Integrand size = 37, antiderivative size = 330 \[ \int \frac {x}{\sqrt {-a-b x^3} \left (-2 \left (5+3 \sqrt {3}\right ) a-b x^3\right )} \, dx=\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}-\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {-a-b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}} \]
1/36*arctan(1/2*3^(1/4)*a^(1/6)*(a^(1/3)+b^(1/3)*x)*(1-3^(1/2))*2^(1/2)/(- b*x^3-a)^(1/2))*(2-3^(1/2))*3^(3/4)/a^(5/6)/b^(2/3)*2^(1/2)+1/18*arctan(1/ 2*3^(1/4)*a^(1/6)*(-2*b^(1/3)*x+a^(1/3)*(1+3^(1/2)))*2^(1/2)/(-b*x^3-a)^(1 /2))*(2-3^(1/2))*3^(3/4)/a^(5/6)/b^(2/3)*2^(1/2)+1/12*arctanh(1/2*3^(1/4)* a^(1/6)*(a^(1/3)+b^(1/3)*x)*(1+3^(1/2))*2^(1/2)/(-b*x^3-a)^(1/2))*(2-3^(1/ 2))*3^(1/4)/a^(5/6)/b^(2/3)*2^(1/2)-1/18*arctanh(1/6*(1-3^(1/2))*(-b*x^3-a )^(1/2)*3^(1/4)*2^(1/2)/a^(1/2))*(2-3^(1/2))*3^(1/4)/a^(5/6)/b^(2/3)*2^(1/ 2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.26 \[ \int \frac {x}{\sqrt {-a-b x^3} \left (-2 \left (5+3 \sqrt {3}\right ) a-b x^3\right )} \, dx=-\frac {x^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b x^3}{10 a+6 \sqrt {3} a}\right )}{\left (20 a+12 \sqrt {3} a\right ) \sqrt {-a-b x^3}} \]
-((x^2*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), -((b*x ^3)/(10*a + 6*Sqrt[3]*a))])/((20*a + 12*Sqrt[3]*a)*Sqrt[-a - b*x^3]))
Time = 0.29 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {990}
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.
| \(\displaystyle \int \frac {x}{\sqrt {-a-b x^3} \left (-2 \left (5+3 \sqrt {3}\right ) a-b x^3\right )} \, dx\) |
| \(\Big \downarrow \) 990 |
| \(\displaystyle \frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}-\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {-a-b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}\) |
((2 - Sqrt[3])*ArcTan[(3^(1/4)*a^(1/6)*((1 + Sqrt[3])*a^(1/3) - 2*b^(1/3)* x))/(Sqrt[2]*Sqrt[-a - b*x^3])])/(3*Sqrt[2]*3^(1/4)*a^(5/6)*b^(2/3)) + ((2 - Sqrt[3])*ArcTan[(3^(1/4)*(1 - Sqrt[3])*a^(1/6)*(a^(1/3) + b^(1/3)*x))/( Sqrt[2]*Sqrt[-a - b*x^3])])/(6*Sqrt[2]*3^(1/4)*a^(5/6)*b^(2/3)) + ((2 - Sq rt[3])*ArcTanh[(3^(1/4)*(1 + Sqrt[3])*a^(1/6)*(a^(1/3) + b^(1/3)*x))/(Sqrt [2]*Sqrt[-a - b*x^3])])/(2*Sqrt[2]*3^(3/4)*a^(5/6)*b^(2/3)) - ((2 - Sqrt[3 ])*ArcTanh[((1 - Sqrt[3])*Sqrt[-a - b*x^3])/(Sqrt[2]*3^(3/4)*Sqrt[a])])/(3 *Sqrt[2]*3^(3/4)*a^(5/6)*b^(2/3))
3.4.53.3.1 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)*( ArcTanh[(1 - r)*(Sqrt[a + b*x^3]/(Sqrt[2]*Rt[-a, 2]*r^(3/2)))]/(3*Sqrt[2]*R t[-a, 2]*d*r^(3/2))), x] + (-Simp[q*(2 - r)*(ArcTanh[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)*(ArcTan[Rt[-a, 2]*Sqrt[r]*((1 + r - 2*q*x)/(Sqrt[2]*S qrt[a + b*x^3]))]/(3*Sqrt[2]*Rt[-a, 2]*d*Sqrt[r])), x] - Simp[q*(2 - r)*(Ar cTan[Rt[-a, 2]*(1 - r)*Sqrt[r]*((1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3]))]/(6*Sq rt[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] && NegQ[a]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.75 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.64
| method | result | size |
| default | \(\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+6 \sqrt {3}\, a +10 a \right )}{\sum }\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i b \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i b \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (-3 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b +4 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}+3 i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}-2 \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b +6 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -6 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-2 \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}-6 i \left (-a \,b^{2}\right )^{\frac {2}{3}}+3 \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b +3 \left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \Pi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -4 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b +2 \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b +2 i \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +2 \sqrt {3}\, a b -3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -2 i a b -3 a b}{6 b a}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \sqrt {-b \,x^{3}-a}}\right )}{27 a \,b^{3}}\) | \(541\) |
