3.1376 \(\int \frac {(3-2 x)^{5/2}}{\sqrt {1-3 x+x^2}} \, dx\)

Optimal. Leaf size=128 \[ -\frac {4}{5} \sqrt {x^2-3 x+1} (3-2 x)^{3/2}+\frac {6 \sqrt [4]{5} \sqrt {-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt {x^2-3 x+1}}-\frac {6 \sqrt [4]{5} \sqrt {-x^2+3 x-1} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt {x^2-3 x+1}} \]

[Out]

-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)

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {692, 691, 690, 307, 221, 1181, 21, 424} \[ -\frac {4}{5} \sqrt {x^2-3 x+1} (3-2 x)^{3/2}+\frac {6 \sqrt [4]{5} \sqrt {-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt {x^2-3 x+1}}-\frac {6 \sqrt [4]{5} \sqrt {-x^2+3 x-1} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt {x^2-3 x+1}} \]

Antiderivative was successfully verified.

[In]

Int[(3 - 2*x)^(5/2)/Sqrt[1 - 3*x + x^2],x]

[Out]

(-4*(3 - 2*x)^(3/2)*Sqrt[1 - 3*x + x^2])/5 - (6*5^(1/4)*Sqrt[-1 + 3*x - x^2]*EllipticE[ArcSin[Sqrt[3 - 2*x]/5^
(1/4)], -1])/Sqrt[1 - 3*x + x^2] + (6*5^(1/4)*Sqrt[-1 + 3*x - x^2]*EllipticF[ArcSin[Sqrt[3 - 2*x]/5^(1/4)], -1
])/Sqrt[1 - 3*x + x^2]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 307

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(b/a), 2]}, -Dist[q^(-1), Int[1/Sqrt[a + b*x^
4], x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 424

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

Rule 690

Int[Sqrt[(d_) + (e_.)*(x_)]/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(4*Sqrt[-(c/(b^2 - 4*a*
c))])/e, 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] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]

Rule 691

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[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] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && EqQ[m^2, 1/4]

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*d*(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1))/(b*(m + 2*p + 1)), x] + Dist[(d^2*(m - 1)*(b^2 - 4*a*c))/(b^2*(m + 2*p + 1)), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && 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])

Rule 1181

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[Sqrt[-c], Int
[(d + e*x^2)/(Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c, 0]

Rubi steps

\begin {align*} \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}+3 \int \frac {\sqrt {3-2 x}}{\sqrt {1-3 x+x^2}} \, dx\\ &=-\frac {4}{5} (3-2 x)^{3/2} \sqrt {1-3 x+x^2}+\frac {\left (3 \sqrt {-1+3 x-x^2}\right ) \int \frac {\sqrt {3-2 x}}{\sqrt {-\frac {1}{5}+\frac {3 x}{5}-\frac {x^2}{5}}} \, dx}{\sqrt {5} \sqrt {1-3 x+x^2}}\\ &=-\frac {4}{5} (3-2 x)^{3/2} \sqrt {1-3 x+x^2}-\frac {\left (6 \sqrt {-1+3 x-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{\sqrt {5} \sqrt {1-3 x+x^2}}\\ &=-\frac {4}{5} (3-2 x)^{3/2} \sqrt {1-3 x+x^2}+\frac {\left (6 \sqrt {-1+3 x-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{\sqrt {1-3 x+x^2}}-\frac {\left (6 \sqrt {-1+3 x-x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {5}}}{\sqrt {1-\frac {x^4}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{\sqrt {1-3 x+x^2}}\\ &=-\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} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt {1-3 x+x^2}}-\frac {\left (6 \sqrt {-1+3 x-x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {5}}}{\sqrt {\frac {1}{\sqrt {5}}-\frac {x^2}{5}} \sqrt {\frac {1}{\sqrt {5}}+\frac {x^2}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{\sqrt {5} \sqrt {1-3 x+x^2}}\\ &=-\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} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt {1-3 x+x^2}}-\frac {\left (6 \sqrt {-1+3 x-x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {\frac {1}{\sqrt {5}}+\frac {x^2}{5}}}{\sqrt {\frac {1}{\sqrt {5}}-\frac {x^2}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{\sqrt {1-3 x+x^2}}\\ &=-\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 .\sin ^{-1}\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} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt {1-3 x+x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.02, size = 76, normalized size = 0.59 \[ -\frac {2 (3-2 x)^{3/2} \left (\sqrt {5} \sqrt {-x^2+3 x-1} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};\frac {1}{5} (3-2 x)^2\right )+2 x^2-6 x+2\right )}{5 \sqrt {x^2-3 x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 - 2*x)^(5/2)/Sqrt[1 - 3*x + x^2],x]

