\(\int \frac {A+B x+C x^2}{\sqrt {8-8 x+4 x^2-x^3}} \, dx\) [132]

Optimal result

Integrand size = 30, antiderivative size = 270 \[ \int \frac {A+B x+C x^2}{\sqrt {8-8 x+4 x^2-x^3}} \, dx=-\frac {2 C (2-x) \left (4-2 x+x^2\right )}{3 \sqrt {8-8 x+4 x^2-x^3}}+\frac {2 (3 B+8 C) (2-x) \left (4-2 x+x^2\right )}{3 (4-x) \sqrt {8-8 x+4 x^2-x^3}}-\frac {2 \sqrt {2} (3 B+8 C) \sqrt {2-x} (4-x) \sqrt {\frac {4-2 x+x^2}{(4-x)^2}} E\left (2 \arctan \left (\frac {\sqrt {2-x}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{3 \sqrt {8-8 x+4 x^2-x^3}}-\frac {(3 A-8 C) \sqrt {2-x} (4-x) \sqrt {\frac {4-2 x+x^2}{(4-x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2-x}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{3 \sqrt {2} \sqrt {8-8 x+4 x^2-x^3}} \] Output:

-2/3*C*(2-x)*(x^2-2*x+4)/(-x^3+4*x^2-8*x+8)^(1/2)+2/3*(3*B+8*C)*(2-x)*(x^2 
-2*x+4)/(4-x)/(-x^3+4*x^2-8*x+8)^(1/2)-2/3*2^(1/2)*(3*B+8*C)*(2-x)^(1/2)*( 
4-x)*((x^2-2*x+4)/(4-x)^2)^(1/2)*EllipticE(sin(2*arctan(1/2*2^(1/2)*(2-x)^ 
(1/2))),1/2*3^(1/2))/(-x^3+4*x^2-8*x+8)^(1/2)-1/6*(3*A-8*C)*(2-x)^(1/2)*(4 
-x)*((x^2-2*x+4)/(4-x)^2)^(1/2)*InverseJacobiAM(2*arctan(1/2*2^(1/2)*(2-x) 
^(1/2)),1/2*3^(1/2))*2^(1/2)/(-x^3+4*x^2-8*x+8)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.71 (sec) , antiderivative size = 655, normalized size of antiderivative = 2.43 \[ \int \frac {A+B x+C x^2}{\sqrt {8-8 x+4 x^2-x^3}} \, dx=\frac {2 \left (C \left (-8+8 x-4 x^2+x^3\right )+\frac {3 B \sqrt {i+\sqrt {3}-i x} \sqrt {\frac {i (-2+x)}{-i+\sqrt {3}}} \left (-1-i \sqrt {3}+x\right ) \left (\left (1+i \sqrt {3}\right ) E\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-i x}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )-2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )\right )}{\sqrt {-i+\sqrt {3}+i x}}+\frac {8 C \sqrt {i+\sqrt {3}-i x} \sqrt {\frac {i (-2+x)}{-i+\sqrt {3}}} \left (-1-i \sqrt {3}+x\right ) \left (\left (1+i \sqrt {3}\right ) E\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-i x}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )-2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )\right )}{\sqrt {-i+\sqrt {3}+i x}}+\frac {3 A \sqrt {-i+\sqrt {3}+i x} \sqrt {\frac {i (-2+x)}{-i+\sqrt {3}}} \left (-1+i \sqrt {3}+x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )}{\sqrt {i+\sqrt {3}-i x}}-\frac {8 C \sqrt {-i+\sqrt {3}+i x} \sqrt {\frac {i (-2+x)}{-i+\sqrt {3}}} \left (-1+i \sqrt {3}+x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )}{\sqrt {i+\sqrt {3}-i x}}\right )}{3 \sqrt {8-8 x+4 x^2-x^3}} \] Input:

