Integrand size = 36, antiderivative size = 768 \[ \int \frac {x \sqrt {-a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\frac {2 \sqrt {-a-b x^3}}{b^{2/3} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {3^{3/4} \sqrt [6]{a} \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 )}{2 \sqrt {2} b^{2/3}}+\frac {\sqrt [6]{a} \arctan \left (\frac {\left (1+\sqrt {3}\right ) \sqrt {-a-b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{\sqrt {2} \sqrt [4]{3} b^{2/3}}+\frac {\sqrt [4]{3} \sqrt [6]{a} \text {arctanh}\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 )}{\sqrt {2} b^{2/3}}+\frac {\sqrt [4]{3} \sqrt [6]{a} \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} b^{2/3}}-\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7+4 \sqrt {3}\right )}{b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}}+\frac {2 \sqrt {2} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}} \]
-1/4*3^(3/4)*a^(1/6)*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))/b^(2/3)*2^(1/2)+1/6*a^(1/6)*arctan(1/6*(1+ 3^(1/2))*(-b*x^3-a)^(1/2)*3^(1/4)*2^(1/2)/a^(1/2))*3^(3/4)/b^(2/3)*2^(1/2) +1/2*3^(1/4)*a^(1/6)*arctanh(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))/b^(2/3)*2^(1/2)+1/4*3^(1/4)*a^(1/6)*ar ctanh(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))/b^(2/3)*2^(1/2)+2*(-b*x^3-a)^(1/2)/b^(2/3)/(b^(1/3)*x+a^(1/3)*(1 -3^(1/2)))+2/3*a^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1 +3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1-3^(1/2))),2*I-I*3^(1/2))*2^(1/2)*((a^(2/3 )-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1-3^(1/2)))^2)^(1/2)* 3^(3/4)/b^(2/3)/(-b*x^3-a)^(1/2)/(-a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+ a^(1/3)*(1-3^(1/2)))^2)^(1/2)-3^(1/4)*a^(1/3)*(a^(1/3)+b^(1/3)*x)*Elliptic E((b^(1/3)*x+a^(1/3)*(1+3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1-3^(1/2))),2*I-I*3^ (1/2))*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1-3^(1 /2)))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))/b^(2/3)/(-b*x^3-a)^(1/2)/(-a^(1/3 )*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1-3^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.12 \[ \int \frac {x \sqrt {-a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=-\frac {x^2 \sqrt {-a-b x^3} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )}{4 \left (-5+3 \sqrt {3}\right ) a \sqrt {1+\frac {b x^3}{a}}} \]
-1/4*(x^2*Sqrt[-a - b*x^3]*AppellF1[2/3, -1/2, 1, 5/3, -((b*x^3)/a), -((b* x^3)/(10*a - 6*Sqrt[3]*a))])/((-5 + 3*Sqrt[3])*a*Sqrt[1 + (b*x^3)/a])
Time = 0.82 (sec) , antiderivative size = 837, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {984, 833, 760, 990, 2418}
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.
