Integrand size = 27, antiderivative size = 358 \[ \int \frac {1-\sqrt {3}+x}{(c+d x) \sqrt {1+x^3}} \, dx=-\frac {\left (c-\left (1-\sqrt {3}\right ) d\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \text {arctanh}\left (\frac {2 \sqrt {2+\sqrt {3}} \sqrt {c^2+c d+d^2} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}}}{\sqrt {c-d} \sqrt {d} \sqrt {7+4 \sqrt {3}+\frac {\left (1+\sqrt {3}+x\right )^2}{\left (1-\sqrt {3}+x\right )^2}}}\right )}{\sqrt {c-d} \sqrt {d} \sqrt {c^2+c d+d^2} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2}{\left (c-\left (1+\sqrt {3}\right ) d\right )^2},\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right ),-7+4 \sqrt {3}\right )}{\left (c-d-\sqrt {3} d\right ) \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \]
-(1+x)*arctanh(2*(c^2+c*d+d^2)^(1/2)*((-1-x)/(1+x-3^(1/2))^2)^(1/2)*(1/2*6 ^(1/2)+1/2*2^(1/2))/(c-d)^(1/2)/d^(1/2)/(7+4*3^(1/2)+(1+x+3^(1/2))^2/(1+x- 3^(1/2))^2)^(1/2))*(c-d*(1-3^(1/2)))*((x^2-x+1)/(1+x-3^(1/2))^2)^(1/2)/(c- d)^(1/2)/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)*(1+x)*EllipticPi((-1-x-3^(1/2))/(1+x-3^(1/2)),(c-d*(1-3^ (1/2)))^2/(c-d*(1+3^(1/2)))^2,2*I-I*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)/(-d*3^(1/2)+c-d)/(x^3+1)^(1/2)/((-1-x)/(1+x -3^(1/2))^2)^(1/2)
Result contains complex when optimal does not.
Time = 10.52 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.59 \[ \int \frac {1-\sqrt {3}+x}{(c+d x) \sqrt {1+x^3}} \, dx=\frac {2 \sqrt {\frac {1+x}{1+\sqrt [3]{-1}}} \left (-\frac {\left (\sqrt [3]{-1}-x\right ) \sqrt {\frac {\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {i \left (c+\left (-1+\sqrt {3}\right ) d\right ) \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {i \sqrt {3} d}{c+\sqrt [3]{-1} d},\arcsin \left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{c+\sqrt [3]{-1} d}\right )}{d \sqrt {1+x^3}} \]
(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))]) + (I*(c + (-1 + Sqrt[3])*d)*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)))/(d*Sqrt[1 + x^3])
Time = 1.07 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.86, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2568, 25, 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.
\(\displaystyle \int \frac {x-\sqrt {3}+1}{\sqrt {x^3+1} (c+d x)} \, dx\) |
\(\Big \downarrow \) 2568 |
\(\displaystyle -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \int -\frac {1}{\left (c-\sqrt {3} d-d-\frac {\left (c-\left (1-\sqrt {3}\right ) d\right ) \left (x+\sqrt {3}+1\right )}{x-\sqrt {3}+1}\right ) \sqrt {1-\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7}}d\left (-\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )}{\sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \int \frac {1}{\left (c-\left (1+\sqrt {3}\right ) d-\frac {\left (c-\left (1-\sqrt {3}\right ) d\right ) \left (x+\sqrt {3}+1\right )}{x-\sqrt {3}+1}\right ) \sqrt {1-\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7}}d\left (-\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )}{\sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 2538 |
\(\displaystyle -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (\left (c-\left (1-\sqrt {3}\right ) d\right ) \int -\frac {x+\sqrt {3}+1}{\left (x-\sqrt {3}+1\right ) \sqrt {1-\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7} \left (\left (c-\left (1+\sqrt {3}\right ) d\right )^2-\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2 \left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}\right )}d\left (-\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )-\left (c-\left (1+\sqrt {3}\right ) d\right ) \int \frac {1}{\sqrt {1-\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7} \left (\left (c-\left (1+\sqrt {3}\right ) d\right )^2-\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2 \left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}\right )}d\left (-\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )\right )}{\sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (\left (c-\left (1-\sqrt {3}\right ) d\right ) \int -\frac {x+\sqrt {3}+1}{\left (x-\sqrt {3}+1\right ) \sqrt {1-\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7} \left (\left (c-\left (1+\sqrt {3}\right ) d\right )^2-\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2 \left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}\right )}d\left (-\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )+\frac {\left (c-\left (1+\sqrt {3}\right ) d\right ) \operatorname {EllipticPi}\left (\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2}{\left (c-\left (1+\sqrt {3}\right ) d\right )^2},\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt {7+4 \sqrt {3}} \left (c-\sqrt {3} d-d\right )^2}\right )}{\sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 435 |
