Integrand size = 24, antiderivative size = 474 \[ \int \frac {e+f x}{(c+d x) \sqrt {1-x^3}} \, dx=-\frac {(d e-c f) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \text {arctanh}\left (\frac {\sqrt {c^2-c d+d^2} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}}}{\sqrt {d} \sqrt {c+d} \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}}\right )}{\sqrt {d} \sqrt {c+d} \sqrt {c^2-c d+d^2} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}}-\frac {2 \sqrt {2+\sqrt {3}} \left (e+f+\sqrt {3} f\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (c+d+\sqrt {3} d\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (d e-c f) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c+d+\sqrt {3} d\right )^2}{\left (c+d-\sqrt {3} d\right )^2},\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{\left (c^2+2 c d-2 d^2\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}} \]
-(-c*f+d*e)*(1-x)*arctanh((c^2-c*d+d^2)^(1/2)*((1-x)/(1-x+3^(1/2))^2)^(1/2 )/d^(1/2)/(c+d)^(1/2)/((x^2+x+1)/(1-x+3^(1/2))^2)^(1/2))*((x^2+x+1)/(1-x+3 ^(1/2))^2)^(1/2)/d^(1/2)/(c+d)^(1/2)/(c^2-c*d+d^2)^(1/2)/(-x^3+1)^(1/2)/(( 1-x)/(1-x+3^(1/2))^2)^(1/2)+4*3^(1/4)*(-c*f+d*e)*(1-x)*EllipticPi((-1+x+3^ (1/2))/(1-x+3^(1/2)),(c+d+d*3^(1/2))^2/(c+d-d*3^(1/2))^2,I*3^(1/2)+2*I)*(1 /2*6^(1/2)+1/2*2^(1/2))*((x^2+x+1)/(1-x+3^(1/2))^2)^(1/2)/(c^2+2*c*d-2*d^2 )/(-x^3+1)^(1/2)/((1-x)/(1-x+3^(1/2))^2)^(1/2)-2/3*(1-x)*EllipticF((1-x-3^ (1/2))/(1-x+3^(1/2)),I*3^(1/2)+2*I)*(e+f+f*3^(1/2))*(1/2*6^(1/2)+1/2*2^(1/ 2))*((x^2+x+1)/(1-x+3^(1/2))^2)^(1/2)*3^(3/4)/(c+d+d*3^(1/2))/(-x^3+1)^(1/ 2)/((1-x)/(1-x+3^(1/2))^2)^(1/2)
Result contains complex when optimal does not.
Time = 10.58 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.49 \[ \int \frac {e+f x}{(c+d x) \sqrt {1-x^3}} \, dx=\frac {2 \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (\frac {3 f \left (\sqrt [3]{-1}+x\right ) \sqrt {\frac {\sqrt [3]{-1}+(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {\sqrt [3]{-1} \sqrt {3} \left (1+\sqrt [3]{-1}\right ) (-d e+c f) \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {i \sqrt {3} d}{-c+\sqrt [3]{-1} d},\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{-c+\sqrt [3]{-1} d}\right )}{3 d \sqrt {1-x^3}} \]
(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((3*f*((-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)) ] + ((-1)^(1/3)*Sqrt[3]*(1 + (-1)^(1/3))*(-(d*e) + c*f)*Sqrt[1 + x + x^2]* EllipticPi[(I*Sqrt[3]*d)/(-c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 - (-1)^(2/3)* x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-c + (-1)^(1/3)*d)))/(3*d*Sqrt[1 - x^ 3])
Time = 1.39 (sec) , antiderivative size = 460, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2569, 759, 2567, 2538, 412, 435, 104, 221}
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.