| \(\displaystyle \int \frac {x \sqrt {-a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx\) |
| \(\Big \downarrow \) 984 |
| \(\displaystyle 3 \left (3-2 \sqrt {3}\right ) a \int \frac {x}{\sqrt {-b x^3-a} \left (b x^3+2 \left (5-3 \sqrt {3}\right ) a\right )}dx-\int \frac {x}{\sqrt {-b x^3-a}}dx\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {-b x^3-a}}dx}{\sqrt [3]{b}}-\frac {\int \frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {-b x^3-a}}dx}{\sqrt [3]{b}}+3 \left (3-2 \sqrt {3}\right ) a \int \frac {x}{\sqrt {-b x^3-a} \left (b x^3+2 \left (5-3 \sqrt {3}\right ) a\right )}dx\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle -\frac {\int \frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {-b x^3-a}}dx}{\sqrt [3]{b}}+3 \left (3-2 \sqrt {3}\right ) a \int \frac {x}{\sqrt {-b x^3-a} \left (b x^3+2 \left (5-3 \sqrt {3}\right ) a\right )}dx+\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}}\) |
| \(\Big \downarrow \) 990 |
| \(\displaystyle -\frac {\int \frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {-b x^3-a}}dx}{\sqrt [3]{b}}+\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}}+3 \left (3-2 \sqrt {3}\right ) a \left (\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\left (1+\sqrt {3}\right ) \sqrt {-a-b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}-\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}\right )\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle 3 \left (3-2 \sqrt {3}\right ) a \left (\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt {2} \sqrt {-b x^3-a}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\left (1+\sqrt {3}\right ) \sqrt {-b x^3-a}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}-\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\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 {-b x^3-a}}\right )}{3 \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]{b} x+\sqrt [3]{a}\right )}{\sqrt {2} \sqrt {-b x^3-a}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}\right )-\frac {\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right ) \sqrt {\frac {b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\left (\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\left (\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \sqrt {-b x^3-a}}-\frac {2 \sqrt {-b x^3-a}}{\sqrt [3]{b} \left (\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}\right )}}{\sqrt [3]{b}}+\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right ) \sqrt {\frac {b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\left (\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\left (\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \sqrt {-b x^3-a}}\) |
3*(3 - 2*Sqrt[3])*a*(((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])])/(2*Sqrt[2]*3^(3/4)*a^( 5/6)*b^(2/3)) - ((2 + Sqrt[3])*ArcTan[((1 + Sqrt[3])*Sqrt[-a - b*x^3])/(Sq rt[2]*3^(3/4)*Sqrt[a])])/(3*Sqrt[2]*3^(3/4)*a^(5/6)*b^(2/3)) - ((2 + Sqrt[ 3])*ArcTanh[(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] )*ArcTanh[(3^(1/4)*(1 + Sqrt[3])*a^(1/6)*(a^(1/3) + b^(1/3)*x))/(Sqrt[2]*S qrt[-a - b*x^3])])/(6*Sqrt[2]*3^(1/4)*a^(5/6)*b^(2/3))) - ((-2*Sqrt[-a - b *x^3])/(b^(1/3)*((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)) + (3^(1/4)*Sqrt[2 + S qrt[3]]*a^(1/3)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 + 4*Sqrt[3]])/(b^(1/3)*Sqrt[-((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 - Sqrt[3]) *a^(1/3) + b^(1/3)*x)^2)]*Sqrt[-a - b*x^3]))/b^(1/3) + (2*Sqrt[2 - Sqrt[3] ]*(1 + Sqrt[3])*a^(1/3)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1 /3)*x + b^(2/3)*x^2)/((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcS in[((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x) ], -7 + 4*Sqrt[3]])/(3^(1/4)*b^(2/3)*Sqrt[-((a^(1/3)*(a^(1/3) + b^(1/3)*x) )/((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)^2)]*Sqrt[-a - b*x^3])