Integrate[(A + B*x + C*x^2)/Sqrt[8 - 8*x + 4*x^2 - x^3],x]
 

Output:

(2*(C*(-8 + 8*x - 4*x^2 + x^3) + (3*B*Sqrt[I + Sqrt[3] - I*x]*Sqrt[(I*(-2 
+ x))/(-I + Sqrt[3])]*(-1 - I*Sqrt[3] + x)*((1 + I*Sqrt[3])*EllipticE[ArcS 
in[Sqrt[I + Sqrt[3] - I*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])] 
 - 2*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - I*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[ 
3])/(-I + Sqrt[3])]))/Sqrt[-I + Sqrt[3] + I*x] + (8*C*Sqrt[I + Sqrt[3] - I 
*x]*Sqrt[(I*(-2 + x))/(-I + Sqrt[3])]*(-1 - I*Sqrt[3] + x)*((1 + I*Sqrt[3] 
)*EllipticE[ArcSin[Sqrt[I + Sqrt[3] - I*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3]) 
/(-I + Sqrt[3])] - 2*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - I*x]/(Sqrt[2]*3^( 
1/4))], (2*Sqrt[3])/(-I + Sqrt[3])]))/Sqrt[-I + Sqrt[3] + I*x] + (3*A*Sqrt 
[-I + Sqrt[3] + I*x]*Sqrt[(I*(-2 + x))/(-I + Sqrt[3])]*(-1 + I*Sqrt[3] + x 
)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - I*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3]) 
/(-I + Sqrt[3])])/Sqrt[I + Sqrt[3] - I*x] - (8*C*Sqrt[-I + Sqrt[3] + I*x]* 
Sqrt[(I*(-2 + x))/(-I + Sqrt[3])]*(-1 + I*Sqrt[3] + x)*EllipticF[ArcSin[Sq 
rt[I + Sqrt[3] - I*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/Sqr 
t[I + Sqrt[3] - I*x]))/(3*Sqrt[8 - 8*x + 4*x^2 - x^3])
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.91 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2526, 25, 2490, 2486, 27, 1269, 1172, 321, 327}

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.

\(\displaystyle \int \frac {A+B x+C x^2}{\sqrt {-x^3+4 x^2-8 x+8}} \, dx\)

\(\Big \downarrow \) 2526

\(\displaystyle -\frac {1}{3} \int -\frac {3 A-8 C+(3 B+8 C) x}{\sqrt {-x^3+4 x^2-8 x+8}}dx-\frac {2}{3} C \sqrt {-x^3+4 x^2-8 x+8}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \int \frac {3 A-8 C+(3 B+8 C) x}{\sqrt {-x^3+4 x^2-8 x+8}}dx-\frac {2}{3} C \sqrt {-x^3+4 x^2-8 x+8}\)

\(\Big \downarrow \) 2490

\(\displaystyle \frac {1}{3} \int \frac {\frac {1}{3} (3 (3 A-8 C)+4 (3 B+8 C))+(3 B+8 C) \left (x-\frac {4}{3}\right )}{\sqrt {-\left (x-\frac {4}{3}\right )^3-\frac {8}{3} \left (x-\frac {4}{3}\right )+\frac {56}{27}}}d\left (x-\frac {4}{3}\right )-\frac {2}{3} C \sqrt {-x^3+4 x^2-8 x+8}\)

