Integrand size = 22, antiderivative size = 128 \[ \int \frac {(3-2 x)^{5/2}}{\sqrt {1-3 x+x^2}} \, dx=-\frac {4}{5} (3-2 x)^{3/2} \sqrt {1-3 x+x^2}-\frac {6 \sqrt [4]{5} \sqrt {-1+3 x-x^2} E\left (\left .\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt {1-3 x+x^2}}+\frac {6 \sqrt [4]{5} \sqrt {-1+3 x-x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right ),-1\right )}{\sqrt {1-3 x+x^2}} \]
-6*5^(1/4)*EllipticE(1/5*(3-2*x)^(1/2)*5^(3/4),I)*(-x^2+3*x-1)^(1/2)/(x^2- 3*x+1)^(1/2)+6*5^(1/4)*EllipticF(1/5*(3-2*x)^(1/2)*5^(3/4),I)*(-x^2+3*x-1) ^(1/2)/(x^2-3*x+1)^(1/2)-4/5*(3-2*x)^(3/2)*(x^2-3*x+1)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.59 \[ \int \frac {(3-2 x)^{5/2}}{\sqrt {1-3 x+x^2}} \, dx=-\frac {2 (3-2 x)^{3/2} \left (2-6 x+2 x^2+\sqrt {5} \sqrt {-1+3 x-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {1}{5} (3-2 x)^2\right )\right )}{5 \sqrt {1-3 x+x^2}} \]
(-2*(3 - 2*x)^(3/2)*(2 - 6*x + 2*x^2 + Sqrt[5]*Sqrt[-1 + 3*x - x^2]*Hyperg eometric2F1[1/2, 3/4, 7/4, (3 - 2*x)^2/5]))/(5*Sqrt[1 - 3*x + x^2])
Time = 0.31 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.81, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1116, 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.
| \(\displaystyle \int \frac {(3-2 x)^{5/2}}{\sqrt {x^2-3 x+1}} \, dx\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle 3 \int \frac {\sqrt {3-2 x}}{\sqrt {x^2-3 x+1}}dx-\frac {4}{5} (3-2 x)^{3/2} \sqrt {x^2-3 x+1}\) |
| \(\Big \downarrow \) 1115 |
| \(\displaystyle \frac {3 \sqrt {-x^2+3 x-1} \int \frac {\sqrt {5} \sqrt {3-2 x}}{\sqrt {-x^2+3 x-1}}dx}{\sqrt {5} \sqrt {x^2-3 x+1}}-\frac {4}{5} (3-2 x)^{3/2} \sqrt {x^2-3 x+1}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \sqrt {-x^2+3 x-1} \int \frac {\sqrt {3-2 x}}{\sqrt {-x^2+3 x-1}}dx}{\sqrt {x^2-3 x+1}}-\frac {4}{5} (3-2 x)^{3/2} \sqrt {x^2-3 x+1}\) |
| \(\Big \downarrow \) 1114 |
| \(\displaystyle -\frac {6 \sqrt {-x^2+3 x-1} \int \frac {\sqrt {5} (3-2 x)}{\sqrt {5-(3-2 x)^2}}d\sqrt {3-2 x}}{\sqrt {5} \sqrt {x^2-3 x+1}}-\frac {4}{5} \sqrt {x^2-3 x+1} (3-2 x)^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {6 \sqrt {-x^2+3 x-1} \int \frac {3-2 x}{\sqrt {5-(3-2 x)^2}}d\sqrt {3-2 x}}{\sqrt {x^2-3 x+1}}-\frac {4}{5} \sqrt {x^2-3 x+1} (3-2 x)^{3/2}\) |
| \(\Big \downarrow \) 836 |
| \(\displaystyle -\frac {6 \sqrt {-x^2+3 x-1} \left (\sqrt {5} \int \frac {-2 x+\sqrt {5}+3}{\sqrt {5} \sqrt {5-(3-2 x)^2}}d\sqrt {3-2 x}-\sqrt {5} \int \frac {1}{\sqrt {5-(3-2 x)^2}}d\sqrt {3-2 x}\right )}{\sqrt {x^2-3 x+1}}-\frac {4}{5} \sqrt {x^2-3 x+1} (3-2 x)^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {6 \sqrt {-x^2+3 x-1} \left (\int \frac {-2 x+\sqrt {5}+3}{\sqrt {5-(3-2 x)^2}}d\sqrt {3-2 x}-\sqrt {5} \int \frac {1}{\sqrt {5-(3-2 x)^2}}d\sqrt {3-2 x}\right )}{\sqrt {x^2-3 x+1}}-\frac {4}{5} \sqrt {x^2-3 x+1} (3-2 x)^{3/2}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle -\frac {6 \sqrt {-x^2+3 x-1} \left (\int \frac {-2 x+\sqrt {5}+3}{\sqrt {5-(3-2 x)^2}}d\sqrt {3-2 x}-\sqrt [4]{5} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right ),-1\right )\right )}{\sqrt {x^2-3 x+1}}-\frac {4}{5} \sqrt {x^2-3 x+1} (3-2 x)^{3/2}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {6 \sqrt {-x^2+3 x-1} \left (\int \frac {\sqrt {-2 x+\sqrt {5}+3}}{\sqrt {2 x+\sqrt {5}-3}}d\sqrt {3-2 x}-\sqrt [4]{5} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right ),-1\right )\right )}{\sqrt {x^2-3 x+1}}-\frac {4}{5} \sqrt {x^2-3 x+1} (3-2 x)^{3/2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {6 \sqrt {-x^2+3 x-1} \left (\sqrt [4]{5} E\left (\left .\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )-\sqrt [4]{5} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right ),-1\right )\right )}{\sqrt {x^2-3 x+1}}-\frac {4}{5} \sqrt {x^2-3 x+1} (3-2 x)^{3/2}\) |
(-4*(3 - 2*x)^(3/2)*Sqrt[1 - 3*x + x^2])/5 - (6*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]))/Sqrt[1 - 3*x + x^2]
3.14.76.3.1 Defintions of rubi rules used
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*d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Simp[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))) 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] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
