### 3.310 $$\int \frac{(2 x-x^2)^{3/2}}{2-2 x} \, dx$$

Optimal. Leaf size=53 $-\frac{1}{6} \left (2 x-x^2\right )^{3/2}-\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right )$

[Out]

-Sqrt[2*x - x^2]/2 - (2*x - x^2)^(3/2)/6 + ArcTanh[Sqrt[2*x - x^2]]/2

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Rubi [A]  time = 0.0241723, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {685, 688, 207} $-\frac{1}{6} \left (2 x-x^2\right )^{3/2}-\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(2*x - x^2)^(3/2)/(2 - 2*x),x]

[Out]

-Sqrt[2*x - x^2]/2 - (2*x - x^2)^(3/2)/6 + ArcTanh[Sqrt[2*x - x^2]]/2

Rule 685

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[(d*p*(b^2 - 4*a*c))/(b*e*(m + 2*p + 1)), Int[(d + e*x)^m*(a +
b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] &&
NeQ[m + 2*p + 3, 0] && GtQ[p, 0] &&  !LtQ[m, -1] &&  !(IGtQ[(m - 1)/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p]))
&& RationalQ[m] && IntegerQ[2*p]

Rule 688

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

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (2 x-x^2\right )^{3/2}}{2-2 x} \, dx &=-\frac{1}{6} \left (2 x-x^2\right )^{3/2}+\int \frac{\sqrt{2 x-x^2}}{2-2 x} \, dx\\ &=-\frac{1}{2} \sqrt{2 x-x^2}-\frac{1}{6} \left (2 x-x^2\right )^{3/2}+\int \frac{1}{(2-2 x) \sqrt{2 x-x^2}} \, dx\\ &=-\frac{1}{2} \sqrt{2 x-x^2}-\frac{1}{6} \left (2 x-x^2\right )^{3/2}-4 \operatorname{Subst}\left (\int \frac{1}{-8+8 x^2} \, dx,x,\sqrt{2 x-x^2}\right )\\ &=-\frac{1}{2} \sqrt{2 x-x^2}-\frac{1}{6} \left (2 x-x^2\right )^{3/2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0600379, size = 48, normalized size = 0.91 $\frac{1}{6} \sqrt{-(x-2) x} \left (x^2-2 x+\frac{6 \tan ^{-1}\left (\sqrt{\frac{x-2}{x}}\right )}{\sqrt{x-2} \sqrt{x}}-3\right )$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[-((-2 + x)*x)]*(-3 - 2*x + x^2 + (6*ArcTan[Sqrt[(-2 + x)/x]])/(Sqrt[-2 + x]*Sqrt[x])))/6

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Maple [A]  time = 0.052, size = 42, normalized size = 0.8 \begin{align*} -{\frac{1}{6} \left ( - \left ( -1+x \right ) ^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{2}\sqrt{- \left ( -1+x \right ) ^{2}+1}}+{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{- \left ( -1+x \right ) ^{2}+1}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(-(-1+x)^2+1)^(3/2)-1/2*(-(-1+x)^2+1)^(1/2)+1/2*arctanh(1/(-(-1+x)^2+1)^(1/2))

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Maxima [A]  time = 1.67784, size = 78, normalized size = 1.47 \begin{align*} -\frac{1}{6} \,{\left (-x^{2} + 2 \, x\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} + \frac{1}{2} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 2 \, x}}{{\left | x - 1 \right |}} + \frac{2}{{\left | x - 1 \right |}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(-x^2 + 2*x)^(3/2) - 1/2*sqrt(-x^2 + 2*x) + 1/2*log(2*sqrt(-x^2 + 2*x)/abs(x - 1) + 2/abs(x - 1))

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Fricas [A]  time = 2.12967, size = 150, normalized size = 2.83 \begin{align*} \frac{1}{6} \,{\left (x^{2} - 2 \, x - 3\right )} \sqrt{-x^{2} + 2 \, x} + \frac{1}{2} \, \log \left (\frac{x + \sqrt{-x^{2} + 2 \, x}}{x}\right ) - \frac{1}{2} \, \log \left (-\frac{x - \sqrt{-x^{2} + 2 \, x}}{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(x^2 - 2*x - 3)*sqrt(-x^2 + 2*x) + 1/2*log((x + sqrt(-x^2 + 2*x))/x) - 1/2*log(-(x - sqrt(-x^2 + 2*x))/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{2 x \sqrt{- x^{2} + 2 x}}{x - 1}\, dx + \int - \frac{x^{2} \sqrt{- x^{2} + 2 x}}{x - 1}\, dx}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(Integral(2*x*sqrt(-x**2 + 2*x)/(x - 1), x) + Integral(-x**2*sqrt(-x**2 + 2*x)/(x - 1), x))/2

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Giac [A]  time = 1.31522, size = 63, normalized size = 1.19 \begin{align*} \frac{1}{6} \,{\left ({\left (x - 2\right )} x - 3\right )} \sqrt{-x^{2} + 2 \, x} - \frac{1}{2} \, \log \left (-\frac{2 \,{\left (\sqrt{-x^{2} + 2 \, x} - 1\right )}}{{\left | -2 \, x + 2 \right |}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((x - 2)*x - 3)*sqrt(-x^2 + 2*x) - 1/2*log(-2*(sqrt(-x^2 + 2*x) - 1)/abs(-2*x + 2))