Integrand size = 22, antiderivative size = 128 \[ \int \frac {1}{(3-2 x)^{3/2} \sqrt {1-3 x+x^2}} \, dx=-\frac {4 \sqrt {1-3 x+x^2}}{5 \sqrt {3-2 x}}+\frac {2 \sqrt {-1+3 x-x^2} E\left (\left .\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt {1-3 x+x^2}}-\frac {2 \sqrt {-1+3 x-x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right ),-1\right )}{5^{3/4} \sqrt {1-3 x+x^2}} \] Output:
-4/5*(x^2-3*x+1)^(1/2)/(3-2*x)^(1/2)+2/5*5^(1/4)*(-x^2+3*x-1)^(1/2)*Ellipt icE(1/5*(3-2*x)^(1/2)*5^(3/4),I)/(x^2-3*x+1)^(1/2)-2/5*5^(1/4)*(-x^2+3*x-1 )^(1/2)*EllipticF(1/5*(3-2*x)^(1/2)*5^(3/4),I)/(x^2-3*x+1)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(3-2 x)^{3/2} \sqrt {1-3 x+x^2}} \, dx=\frac {2 \sqrt {-1+3 x-x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\frac {1}{5} (3-2 x)^2\right )}{\sqrt {5} \sqrt {3-2 x} \sqrt {1-3 x+x^2}} \] Input:
Integrate[1/((3 - 2*x)^(3/2)*Sqrt[1 - 3*x + x^2]),x]
Output:
(2*Sqrt[-1 + 3*x - x^2]*Hypergeometric2F1[-1/4, 1/2, 3/4, (3 - 2*x)^2/5])/ (Sqrt[5]*Sqrt[3 - 2*x]*Sqrt[1 - 3*x + x^2])
Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.83, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1117, 1115, 27, 1114, 27, 836, 27, 762, 1388, 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.
| \(\displaystyle \int \frac {1}{(3-2 x)^{3/2} \sqrt {x^2-3 x+1}} \, dx\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle -\frac {1}{5} \int \frac {\sqrt {3-2 x}}{\sqrt {x^2-3 x+1}}dx-\frac {4 \sqrt {x^2-3 x+1}}{5 \sqrt {3-2 x}}\) |
| \(\Big \downarrow \) 1115 |
| \(\displaystyle -\frac {\sqrt {-x^2+3 x-1} \int \frac {\sqrt {5} \sqrt {3-2 x}}{\sqrt {-x^2+3 x-1}}dx}{5 \sqrt {5} \sqrt {x^2-3 x+1}}-\frac {4 \sqrt {x^2-3 x+1}}{5 \sqrt {3-2 x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {-x^2+3 x-1} \int \frac {\sqrt {3-2 x}}{\sqrt {-x^2+3 x-1}}dx}{5 \sqrt {x^2-3 x+1}}-\frac {4 \sqrt {x^2-3 x+1}}{5 \sqrt {3-2 x}}\) |
| \(\Big \downarrow \) 1114 |
| \(\displaystyle \frac {2 \sqrt {-x^2+3 x-1} \int \frac {\sqrt {5} (3-2 x)}{\sqrt {5-(3-2 x)^2}}d\sqrt {3-2 x}}{5 \sqrt {5} \sqrt {x^2-3 x+1}}-\frac {4 \sqrt {x^2-3 x+1}}{5 \sqrt {3-2 x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {-x^2+3 x-1} \int \frac {3-2 x}{\sqrt {5-(3-2 x)^2}}d\sqrt {3-2 x}}{5 \sqrt {x^2-3 x+1}}-\frac {4 \sqrt {x^2-3 x+1}}{5 \sqrt {3-2 x}}\) |
| \(\Big \downarrow \) 836 |
| \(\displaystyle \frac {2 \sqrt {-x^2+3 x-1} \left (\sqrt {5} \int \frac {-2 x+\sqrt {5}+3}{\sqrt {5} \sqrt {5-(3-2 x)^2}}d\sqrt {3-2 x}-\sqrt {5} \int \frac {1}{\sqrt {5-(3-2 x)^2}}d\sqrt {3-2 x}\right )}{5 \sqrt {x^2-3 x+1}}-\frac {4 \sqrt {x^2-3 x+1}}{5 \sqrt {3-2 x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {-x^2+3 x-1} \left (\int \frac {-2 x+\sqrt {5}+3}{\sqrt {5-(3-2 x)^2}}d\sqrt {3-2 x}-\sqrt {5} \int \frac {1}{\sqrt {5-(3-2 x)^2}}d\sqrt {3-2 x}\right )}{5 \sqrt {x^2-3 x+1}}-\frac {4 \sqrt {x^2-3 x+1}}{5 \sqrt {3-2 x}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {2 \sqrt {-x^2+3 x-1} \left (\int \frac {-2 x+\sqrt {5}+3}{\sqrt {5-(3-2 x)^2}}d\sqrt {3-2 x}-\sqrt [4]{5} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right ),-1\right )\right )}{5 \sqrt {x^2-3 x+1}}-\frac {4 \sqrt {x^2-3 x+1}}{5 \sqrt {3-2 x}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \frac {2 \sqrt {-x^2+3 x-1} \left (\int \frac {\sqrt {-2 x+\sqrt {5}+3}}{\sqrt {2 x+\sqrt {5}-3}}d\sqrt {3-2 x}-\sqrt [4]{5} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right ),-1\right )\right )}{5 \sqrt {x^2-3 x+1}}-\frac {4 \sqrt {x^2-3 x+1}}{5 \sqrt {3-2 x}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 \sqrt {-x^2+3 x-1} \left (\sqrt [4]{5} E\left (\left .\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )-\sqrt [4]{5} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right ),-1\right )\right )}{5 \sqrt {x^2-3 x+1}}-\frac {4 \sqrt {x^2-3 x+1}}{5 \sqrt {3-2 x}}\) |
Input:
Int[1/((3 - 2*x)^(3/2)*Sqrt[1 - 3*x + x^2]),x]
Output:
(-4*Sqrt[1 - 3*x + x^2])/(5*Sqrt[3 - 2*x]) + (2*Sqrt[-1 + 3*x - x^2]*(5^(1 /4)*EllipticE[ArcSin[Sqrt[3 - 2*x]/5^(1/4)], -1] - 5^(1/4)*EllipticF[ArcSi n[Sqrt[3 - 2*x]/5^(1/4)], -1]))/(5*Sqrt[1 - 3*x + x^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-q^(-1) Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q Int[(1 + q*x^2)/S qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
Int[Sqrt[(d_) + (e_.)*(x_)]/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symb ol] :> Simp[(4/e)*Sqrt[-c/(b^2 - 4*a*c)] Subst[Int[x^2/Sqrt[Simp[1 - b^2* (x^4/(d^2*(b^2 - 4*a*c))), x]], x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c , d, e}, x] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]
Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sym bol] :> Simp[Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/Sqrt[a + b*x + c* x^2] Int[(d + e*x)^m/Sqrt[(-a)*(c/(b^2 - 4*a*c)) - b*c*(x/(b^2 - 4*a*c)) - c^2*(x^2/(b^2 - 4*a*c))], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c* d - b*e, 0] && EqQ[m^2, 1/4]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[-2*b*d*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(d^2*(m + 1)*(b^2 - 4*a*c))), x] + Simp[b^2*((m + 2*p + 3)/(d^2*(m + 1)*(b^2 - 4*a* c))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] & & (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3) /2])
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Time = 0.84 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91
| method | result | size |
| default | \(\frac {\sqrt {3-2 x}\, \sqrt {x^{2}-3 x +1}\, \left (\sqrt {\left (2 x -3\right ) \sqrt {5}}\, \sqrt {\left (2 x -3+\sqrt {5}\right ) \sqrt {5}}\, \operatorname {EllipticE}\left (\frac {\sqrt {5}\, \sqrt {2}\, \sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}}{10}, \sqrt {2}\right ) \sqrt {5}\, \sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}+20 x^{2}-60 x +20\right )}{50 x^{3}-225 x^{2}+275 x -75}\) | \(116\) |
