Integrand size = 33, antiderivative size = 738 \[ \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 {\sqrt [4]{3} \sqrt [6]{a} \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 )}{\sqrt {2} b^{2/3}}-\frac {\sqrt [4]{3} \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 {3^{3/4} \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 [6]{a} \text {arctanh}\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 {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}} \]
-1/2*3^(1/4)*a^(1/6)*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))/b^(2/3)*2^(1/2)-1/4*3^(1/4)*a^(1/6)*arct an(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)+1/4*3^(3/4)*a^(1/6)*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))/b^(2/3)*2^(1/2)+1/6*a ^(1/6)*arctanh(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)+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))),I*3^(1/2)+2*I)*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)*EllipticE((b^(1 /3)*x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)* (1/2*6^(1/2)-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)/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 = 10.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {a+b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\frac {x^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )}{\left (20-12 \sqrt {3}\right ) \sqrt {a+b x^3}} \]
(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 - 12*Sqrt[3])*Sqrt[a + b*x^3])
Time = 0.79 (sec) , antiderivative size = 806, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {984, 832, 759, 989, 2416}
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 |
| \(\displaystyle \int \frac {x}{\sqrt {b x^3+a}}dx-3 \left (3-2 \sqrt {3}\right ) a \int \frac {x}{\sqrt {b x^3+a} \left (b x^3+2 \left (5-3 \sqrt {3}\right ) a\right )}dx\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle -\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}+\frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-3 \left (3-2 \sqrt {3}\right ) a \int \frac {x}{\sqrt {b x^3+a} \left (b x^3+2 \left (5-3 \sqrt {3}\right ) a\right )}dx\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-3 \left (3-2 \sqrt {3}\right ) a \int \frac {x}{\sqrt {b x^3+a} \left (b x^3+2 \left (5-3 \sqrt {3}\right ) a\right )}dx-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 989 |
| \(\displaystyle \frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-3 \left (3-2 \sqrt {3}\right ) a \left (-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt {a+b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}\right )\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle -3 \left (3-2 \sqrt {3}\right ) a \left (-\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 {b x^3+a}}\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]{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}}+\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 )}{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 {b x^3+a}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}\right )+\frac {\frac {2 \sqrt {b x^3+a}}{\sqrt [3]{b} \left (\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )}-\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}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \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}} \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*(-1/3*((2 + Sqrt[3])*ArcTan[(3^(1/4)*a^(1/6)*((1 - Sq rt[3])*a^(1/3) - 2*b^(1/3)*x))/(Sqrt[2]*Sqrt[a + b*x^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 + Sqrt[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))) + ((2*Sqrt[a + b*x ^3])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (3^(1/4)*Sqrt[2 - Sqr t[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 - S qrt[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*(1 - Sqrt[3])*Sqrt[2 + 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[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]])/(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])
3.4.46.3.1 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] & & PosQ[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] && PosQ[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 )*(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]
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 + Sqrt [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] && PosQ[a] && Eq Q[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 = 5.29 (sec) , antiderivative size = 977, normalized size of antiderivative = 1.32
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(977\) |
| default | \(\text {Expression too large to display}\) | \(995\) |
-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)^(1...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.39 (sec) , antiderivative size = 4931, normalized size of antiderivative = 6.68 \[ \int \frac {x \sqrt {a+b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\text {Too large to display} \]
1/8*((1/72)^(1/6)*(sqrt(-3)*b + b)*(sqrt(3)*a/b^4)^(1/6)*log((12*(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 - 1 44*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(b*x^3 + a)*(sqrt(3)*a/b^4)^(2/3) + 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^1 0 + 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) + 6*(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 + 2 40*a^2*b^2*x^4 + 160*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(b*x^3 + a)*(sqrt(3)*a/b^4)^(1/3) - 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) + 24*(3*a*b^2*x^8 - 12*a^2* b*x^5 - 96*a^3*x^2 - 2*sqrt(3)*(a*b^2*x^8 + 2*a^2*b*x^5 + 28*a^3*x^2))*sqr t(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 + 256*a^4 + sqrt(-3)*(b^4*x^12 + 124*a*b^3*x^9 + 7 44*a^2*b^2*x^6 + 1120*a^3*b*x^3 + 256*a^4)) + 3*sqrt(-3)*(b^4*x^12 - 4*...
\[ \int \frac {x \sqrt {a+b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\int \frac {x \sqrt {a + b x^{3}}}{- 6 \sqrt {3} a + 10 a + b x^{3}}\, dx \]
\[ \int \frac {x \sqrt {a+b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\int { \frac {\sqrt {b x^{3} + a} x}{b x^{3} - 2 \, a {\left (3 \, \sqrt {3} - 5\right )}} \,d x } \]
\[ \int \frac {x \sqrt {a+b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\int { \frac {\sqrt {b x^{3} + a} x}{b x^{3} - 2 \, a {\left (3 \, \sqrt {3} - 5\right )}} \,d x } \]
Timed out. \[ \int \frac {x \sqrt {a+b x^3}}{2 \left (5-3 \sqrt {3}\right ) a+b x^3} \, dx=\int \frac {x\,\sqrt {b\,x^3+a}}{b\,x^3-2\,a\,\left (3\,\sqrt {3}-5\right )} \,d x \]