| \(\displaystyle \int \frac {e+f x}{\sqrt {1-x^3} (c+d x)} \, dx\) |
| \(\Big \downarrow \) 2569 |
| \(\displaystyle \frac {\left (e+\sqrt {3} f+f\right ) \int \frac {1}{\sqrt {1-x^3}}dx}{c+\sqrt {3} d+d}+\frac {(d e-c f) \int \frac {-x+\sqrt {3}+1}{(c+d x) \sqrt {1-x^3}}dx}{c+\sqrt {3} d+d}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {(d e-c f) \int \frac {-x+\sqrt {3}+1}{(c+d x) \sqrt {1-x^3}}dx}{c+\sqrt {3} d+d}-\frac {2 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (e+\sqrt {3} f+f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )}\) |
| \(\Big \downarrow \) 2567 |
| \(\displaystyle \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} (d e-c f) \int \frac {1}{\sqrt {1-\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (c-\sqrt {3} d+d-\frac {\left (c+\sqrt {3} d+d\right ) \left (-x-\sqrt {3}+1\right )}{-x+\sqrt {3}+1}\right )}d\left (-\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )}-\frac {2 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (e+\sqrt {3} f+f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )}\) |
| \(\Big \downarrow \) 2538 |
| \(\displaystyle \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} (d e-c f) \left (\left (c-\sqrt {3} d+d\right ) \int \frac {1}{\sqrt {1-\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (\left (c-\sqrt {3} d+d\right )^2-\frac {\left (c+\sqrt {3} d+d\right )^2 \left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}\right )}d\left (-\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )-\left (c+\sqrt {3} d+d\right ) \int -\frac {-x-\sqrt {3}+1}{\sqrt {1-\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (\left (c-\sqrt {3} d+d\right )^2-\frac {\left (c+\sqrt {3} d+d\right )^2 \left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}\right ) \left (-x+\sqrt {3}+1\right )}d\left (-\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )}-\frac {2 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (e+\sqrt {3} f+f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} (d e-c f) \left (-\left (c+\sqrt {3} d+d\right ) \int -\frac {-x-\sqrt {3}+1}{\sqrt {1-\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (\left (c-\sqrt {3} d+d\right )^2-\frac {\left (c+\sqrt {3} d+d\right )^2 \left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}\right ) \left (-x+\sqrt {3}+1\right )}d\left (-\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )-\frac {\operatorname {EllipticPi}\left (\frac {\left (c+\sqrt {3} d+d\right )^2}{\left (c-\sqrt {3} d+d\right )^2},\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \left (c-\sqrt {3} d+d\right )}\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )}-\frac {2 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (e+\sqrt {3} f+f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )}\) |
| \(\Big \downarrow \) 435 |
| \(\displaystyle \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} (d e-c f) \left (-\frac {1}{2} \left (c+\sqrt {3} d+d\right ) \int \frac {1}{\sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \sqrt {\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}+1} \left (\left (c-\sqrt {3} d+d\right )^2+\frac {\left (c+\sqrt {3} d+d\right )^2 \left (-x-\sqrt {3}+1\right )}{-x+\sqrt {3}+1}\right )}d\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-\frac {\operatorname {EllipticPi}\left (\frac {\left (c+\sqrt {3} d+d\right )^2}{\left (c-\sqrt {3} d+d\right )^2},\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \left (c-\sqrt {3} d+d\right )}\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )}-\frac {2 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (e+\sqrt {3} f+f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} (d e-c f) \left (-\left (c+\sqrt {3} d+d\right ) \int \frac {1}{4 \sqrt {3} d (c+d)-\frac {4 \left (2-\sqrt {3}\right ) \left (c^2-d c+d^2\right ) \sqrt {\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}+1}}{\sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7}}}d\frac {\sqrt {\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}+1}}{\sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7}}-\frac {\operatorname {EllipticPi}\left (\frac {\left (c+\sqrt {3} d+d\right )^2}{\left (c-\sqrt {3} d+d\right )^2},\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \left (c-\sqrt {3} d+d\right )}\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )}-\frac {2 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (e+\sqrt {3} f+f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} (d e-c f) \left (\frac {\left (c+\sqrt {3} d+d\right ) \text {arctanh}\left (\frac {\sqrt {2-\sqrt {3}} \left (-x-\sqrt {3}+1\right ) \sqrt {c^2-c d+d^2}}{\sqrt [4]{3} \sqrt {d} \left (-x+\sqrt {3}+1\right ) \sqrt {c+d}}\right )}{4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {d} \sqrt {c+d} \sqrt {c^2-c d+d^2}}-\frac {\operatorname {EllipticPi}\left (\frac {\left (c+\sqrt {3} d+d\right )^2}{\left (c-\sqrt {3} d+d\right )^2},\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \left (c-\sqrt {3} d+d\right )}\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )}-\frac {2 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (e+\sqrt {3} f+f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )}\) |
(-2*Sqrt[2 + Sqrt[3]]*(e + f + Sqrt[3]*f)*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(3^(1/4)*(c + d + Sqrt[3]*d)*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^ 2]*Sqrt[1 - x^3]) + (4*3^(1/4)*Sqrt[2 - Sqrt[3]]*(d*e - c*f)*(1 - x)*Sqrt[ (1 + x + x^2)/(1 + Sqrt[3] - x)^2]*(((c + d + Sqrt[3]*d)*ArcTanh[(Sqrt[2 - Sqrt[3]]*Sqrt[c^2 - c*d + d^2]*(1 - Sqrt[3] - x))/(3^(1/4)*Sqrt[d]*Sqrt[c + d]*(1 + Sqrt[3] - x))])/(4*3^(1/4)*Sqrt[2 - Sqrt[3]]*Sqrt[d]*Sqrt[c + d ]*Sqrt[c^2 - c*d + d^2]) - EllipticPi[(c + d + Sqrt[3]*d)^2/(c + d - Sqrt[ 3]*d)^2, ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]]/(Sqr t[7 - 4*Sqrt[3]]*(c + d - Sqrt[3]*d))))/((c + d + Sqrt[3]*d)*Sqrt[(1 - x)/ (1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3])
3.2.65.3.1 Defintions of rubi rules used
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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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 = 1.02 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.56
| method | result | size |
| default | \(-\frac {2 i f \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d \sqrt {-x^{3}+1}}-\frac {2 i \left (-c f +e d \right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d^{2} \sqrt {-x^{3}+1}\, \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}\right )}\) | \(265\) |
| elliptic | \(-\frac {2 i f \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d \sqrt {-x^{3}+1}}+\frac {2 i \left (c f -e d \right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d^{2} \sqrt {-x^{3}+1}\, \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}\right )}\) | \(265\) |
-2/3*I*f/d*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 )*EllipticF(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))-2/3*I*(-c*f+d*e)/d^2*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)/(-1/2+1/2*I*3^(1/2)+c/d)*Ellipti cPi(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(-1/2+1/ 2*I*3^(1/2)+c/d),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2))
Timed out. \[ \int \frac {e+f x}{(c+d x) \sqrt {1-x^3}} \, dx=\text {Timed out} \]
\[ \int \frac {e+f x}{(c+d x) \sqrt {1-x^3}} \, dx=\int \frac {e + f x}{\sqrt {- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (c + d x\right )}\, dx \]
\[ \int \frac {e+f x}{(c+d x) \sqrt {1-x^3}} \, dx=\int { \frac {f x + e}{\sqrt {-x^{3} + 1} {\left (d x + c\right )}} \,d x } \]
\[ \int \frac {e+f x}{(c+d x) \sqrt {1-x^3}} \, dx=\int { \frac {f x + e}{\sqrt {-x^{3} + 1} {\left (d x + c\right )}} \,d x } \]
Time = 20.11 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.82 \[ \int \frac {e+f x}{(c+d x) \sqrt {1-x^3}} \, dx=-\frac {2\,f\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\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}}}\,\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 )}{d\,\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 )}}+\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\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}}}\,\left (c\,f-d\,e\right )\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {c}{d}+1};\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 )}{d^2\,\sqrt {1-x^3}\,\left (\frac {c}{d}+1\right )\,\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 )}} \]
(2*((3^(1/2)*1i)/2 + 3/2)*(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)*(c*f - d*e)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi( ((3^(1/2)*1i)/2 + 3/2)/(c/d + 1), 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)))/(d^2*(1 - x^3)^(1/ 2)*(c/d + 1)*(((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)) - (2*f*((3^(1/2)*1 i)/2 + 3/2)*(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)*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)))/(d*(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))