| elliptic | \(\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+6 \sqrt {3}\, a +10 a \right )}{\sum }\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i b \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i b \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (-3 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b +4 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}+3 i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}-2 \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b +6 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -6 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-2 \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}-6 i \left (-a \,b^{2}\right )^{\frac {2}{3}}+3 \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b +3 \left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \Pi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -4 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b +2 \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b +2 i \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +2 \sqrt {3}\, a b -3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -2 i a b -3 a b}{6 b a}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \sqrt {-b \,x^{3}-a}}\right )}{27 a \,b^{3}}\) | \(541\) |
1/27*I/a/b^3*2^(1/2)*sum(1/_alpha*(-a*b^2)^(1/3)*(1/2*I*b*(2*x+1/b*((-a*b^ 2)^(1/3)-I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)*(b*(x-1/b*(-a*b^ 2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))^(1/2)*(-1/2*I*b*(2 *x+1/b*((-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)/(- b*x^3-a)^(1/2)*(-3*I*(-a*b^2)^(1/3)*_alpha*3^(1/2)*b+4*b^2*_alpha^2*3^(1/2 )+3*I*(-a*b^2)^(2/3)*3^(1/2)-2*3^(1/2)*(-a*b^2)^(1/3)*_alpha*b+6*I*(-a*b^2 )^(1/3)*_alpha*b-6*b^2*_alpha^2-2*3^(1/2)*(-a*b^2)^(2/3)-6*I*(-a*b^2)^(2/3 )+3*(-a*b^2)^(1/3)*_alpha*b+3*(-a*b^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x +1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^( 1/3))^(1/2),-1/6/b*(2*I*3^(1/2)*(-a*b^2)^(1/3)*_alpha^2*b-I*3^(1/2)*(-a*b^ 2)^(2/3)*_alpha-4*I*(-a*b^2)^(1/3)*_alpha^2*b+2*3^(1/2)*(-a*b^2)^(2/3)*_al pha+I*3^(1/2)*a*b+2*I*(-a*b^2)^(2/3)*_alpha+2*3^(1/2)*a*b-3*(-a*b^2)^(2/3) *_alpha-2*I*a*b-3*a*b)/a,(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3 )+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)),_alpha=RootOf(b*_Z^3+6*3^(1/2)*a +10*a))
Leaf count of result is larger than twice the leaf count of optimal. 5679 vs. \(2 (223) = 446\).
Time = 3.55 (sec) , antiderivative size = 5679, normalized size of antiderivative = 17.21 \[ \int \frac {x}{\sqrt {-a-b x^3} \left (-2 \left (5+3 \sqrt {3}\right ) a-b x^3\right )} \, dx=\text {Too large to display} \]
\[ \int \frac {x}{\sqrt {-a-b x^3} \left (-2 \left (5+3 \sqrt {3}\right ) a-b x^3\right )} \, dx=- \int \frac {x}{10 a \sqrt {- a - b x^{3}} + 6 \sqrt {3} a \sqrt {- a - b x^{3}} + b x^{3} \sqrt {- a - b x^{3}}}\, dx \]
-Integral(x/(10*a*sqrt(-a - b*x**3) + 6*sqrt(3)*a*sqrt(-a - b*x**3) + b*x* *3*sqrt(-a - b*x**3)), x)
\[ \int \frac {x}{\sqrt {-a-b x^3} \left (-2 \left (5+3 \sqrt {3}\right ) a-b x^3\right )} \, dx=\int { -\frac {x}{{\left (b x^{3} + 2 \, a {\left (3 \, \sqrt {3} + 5\right )}\right )} \sqrt {-b x^{3} - a}} \,d x } \]
Exception generated. \[ \int \frac {x}{\sqrt {-a-b x^3} \left (-2 \left (5+3 \sqrt {3}\right ) a-b x^3\right )} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:index.cc index_m operator + Error: Bad Argument Value
Timed out. \[ \int \frac {x}{\sqrt {-a-b x^3} \left (-2 \left (5+3 \sqrt {3}\right ) a-b x^3\right )} \, dx=\int -\frac {x}{\sqrt {-b\,x^3-a}\,\left (b\,x^3+2\,a\,\left (3\,\sqrt {3}+5\right )\right )} \,d x \]