Integrand size = 35, antiderivative size = 758 \[ \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} \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 [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 {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}} \] Output:
2*(-b*x^3+a)^(1/2)/b^(2/3)/((1+3^(1/2))*a^(1/3)-b^(1/3)*x)-1/4*3^(1/4)*a^( 1/6)*arctan(1/2*3^(1/4)*(1+3^(1/2))*a^(1/6)*(a^(1/3)-b^(1/3)*x)*2^(1/2)/(- b*x^3+a)^(1/2))*2^(1/2)/b^(2/3)-1/2*3^(1/4)*a^(1/6)*arctan(1/2*3^(1/4)*a^( 1/6)*((1-3^(1/2))*a^(1/3)+2*b^(1/3)*x)*2^(1/2)/(-b*x^3+a)^(1/2))*2^(1/2)/b ^(2/3)+1/4*3^(3/4)*a^(1/6)*arctanh(1/2*3^(1/4)*(1-3^(1/2))*a^(1/6)*(a^(1/3 )-b^(1/3)*x)*2^(1/2)/(-b*x^3+a)^(1/2))*2^(1/2)/b^(2/3)+1/6*a^(1/6)*arctanh (1/6*(1+3^(1/2))*(-b*x^3+a)^(1/2)*2^(1/2)*3^(1/4)/a^(1/2))*2^(1/2)*3^(3/4) /b^(2/3)-3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*a^(1/3)*(a^(1/3)-b^(1/3)*x)*((a ^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+3^(1/2))*a^(1/3)-b^(1/3)*x)^2)^( 1/2)*EllipticE(((1-3^(1/2))*a^(1/3)-b^(1/3)*x)/((1+3^(1/2))*a^(1/3)-b^(1/3 )*x),I*3^(1/2)+2*I)/b^(2/3)/(a^(1/3)*(a^(1/3)-b^(1/3)*x)/((1+3^(1/2))*a^(1 /3)-b^(1/3)*x)^2)^(1/2)/(-b*x^3+a)^(1/2)+2/3*2^(1/2)*a^(1/3)*(a^(1/3)-b^(1 /3)*x)*((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+3^(1/2))*a^(1/3)-b^(1/ 3)*x)^2)^(1/2)*EllipticF(((1-3^(1/2))*a^(1/3)-b^(1/3)*x)/((1+3^(1/2))*a^(1 /3)-b^(1/3)*x),I*3^(1/2)+2*I)*3^(3/4)/b^(2/3)/(a^(1/3)*(a^(1/3)-b^(1/3)*x) /((1+3^(1/2))*a^(1/3)-b^(1/3)*x)^2)^(1/2)/(-b*x^3+a)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a-b x^3} \, dx=\frac {x^2 \sqrt {1-\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {1}{2},1,\frac {5}{3},\frac {b x^3}{a},\frac {b x^3}{10 a-6 \sqrt {3} a}\right )}{\left (20-12 \sqrt {3}\right ) \sqrt {a-b x^3}} \] Input:
Integrate[(x*Sqrt[a - b*x^3])/(2*(5 - 3*Sqrt[3])*a - b*x^3),x]
Output:
(x^2*Sqrt[1 - (b*x^3)/a]*AppellF1[2/3, -1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(1 0*a - 6*Sqrt[3]*a)])/((20 - 12*Sqrt[3])*Sqrt[a - b*x^3])
Time = 1.43 (sec) , antiderivative size = 826, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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 |
| \(\displaystyle \int \frac {x}{\sqrt {a-b x^3}}dx-3 \left (3-2 \sqrt {3}\right ) a \int \frac {x}{\sqrt {a-b x^3} \left (2 \left (5-3 \sqrt {3}\right ) a-b x^3\right )}dx\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {a-b x^3}}dx}{\sqrt [3]{b}}-\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt {a-b x^3}}dx}{\sqrt [3]{b}}-3 \left (3-2 \sqrt {3}\right ) a \int \frac {x}{\sqrt {a-b x^3} \left (2 \left (5-3 \sqrt {3}\right ) a-b x^3\right )}dx\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle -\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt {a-b x^3}}dx}{\sqrt [3]{b}}-3 \left (3-2 \sqrt {3}\right ) a \int \frac {x}{\sqrt {a-b x^3} \left (2 \left (5-3 \sqrt {3}\right ) a-b x^3\right )}dx-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}\) |
| \(\Big \downarrow \) 989 |
| \(\displaystyle -\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt {a-b x^3}}dx}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}-3 \left (3-2 \sqrt {3}\right ) a \left (-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a-b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a-b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a-b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt {a-b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}\right )\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle -3 \left (3-2 \sqrt {3}\right ) a \left (-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \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 ) \arctan \left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (2 \sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}\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 )}{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 )-\frac {\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\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 )}{\sqrt [3]{b} \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 {a-b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}}{\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 {b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\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}}\) |
Input:
Int[(x*Sqrt[a - b*x^3])/(2*(5 - 3*Sqrt[3])*a - b*x^3),x]
Output:
-3*(3 - 2*Sqrt[3])*a*(-1/6*((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])])/(Sqrt[2]*3^(1/4)* 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])*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 - Sq rt[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*(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])
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 = 2.70 (sec) , antiderivative size = 924, normalized size of antiderivative = 1.22
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(924\) |
| default | \(\text {Expression too large to display}\) | \(942\) |
Input:
int(x*(-b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a-b*x^3),x,method=_RETURNVERBOSE)
Output:
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*(-3 +2*3^(1/2))*(a*b^2)^(1/3)*(-1/2*I*b*(2*x+1/b*(I*3^(1/2)*(a*b^2)^(1/3)+(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*(-I*3^(1/2)*(a*b^2)^(1 /3)+(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*(a*b^2)^(1/3)*_alpha*3^(1/2)*b+6*_alpha^2*b^2+6*I *(a*b^2)^(2/3)-2*(a*b^2)^(2/3)*3^(1/2)-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*(a*b^2)^(1/3)*...