\(\Big \downarrow \) 2486

\(\displaystyle \frac {\sqrt {2-3 \left (x-\frac {4}{3}\right )} \sqrt {9 \left (x-\frac {4}{3}\right )^2+6 \left (x-\frac {4}{3}\right )+28} \int \frac {\sqrt {3} \left (9 A+12 B+8 C+3 (3 B+8 C) \left (x-\frac {4}{3}\right )\right )}{\sqrt {2-3 \left (x-\frac {4}{3}\right )} \sqrt {9 \left (x-\frac {4}{3}\right )^2+6 \left (x-\frac {4}{3}\right )+28}}d\left (x-\frac {4}{3}\right )}{3 \sqrt {-27 \left (x-\frac {4}{3}\right )^3-72 \left (x-\frac {4}{3}\right )+56}}-\frac {2}{3} C \sqrt {-x^3+4 x^2-8 x+8}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {2-3 \left (x-\frac {4}{3}\right )} \sqrt {9 \left (x-\frac {4}{3}\right )^2+6 \left (x-\frac {4}{3}\right )+28} \int \frac {9 A+12 B+8 C+3 (3 B+8 C) \left (x-\frac {4}{3}\right )}{\sqrt {2-3 \left (x-\frac {4}{3}\right )} \sqrt {9 \left (x-\frac {4}{3}\right )^2+6 \left (x-\frac {4}{3}\right )+28}}d\left (x-\frac {4}{3}\right )}{\sqrt {3} \sqrt {-27 \left (x-\frac {4}{3}\right )^3-72 \left (x-\frac {4}{3}\right )+56}}-\frac {2}{3} C \sqrt {-x^3+4 x^2-8 x+8}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\sqrt {2-3 \left (x-\frac {4}{3}\right )} \sqrt {9 \left (x-\frac {4}{3}\right )^2+6 \left (x-\frac {4}{3}\right )+28} \left (3 (3 A+6 B+8 C) \int \frac {1}{\sqrt {2-3 \left (x-\frac {4}{3}\right )} \sqrt {9 \left (x-\frac {4}{3}\right )^2+6 \left (x-\frac {4}{3}\right )+28}}d\left (x-\frac {4}{3}\right )-(3 B+8 C) \int \frac {\sqrt {2-3 \left (x-\frac {4}{3}\right )}}{\sqrt {9 \left (x-\frac {4}{3}\right )^2+6 \left (x-\frac {4}{3}\right )+28}}d\left (x-\frac {4}{3}\right )\right )}{\sqrt {3} \sqrt {-27 \left (x-\frac {4}{3}\right )^3-72 \left (x-\frac {4}{3}\right )+56}}-\frac {2}{3} C \sqrt {-x^3+4 x^2-8 x+8}\)

