x^3\right )^{2/3}+\frac {1}{6} a x^4 \left (-x+x^3\right )^{2/3}+\frac {\sqrt {3} b \sqrt [3]{x} \sqrt [3]{-1+x^2} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-x+x^3}}-\frac {3 b \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 \sqrt [3]{-x+x^3}}+\frac {\left (7 a \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{27 \sqrt [3]{-x+x^3}} \\ & = \frac {7}{27} a \left (-x+x^3\right )^{2/3}+\frac {7}{36} a x^2 \left (-x+x^3\right )^{2/3}+\frac {1}{6} a x^4 \left (-x+x^3\right )^{2/3}+\frac {7 a \sqrt [3]{x} \sqrt [3]{-1+x^2} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{27 \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt {3} b \sqrt [3]{x} \sqrt [3]{-1+x^2} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-x+x^3}}-\frac {7 a \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{54 \sqrt [3]{-x+x^3}}-\frac {3 b \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 \sqrt [3]{-x+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.12 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.57 \[ \int \frac {b+a x^6}{\sqrt [3]{-x+x^3}} \, dx=\frac {a x \left (-1+x^2\right ) \left (28+21 x^2+18 x^4\right )}{108 \sqrt [3]{x \left (-1+x^2\right )}}+\frac {(14 a+81 b) \sqrt [3]{x} \sqrt [3]{-1+x^2} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )}{54 \sqrt {3} \sqrt [3]{x \left (-1+x^2\right )}}+\frac {(-14 a-81 b) \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )}{162 \sqrt [3]{x \left (-1+x^2\right )}}+\frac {(14 a+81 b) \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )}{324 \sqrt [3]{x \left (-1+x^2\right )}} \]

[In]

Integrate[(b + a*x^6)/(-x + x^3)^(1/3),x]

[Out]

(a*x*(-1 + x^2)*(28 + 21*x^2 + 18*x^4))/(108*(x*(-1 + x^2))^(1/3)) + ((14*a + 81*b)*x^(1/3)*(-1 + x^2)^(1/3)*A
rcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2*(-1 + x^2)^(1/3))])/(54*Sqrt[3]*(x*(-1 + x^2))^(1/3)) + ((-14*a - 81*b)*x
^(1/3)*(-1 + x^2)^(1/3)*Log[-x^(2/3) + (-1 + x^2)^(1/3)])/(162*(x*(-1 + x^2))^(1/3)) + ((14*a + 81*b)*x^(1/3)*
(-1 + x^2)^(1/3)*Log[x^(4/3) + x^(2/3)*(-1 + x^2)^(1/3) + (-1 + x^2)^(2/3)])/(324*(x*(-1 + x^2))^(1/3))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.47

method	result	size
meijerg	\(\frac {3 a {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}} x^{\frac {20}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {10}{3}\right ], \left [\frac {13}{3}\right ], x^{2}\right )}{20 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}+\frac {3 b {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}} x^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{2}\right )}{2 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}\)	\(68\)
risch	\(\frac {a \left (18 x^{4}+21 x^{2}+28\right ) x \left (x^{2}-1\right )}{108 {\left (x \left (x^{2}-1\right )\right )}^{\frac {1}{3}}}+\frac {3 b {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}} x^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{2}\right )}{2 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}+\frac {7 a {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}} x^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{2}\right )}{27 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}\)	\(98\)
pseudoelliptic	\(-\frac {7 x^{3} \left (3 a \left (-1-\frac {9}{14} x^{4}-\frac {3}{4} x^{2}\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+\left (a +\frac {81 b}{14}\right ) \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right )+\ln \left (\frac {-x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right )\right )}{81 {\left (\left (x^{3}-x \right )^{\frac {2}{3}}+x \left (x +\left (x^{3}-x \right )^{\frac {1}{3}}\right )\right )}^{3} {\left (-\left (x^{3}-x \right )^{\frac {1}{3}}+x \right )}^{3}}\)	\(153\)
trager	\(\frac {a \left (18 x^{4}+21 x^{2}+28\right ) \left (x^{3}-x \right )^{\frac {2}{3}}}{108}-\frac {\left (14 a +81 b \right ) \left (18 \ln \left (-228744 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )^{2} x^{2}+40608 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+96390 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -124290 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) x^{2}+914976 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}-7611 x \left (x^{3}-x \right )^{\frac {1}{3}}+1550 x^{2}+26964 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )-465\right ) \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )+\ln \left (178524 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )^{2} x^{2}-40608 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+136998 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -106308 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) x^{2}-714096 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )^{2}+7611 \left (x^{3}-x \right )^{\frac {2}{3}}-5355 x \left (x^{3}-x \right )^{\frac {1}{3}}-1705 x^{2}+58266 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )+1085\right )-\ln \left (-228744 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )^{2} x^{2}+40608 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+96390 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -124290 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) x^{2}+914976 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}-7611 x \left (x^{3}-x \right )^{\frac {1}{3}}+1550 x^{2}+26964 \operatorname {RootOf}\left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )-465\right )\right )}{162}\)	\(465\)

