Integrand size = 25, antiderivative size = 218 \[ \int \frac {x}{\sqrt {1+x^3} \left (10+6 \sqrt {3}+x^3\right )} \, dx=-\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1+x)}{\sqrt {2} \sqrt {1+x^3}}\right )}{2 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\left (1-\sqrt {3}\right ) \sqrt {1+x^3}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}-2 x\right )}{\sqrt {2} \sqrt {1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}}-\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1+x)}{\sqrt {2} \sqrt {1+x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3}} \]
-1/12*arctan(1/2*3^(1/4)*(1+x)*(1+3^(1/2))*2^(1/2)/(x^3+1)^(1/2))*(2-3^(1/ 2))*3^(1/4)*2^(1/2)-1/18*arctan(1/6*(1-3^(1/2))*(x^3+1)^(1/2)*3^(1/4)*2^(1 /2))*(2-3^(1/2))*3^(1/4)*2^(1/2)-1/36*arctanh(1/2*3^(1/4)*(1+x)*(1-3^(1/2) )*2^(1/2)/(x^3+1)^(1/2))*(2-3^(1/2))*3^(3/4)*2^(1/2)-1/18*arctanh(1/2*3^(1 /4)*(1-2*x+3^(1/2))*2^(1/2)/(x^3+1)^(1/2))*(2-3^(1/2))*3^(3/4)*2^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.22 \[ \int \frac {x}{\sqrt {1+x^3} \left (10+6 \sqrt {3}+x^3\right )} \, dx=\frac {x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-x^3,-\frac {x^3}{10+6 \sqrt {3}}\right )}{20+12 \sqrt {3}} \]
Time = 0.23 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {989}
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.
| \(\displaystyle \int \frac {x}{\sqrt {x^3+1} \left (x^3+6 \sqrt {3}+10\right )} \, dx\) |
| \(\Big \downarrow \) 989 |
| \(\displaystyle -\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (x+1)}{\sqrt {2} \sqrt {x^3+1}}\right )}{2 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\left (1-\sqrt {3}\right ) \sqrt {x^3+1}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (-2 x+\sqrt {3}+1\right )}{\sqrt {2} \sqrt {x^3+1}}\right )}{3 \sqrt {2} \sqrt [4]{3}}-\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (x+1)}{\sqrt {2} \sqrt {x^3+1}}\right )}{6 \sqrt {2} \sqrt [4]{3}}\) |
-1/2*((2 - Sqrt[3])*ArcTan[(3^(1/4)*(1 + Sqrt[3])*(1 + x))/(Sqrt[2]*Sqrt[1 + x^3])])/(Sqrt[2]*3^(3/4)) - ((2 - Sqrt[3])*ArcTan[((1 - Sqrt[3])*Sqrt[1 + x^3])/(Sqrt[2]*3^(3/4))])/(3*Sqrt[2]*3^(3/4)) - ((2 - Sqrt[3])*ArcTanh[ (3^(1/4)*(1 + Sqrt[3] - 2*x))/(Sqrt[2]*Sqrt[1 + x^3])])/(3*Sqrt[2]*3^(1/4) ) - ((2 - Sqrt[3])*ArcTanh[(3^(1/4)*(1 - Sqrt[3])*(1 + x))/(Sqrt[2]*Sqrt[1 + x^3])])/(6*Sqrt[2]*3^(1/4))
3.1.86.3.1 Defintions of rubi rules used
Int[(x_)/(Sqrt[(a_) + (b_.)*(x_)^3]*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wi th[{q = Rt[b/a, 3], r = Simplify[(b*c - 10*a*d)/(6*a*d)]}, Simp[(-q)*(2 - r )*(ArcTan[(1 - r)*(Sqrt[a + b*x^3]/(Sqrt[2]*Rt[a, 2]*r^(3/2)))]/(3*Sqrt[2]* Rt[a, 2]*d*r^(3/2))), x] + (-Simp[q*(2 - r)*(ArcTan[Rt[a, 2]*Sqrt[r]*(1 + r )*((1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3]))]/(2*Sqrt[2]*Rt[a, 2]*d*r^(3/2))), x ] - Simp[q*(2 - r)*(ArcTanh[Rt[a, 2]*Sqrt[r]*((1 + r - 2*q*x)/(Sqrt[2]*Sqrt [a + b*x^3]))]/(3*Sqrt[2]*Rt[a, 2]*d*Sqrt[r])), x] - Simp[q*(2 - r)*(ArcTan h[Rt[a, 2]*(1 - r)*Sqrt[r]*((1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3]))]/(6*Sqrt[2 ]*Rt[a, 2]*d*Sqrt[r])), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0] && PosQ[a]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 48.35 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.63
| method | result | size |