\(\Big \downarrow \) 1172

\(\displaystyle -\frac {2}{3} C \sqrt {-x^3+4 x^2-8 x+8}+\frac {\sqrt {2-3 \left (x-\frac {4}{3}\right )} \sqrt {9 \left (x-\frac {4}{3}\right )^2+6 \left (x-\frac {4}{3}\right )+28} \left (\frac {2 i \sqrt {\frac {2-3 \left (x-\frac {4}{3}\right )}{1+i \sqrt {3}}} (3 A+6 B+8 C) \int \frac {1}{\sqrt {\frac {i \left (3 \left (x-\frac {4}{3}\right )+3 i \sqrt {3}+1\right )}{6 \sqrt {3}}+1} \sqrt {\frac {i \left (3 \left (x-\frac {4}{3}\right )+3 i \sqrt {3}+1\right )}{\sqrt {3} \left (3-i \sqrt {3}\right )}+1}}d\frac {\sqrt {-i \left (3 \left (x-\frac {4}{3}\right )+3 i \sqrt {3}+1\right )}}{\sqrt {2} 3^{3/4}}}{\sqrt {3} \sqrt {2-3 \left (x-\frac {4}{3}\right )}}-\frac {2 i \sqrt {2-3 \left (x-\frac {4}{3}\right )} (3 B+8 C) \int \frac {\sqrt {\frac {i \left (3 \left (x-\frac {4}{3}\right )+3 i \sqrt {3}+1\right )}{\sqrt {3} \left (3-i \sqrt {3}\right )}+1}}{\sqrt {\frac {i \left (3 \left (x-\frac {4}{3}\right )+3 i \sqrt {3}+1\right )}{6 \sqrt {3}}+1}}d\frac {\sqrt {-i \left (3 \left (x-\frac {4}{3}\right )+3 i \sqrt {3}+1\right )}}{\sqrt {2} 3^{3/4}}}{\sqrt {3} \sqrt {\frac {2-3 \left (x-\frac {4}{3}\right )}{1+i \sqrt {3}}}}\right )}{\sqrt {3} \sqrt {-27 \left (x-\frac {4}{3}\right )^3-72 \left (x-\frac {4}{3}\right )+56}}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2}{3} C \sqrt {-x^3+4 x^2-8 x+8}+\frac {\sqrt {2-3 \left (x-\frac {4}{3}\right )} \sqrt {9 \left (x-\frac {4}{3}\right )^2+6 \left (x-\frac {4}{3}\right )+28} \left (-\frac {2 i \sqrt {2-3 \left (x-\frac {4}{3}\right )} (3 B+8 C) \int \frac {\sqrt {\frac {i \left (3 \left (x-\frac {4}{3}\right )+3 i \sqrt {3}+1\right )}{\sqrt {3} \left (3-i \sqrt {3}\right )}+1}}{\sqrt {\frac {i \left (3 \left (x-\frac {4}{3}\right )+3 i \sqrt {3}+1\right )}{6 \sqrt {3}}+1}}d\frac {\sqrt {-i \left (3 \left (x-\frac {4}{3}\right )+3 i \sqrt {3}+1\right )}}{\sqrt {2} 3^{3/4}}}{\sqrt {3} \sqrt {\frac {2-3 \left (x-\frac {4}{3}\right )}{1+i \sqrt {3}}}}-\frac {2 i \sqrt {\frac {2-3 \left (x-\frac {4}{3}\right )}{1+i \sqrt {3}}} (3 A+6 B+8 C) \operatorname {EllipticF}\left (\arcsin \left (\frac {4}{3}-x\right ),\frac {6}{3-i \sqrt {3}}\right )}{\sqrt {3} \sqrt {2-3 \left (x-\frac {4}{3}\right )}}\right )}{\sqrt {3} \sqrt {-27 \left (x-\frac {4}{3}\right )^3-72 \left (x-\frac {4}{3}\right )+56}}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {2}{3} C \sqrt {-x^3+4 x^2-8 x+8}+\frac {\sqrt {2-3 \left (x-\frac {4}{3}\right )} \sqrt {9 \left (x-\frac {4}{3}\right )^2+6 \left (x-\frac {4}{3}\right )+28} \left (\frac {2 i \sqrt {2-3 \left (x-\frac {4}{3}\right )} (3 B+8 C) E\left (\arcsin \left (\frac {4}{3}-x\right )|\frac {6}{3-i \sqrt {3}}\right )}{\sqrt {3} \sqrt {\frac {2-3 \left (x-\frac {4}{3}\right )}{1+i \sqrt {3}}}}-\frac {2 i \sqrt {\frac {2-3 \left (x-\frac {4}{3}\right )}{1+i \sqrt {3}}} (3 A+6 B+8 C) \operatorname {EllipticF}\left (\arcsin \left (\frac {4}{3}-x\right ),\frac {6}{3-i \sqrt {3}}\right )}{\sqrt {3} \sqrt {2-3 \left (x-\frac {4}{3}\right )}}\right )}{\sqrt {3} \sqrt {-27 \left (x-\frac {4}{3}\right )^3-72 \left (x-\frac {4}{3}\right )+56}}\)

Input:

Int[(A + B*x + C*x^2)/Sqrt[8 - 8*x + 4*x^2 - x^3],x]
 

Output:

(-2*C*Sqrt[8 - 8*x + 4*x^2 - x^3])/3 + (Sqrt[2 - 3*(-4/3 + x)]*Sqrt[28 + 6 
*(-4/3 + x) + 9*(-4/3 + x)^2]*(((2*I)*(3*B + 8*C)*Sqrt[2 - 3*(-4/3 + x)]*E 
llipticE[ArcSin[4/3 - x], 6/(3 - I*Sqrt[3])])/(Sqrt[3]*Sqrt[(2 - 3*(-4/3 + 
 x))/(1 + I*Sqrt[3])]) - ((2*I)*(3*A + 6*B + 8*C)*Sqrt[(2 - 3*(-4/3 + x))/ 
(1 + I*Sqrt[3])]*EllipticF[ArcSin[4/3 - x], 6/(3 - I*Sqrt[3])])/(Sqrt[3]*S 
qrt[2 - 3*(-4/3 + x)])))/(Sqrt[3]*Sqrt[56 - 72*(-4/3 + x) - 27*(-4/3 + x)^ 
3])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 2486

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_S 
ymbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]} 
, Simp[(a + b*x + d*x^3)^p/(Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x 
]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/ 
3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p)   Int[(e + f*x)^m*Sim 
p[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2 
*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/ 
3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && NeQ[4* 
b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]
 

rule 2490

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 
, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su 
bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 
*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c 
, 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
 

rule 2526

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] 
}, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n])), x] + Simp 
[1/(n*Coeff[Qn, x, n])   Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x 
, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm 
, x] && PolyQ[Qn, x] && NeQ[p, -1]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.28 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.20

method	result	size
elliptic	\(-\frac {2 C \sqrt {-x^{3}+4 x^{2}-8 x +8}}{3}-\frac {2 i \left (A -\frac {8 C}{3}\right ) \sqrt {3}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}\, \sqrt {\frac {x -2}{-1+i \sqrt {3}}}\, \sqrt {-i \left (x -1+i \sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {6}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-1+i \sqrt {3}}}\right )}{3 \sqrt {-x^{3}+4 x^{2}-8 x +8}}-\frac {2 i \left (B +\frac {8 C}{3}\right ) \sqrt {3}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}\, \sqrt {\frac {x -2}{-1+i \sqrt {3}}}\, \sqrt {-i \left (x -1+i \sqrt {3}\right ) \sqrt {3}}\, \left (\left (-1+i \sqrt {3}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {6}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-1+i \sqrt {3}}}\right )+2 \operatorname {EllipticF}\left (\frac {\sqrt {6}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-1+i \sqrt {3}}}\right )\right )}{3 \sqrt {-x^{3}+4 x^{2}-8 x +8}}\)	\(325\)
risch	\(\frac {2 C \left (x^{3}-4 x^{2}+8 x -8\right )}{3 \sqrt {-x^{3}+4 x^{2}-8 x +8}}-\frac {2 i \left (3 B +8 C \right ) \sqrt {3}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}\, \sqrt {\frac {x -2}{-1+i \sqrt {3}}}\, \sqrt {-i \left (x -1+i \sqrt {3}\right ) \sqrt {3}}\, \left (\left (-1+i \sqrt {3}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {6}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-1+i \sqrt {3}}}\right )+2 \operatorname {EllipticF}\left (\frac {\sqrt {6}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-1+i \sqrt {3}}}\right )\right )}{9 \sqrt {-x^{3}+4 x^{2}-8 x +8}}-\frac {2 i A \sqrt {3}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}\, \sqrt {\frac {x -2}{-1+i \sqrt {3}}}\, \sqrt {-i \left (x -1+i \sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {6}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-1+i \sqrt {3}}}\right )}{3 \sqrt {-x^{3}+4 x^{2}-8 x +8}}+\frac {16 i C \sqrt {3}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}\, \sqrt {\frac {x -2}{-1+i \sqrt {3}}}\, \sqrt {-i \left (x -1+i \sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {6}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-1+i \sqrt {3}}}\right )}{9 \sqrt {-x^{3}+4 x^{2}-8 x +8}}\)	\(455\)
default	\(-\frac {2 i A \sqrt {3}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}\, \sqrt {\frac {x -2}{-1+i \sqrt {3}}}\, \sqrt {-i \left (x -1+i \sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {6}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-1+i \sqrt {3}}}\right )}{3 \sqrt {-x^{3}+4 x^{2}-8 x +8}}-\frac {2 i B \sqrt {3}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}\, \sqrt {\frac {x -2}{-1+i \sqrt {3}}}\, \sqrt {-i \left (x -1+i \sqrt {3}\right ) \sqrt {3}}\, \left (\left (-1+i \sqrt {3}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {6}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-1+i \sqrt {3}}}\right )+2 \operatorname {EllipticF}\left (\frac {\sqrt {6}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-1+i \sqrt {3}}}\right )\right )}{3 \sqrt {-x^{3}+4 x^{2}-8 x +8}}+C \left (-\frac {2 \sqrt {-x^{3}+4 x^{2}-8 x +8}}{3}+\frac {16 i \sqrt {3}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}\, \sqrt {\frac {x -2}{-1+i \sqrt {3}}}\, \sqrt {-i \left (x -1+i \sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {6}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-1+i \sqrt {3}}}\right )}{9 \sqrt {-x^{3}+4 x^{2}-8 x +8}}-\frac {16 i \sqrt {3}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}\, \sqrt {\frac {x -2}{-1+i \sqrt {3}}}\, \sqrt {-i \left (x -1+i \sqrt {3}\right ) \sqrt {3}}\, \left (\left (-1+i \sqrt {3}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {6}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-1+i \sqrt {3}}}\right )+2 \operatorname {EllipticF}\left (\frac {\sqrt {6}\, \sqrt {i \left (x -1-i \sqrt {3}\right ) \sqrt {3}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-1+i \sqrt {3}}}\right )\right )}{9 \sqrt {-x^{3}+4 x^{2}-8 x +8}}\right )\)	\(612\)