[Out]

(-2*(3 - 2*x)^(3/2)*(2 - 6*x + 2*x^2 + Sqrt[5]*Sqrt[-1 + 3*x - x^2]*Hypergeometric2F1[1/2, 3/4, 7/4, (3 - 2*x)
^2/5]))/(5*Sqrt[1 - 3*x + x^2])

________________________________________________________________________________________

fricas [F]  time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 12 \, x + 9\right )} \sqrt {-2 \, x + 3}}{\sqrt {x^{2} - 3 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-2*x)^(5/2)/(x^2-3*x+1)^(1/2),x, algorithm="fricas")

[Out]

integral((4*x^2 - 12*x + 9)*sqrt(-2*x + 3)/sqrt(x^2 - 3*x + 1), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-2 \, x + 3\right )}^{\frac {5}{2}}}{\sqrt {x^{2} - 3 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-2*x)^(5/2)/(x^2-3*x+1)^(1/2),x, algorithm="giac")

[Out]

integrate((-2*x + 3)^(5/2)/sqrt(x^2 - 3*x + 1), x)

________________________________________________________________________________________

maple [A]  time = 0.19, size = 127, normalized size = 0.99 \[ -\frac {\sqrt {-2 x +3}\, \sqrt {x^{2}-3 x +1}\, \left (-16 x^{4}+96 x^{3}-196 x^{2}+156 x +3 \sqrt {\left (2 x -3\right ) \sqrt {5}}\, \sqrt {\left (2 x -3+\sqrt {5}\right ) \sqrt {5}}\, \sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}\, \sqrt {5}\, \EllipticE \left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}}{10}, \sqrt {2}\right )-36\right )}{5 \left (2 x^{3}-9 x^{2}+11 x -3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+3)^(5/2)/(x^2-3*x+1)^(1/2),x)

[Out]

-1/5*(-2*x+3)^(1/2)*(x^2-3*x+1)^(1/2)*(3*((2*x-3)*5^(1/2))^(1/2)*((2*x-3+5^(1/2))*5^(1/2))^(1/2)*EllipticE(1/1
0*2^(1/2)*5^(1/2)*((-2*x+3+5^(1/2))*5^(1/2))^(1/2),2^(1/2))*((-2*x+3+5^(1/2))*5^(1/2))^(1/2)*5^(1/2)-16*x^4+96
*x^3-196*x^2+156*x-36)/(2*x^3-9*x^2+11*x-3)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-2 \, x + 3\right )}^{\frac {5}{2}}}{\sqrt {x^{2} - 3 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-2*x)^(5/2)/(x^2-3*x+1)^(1/2),x, algorithm="maxima")

[Out]

integrate((-2*x + 3)^(5/2)/sqrt(x^2 - 3*x + 1), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3-2\,x\right )}^{5/2}}{\sqrt {x^2-3\,x+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3 - 2*x)^(5/2)/(x^2 - 3*x + 1)^(1/2),x)

[Out]

int((3 - 2*x)^(5/2)/(x^2 - 3*x + 1)^(1/2), x)

________________________________________________________________________________________

sympy [A]  time = 19.80, size = 41, normalized size = 0.32 \[ \frac {\sqrt {5} i \left (3 - 2 x\right )^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {\left (3 - 2 x\right )^{2}}{5}} \right )}}{10 \Gamma \left (\frac {11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-2*x)**(5/2)/(x**2-3*x+1)**(1/2),x)

[Out]

sqrt(5)*I*(3 - 2*x)**(7/2)*gamma(7/4)*hyper((1/2, 7/4), (11/4,), (3 - 2*x)**2/5)/(10*gamma(11/4))

________________________________________________________________________________________