Integrand size = 27, antiderivative size = 325 \[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=-\frac {\left (c+d+\sqrt {3} d\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \text {arctanh}\left (\frac {\sqrt {c^2-c d+d^2} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}}}{\sqrt {d} \sqrt {c+d} \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}}\right )}{\sqrt {d} \sqrt {c+d} \sqrt {c^2-c d+d^2} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c+d+\sqrt {3} d\right )^2}{\left (c+d-\sqrt {3} d\right )^2},\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{\left (c+d-\sqrt {3} d\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \] Output:
-(c+d+3^(1/2)*d)*(1-x)*((x^2+x+1)/(1+3^(1/2)-x)^2)^(1/2)*arctanh((c^2-c*d+ d^2)^(1/2)*((1-x)/(1+3^(1/2)-x)^2)^(1/2)/d^(1/2)/(c+d)^(1/2)/((x^2+x+1)/(1 +3^(1/2)-x)^2)^(1/2))/d^(1/2)/(c+d)^(1/2)/(c^2-c*d+d^2)^(1/2)/((1-x)/(1+3^ (1/2)-x)^2)^(1/2)/(x^3-1)^(1/2)-4*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)*EllipticPi((1-3^(1/2)-x)/(1+3^(1/2)-x),( c+d+3^(1/2)*d)^2/(c+d-3^(1/2)*d)^2,I*3^(1/2)+2*I)/(c+d-3^(1/2)*d)/((1-x)/( 1+3^(1/2)-x)^2)^(1/2)/(x^3-1)^(1/2)
Result contains complex when optimal does not.
Time = 10.24 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.72 \[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=\frac {2 \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (-\frac {3 \left (\sqrt [3]{-1}+x\right ) \sqrt {\frac {\sqrt [3]{-1}+(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (\sqrt {3} c+\left (3+\sqrt {3}\right ) d\right ) \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {i \sqrt {3} d}{-c+\sqrt [3]{-1} d},\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{c-\sqrt [3]{-1} d}\right )}{3 d \sqrt {-1+x^3}} \] Input:
Integrate[(1 + Sqrt[3] - x)/((c + d*x)*Sqrt[-1 + x^3]),x]
Output:
(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((-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))] + ((-1)^(1/3)*(1 + (-1)^(1/3))*(Sqrt[3]*c + (3 + 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)))/(3*d*Sqrt[- 1 + x^3])
Time = 1.54 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {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.
\(\displaystyle \int \frac {-x+\sqrt {3}+1}{\sqrt {x^3-1} (c+d x)} \, dx\) |
\(\Big \downarrow \) 2567 |
\(\displaystyle \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \int \frac {1}{\sqrt {1-\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (c-\sqrt {3} d+d-\frac {\left (c+\sqrt {3} d+d\right ) \left (-x-\sqrt {3}+1\right )}{-x+\sqrt {3}+1}\right )}d\left (-\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 2538 |
\(\displaystyle \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (\left (c-\sqrt {3} d+d\right ) \int \frac {1}{\sqrt {1-\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (\left (c-\sqrt {3} d+d\right )^2-\frac {\left (c+\sqrt {3} d+d\right )^2 \left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}\right )}d\left (-\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )-\left (c+\sqrt {3} d+d\right ) \int -\frac {-x-\sqrt {3}+1}{\sqrt {1-\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (\left (c-\sqrt {3} d+d\right )^2-\frac {\left (c+\sqrt {3} d+d\right )^2 \left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}\right ) \left (-x+\sqrt {3}+1\right )}d\left (-\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (-\left (c+\sqrt {3} d+d\right ) \int -\frac {-x-\sqrt {3}+1}{\sqrt {1-\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (\left (c-\sqrt {3} d+d\right )^2-\frac {\left (c+\sqrt {3} d+d\right )^2 \left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}\right ) \left (-x+\sqrt {3}+1\right )}d\left (-\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )-\frac {\operatorname {EllipticPi}\left (\frac {\left (c+\sqrt {3} d+d\right )^2}{\left (c-\sqrt {3} d+d\right )^2},\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \left (c-\sqrt {3} d+d\right )}\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 435 |