Input:

int((C*x^2+B*x+A)/(-x^3+4*x^2-8*x+8)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-2/3*C*(-x^3+4*x^2-8*x+8)^(1/2)-2/3*I*(A-8/3*C)*3^(1/2)*(I*(x-1-I*3^(1/2)) 
*3^(1/2))^(1/2)*((x-2)/(-1+I*3^(1/2)))^(1/2)*(-I*(x-1+I*3^(1/2))*3^(1/2))^ 
(1/2)/(-x^3+4*x^2-8*x+8)^(1/2)*EllipticF(1/6*6^(1/2)*(I*(x-1-I*3^(1/2))*3^ 
(1/2))^(1/2),2^(1/2)*(I*3^(1/2)/(-1+I*3^(1/2)))^(1/2))-2/3*I*(B+8/3*C)*3^( 
1/2)*(I*(x-1-I*3^(1/2))*3^(1/2))^(1/2)*((x-2)/(-1+I*3^(1/2)))^(1/2)*(-I*(x 
-1+I*3^(1/2))*3^(1/2))^(1/2)/(-x^3+4*x^2-8*x+8)^(1/2)*((-1+I*3^(1/2))*Elli 
pticE(1/6*6^(1/2)*(I*(x-1-I*3^(1/2))*3^(1/2))^(1/2),2^(1/2)*(I*3^(1/2)/(-1 
+I*3^(1/2)))^(1/2))+2*EllipticF(1/6*6^(1/2)*(I*(x-1-I*3^(1/2))*3^(1/2))^(1 
/2),2^(1/2)*(I*3^(1/2)/(-1+I*3^(1/2)))^(1/2)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.21 \[ \int \frac {A+B x+C x^2}{\sqrt {8-8 x+4 x^2-x^3}} \, dx=-\frac {2}{9} \, {\left (9 i \, A + 12 i \, B + 8 i \, C\right )} {\rm weierstrassPInverse}\left (-\frac {32}{3}, \frac {224}{27}, x - \frac {4}{3}\right ) - \frac {2}{3} \, {\left (-3 i \, B - 8 i \, C\right )} {\rm weierstrassZeta}\left (-\frac {32}{3}, \frac {224}{27}, {\rm weierstrassPInverse}\left (-\frac {32}{3}, \frac {224}{27}, x - \frac {4}{3}\right )\right ) - \frac {2}{3} \, \sqrt {-x^{3} + 4 \, x^{2} - 8 \, x + 8} C \] Input:

integrate((C*x^2+B*x+A)/(-x^3+4*x^2-8*x+8)^(1/2),x, algorithm="fricas")
 

