Integrand size = 18, antiderivative size = 377 \[ \int \frac {x}{(3+x) \sqrt {1-x^3}} \, dx=\frac {3 (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \text {arctanh}\left (\frac {\sqrt {7} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}}}{2 \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}}\right )}{2 \sqrt {7} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}}-\frac {2 \sqrt {2 \left (37+20 \sqrt {3}\right )} (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{13 \sqrt [4]{3} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}}+\frac {12 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{169} \left (553+304 \sqrt {3}\right ),\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{13 \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}} \] Output:
3/14*(1-x)*((x^2+x+1)/(1+3^(1/2)-x)^2)^(1/2)*arctanh(1/2*7^(1/2)*((1-x)/(1 +3^(1/2)-x)^2)^(1/2)/((x^2+x+1)/(1+3^(1/2)-x)^2)^(1/2))*7^(1/2)/((1-x)/(1+ 3^(1/2)-x)^2)^(1/2)/(-x^3+1)^(1/2)-2/39*(5*2^(1/2)+2*6^(1/2))*(1-x)*((x^2+ x+1)/(1+3^(1/2)-x)^2)^(1/2)*EllipticF((1-3^(1/2)-x)/(1+3^(1/2)-x),I*3^(1/2 )+2*I)*3^(3/4)/((1-x)/(1+3^(1/2)-x)^2)^(1/2)/(-x^3+1)^(1/2)+12/13*3^(1/4)* (1/2*6^(1/2)+1/2*2^(1/2))*(1-x)*((x^2+x+1)/(1+3^(1/2)-x)^2)^(1/2)*Elliptic Pi((1-3^(1/2)-x)/(1+3^(1/2)-x),553/169+304/169*3^(1/2),I*3^(1/2)+2*I)/((1- x)/(1+3^(1/2)-x)^2)^(1/2)/(-x^3+1)^(1/2)
Result contains complex when optimal does not.
Time = 10.40 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.52 \[ \int \frac {x}{(3+x) \sqrt {1-x^3}} \, dx=\frac {2 \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (\frac {\left (\sqrt [3]{-1}+x\right ) \sqrt {\frac {\sqrt [3]{-1}+(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {3 i \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{5 i+\sqrt {3}},\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{-3+\sqrt [3]{-1}}\right )}{\sqrt {1-x^3}} \] Input:
Integrate[x/((3 + x)*Sqrt[1 - x^3]),x]
Output:
(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (- 1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((3*I)*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(5*I + Sqrt[3]), ArcSin[Sq rt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-3 + (-1)^(1/3)))) /Sqrt[1 - x^3]
Time = 1.60 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2569, 759, 2567, 2538, 412, 435, 104, 219}
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.
| \(\displaystyle \int \frac {x}{(x+3) \sqrt {1-x^3}} \, dx\) |
| \(\Big \downarrow \) 2569 |
| \(\displaystyle \frac {\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {1-x^3}}dx}{4+\sqrt {3}}-\frac {3 \int \frac {-x+\sqrt {3}+1}{(x+3) \sqrt {1-x^3}}dx}{4+\sqrt {3}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle -\frac {3 \int \frac {-x+\sqrt {3}+1}{(x+3) \sqrt {1-x^3}}dx}{4+\sqrt {3}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (4+\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 2567 |
| \(\displaystyle -\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \int \frac {1}{\sqrt {1-\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (-\frac {\left (4+\sqrt {3}\right ) \left (-x-\sqrt {3}+1\right )}{-x+\sqrt {3}+1}-\sqrt {3}+4\right )}d\left (-\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )}{\left (4+\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (4+\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 2538 |
| \(\displaystyle -\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (\left (4-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (-\frac {\left (4+\sqrt {3}\right )^2 \left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-8 \sqrt {3}+19\right )}d\left (-\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )-\left (4+\sqrt {3}\right ) \int -\frac {-x-\sqrt {3}+1}{\sqrt {1-\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (-\frac {\left (4+\sqrt {3}\right )^2 \left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-8 \sqrt {3}+19\right ) \left (-x+\sqrt {3}+1\right )}d\left (-\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )\right )}{\left (4+\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (4+\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle -\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (-\left (4+\sqrt {3}\right ) \int -\frac {-x-\sqrt {3}+1}{\sqrt {1-\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (-\frac {\left (4+\sqrt {3}\right )^2 \left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-8 \sqrt {3}+19\right ) \left (-x+\sqrt {3}+1\right )}d\left (-\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )-\frac {1}{169} \left (4-\sqrt {3}\right ) \sqrt {7519+4340 \sqrt {3}} \operatorname {EllipticPi}\left (\frac {1}{169} \left (553+304 \sqrt {3}\right ),\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )\right )}{\left (4+\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (4+\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 435 |
| \(\displaystyle -\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (-\frac {1}{2} \left (4+\sqrt {3}\right ) \int \frac {1}{\sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \sqrt {\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}+1} \left (\frac {\left (4+\sqrt {3}\right )^2 \left (-x-\sqrt {3}+1\right )}{-x+\sqrt {3}+1}-8 \sqrt {3}+19\right )}d\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-\frac {1}{169} \left (4-\sqrt {3}\right ) \sqrt {7519+4340 \sqrt {3}} \operatorname {EllipticPi}\left (\frac {1}{169} \left (553+304 \sqrt {3}\right ),\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )\right )}{\left (4+\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (4+\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (-\left (4+\sqrt {3}\right ) \int \frac {1}{16 \sqrt {3}-\frac {28 \left (2-\sqrt {3}\right ) \sqrt {\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}+1}}{\sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7}}}d\frac {\sqrt {\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}+1}}{\sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7}}-\frac {1}{169} \left (4-\sqrt {3}\right ) \sqrt {7519+4340 \sqrt {3}} \operatorname {EllipticPi}\left (\frac {1}{169} \left (553+304 \sqrt {3}\right ),\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )\right )}{\left (4+\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (4+\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (\frac {\left (4+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt {7 \left (2-\sqrt {3}\right )} \left (-x-\sqrt {3}+1\right )}{2 \sqrt [4]{3} \left (-x+\sqrt {3}+1\right )}\right )}{8 \sqrt [4]{3} \sqrt {7 \left (2-\sqrt {3}\right )}}-\frac {1}{169} \left (4-\sqrt {3}\right ) \sqrt {7519+4340 \sqrt {3}} \operatorname {EllipticPi}\left (\frac {1}{169} \left (553+304 \sqrt {3}\right ),\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )\right )}{\left (4+\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (4+\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
Input:
Int[x/((3 + x)*Sqrt[1 - x^3]),x]
Output:
(-2*(1 + Sqrt[3])*Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3 ] - x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sq rt[3]])/(3^(1/4)*(4 + Sqrt[3])*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3]) - (12*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt [3] - x)^2]*(((4 + Sqrt[3])*ArcTanh[(Sqrt[7*(2 - Sqrt[3])]*(1 - Sqrt[3] - x))/(2*3^(1/4)*(1 + Sqrt[3] - x))])/(8*3^(1/4)*Sqrt[7*(2 - Sqrt[3])]) - (( 4 - Sqrt[3])*Sqrt[7519 + 4340*Sqrt[3]]*EllipticPi[(553 + 304*Sqrt[3])/169, ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/169))/((4 + Sqrt[3])*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3])
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2) *(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