\(\displaystyle \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (-\frac {1}{2} \left (c+\sqrt {3} d+d\right ) \int \frac {1}{\sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \sqrt {\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}+1} \left (\left (c-\sqrt {3} d+d\right )^2+\frac {\left (c+\sqrt {3} d+d\right )^2 \left (-x-\sqrt {3}+1\right )}{-x+\sqrt {3}+1}\right )}d\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-\frac {\operatorname {EllipticPi}\left (\frac {\left (c+\sqrt {3} d+d\right )^2}{\left (c-\sqrt {3} d+d\right )^2},\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \left (c-\sqrt {3} d+d\right )}\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (-\left (c+\sqrt {3} d+d\right ) \int \frac {1}{4 \sqrt {3} d (c+d)-\frac {4 \left (2-\sqrt {3}\right ) \left (c^2-d c+d^2\right ) \sqrt {\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}+1}}{\sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7}}}d\frac {\sqrt {\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}+1}}{\sqrt {\frac {\left (-x-\sqrt {3}+1\right )^2}{\left (-x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7}}-\frac {\operatorname {EllipticPi}\left (\frac {\left (c+\sqrt {3} d+d\right )^2}{\left (c-\sqrt {3} d+d\right )^2},\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \left (c-\sqrt {3} d+d\right )}\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (\frac {\left (c+\sqrt {3} d+d\right ) \text {arctanh}\left (\frac {\sqrt {2-\sqrt {3}} \left (-x-\sqrt {3}+1\right ) \sqrt {c^2-c d+d^2}}{\sqrt [4]{3} \sqrt {d} \left (-x+\sqrt {3}+1\right ) \sqrt {c+d}}\right )}{4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {d} \sqrt {c+d} \sqrt {c^2-c d+d^2}}-\frac {\operatorname {EllipticPi}\left (\frac {\left (c+\sqrt {3} d+d\right )^2}{\left (c-\sqrt {3} d+d\right )^2},\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \left (c-\sqrt {3} d+d\right )}\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}\) |
Input:
Int[(1 + Sqrt[3] - x)/((c + d*x)*Sqrt[-1 + x^3]),x]
Output:
(4*3^(1/4)*Sqrt[2 - Sqrt[3]]*(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]) - Ellip ticPi[(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]]/(Sqrt[7 - 4*Sqrt[3]]*(c + d - Sqrt[3 ]*d))))/(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[(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*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]
Time = 0.52 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {2 \left (c +d +\sqrt {3}\, d \right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{1+\frac {c}{d}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d^{2} \sqrt {x^{3}-1}\, \left (1+\frac {c}{d}\right )}-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d \sqrt {x^{3}-1}}\) | \(273\) |
elliptic | \(\frac {2 \left (c +d +\sqrt {3}\, d \right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{1+\frac {c}{d}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d^{2} \sqrt {x^{3}-1}\, \left (1+\frac {c}{d}\right )}-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d \sqrt {x^{3}-1}}\) | \(273\) |
Input:
int((1+3^(1/2)-x)/(d*x+c)/(x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
2*(c+d+3^(1/2)*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))-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))
Timed out. \[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=\text {Timed out} \] Input:
integrate((1+3^(1/2)-x)/(d*x+c)/(x^3-1)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=- \int \left (- \frac {\sqrt {3}}{c \sqrt {x^{3} - 1} + d x \sqrt {x^{3} - 1}}\right )\, dx - \int \frac {x}{c \sqrt {x^{3} - 1} + d x \sqrt {x^{3} - 1}}\, dx - \int \left (- \frac {1}{c \sqrt {x^{3} - 1} + d x \sqrt {x^{3} - 1}}\right )\, dx \] Input:
integrate((1+3**(1/2)-x)/(d*x+c)/(x**3-1)**(1/2),x)
Output:
-Integral(-sqrt(3)/(c*sqrt(x**3 - 1) + d*x*sqrt(x**3 - 1)), x) - Integral( x/(c*sqrt(x**3 - 1) + d*x*sqrt(x**3 - 1)), x) - Integral(-1/(c*sqrt(x**3 - 1) + d*x*sqrt(x**3 - 1)), x)
\[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=\int { -\frac {x - \sqrt {3} - 1}{\sqrt {x^{3} - 1} {\left (d x + c\right )}} \,d x } \] Input:
integrate((1+3^(1/2)-x)/(d*x+c)/(x^3-1)^(1/2),x, algorithm="maxima")
Output:
-integrate((x - sqrt(3) - 1)/(sqrt(x^3 - 1)*(d*x + c)), x)
\[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=\int { -\frac {x - \sqrt {3} - 1}{\sqrt {x^{3} - 1} {\left (d x + c\right )}} \,d x } \] Input:
integrate((1+3^(1/2)-x)/(d*x+c)/(x^3-1)^(1/2),x, algorithm="giac")
Output:
integrate(-(x - sqrt(3) - 1)/(sqrt(x^3 - 1)*(d*x + c)), x)
Timed out. \[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=\text {Hanged} \] Input:
int((3^(1/2) - x + 1)/((x^3 - 1)^(1/2)*(c + d*x)),x)
Output:
\text{Hanged}
\[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=\int \frac {1+\sqrt {3}-x}{\left (d x +c \right ) \sqrt {x^{3}-1}}d x \] Input:
int((1+3^(1/2)-x)/(d*x+c)/(x^3-1)^(1/2),x)
Output:
int((1+3^(1/2)-x)/(d*x+c)/(x^3-1)^(1/2),x)