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 = 2.39 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(-\frac {\sqrt {3-2 x}\, \sqrt {x^{2}-3 x +1}\, \left (3 \sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}\, \sqrt {5}\, \sqrt {\left (-3+2 x \right ) \sqrt {5}}\, \sqrt {\left (2 x -3+\sqrt {5}\right ) \sqrt {5}}\, E\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}}{10}, \sqrt {2}\right )-16 x^{4}+96 x^{3}-196 x^{2}+156 x -36\right )}{5 \left (2 x^{3}-9 x^{2}+11 x -3\right )}\) | \(127\) |
| elliptic | \(\frac {\sqrt {-\left (-3+2 x \right ) \left (x^{2}-3 x +1\right )}\, \left (\frac {8 x \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}{5}-\frac {12 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}{5}-\frac {18 \sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {10}\, \sqrt {\left (x -\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {\left (x -\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, F\left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{25 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}+\frac {12 \sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {10}\, \sqrt {\left (x -\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {\left (x -\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \left (\frac {\sqrt {5}\, E\left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{2}+\frac {3 F\left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{2}\right )}{25 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}\right )}{\sqrt {3-2 x}\, \sqrt {x^{2}-3 x +1}}\) | \(267\) |
| risch | \(-\frac {4 \left (-3+2 x \right )^{2} \sqrt {x^{2}-3 x +1}\, \sqrt {\left (3-2 x \right ) \left (x^{2}-3 x +1\right )}}{5 \sqrt {-\left (-3+2 x \right ) \left (x^{2}-3 x +1\right )}\, \sqrt {3-2 x}}-\frac {\left (\frac {18 \sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {10}\, \sqrt {\left (x -\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {\left (x -\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, F\left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{25 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}-\frac {12 \sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {10}\, \sqrt {\left (x -\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {\left (x -\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \left (\frac {\sqrt {5}\, E\left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{2}+\frac {3 F\left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{2}\right )}{25 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}\right ) \sqrt {\left (3-2 x \right ) \left (x^{2}-3 x +1\right )}}{\sqrt {3-2 x}\, \sqrt {x^{2}-3 x +1}}\) | \(288\) |
-1/5*(3-2*x)^(1/2)*(x^2-3*x+1)^(1/2)*(3*((-2*x+3+5^(1/2))*5^(1/2))^(1/2)*5 ^(1/2)*((-3+2*x)*5^(1/2))^(1/2)*((2*x-3+5^(1/2))*5^(1/2))^(1/2)*EllipticE( 1/10*2^(1/2)*5^(1/2)*((-2*x+3+5^(1/2))*5^(1/2))^(1/2),2^(1/2))-16*x^4+96*x ^3-196*x^2+156*x-36)/(2*x^3-9*x^2+11*x-3)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.30 \[ \int \frac {(3-2 x)^{5/2}}{\sqrt {1-3 x+x^2}} \, dx=\frac {4}{5} \, \sqrt {x^{2} - 3 \, x + 1} {\left (2 \, x - 3\right )} \sqrt {-2 \, x + 3} - 6 \, \sqrt {-2} {\rm weierstrassZeta}\left (5, 0, {\rm weierstrassPInverse}\left (5, 0, x - \frac {3}{2}\right )\right ) \]
4/5*sqrt(x^2 - 3*x + 1)*(2*x - 3)*sqrt(-2*x + 3) - 6*sqrt(-2)*weierstrassZ eta(5, 0, weierstrassPInverse(5, 0, x - 3/2))
\[ \int \frac {(3-2 x)^{5/2}}{\sqrt {1-3 x+x^2}} \, dx=\int \frac {\left (3 - 2 x\right )^{\frac {5}{2}}}{\sqrt {x^{2} - 3 x + 1}}\, dx \]
\[ \int \frac {(3-2 x)^{5/2}}{\sqrt {1-3 x+x^2}} \, dx=\int { \frac {{\left (-2 \, x + 3\right )}^{\frac {5}{2}}}{\sqrt {x^{2} - 3 \, x + 1}} \,d x } \]
\[ \int \frac {(3-2 x)^{5/2}}{\sqrt {1-3 x+x^2}} \, dx=\int { \frac {{\left (-2 \, x + 3\right )}^{\frac {5}{2}}}{\sqrt {x^{2} - 3 \, x + 1}} \,d x } \]
Timed out. \[ \int \frac {(3-2 x)^{5/2}}{\sqrt {1-3 x+x^2}} \, dx=\int \frac {{\left (3-2\,x\right )}^{5/2}}{\sqrt {x^2-3\,x+1}} \,d x \]