### 3.1373 $$\int \frac{(3-2 x)^{3/2}}{\sqrt{1-3 x+x^2}} \, dx$$

Optimal. Leaf size=79 $-\frac{2\ 5^{3/4} \sqrt{-x^2+3 x-1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right ),-1\right )}{3 \sqrt{x^2-3 x+1}}-\frac{4}{3} \sqrt{3-2 x} \sqrt{x^2-3 x+1}$

[Out]

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

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Rubi [A]  time = 0.0391474, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.182, Rules used = {692, 691, 689, 221} $-\frac{4}{3} \sqrt{3-2 x} \sqrt{x^2-3 x+1}-\frac{2\ 5^{3/4} \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{3 \sqrt{x^2-3 x+1}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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 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 689

Int[1/(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[1/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 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]

Rubi steps

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

Mathematica [C]  time = 0.0276804, size = 76, normalized size = 0.96 $-\frac{2 \sqrt{3-2 x} \left (\sqrt{5} \sqrt{-x^2+3 x-1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{1}{5} (3-2 x)^2\right )+2 x^2-6 x+2\right )}{3 \sqrt{x^2-3 x+1}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.167, size = 118, normalized size = 1.5 \begin{align*}{\frac{1}{6\,{x}^{3}-27\,{x}^{2}+33\,x-9}\sqrt{3-2\,x}\sqrt{{x}^{2}-3\,x+1} \left ( \sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}\sqrt{ \left ( -3+2\,x \right ) \sqrt{5}}\sqrt{ \left ( 2\,x-3+\sqrt{5} \right ) \sqrt{5}}{\it EllipticF} \left ({\frac{\sqrt{2}\sqrt{5}}{10}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}},\sqrt{2} \right ) -8\,{x}^{3}+36\,{x}^{2}-44\,x+12 \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3-2*x)^(1/2)*(x^2-3*x+1)^(1/2)*(((-2*x+3+5^(1/2))*5^(1/2))^(1/2)*((-3+2*x)*5^(1/2))^(1/2)*((2*x-3+5^(1/2)
)*5^(1/2))^(1/2)*EllipticF(1/10*2^(1/2)*5^(1/2)*((-2*x+3+5^(1/2))*5^(1/2))^(1/2),2^(1/2))-8*x^3+36*x^2-44*x+12
)/(2*x^3-9*x^2+11*x-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-2 \, x + 3\right )}^{\frac{3}{2}}}{\sqrt{x^{2} - 3 \, x + 1}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-2 \, x + 3\right )}^{\frac{3}{2}}}{\sqrt{x^{2} - 3 \, x + 1}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 10.5448, size = 41, normalized size = 0.52 \begin{align*} \frac{\sqrt{5} i \left (3 - 2 x\right )^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{\left (3 - 2 x\right )^{2}}{5}} \right )}}{10 \Gamma \left (\frac{9}{4}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(5)*I*(3 - 2*x)**(5/2)*gamma(5/4)*hyper((1/2, 5/4), (9/4,), (3 - 2*x)**2/5)/(10*gamma(9/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-2 \, x + 3\right )}^{\frac{3}{2}}}{\sqrt{x^{2} - 3 \, x + 1}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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