3.4.49.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) *s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x ] && NegQ[a]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && NegQ[a]
Int[((x_)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol ] :> Simp[b/d Int[x*(a + b*x^n)^(p - 1), x], x] - Simp[(b*c - a*d)/d In t[x*((a + b*x^n)^(p - 1)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, 1, 1, n, p, -1, x]
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]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.77 (sec) , antiderivative size = 983, normalized size of antiderivative = 1.28
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(983\) |
| default | \(\text {Expression too large to display}\) | \(1001\) |
2/3*I*3^(1/2)/b*(-a*b^2)^(1/3)*(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)*((x-1/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)*(-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)/(-b*x^3-a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3 ))*EllipticE(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),(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))+1/b*(-a*b^2)^(1 /3)*EllipticF(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),(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)))-1/9*I/b^3*2^( 1/2)*sum(1/_alpha*(2*3^(1/2)-3)*(-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)+6*I*(-a*b^2)^(1/3)*_alpha*b-2*3^(1/2)*(-a*b^2)^( 1/3)*_alpha*b+6*b^2*_alpha^2-6*I*(-a*b^2)^(2/3)-2*3^(1/2)*(-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)^(...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.41 (sec) , antiderivative size = 4875, normalized size of antiderivative = 6.35 \[ \int \frac {x \sqrt {-a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\text {Too large to display} \]
-1/8*((1/72)^(1/6)*(sqrt(-3)*b - b)*(-sqrt(3)*a/b^4)^(1/6)*log((72*(1/72)^ (5/6)*(7*b^6*x^10 + 12*a*b^5*x^7 + 408*a^2*b^4*x^4 + 160*a^3*b^3*x + sqrt( -3)*(7*b^6*x^10 + 12*a*b^5*x^7 + 408*a^2*b^4*x^4 + 160*a^3*b^3*x) - 3*sqrt (3)*(b^6*x^10 - 12*a*b^5*x^7 - 72*a^2*b^4*x^4 - 32*a^3*b^3*x + sqrt(-3)*(b ^6*x^10 - 12*a*b^5*x^7 - 72*a^2*b^4*x^4 - 32*a^3*b^3*x)))*(-sqrt(3)*a/b^4) ^(5/6) + 4*sqrt(1/2)*(3*b^5*x^11 - 18*a*b^4*x^8 + 360*a^2*b^3*x^5 + 624*a^ 3*b^2*x^2 - sqrt(3)*(b^5*x^11 - 42*a*b^4*x^8 - 168*a^2*b^3*x^5 - 368*a^3*b ^2*x^2))*sqrt(-sqrt(3)*a/b^4) + 6*(12*a*b^2*x^8 - 48*a^2*b*x^5 - 384*a^3*x ^2 + 2*(1/9)^(2/3)*(3*b^5*x^9 + 288*a^2*b^3*x^3 + 48*a^3*b^2 - sqrt(3)*(b^ 5*x^9 - 30*a*b^4*x^6 - 144*a^2*b^3*x^3 - 32*a^3*b^2 - sqrt(-3)*(b^5*x^9 - 30*a*b^4*x^6 - 144*a^2*b^3*x^3 - 32*a^3*b^2)) - 3*sqrt(-3)*(b^5*x^9 + 96*a ^2*b^3*x^3 + 16*a^3*b^2))*(-sqrt(3)*a/b^4)^(2/3) - (1/9)^(1/3)*(b^4*x^10 + 240*a^2*b^2*x^4 + 160*a^3*b*x + sqrt(-3)*(b^4*x^10 + 240*a^2*b^2*x^4 + 16 0*a^3*b*x) + 24*sqrt(3)*(a*b^3*x^7 + 5*a^2*b^2*x^4 + 4*a^3*b*x + sqrt(-3)* (a*b^3*x^7 + 5*a^2*b^2*x^4 + 4*a^3*b*x)))*(-sqrt(3)*a/b^4)^(1/3) - 8*sqrt( 3)*(a*b^2*x^8 + 2*a^2*b*x^5 + 28*a^3*x^2))*sqrt(-b*x^3 - a) + (1/72)^(1/6) *(3*b^4*x^12 - 12*a*b^3*x^9 + 1080*a^2*b^2*x^6 + 2208*a^3*b*x^3 + 384*a^4 + sqrt(3)*(b^4*x^12 + 124*a*b^3*x^9 + 744*a^2*b^2*x^6 + 1120*a^3*b*x^3 + 2 56*a^4 - sqrt(-3)*(b^4*x^12 + 124*a*b^3*x^9 + 744*a^2*b^2*x^6 + 1120*a^3*b *x^3 + 256*a^4)) - 3*sqrt(-3)*(b^4*x^12 - 4*a*b^3*x^9 + 360*a^2*b^2*x^6...
\[ \int \frac {x \sqrt {-a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\int \frac {x \sqrt {- a - b x^{3}}}{- 6 \sqrt {3} a + 10 a + b x^{3}}\, dx \]
\[ \int \frac {x \sqrt {-a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\int { \frac {\sqrt {-b x^{3} - a} x}{b x^{3} - 2 \, a {\left (3 \, \sqrt {3} - 5\right )}} \,d x } \]
\[ \int \frac {x \sqrt {-a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\int { \frac {\sqrt {-b x^{3} - a} x}{b x^{3} - 2 \, a {\left (3 \, \sqrt {3} - 5\right )}} \,d x } \]
Timed out. \[ \int \frac {x \sqrt {-a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\int \frac {x\,\sqrt {-b\,x^3-a}}{b\,x^3-2\,a\,\left (3\,\sqrt {3}-5\right )} \,d x \]