[In]

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

[Out]

3/20*a/signum(x^2-1)^(1/3)*(-signum(x^2-1))^(1/3)*x^(20/3)*hypergeom([1/3,10/3],[13/3],x^2)+3/2*b/signum(x^2-1
)^(1/3)*(-signum(x^2-1))^(1/3)*x^(2/3)*hypergeom([1/3,1/3],[4/3],x^2)

Fricas [A] (verification not implemented)

none

Time = 90.85 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.88 \[ \int \frac {b+a x^6}{\sqrt [3]{-x+x^3}} \, dx=\frac {1}{162} \, \sqrt {3} {\left (14 \, a + 81 \, b\right )} \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) - \frac {1}{324} \, {\left (14 \, a + 81 \, b\right )} \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) + \frac {1}{108} \, {\left (18 \, a x^{4} + 21 \, a x^{2} + 28 \, a\right )} {\left (x^{3} - x\right )}^{\frac {2}{3}} \]

[In]

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

[Out]

1/162*sqrt(3)*(14*a + 81*b)*arctan(-(44032959556*sqrt(3)*(x^3 - x)^(1/3)*x + sqrt(3)*(16754327161*x^2 - 270720
4793) - 10524305234*sqrt(3)*(x^3 - x)^(2/3))/(81835897185*x^2 - 1102302937)) - 1/324*(14*a + 81*b)*log(-3*(x^3
- x)^(1/3)*x + 3*(x^3 - x)^(2/3) + 1) + 1/108*(18*a*x^4 + 21*a*x^2 + 28*a)*(x^3 - x)^(2/3)

Sympy [F]

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

[In]

integrate((a*x**6+b)/(x**3-x)**(1/3),x)

[Out]

Integral((a*x**6 + b)/(x*(x - 1)*(x + 1))**(1/3), x)

Maxima [F]

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

[In]

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

[Out]

integrate((a*x^6 + b)/(x^3 - x)^(1/3), x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.92 \[ \int \frac {b+a x^6}{\sqrt [3]{-x+x^3}} \, dx=\frac {1}{108} \, {\left (28 \, a {\left (\frac {1}{x^{2}} - 1\right )}^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} - 77 \, a {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {5}{3}} + 67 \, a {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}}\right )} x^{6} - \frac {1}{162} \, \sqrt {3} {\left (14 \, a + 81 \, b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{324} \, {\left (14 \, a + 81 \, b\right )} \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{162} \, {\left (14 \, a + 81 \, b\right )} \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]

[In]

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

[Out]

1/108*(28*a*(1/x^2 - 1)^2*(-1/x^2 + 1)^(2/3) - 77*a*(-1/x^2 + 1)^(5/3) + 67*a*(-1/x^2 + 1)^(2/3))*x^6 - 1/162*
sqrt(3)*(14*a + 81*b)*arctan(1/3*sqrt(3)*(2*(-1/x^2 + 1)^(1/3) + 1)) + 1/324*(14*a + 81*b)*log((-1/x^2 + 1)^(2
/3) + (-1/x^2 + 1)^(1/3) + 1) - 1/162*(14*a + 81*b)*log(abs((-1/x^2 + 1)^(1/3) - 1))

Mupad [F(-1)]

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

[In]

int((b + a*x^6)/(x^3 - x)^(1/3),x)

[Out]

int((b + a*x^6)/(x^3 - x)^(1/3), x)