| elliptic | \(\frac {\sqrt {-\left (2 x -3\right ) \left (x^{2}-3 x +1\right )}\, \left (\frac {-\frac {4}{5} x^{2}+\frac {12}{5} x -\frac {4}{5}}{\sqrt {\left (x -\frac {3}{2}\right ) \left (-2 x^{2}+6 x -2\right )}}+\frac {6 \sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {10}\, \sqrt {\left (x -\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {\left (x -\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{125 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}-\frac {4 \sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {10}\, \sqrt {\left (x -\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {\left (x -\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \left (\frac {\sqrt {5}\, \operatorname {EllipticE}\left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{2}+\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{2}\right )}{125 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}\right )}{\sqrt {3-2 x}\, \sqrt {x^{2}-3 x +1}}\) | \(256\) |
Input:
int(1/(3-2*x)^(3/2)/(x^2-3*x+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/25*(3-2*x)^(1/2)*(x^2-3*x+1)^(1/2)*(((2*x-3)*5^(1/2))^(1/2)*((2*x-3+5^(1 /2))*5^(1/2))^(1/2)*EllipticE(1/10*5^(1/2)*2^(1/2)*((-2*x+3+5^(1/2))*5^(1/ 2))^(1/2),2^(1/2))*5^(1/2)*((-2*x+3+5^(1/2))*5^(1/2))^(1/2)+20*x^2-60*x+20 )/(2*x^3-9*x^2+11*x-3)
Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.37 \[ \int \frac {1}{(3-2 x)^{3/2} \sqrt {1-3 x+x^2}} \, dx=\frac {2 \, {\left (\sqrt {-2} {\left (2 \, x - 3\right )} {\rm weierstrassZeta}\left (5, 0, {\rm weierstrassPInverse}\left (5, 0, x - \frac {3}{2}\right )\right ) + 2 \, \sqrt {x^{2} - 3 \, x + 1} \sqrt {-2 \, x + 3}\right )}}{5 \, {\left (2 \, x - 3\right )}} \] Input:
integrate(1/(3-2*x)^(3/2)/(x^2-3*x+1)^(1/2),x, algorithm="fricas")
Output:
2/5*(sqrt(-2)*(2*x - 3)*weierstrassZeta(5, 0, weierstrassPInverse(5, 0, x - 3/2)) + 2*sqrt(x^2 - 3*x + 1)*sqrt(-2*x + 3))/(2*x - 3)
\[ \int \frac {1}{(3-2 x)^{3/2} \sqrt {1-3 x+x^2}} \, dx=\int \frac {1}{\left (3 - 2 x\right )^{\frac {3}{2}} \sqrt {x^{2} - 3 x + 1}}\, dx \] Input:
integrate(1/(3-2*x)**(3/2)/(x**2-3*x+1)**(1/2),x)
Output:
Integral(1/((3 - 2*x)**(3/2)*sqrt(x**2 - 3*x + 1)), x)
\[ \int \frac {1}{(3-2 x)^{3/2} \sqrt {1-3 x+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} - 3 \, x + 1} {\left (-2 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(3-2*x)^(3/2)/(x^2-3*x+1)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(x^2 - 3*x + 1)*(-2*x + 3)^(3/2)), x)
\[ \int \frac {1}{(3-2 x)^{3/2} \sqrt {1-3 x+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} - 3 \, x + 1} {\left (-2 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(3-2*x)^(3/2)/(x^2-3*x+1)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(x^2 - 3*x + 1)*(-2*x + 3)^(3/2)), x)
Timed out. \[ \int \frac {1}{(3-2 x)^{3/2} \sqrt {1-3 x+x^2}} \, dx=\int \frac {1}{{\left (3-2\,x\right )}^{3/2}\,\sqrt {x^2-3\,x+1}} \,d x \] Input:
int(1/((3 - 2*x)^(3/2)*(x^2 - 3*x + 1)^(1/2)),x)
Output:
int(1/((3 - 2*x)^(3/2)*(x^2 - 3*x + 1)^(1/2)), x)
\[ \int \frac {1}{(3-2 x)^{3/2} \sqrt {1-3 x+x^2}} \, dx=\int \frac {\sqrt {-2 x +3}\, \sqrt {x^{2}-3 x +1}}{4 x^{4}-24 x^{3}+49 x^{2}-39 x +9}d x \] Input:
int(1/(3-2*x)^(3/2)/(x^2-3*x+1)^(1/2),x)
Output:
int((sqrt( - 2*x + 3)*sqrt(x**2 - 3*x + 1))/(4*x**4 - 24*x**3 + 49*x**2 - 39*x + 9),x)