\(\displaystyle -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (\frac {1}{2} \left (c-\left (1-\sqrt {3}\right ) d\right ) \int \frac {1}{\sqrt {\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}+1} \left (\frac {\left (x+\sqrt {3}+1\right ) \left (c-\left (1-\sqrt {3}\right ) d\right )^2}{x-\sqrt {3}+1}+\left (c-\left (1+\sqrt {3}\right ) d\right )^2\right ) \sqrt {\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7}}d\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}+\frac {\left (c-\left (1+\sqrt {3}\right ) d\right ) \operatorname {EllipticPi}\left (\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2}{\left (c-\left (1+\sqrt {3}\right ) d\right )^2},\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt {7+4 \sqrt {3}} \left (c-\sqrt {3} d-d\right )^2}\right )}{\sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (\left (c-\left (1-\sqrt {3}\right ) d\right ) \int \frac {1}{4 \sqrt {3} (c-d) d-\frac {4 \left (2+\sqrt {3}\right ) \left (c^2+d c+d^2\right ) \sqrt {\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}+1}}{\sqrt {\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7}}}d\frac {\sqrt {\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}+1}}{\sqrt {\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7}}+\frac {\left (c-\left (1+\sqrt {3}\right ) d\right ) \operatorname {EllipticPi}\left (\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2}{\left (c-\left (1+\sqrt {3}\right ) d\right )^2},\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt {7+4 \sqrt {3}} \left (c-\sqrt {3} d-d\right )^2}\right )}{\sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (\frac {\left (c-\left (1+\sqrt {3}\right ) d\right ) \operatorname {EllipticPi}\left (\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2}{\left (c-\left (1+\sqrt {3}\right ) d\right )^2},\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt {7+4 \sqrt {3}} \left (c-\sqrt {3} d-d\right )^2}-\frac {\left (c-\left (1-\sqrt {3}\right ) d\right ) \text {arctanh}\left (\frac {\sqrt {2+\sqrt {3}} \left (x+\sqrt {3}+1\right ) \sqrt {c^2+c d+d^2}}{\sqrt [4]{3} \sqrt {d} \left (x-\sqrt {3}+1\right ) \sqrt {c-d}}\right )}{4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {d} \sqrt {c-d} \sqrt {c^2+c d+d^2}}\right )}{\sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
(-4*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x) ^2]*(-1/4*((c - (1 - Sqrt[3])*d)*ArcTanh[(Sqrt[2 + Sqrt[3]]*Sqrt[c^2 + c*d + d^2]*(1 + Sqrt[3] + x))/(3^(1/4)*Sqrt[c - d]*Sqrt[d]*(1 - Sqrt[3] + x)) ])/(3^(1/4)*Sqrt[2 + Sqrt[3]]*Sqrt[c - d]*Sqrt[d]*Sqrt[c^2 + c*d + d^2]) + ((c - (1 + Sqrt[3])*d)*EllipticPi[(c - (1 - Sqrt[3])*d)^2/(c - (1 + Sqrt[ 3])*d)^2, ArcSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/(S qrt[7 + 4*Sqrt[3]]*(c - d - Sqrt[3]*d)^2)))/(Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[1 + x^3])
3.2.48.3.1 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/(((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* Sqrt[3] + x^2]), x], x, (1 + Sqrt[3] - q*x)/(-1 + Sqrt[3] + q*x)], x]] /; F reeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*e^3 - 2*(5 - 3*Sq rt[3])*a*f^3, 0] && NeQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
Time = 1.23 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d \sqrt {x^{3}+1}}-\frac {2 \left (d \sqrt {3}+c -d \right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \Pi \left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-1+\frac {c}{d}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d^{2} \sqrt {x^{3}+1}\, \left (-1+\frac {c}{d}\right )}\) | \(275\) |
elliptic | \(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d \sqrt {x^{3}+1}}-\frac {2 \left (d \sqrt {3}+c -d \right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \Pi \left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-1+\frac {c}{d}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d^{2} \sqrt {x^{3}+1}\, \left (-1+\frac {c}{d}\right )}\) | \(275\) |
2/d*(3/2-1/2*I*3^(1/2))*((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^ (1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1 /2)))^(1/2)/(x^3+1)^(1/2)*EllipticF(((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2),((-3 /2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))-2*(d*3^(1/2)+c-d)/d^2*(3/2- 1/2*I*3^(1/2))*((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^(1/2))/(- 3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/ 2)/(x^3+1)^(1/2)/(-1+c/d)*EllipticPi(((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2),(-3 /2+1/2*I*3^(1/2))/(-1+c/d),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/ 2))
\[ \int \frac {1-\sqrt {3}+x}{(c+d x) \sqrt {1+x^3}} \, dx=\int { \frac {x - \sqrt {3} + 1}{\sqrt {x^{3} + 1} {\left (d x + c\right )}} \,d x } \]
\[ \int \frac {1-\sqrt {3}+x}{(c+d x) \sqrt {1+x^3}} \, dx=\int \frac {x - \sqrt {3} + 1}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (c + d x\right )}\, dx \]
\[ \int \frac {1-\sqrt {3}+x}{(c+d x) \sqrt {1+x^3}} \, dx=\int { \frac {x - \sqrt {3} + 1}{\sqrt {x^{3} + 1} {\left (d x + c\right )}} \,d x } \]
\[ \int \frac {1-\sqrt {3}+x}{(c+d x) \sqrt {1+x^3}} \, dx=\int { \frac {x - \sqrt {3} + 1}{\sqrt {x^{3} + 1} {\left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1-\sqrt {3}+x}{(c+d x) \sqrt {1+x^3}} \, dx=\text {Hanged} \]