| default | \(\frac {2 \left (-1-\sqrt {3}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \Pi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \left (12+6 \sqrt {3}\right ) \sqrt {x^{3}+1}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}+\left (-1-\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha \sqrt {3}+\underline {\hspace {1.25 ex}}\alpha -2\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x -1}{-3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \left (-1+2 \underline {\hspace {1.25 ex}}\alpha -\underline {\hspace {1.25 ex}}\alpha \sqrt {3}\right ) \Pi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {i \underline {\hspace {1.25 ex}}\alpha }{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{3}+\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{2}-\underline {\hspace {1.25 ex}}\alpha -\frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\left (-1+2 \underline {\hspace {1.25 ex}}\alpha -\sqrt {3}\right ) \sqrt {x^{3}+1}}\right )}{18}\) | \(355\) |
| elliptic | \(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {\left (-1-\sqrt {3}\right )^{2}}{3}+\frac {2 \left (-1-\sqrt {3}\right )^{2} \sqrt {3}}{9}-\frac {2}{3}-\frac {\sqrt {3}}{9}-\frac {2 \left (-1-\sqrt {3}\right ) \sqrt {3}}{9}\right ) \Pi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \left (-1-\sqrt {3}\right )^{2}}{3}-\frac {i \left (-1-\sqrt {3}\right )^{2} \sqrt {3}}{6}-\frac {\left (-1-\sqrt {3}\right )^{2} \sqrt {3}}{3}+\frac {\left (-1-\sqrt {3}\right )^{2}}{2}-\frac {i \left (-1-\sqrt {3}\right )}{3}+\frac {i \left (-1-\sqrt {3}\right ) \sqrt {3}}{6}+\frac {\left (-1-\sqrt {3}\right ) \sqrt {3}}{3}+1+\frac {i}{3}+\frac {\sqrt {3}}{6}-\frac {i \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \left (-1-\sqrt {3}\right ) \sqrt {x^{3}+1}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}+\left (-1-\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )}{\sum }\frac {\left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x -1}{-3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+3 \underline {\hspace {1.25 ex}}\alpha -3+2 \sqrt {3}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +1\right )\right ) \Pi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha ^{2}}{3}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{6}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{3}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}-\frac {i \underline {\hspace {1.25 ex}}\alpha }{3}+\frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{6}+\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{3}-\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}+\frac {i}{3}-\frac {i \sqrt {3}}{6}-\frac {\sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\underline {\hspace {1.25 ex}}\alpha \sqrt {x^{3}+1}}\right )}{27}\) | \(507\) |
| trager | \(\text {Expression too large to display}\) | \(4181\) |
2/3*(-1-3^(1/2))/(12+6*3^(1/2))*(3/2-1/2*I*3^(1/2))*((1+x)/(3/2-1/2*I*3^(1 /2)))^(1/2)*((x-1/2-1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2 *I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(x^3+1)^(1/2)*3^(1/2)*EllipticPi(( (1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),1/3*(-3/2+1/2*I*3^(1/2))*3^(1/2),((-3/2+1 /2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))-1/18*2^(1/2)*sum((-_alpha*3^(1/ 2)+_alpha-2)/(-1+2*_alpha-3^(1/2))*(3-I*3^(1/2))*((1+x)/(3-I*3^(1/2)))^(1/ 2)*((-I*3^(1/2)+2*x-1)/(-3-I*3^(1/2)))^(1/2)*((I*3^(1/2)+2*x-1)/(I*3^(1/2) -3))^(1/2)/(x^3+1)^(1/2)*(-1+2*_alpha-_alpha*3^(1/2))*EllipticPi(((1+x)/(3 /2-1/2*I*3^(1/2)))^(1/2),-1/2*I*_alpha+1/3*I*_alpha*3^(1/2)+1/2*_alpha*3^( 1/2)-_alpha-1/6*I*3^(1/2)+1/2,((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^ (1/2)),_alpha=RootOf(_Z^2+(-1-3^(1/2))*_Z+2*3^(1/2)+4))
Leaf count of result is larger than twice the leaf count of optimal. 1569 vs. \(2 (148) = 296\).