Leaf count of result is larger than twice the leaf count of optimal. 4953 vs. \(2 (544) = 1088\).
Time = 4.30 (sec) , antiderivative size = 4953, normalized size of antiderivative = 6.53 \[ \int \frac {x \sqrt {a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a-b x^3} \, dx=\text {Too large to display} \] Input:
integrate(x*(-b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a-b*x^3),x, algorithm="frica s")
Output:
Too large to include
\[ \int \frac {x \sqrt {a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a-b x^3} \, dx=- \int \frac {x \sqrt {a - b x^{3}}}{- 10 a + 6 \sqrt {3} a + b x^{3}}\, dx \] Input:
integrate(x*(-b*x**3+a)**(1/2)/(2*(5-3*3**(1/2))*a-b*x**3),x)
Output:
-Integral(x*sqrt(a - b*x**3)/(-10*a + 6*sqrt(3)*a + b*x**3), x)
\[ \int \frac {x \sqrt {a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a-b x^3} \, dx=\int { -\frac {\sqrt {-b x^{3} + a} x}{b x^{3} + 2 \, a {\left (3 \, \sqrt {3} - 5\right )}} \,d x } \] Input:
integrate(x*(-b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a-b*x^3),x, algorithm="maxim a")
Output:
-integrate(sqrt(-b*x^3 + a)*x/(b*x^3 + 2*a*(3*sqrt(3) - 5)), x)
\[ \int \frac {x \sqrt {a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a-b x^3} \, dx=\int { -\frac {\sqrt {-b x^{3} + a} x}{b x^{3} + 2 \, a {\left (3 \, \sqrt {3} - 5\right )}} \,d x } \] Input:
integrate(x*(-b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a-b*x^3),x, algorithm="giac" )
Output:
integrate(-sqrt(-b*x^3 + a)*x/(b*x^3 + 2*a*(3*sqrt(3) - 5)), x)
Timed out. \[ \int \frac {x \sqrt {a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a-b x^3} \, dx=\int -\frac {x\,\sqrt {a-b\,x^3}}{b\,x^3+2\,a\,\left (3\,\sqrt {3}-5\right )} \,d x \] Input:
int(-(x*(a - b*x^3)^(1/2))/(b*x^3 + 2*a*(3*3^(1/2) - 5)),x)
Output:
int(-(x*(a - b*x^3)^(1/2))/(b*x^3 + 2*a*(3*3^(1/2) - 5)), x)
\[ \int \frac {x \sqrt {a-b x^3}}{2 \left (5-3 \sqrt {3}\right ) a-b x^3} \, dx=-6 \sqrt {3}\, \left (\int \frac {\sqrt {-b \,x^{3}+a}\, x}{-b^{2} x^{6}+20 a b \,x^{3}+8 a^{2}}d x \right ) a +\left (\int \frac {\sqrt {-b \,x^{3}+a}\, x^{4}}{-b^{2} x^{6}+20 a b \,x^{3}+8 a^{2}}d x \right ) b -10 \left (\int \frac {\sqrt {-b \,x^{3}+a}\, x}{-b^{2} x^{6}+20 a b \,x^{3}+8 a^{2}}d x \right ) a \] Input:
int(x*(-b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a-b*x^3),x)
Output:
- 6*sqrt(3)*int((sqrt(a - b*x**3)*x)/(8*a**2 + 20*a*b*x**3 - b**2*x**6),x )*a + int((sqrt(a - b*x**3)*x**4)/(8*a**2 + 20*a*b*x**3 - b**2*x**6),x)*b - 10*int((sqrt(a - b*x**3)*x)/(8*a**2 + 20*a*b*x**3 - b**2*x**6),x)*a