3.2368 $$\int (7-2 x) \sqrt{9+16 x-4 x^2} \, dx$$

Optimal. Leaf size=56 $\frac{1}{6} \left (-4 x^2+16 x+9\right )^{3/2}-\frac{3}{2} (2-x) \sqrt{-4 x^2+16 x+9}-\frac{75}{4} \sin ^{-1}\left (\frac{2 (2-x)}{5}\right )$

[Out]

(-3*(2 - x)*Sqrt[9 + 16*x - 4*x^2])/2 + (9 + 16*x - 4*x^2)^(3/2)/6 - (75*ArcSin[(2*(2 - x))/5])/4

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Rubi [A]  time = 0.0171845, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {640, 612, 619, 216} $\frac{1}{6} \left (-4 x^2+16 x+9\right )^{3/2}-\frac{3}{2} (2-x) \sqrt{-4 x^2+16 x+9}-\frac{75}{4} \sin ^{-1}\left (\frac{2 (2-x)}{5}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(7 - 2*x)*Sqrt[9 + 16*x - 4*x^2],x]

[Out]

(-3*(2 - x)*Sqrt[9 + 16*x - 4*x^2])/2 + (9 + 16*x - 4*x^2)^(3/2)/6 - (75*ArcSin[(2*(2 - x))/5])/4

Rule 640

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

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 216

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

Rubi steps

\begin{align*} \int (7-2 x) \sqrt{9+16 x-4 x^2} \, dx &=\frac{1}{6} \left (9+16 x-4 x^2\right )^{3/2}+3 \int \sqrt{9+16 x-4 x^2} \, dx\\ &=-\frac{3}{2} (2-x) \sqrt{9+16 x-4 x^2}+\frac{1}{6} \left (9+16 x-4 x^2\right )^{3/2}+\frac{75}{2} \int \frac{1}{\sqrt{9+16 x-4 x^2}} \, dx\\ &=-\frac{3}{2} (2-x) \sqrt{9+16 x-4 x^2}+\frac{1}{6} \left (9+16 x-4 x^2\right )^{3/2}-\frac{15}{16} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{400}}} \, dx,x,16-8 x\right )\\ &=-\frac{3}{2} (2-x) \sqrt{9+16 x-4 x^2}+\frac{1}{6} \left (9+16 x-4 x^2\right )^{3/2}-\frac{75}{4} \sin ^{-1}\left (\frac{2 (2-x)}{5}\right )\\ \end{align*}

Mathematica [A]  time = 0.0264351, size = 43, normalized size = 0.77 $-\frac{1}{6} \sqrt{-4 x^2+16 x+9} \left (4 x^2-25 x+9\right )-\frac{75}{4} \sin ^{-1}\left (\frac{4}{5}-\frac{2 x}{5}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(7 - 2*x)*Sqrt[9 + 16*x - 4*x^2],x]

[Out]

-(Sqrt[9 + 16*x - 4*x^2]*(9 - 25*x + 4*x^2))/6 - (75*ArcSin[4/5 - (2*x)/5])/4

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Maple [A]  time = 0.046, size = 43, normalized size = 0.8 \begin{align*} -{\frac{-24\,x+48}{16}\sqrt{-4\,{x}^{2}+16\,x+9}}+{\frac{75}{4}\arcsin \left ( -{\frac{4}{5}}+{\frac{2\,x}{5}} \right ) }+{\frac{1}{6} \left ( -4\,{x}^{2}+16\,x+9 \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((7-2*x)*(-4*x^2+16*x+9)^(1/2),x)

[Out]

-3/16*(-8*x+16)*(-4*x^2+16*x+9)^(1/2)+75/4*arcsin(-4/5+2/5*x)+1/6*(-4*x^2+16*x+9)^(3/2)

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Maxima [A]  time = 1.49151, size = 70, normalized size = 1.25 \begin{align*} \frac{1}{6} \,{\left (-4 \, x^{2} + 16 \, x + 9\right )}^{\frac{3}{2}} + \frac{3}{2} \, \sqrt{-4 \, x^{2} + 16 \, x + 9} x - 3 \, \sqrt{-4 \, x^{2} + 16 \, x + 9} - \frac{75}{4} \, \arcsin \left (-\frac{2}{5} \, x + \frac{4}{5}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((7-2*x)*(-4*x^2+16*x+9)^(1/2),x, algorithm="maxima")

[Out]

1/6*(-4*x^2 + 16*x + 9)^(3/2) + 3/2*sqrt(-4*x^2 + 16*x + 9)*x - 3*sqrt(-4*x^2 + 16*x + 9) - 75/4*arcsin(-2/5*x
+ 4/5)

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Fricas [A]  time = 2.39584, size = 135, normalized size = 2.41 \begin{align*} -\frac{1}{6} \,{\left (4 \, x^{2} - 25 \, x + 9\right )} \sqrt{-4 \, x^{2} + 16 \, x + 9} - \frac{75}{2} \, \arctan \left (\frac{\sqrt{-4 \, x^{2} + 16 \, x + 9} - 3}{2 \, x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((7-2*x)*(-4*x^2+16*x+9)^(1/2),x, algorithm="fricas")

[Out]

-1/6*(4*x^2 - 25*x + 9)*sqrt(-4*x^2 + 16*x + 9) - 75/2*arctan(1/2*(sqrt(-4*x^2 + 16*x + 9) - 3)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int 2 x \sqrt{- 4 x^{2} + 16 x + 9}\, dx - \int - 7 \sqrt{- 4 x^{2} + 16 x + 9}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((7-2*x)*(-4*x**2+16*x+9)**(1/2),x)

[Out]

-Integral(2*x*sqrt(-4*x**2 + 16*x + 9), x) - Integral(-7*sqrt(-4*x**2 + 16*x + 9), x)

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Giac [A]  time = 1.08761, size = 43, normalized size = 0.77 \begin{align*} -\frac{1}{6} \,{\left ({\left (4 \, x - 25\right )} x + 9\right )} \sqrt{-4 \, x^{2} + 16 \, x + 9} + \frac{75}{4} \, \arcsin \left (\frac{2}{5} \, x - \frac{4}{5}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((7-2*x)*(-4*x^2+16*x+9)^(1/2),x, algorithm="giac")

[Out]

-1/6*((4*x - 25)*x + 9)*sqrt(-4*x^2 + 16*x + 9) + 75/4*arcsin(2/5*x - 4/5)