Time = 0.43 (sec) , antiderivative size = 1569, normalized size of antiderivative = 7.20 \[ \int \frac {x}{\sqrt {1+x^3} \left (10+6 \sqrt {3}+x^3\right )} \, dx=\text {Too large to display} \]
1/36*sqrt(14*sqrt(3) - 24)*arctan(1/12*(3*x^2 + sqrt(3)*(x^2 - 10*x - 8) - 18*x - 12)*sqrt(14*sqrt(3) - 24)/sqrt(x^3 + 1)) + 1/72*sqrt(7*sqrt(3) + 3 *sqrt(56*sqrt(3) - 97) - 12)*log((x^8 - x^7 - 11*x^6 - 16*x^5 - 20*x^4 + 3 2*x^3 - 44*x^2 + 2*sqrt(3)*(x^7 - 8*x^6 - 7*x^4 - 16*x^3 - 8*x - 8) + 3*(2 6*x^7 + 12*x^6 - 48*x^5 - 98*x^4 - 96*x^3 - 48*x^2 + sqrt(3)*(15*x^7 + 7*x ^6 - 28*x^5 - 56*x^4 - 56*x^3 - 28*x^2 - 8*x) - 16*x)*sqrt(56*sqrt(3) - 97 ) + ((336*x^5 + 33*x^4 - 132*x^3 - 474*x^2 + sqrt(3)*(194*x^5 + 19*x^4 - 7 6*x^3 - 274*x^2 - 152*x - 76) - 264*x - 132)*sqrt(x^3 + 1)*sqrt(56*sqrt(3) - 97) + (5*x^6 - 6*x^5 - 33*x^4 - 44*x^3 - 42*x^2 + sqrt(3)*(3*x^6 - 4*x^ 5 - 17*x^4 - 28*x^3 - 22*x^2 - 8*x - 4) - 24*x - 4)*sqrt(x^3 + 1))*sqrt(7* sqrt(3) + 3*sqrt(56*sqrt(3) - 97) - 12) + 8*x + 16)/(x^8 - 4*x^7 + 16*x^6 - 16*x^5 + 28*x^4 + 32*x^3 + 64*x^2 + 32*x + 16)) - 1/72*sqrt(7*sqrt(3) + 3*sqrt(56*sqrt(3) - 97) - 12)*log((x^8 - x^7 - 11*x^6 - 16*x^5 - 20*x^4 + 32*x^3 - 44*x^2 + 2*sqrt(3)*(x^7 - 8*x^6 - 7*x^4 - 16*x^3 - 8*x - 8) + 3*( 26*x^7 + 12*x^6 - 48*x^5 - 98*x^4 - 96*x^3 - 48*x^2 + sqrt(3)*(15*x^7 + 7* x^6 - 28*x^5 - 56*x^4 - 56*x^3 - 28*x^2 - 8*x) - 16*x)*sqrt(56*sqrt(3) - 9 7) - ((336*x^5 + 33*x^4 - 132*x^3 - 474*x^2 + sqrt(3)*(194*x^5 + 19*x^4 - 76*x^3 - 274*x^2 - 152*x - 76) - 264*x - 132)*sqrt(x^3 + 1)*sqrt(56*sqrt(3 ) - 97) + (5*x^6 - 6*x^5 - 33*x^4 - 44*x^3 - 42*x^2 + sqrt(3)*(3*x^6 - 4*x ^5 - 17*x^4 - 28*x^3 - 22*x^2 - 8*x - 4) - 24*x - 4)*sqrt(x^3 + 1))*sqr...
\[ \int \frac {x}{\sqrt {1+x^3} \left (10+6 \sqrt {3}+x^3\right )} \, dx=\int \frac {x}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x^{3} + 10 + 6 \sqrt {3}\right )}\, dx \]
\[ \int \frac {x}{\sqrt {1+x^3} \left (10+6 \sqrt {3}+x^3\right )} \, dx=\int { \frac {x}{{\left (x^{3} + 6 \, \sqrt {3} + 10\right )} \sqrt {x^{3} + 1}} \,d x } \]
\[ \int \frac {x}{\sqrt {1+x^3} \left (10+6 \sqrt {3}+x^3\right )} \, dx=\int { \frac {x}{{\left (x^{3} + 6 \, \sqrt {3} + 10\right )} \sqrt {x^{3} + 1}} \,d x } \]
Timed out. \[ \int \frac {x}{\sqrt {1+x^3} \left (10+6 \sqrt {3}+x^3\right )} \, dx=\int \frac {x}{\sqrt {x^3+1}\,\left (x^3+6\,\sqrt {3}+10\right )} \,d x \]
\[ \int \frac {x}{\sqrt {1+x^3} \left (10+6 \sqrt {3}+x^3\right )} \, dx=-6 \sqrt {3}\, \left (\int \frac {\sqrt {x^{3}+1}\, x}{x^{9}+21 x^{6}+12 x^{3}-8}d x \right )+\int \frac {\sqrt {x^{3}+1}\, x^{4}}{x^{9}+21 x^{6}+12 x^{3}-8}d x +10 \left (\int \frac {\sqrt {x^{3}+1}\, x}{x^{9}+21 x^{6}+12 x^{3}-8}d x \right ) \]