Integrand size = 37, 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 {\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}} \] Output:
-2*(-b*x^3-a)^(1/2)/b^(2/3)/((1-3^(1/2))*a^(1/3)+b^(1/3)*x)+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^(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)+1/4*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)*arctan h(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),2*I-I*3^(1/2))/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),2*I-I*3^(1/2))*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 = 10.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.12 \[ \int \frac {x \sqrt {-a-b x^3}}{-2 \left (5+3 \sqrt {3}\right ) a-b x^3} \, dx=-\frac {x^2 \sqrt {-a-b x^3} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b x^3}{10 a+6 \sqrt {3} a}\right )}{4 \left (5+3 \sqrt {3}\right ) a \sqrt {\frac {a+b x^3}{a}}} \] Input:
Integrate[(x*Sqrt[-a - b*x^3])/(-2*(5 + 3*Sqrt[3])*a - b*x^3),x]
Output:
-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[(a + b*x^3)/a])
Time = 1.45 (sec) , antiderivative size = 844, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {984, 25, 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 |
| \(\displaystyle \int \frac {x}{\sqrt {-b x^3-a}}dx+3 \left (3+2 \sqrt {3}\right ) a \int -\frac {x}{\sqrt {-b x^3-a} \left (b x^3+2 \left (5+3 \sqrt {3}\right ) a\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {x}{\sqrt {-b x^3-a}}dx-3 \left (3+2 \sqrt {3}\right ) a \int \frac {x}{\sqrt {-b x^3-a} \left (b x^3+2 \left (5+3 \sqrt {3}\right ) a\right )}dx\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle \frac {\int \frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {-b x^3-a}}dx}{\sqrt [3]{b}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}}-3 \left (3+2 \sqrt {3}\right ) a \left (-\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}+\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {-a-b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}\right )\) |
| \(\Big \downarrow \) 2418 |
| \(\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 {\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}}\) |
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*(-1/3*((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 + 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]*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 - Sqr t[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[ 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 ] && 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 = 2.00 (sec) , antiderivative size = 983, normalized size of antiderivative = 1.28
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(983\) |
| default | \(\text {Expression too large to display}\) | \(1001\) |
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*((-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*3^(1/2)*(-a*b^2)^(2/3)+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*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)...
Leaf count of result is larger than twice the leaf count of optimal. 4981 vs. \(2 (546) = 1092\).
Time = 4.21 (sec) , antiderivative size = 4981, normalized size of antiderivative = 6.49 \[ \int \frac {x \sqrt {-a-b x^3}}{-2 \left (5+3 \sqrt {3}\right ) a-b x^3} \, dx=\text {Too large to display} \] Input:
integrate(x*(-b*x^3-a)^(1/2)/(-2*(5+3*3^(1/2))*a-b*x^3),x, algorithm="fric as")
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="maxi ma")
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. \[ \int \frac {x \sqrt {-a-b x^3}}{-2 \left (5+3 \sqrt {3}\right ) a-b x^3} \, dx=\int -\frac {x\,\sqrt {-b\,x^3-a}}{b\,x^3+2\,a\,\left (3\,\sqrt {3}+5\right )} \,d x \] 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)
\[ \int \frac {x \sqrt {-a-b x^3}}{-2 \left (5+3 \sqrt {3}\right ) a-b x^3} \, dx=i \left (-6 \sqrt {3}\, \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{-b^{2} x^{6}-20 a b \,x^{3}+8 a^{2}}d x \right ) a +\left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{4}}{-b^{2} x^{6}-20 a b \,x^{3}+8 a^{2}}d x \right ) b +10 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{-b^{2} x^{6}-20 a b \,x^{3}+8 a^{2}}d x \right ) a \right ) \] Input:
int(x*(-b*x^3-a)^(1/2)/(-2*(5+3*3^(1/2))*a-b*x^3),x)
Output:
i*( - 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)