Output:

-2/9*(9*I*A + 12*I*B + 8*I*C)*weierstrassPInverse(-32/3, 224/27, x - 4/3) 
- 2/3*(-3*I*B - 8*I*C)*weierstrassZeta(-32/3, 224/27, weierstrassPInverse( 
-32/3, 224/27, x - 4/3)) - 2/3*sqrt(-x^3 + 4*x^2 - 8*x + 8)*C
 

Sympy [F]

\[ \int \frac {A+B x+C x^2}{\sqrt {8-8 x+4 x^2-x^3}} \, dx=\int \frac {A + B x + C x^{2}}{\sqrt {- \left (x - 2\right ) \left (x^{2} - 2 x + 4\right )}}\, dx \] Input:

integrate((C*x**2+B*x+A)/(-x**3+4*x**2-8*x+8)**(1/2),x)
 

Output:

Integral((A + B*x + C*x**2)/sqrt(-(x - 2)*(x**2 - 2*x + 4)), x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {A+B x+C x^2}{\sqrt {8-8 x+4 x^2-x^3}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {-x^{3} + 4 \, x^{2} - 8 \, x + 8}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(-x^3+4*x^2-8*x+8)^(1/2),x, algorithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)/sqrt(-x^3 + 4*x^2 - 8*x + 8), x)
 

Giac [F]

\[ \int \frac {A+B x+C x^2}{\sqrt {8-8 x+4 x^2-x^3}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {-x^{3} + 4 \, x^{2} - 8 \, x + 8}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(-x^3+4*x^2-8*x+8)^(1/2),x, algorithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)/sqrt(-x^3 + 4*x^2 - 8*x + 8), x)
 

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 879, normalized size of antiderivative = 3.26 \[ \int \frac {A+B x+C x^2}{\sqrt {8-8 x+4 x^2-x^3}} \, dx=\text {Too large to display} \] Input:

int((A + B*x + C*x^2)/(4*x^2 - 8*x - x^3 + 8)^(1/2),x)
 

Output:

