Integrand size = 19, antiderivative size = 145 \[ \int \frac {b+a x^6}{\sqrt [3]{-x+x^3}} \, dx=\frac {1}{108} \left (-x+x^3\right )^{2/3} \left (28 a+21 a x^2+18 a x^4\right )+\frac {1}{162} \left (14 \sqrt {3} a+81 \sqrt {3} b\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )+\frac {1}{162} (-14 a-81 b) \log \left (-x+\sqrt [3]{-x+x^3}\right )+\frac {1}{324} (14 a+81 b) \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \]
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))
Time = 8.12 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.57 \[ \int \frac {b+a x^6}{\sqrt [3]{-x+x^3}} \, dx=\frac {a x \left (-1+x^2\right ) \left (28+21 x^2+18 x^4\right )}{108 \sqrt [3]{x \left (-1+x^2\right )}}+\frac {(14 a+81 b) \sqrt [3]{x} \sqrt [3]{-1+x^2} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )}{54 \sqrt {3} \sqrt [3]{x \left (-1+x^2\right )}}+\frac {(-14 a-81 b) \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )}{162 \sqrt [3]{x \left (-1+x^2\right )}}+\frac {(14 a+81 b) \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )}{324 \sqrt [3]{x \left (-1+x^2\right )}} \]
(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)*ArcTan[(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))
Time = 0.43 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.87, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2450, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a x^6+b}{\sqrt [3]{x^3-x}} \, dx\) |
| \(\Big \downarrow \) 2450 |
| \(\displaystyle \int \left (\frac {a x^6}{\sqrt [3]{x^3-x}}+\frac {b}{\sqrt [3]{x^3-x}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {7 a \sqrt [3]{x^2-1} \sqrt [3]{x} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{27 \sqrt {3} \sqrt [3]{x^3-x}}+\frac {7}{27} a \left (x^3-x\right )^{2/3}+\frac {1}{6} a \left (x^3-x\right )^{2/3} x^4+\frac {7}{36} a \left (x^3-x\right )^{2/3} x^2-\frac {7 a \sqrt [3]{x^2-1} \sqrt [3]{x} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{54 \sqrt [3]{x^3-x}}+\frac {\sqrt {3} b \sqrt [3]{x^2-1} \sqrt [3]{x} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{x^3-x}}-\frac {3 b \sqrt [3]{x^2-1} \sqrt [3]{x} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{x^3-x}}\) |
(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))
3.21.38.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[Expan dIntegrand[Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && (Po lyQ[Pq, x] || PolyQ[Pq, x^n]) && !IntegerQ[p] && NeQ[n, j]
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}\) | \(465\) |
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)
Time = 90.85 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.88 \[ \int \frac {b+a x^6}{\sqrt [3]{-x+x^3}} \, dx=\frac {1}{162} \, \sqrt {3} {\left (14 \, a + 81 \, b\right )} \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) - \frac {1}{324} \, {\left (14 \, a + 81 \, b\right )} \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) + \frac {1}{108} \, {\left (18 \, a x^{4} + 21 \, a x^{2} + 28 \, a\right )} {\left (x^{3} - x\right )}^{\frac {2}{3}} \]
1/162*sqrt(3)*(14*a + 81*b)*arctan(-(44032959556*sqrt(3)*(x^3 - x)^(1/3)*x + sqrt(3)*(16754327161*x^2 - 2707204793) - 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)
\[ \int \frac {b+a x^6}{\sqrt [3]{-x+x^3}} \, dx=\int \frac {a x^{6} + b}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
\[ \int \frac {b+a x^6}{\sqrt [3]{-x+x^3}} \, dx=\int { \frac {a x^{6} + b}{{\left (x^{3} - x\right )}^{\frac {1}{3}}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.92 \[ \int \frac {b+a x^6}{\sqrt [3]{-x+x^3}} \, dx=\frac {1}{108} \, {\left (28 \, a {\left (\frac {1}{x^{2}} - 1\right )}^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} - 77 \, a {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {5}{3}} + 67 \, a {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}}\right )} x^{6} - \frac {1}{162} \, \sqrt {3} {\left (14 \, a + 81 \, b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{324} \, {\left (14 \, a + 81 \, b\right )} \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{162} \, {\left (14 \, a + 81 \, b\right )} \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
1/108*(28*a*(1/x^2 - 1)^2*(-1/x^2 + 1)^(2/3) - 77*a*(-1/x^2 + 1)^(5/3) + 6 7*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))
Timed out. \[ \int \frac {b+a x^6}{\sqrt [3]{-x+x^3}} \, dx=\int \frac {a\,x^6+b}{{\left (x^3-x\right )}^{1/3}} \,d x \]