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[1/(((a_) + (b_.)*(x_))*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[a Int[1/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] - Simp[b Int[x/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> With[{q = Simplify[(1 + Sqrt[3])*(f/e)]}, Simp[4*3^(1/4)*Sqrt[2 - Sqrt[3]]*f*(1 + q*x)*(Sqrt[(1 - q*x + q^2*x^2)/(1 + Sqrt[3] + q*x)^2]/(q* Sqrt[a + b*x^3]*Sqrt[(1 + q*x)/(1 + Sqrt[3] + q*x)^2])) Subst[Int[1/(((1 - Sqrt[3])*d - c*q + ((1 + Sqrt[3])*d - c*q)*x)*Sqrt[1 - x^2]*Sqrt[7 - 4*Sq rt[3] + x^2]), x], x, (-1 + Sqrt[3] - q*x)/(1 + Sqrt[3] + q*x)], x]] /; Fre eQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*e^3 - 2*(5 + 3*Sqrt [3])*a*f^3, 0] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x _Symbol] :> With[{q = Rt[b/a, 3]}, Simp[((1 + Sqrt[3])*f - e*q)/((1 + Sqrt[ 3])*d - c*q) Int[1/Sqrt[a + b*x^3], x], x] + Simp[(d*e - c*f)/((1 + Sqrt[ 3])*d - c*q) Int[(1 + Sqrt[3] + q*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && NeQ[b^2*c^6 - 20*a *b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[b^2*e^6 - 20*a*b*e^3*f^3 - 8*a^2*f^6, 0]
Time = 0.41 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.64
| method | result | size |
| default | \(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}+\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {5}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}+1}\, \left (\frac {5}{2}+\frac {i \sqrt {3}}{2}\right )}\) | \(240\) |
| elliptic | \(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}+\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {5}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}+1}\, \left (\frac {5}{2}+\frac {i \sqrt {3}}{2}\right )}\) | \(240\) |
Input:
int(x/(3+x)/(-x^3+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/3*I*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((x-1)/(-3/2+1/2*I* 3^(1/2)))^(1/2)*(-I*(x+1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3+1)^(1/2)*El lipticF(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(-3 /2+1/2*I*3^(1/2)))^(1/2))+2*I*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1 /2)*((x-1)/(-3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x+1/2+1/2*I*3^(1/2))*3^(1/2))^ (1/2)/(-x^3+1)^(1/2)/(5/2+1/2*I*3^(1/2))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2- 1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(5/2+1/2*I*3^(1/2)),(I*3^(1/2)/(-3 /2+1/2*I*3^(1/2)))^(1/2))
\[ \int \frac {x}{(3+x) \sqrt {1-x^3}} \, dx=\int { \frac {x}{\sqrt {-x^{3} + 1} {\left (x + 3\right )}} \,d x } \] Input:
integrate(x/(3+x)/(-x^3+1)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-x^3 + 1)*x/(x^4 + 3*x^3 - x - 3), x)
\[ \int \frac {x}{(3+x) \sqrt {1-x^3}} \, dx=\int \frac {x}{\sqrt {- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 3\right )}\, dx \] Input:
integrate(x/(3+x)/(-x**3+1)**(1/2),x)
Output:
Integral(x/(sqrt(-(x - 1)*(x**2 + x + 1))*(x + 3)), x)
\[ \int \frac {x}{(3+x) \sqrt {1-x^3}} \, dx=\int { \frac {x}{\sqrt {-x^{3} + 1} {\left (x + 3\right )}} \,d x } \] Input:
integrate(x/(3+x)/(-x^3+1)^(1/2),x, algorithm="maxima")
Output:
integrate(x/(sqrt(-x^3 + 1)*(x + 3)), x)
\[ \int \frac {x}{(3+x) \sqrt {1-x^3}} \, dx=\int { \frac {x}{\sqrt {-x^{3} + 1} {\left (x + 3\right )}} \,d x } \] Input:
integrate(x/(3+x)/(-x^3+1)^(1/2),x, algorithm="giac")
Output:
integrate(x/(sqrt(-x^3 + 1)*(x + 3)), x)
Time = 23.42 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.59 \[ \int \frac {x}{(3+x) \sqrt {1-x^3}} \, dx=-\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {x^3-1}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (4\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-3\,\Pi \left (\frac {3}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{4\,\sqrt {1-x^3}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(x/((1 - x^3)^(1/2)*(x + 3)),x)
Output:
-((3^(1/2)*1i + 3)*(x^3 - 1)^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)* 1i)/2 - 3/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1 /2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(4*ellipticF(asin((-(x - 1)/(( 3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/ 2)) - 3*ellipticPi((3^(1/2)*1i)/8 + 3/8, asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))))/(4*(1 - x^ 3)^(1/2)*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i) /2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2))
\[ \int \frac {x}{(3+x) \sqrt {1-x^3}} \, dx=-\left (\int \frac {\sqrt {-x^{3}+1}\, x}{x^{4}+3 x^{3}-x -3}d x \right ) \] Input:
int(x/(3+x)/(-x^3+1)^(1/2),x)
Output:
- int((sqrt( - x**3 + 1)*x)/(x**4 + 3*x**3 - x - 3),x)