(C*((2*(8*x - 4*x^2 + x^3 - 8)^(1/2))/3 - (16*(3^(1/2)*1i - 1)*((x + 3^(1/ 
2)*1i - 1)/(3^(1/2)*1i + 1))^(1/2)*((x - 2)/(3^(1/2)*1i - 1))^(1/2)*((3^(1 
/2)*1i - x + 1)/(3^(1/2)*1i - 1))^(1/2)*ellipticF(asin(((x - 2)/(3^(1/2)*1 
i - 1))^(1/2)), -(3^(1/2)*1i - 1)/(3^(1/2)*1i + 1)))/(3*(2*(3^(1/2)*1i - 1 
)*(3^(1/2)*1i + 1) - x*((3^(1/2)*1i - 1)*(3^(1/2)*1i + 1) - 4) - 4*x^2 + x 
^3)^(1/2)) + (16*((3^(1/2)*1i + 1)*ellipticE(asin(((x - 2)/(3^(1/2)*1i - 1 
))^(1/2)), -(3^(1/2)*1i - 1)/(3^(1/2)*1i + 1)) - (3^(1/2)*1i - 1)*elliptic 
F(asin(((x - 2)/(3^(1/2)*1i - 1))^(1/2)), -(3^(1/2)*1i - 1)/(3^(1/2)*1i + 
1)))*(3^(1/2)*1i - 1)*((x + 3^(1/2)*1i - 1)/(3^(1/2)*1i + 1))^(1/2)*((x - 
2)/(3^(1/2)*1i - 1))^(1/2)*((3^(1/2)*1i - x + 1)/(3^(1/2)*1i - 1))^(1/2))/ 
(3*(2*(3^(1/2)*1i - 1)*(3^(1/2)*1i + 1) - x*((3^(1/2)*1i - 1)*(3^(1/2)*1i 
+ 1) - 4) - 4*x^2 + x^3)^(1/2)))*(8*x - 4*x^2 + x^3 - 8)^(1/2))/(4*x^2 - 8 
*x - x^3 + 8)^(1/2) + (2*A*(3^(1/2)*1i - 1)*((x + 3^(1/2)*1i - 1)/(3^(1/2) 
*1i + 1))^(1/2)*((x - 2)/(3^(1/2)*1i - 1))^(1/2)*((3^(1/2)*1i - x + 1)/(3^ 
(1/2)*1i - 1))^(1/2)*ellipticF(asin(((x - 2)/(3^(1/2)*1i - 1))^(1/2)), -(3 
^(1/2)*1i - 1)/(3^(1/2)*1i + 1))*(8*x - 4*x^2 + x^3 - 8)^(1/2))/((4*x^2 - 
8*x - x^3 + 8)^(1/2)*(2*(3^(1/2)*1i - 1)*(3^(1/2)*1i + 1) - x*((3^(1/2)*1i 
 - 1)*(3^(1/2)*1i + 1) - 4) - 4*x^2 + x^3)^(1/2)) + (2*B*((3^(1/2)*1i + 1) 
*ellipticE(asin(((x - 2)/(3^(1/2)*1i - 1))^(1/2)), -(3^(1/2)*1i - 1)/(3^(1 
/2)*1i + 1)) - (3^(1/2)*1i - 1)*ellipticF(asin(((x - 2)/(3^(1/2)*1i - 1...
 

Reduce [F]

\[ \int \frac {A+B x+C x^2}{\sqrt {8-8 x+4 x^2-x^3}} \, dx=\frac {\sqrt {-x^{3}+4 x^{2}-8 x +8}\, b}{4}-\left (\int \frac {\sqrt {-x^{3}+4 x^{2}-8 x +8}}{x^{3}-4 x^{2}+8 x -8}d x \right ) a -\left (\int \frac {\sqrt {-x^{3}+4 x^{2}-8 x +8}}{x^{3}-4 x^{2}+8 x -8}d x \right ) b -\frac {3 \left (\int \frac {\sqrt {-x^{3}+4 x^{2}-8 x +8}\, x^{2}}{x^{3}-4 x^{2}+8 x -8}d x \right ) b}{8}-\left (\int \frac {\sqrt {-x^{3}+4 x^{2}-8 x +8}\, x^{2}}{x^{3}-4 x^{2}+8 x -8}d x \right ) c \] Input:

int((C*x^2+B*x+A)/(-x^3+4*x^2-8*x+8)^(1/2),x)
 

Output:

(2*sqrt( - x**3 + 4*x**2 - 8*x + 8)*b - 8*int(sqrt( - x**3 + 4*x**2 - 8*x 
+ 8)/(x**3 - 4*x**2 + 8*x - 8),x)*a - 8*int(sqrt( - x**3 + 4*x**2 - 8*x + 
8)/(x**3 - 4*x**2 + 8*x - 8),x)*b - 3*int((sqrt( - x**3 + 4*x**2 - 8*x + 8 
)*x**2)/(x**3 - 4*x**2 + 8*x - 8),x)*b - 8*int((sqrt( - x**3 + 4*x**2 - 8* 
x + 8)*x**2)/(x**3 - 4*x**2 + 8*x - 8),x)*c)/8